Stretch Processing Radar RFIC System Analysis and Front-end Design by Zachary Hubbard A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama May 7, 2012 Keywords: stretch processing, chirp, LFM, group delay, radar, LNA Copyright 2012 by Zachary Hubbard Approved by Fa Foster Dai, Chair, Professor of Electrical and Computer Engineering Michael Baginski, Associate Professor of Electrical and Computer Engineering Stuart Wentworth, Associate Professor of Electrical and Computer Engineering Abstract Current 0.13 m SiGe BiCMOS technology allows for most of the components for an X- band radar to be fabricated on a single IC. This paper discusses system design and analysis of a stretch processing radar. The design uses a single ADC, no matched lters, and wide- band waveforms. Chapters include an in-depth analysis of stretch processing, LNA design concepts, and test software design for the IC. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Integrated Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Radar Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Pulsed Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Stretch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Cadence Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 TX Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 RX Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Stretch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Pulsewidth and LFM Start Frequency . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Pulsewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.2 O set Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.3 Range Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Doppler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 iii 3.3 Group Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 GD Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.2 GD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.3 GD Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.4 GD Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.5 GD Range Resolution E ect . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 IQ Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 Phasor Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.2 Linear Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Processing Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5.1 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.2 Processing Gain Derivation . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.3 ROC Processing Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Correct Stretch Processing on ROC . . . . . . . . . . . . . . . . . . . . . . . 59 3.6.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1 Group Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1.2 LNA with Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Common-Base Ampli er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.1 CB-CC Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.2 Integrated Duplexer . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Software Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 Platform Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 iv 5.4 Code Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.1 GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 ROCDriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4.3 Serial Communication . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4.4 Load and Save ROC States . . . . . . . . . . . . . . . . . . . . . . . 97 5.4.5 Tools Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4.6 ADC Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4.7 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Stretch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 v List of Figures 1.1 PRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Two di erent waveforms with equal average power . . . . . . . . . . . . . . . . 5 1.3 Frequency vs. time of an LFM waveform . . . . . . . . . . . . . . . . . . . . . . 8 1.4 a) Time domain of LFM, b) Frequency domain of LFM . . . . . . . . . . . . . . 9 1.5 Stetch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 ROC block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 LFM waveform at IF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Single-sideband LFM waveform at IF . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Filtered RF waveform in the frequency domain . . . . . . . . . . . . . . . . . . 17 2.5 Received BB waveform in the frequency domain . . . . . . . . . . . . . . . . . . 19 2.6 Destretching with a return at 3 s . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 Example of a received waveform after destretching . . . . . . . . . . . . . . . . 21 2.8 Destretched waveform return mixed with the o set frequency . . . . . . . . . . 23 2.9 Result of subtracting the two channels . . . . . . . . . . . . . . . . . . . . . . . 24 2.10 45 MHz transmitter simulation result . . . . . . . . . . . . . . . . . . . . . . . . 26 vi 2.11 120 MHz transmitter simulation result . . . . . . . . . . . . . . . . . . . . . . . 27 2.12 195 MHz transmitter simulation result . . . . . . . . . . . . . . . . . . . . . . . 28 2.13 RX IF VGA output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.14 RX simulation results for low gain . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.15 RX simulation results for low gain . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 LFM with frequency o set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 GD Veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 The e ect of GD on range resolution . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Contour plot for the power of the image frequency . . . . . . . . . . . . . . . . 52 3.5 Plot showing chirp-squared term . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6 a)Chirp-squared term hiding a smaller tone b)Ideal stretch Processing . . . . . . 59 3.7 Updated Stretch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Fifth order Butterworth LPF with GD response . . . . . . . . . . . . . . . . . . 66 4.2 Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Equalizer plots a) impedance b) phase c) group delay . . . . . . . . . . . . . . 68 4.4 Cascode LNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 LNA gain with no equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.6 LNA gain with equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 vii 4.7 LNA NF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.8 a) S22 with no equalizer b) S22 with equalizer . . . . . . . . . . . . . . . . . . . 74 4.9 Common-base ampli er with voltage-voltage feedback . . . . . . . . . . . . . . . 76 4.10 NFmin vs. IEE for determining Jc . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.11 NFmin vs. emitter length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.12 The interface between CB and CC . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.13 Schematic of CB-CC LNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.14 S11 and S22 are simultaneously matched . . . . . . . . . . . . . . . . . . . . . . 81 4.15 Input impedance of CB-CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.16 NF and NFmin comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.17 CB-CC Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.18 Autotransformer used as hybrid transformer . . . . . . . . . . . . . . . . . . . . 84 4.19 Duplexer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.20 S-parameters for LNA with duplexer . . . . . . . . . . . . . . . . . . . . . . . . 86 4.21 Duplexer isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.22 Duplexer and LNA NF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.1 SPI diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 ROC communication diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 viii 5.3 ROC Interface SPI tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 ROC Interface communication tab . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 ROC Interface Excel tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6 ROC Interface ADC tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1 Updated Stretch Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ix List of Tables 2.1 Delay and frequency values for RX simulation . . . . . . . . . . . . . . . . . . . 32 3.1 Varying to adjust the range window with constant range resolution . . . . . . 36 3.2 GD Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1 CB-CC performance summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 x List of Abbreviations BJT bipolar junction transistor CW continuous wave GD group delay HBT heterojunction bipolar transistor IF intermediate frequency LFM linear frequency modulation LO local oscillator LPF low pass lter PLL phase-lock loop PRF pulse repetition frequency PRI pulse repetition interval ROC radar-on-chip SPI serial peripheral interface xi Chapter 1 Introduction As transistor technologies continue to improve and decrease feature size, fabricating entire transceivers on a single chip is possible. Some radios in the cellular or GPS range can be fabricated on a single chip using high performance CMOS technology. High frequency analog applications are still more suitable to bipolar junction transistors (BJTs) . Traditionally, BJTs are optimal for high frequency analog operations because of their high speed, high gain, and low noise. CMOS transistors, on the other hand, are suited for digital circuits because of their low power consumption, high input impedance, and larger noise. As CMOS technologies have improved, they have taken the place of BJTs in some analog circuitry. When possible, mixed-signal circuits fabricated entirely in CMOS is desirable because of the price. For mixed signal ICs that require BJT performance, BiCMOS is the solution. By using several extra steps in the fabrication, BJT transistors are available for use [7]. The transition frequency, fT is a measure of the high frequency capabilities of a transis- tor. It is the frequency at which the short-circuit gain of the transistor is equal to unity. In practice, ampli ers can be fabricated for operating frequencies of fT divided by ten or fteen. The transition frequency is limited by the base charging capacitance and the emitter-base depletion layer capacitance. A large portion of this capacitance is determined by the base transit time. The base transit time can be reduced by reducing the size of the base. This is one reason that transistor scaling increases performance. Transistor scaling has performance limits though. As the base width is reduced, the base resistance increases. When the base resistance increases enough, a time constant forms that limits frequency performance. 1 Silicon BJT technology would have reached its peak if not for heterojunction devices. A heterojunction is a pn junction made of two di erent materials. SiGe devices use SiGe in the base of a BJT to increase device performance. Without going into too much detail, heterojunction bipolar transistors (HBTs ) work because of the di erent band gaps of the two materials. The main bene t of SiGe HBTs is that the base width can be reduced while the base resistance remains constant. Therefore, the maximum operating frequency is increased. SiGe HBTs also have better noise performance than comparable BJTs. Another big advantage of SiGe HBTs over III-V materials such as GaAs is that they can be fabricated with a BiCMOS process. The system discussed in this paper was fabricated using a 0.13 m SiGe BiCMOS technology. 1.1 Integrated Radar The mixed signal capabilities a orded by SiGe HBT BiCMOS technology allows for most of the components of an X-band (8-12 GHz) radar to be fabricated on a single IC. The only components that cannot be fabricated on the IC practically are the IF lters, power ampli er, and antenna. An integrated radar has many advantages. The cost and mass producibility alone make it a worthy endeavor. An integrated radar would also weigh less and consume less power than otherwise. Another common radar component that is hard to integrate for many waveforms is the matched lter. A matched lter correlates a received signal with the waveform that was transmitted. A matched lter has an optimum SNR. Matched lters are usually implemented with SAW lters or in digital processing. They work only for the speci c waveform that they are designed for. If the radar uses multiple waveforms, a bank of matched lters is needed. A bank of custom designed o -chip components is not good for an integrated radar. Implementing a matched lter digitally also has a disadvantage. The bandwidth of the waveform is limited to the capabilities of the ADC. To overcome the di culties of a matched lter, an alternative signal processing method is used. Stretch processing was rst suggested 2 by W.J. Caputi in 1971 [4]. Stretch processing can be implemented on the IC and reduces the required bandwidth of the ADC. Much of this thesis has to do with implementation and analysis of stretch processing. 1.2 Radar Overview An introduction to radar is prerequisite to fully explain the following radar system analysis. A comprehensive introduction is not needed. Only the basic principles to establish the foundation for the rest of the thesis are given. The introduction will focus on pulsed radar waveforms. Radar waveforms can be broadly classi ed as either pulsed or continuous wave (CW). A CW radar continuously transmits a signal. A pulsed radar emits a signal for a time, usually on the order of s. It then quits transmitting and \listens" for the return signal for a set time before transmitting again. A CW radar is normally bistatic. Bistatic means that there are separate antennas for receiving and transmitting the waveform. Bistatic also implies that the two antennas are physically separated. This is usually necessary in a CW radar to avoid the high-power transmitter interfering or damaging the sensitive receiver. The alternative to a bistatic con guration is monostatic con guration. Monostatic radars are pulsed. A monostatic radar shares the same antenna for receiving and transmitting. When the radar is transmitting, the receiver is usually turned o . When the radar is receiving or \listening," the transmitter is turned o . This document will focus on a pulsed monostatic con guration. 1.2.1 Pulsed Radar The rate at which a radar pulses is the pulse repetition frequency (PRF ).The PRF is measured in Hz. The period of the PRF is the pulse repetition interval (PRI), measured in seconds. Figure 1.1 shows the PRI of a pulsed sine wave. The duty cycle of the pulse is the fraction of the PRI that the wave is transmitting. The equation for duty cycle is 3 Figure 1.1: PRI dt = PRI = PRF (1.1) The average power of a radar pulse is important. The average power determines the strength of a received signal. The average power has to increase as the target re ects less power or the desired target range is increased. It is calculated as the product of the duty cycle and the peak power, Pt: Pavg = Pt dt (1.2) The average power can be boosted by increasing the peak power or increasing . For illustra- tion, Figure 1.2 demonstrates two waveforms with equal energy content and di erent s and peak powers. Often hardware limitations create the ceiling for the peak power. There are also problems associated with extending the transmitting time of the pulse. The minimum detectable range of the radar system is determined by the length of because the radar cannot start receiving until it has stopped transmitting. A target cannot be detected if its 4 Figure 1.2: Two di erent waveforms with equal average power return arrives while the radar is still transmitting. Range resolution is also an important metric that is a ected by the length of . Range resolution describes a radar?s ability to distinguish between two closely spaced objects. It is the minimum separation between targets that the radar can still resolve distinct targets. The range resolution of an unmodulated pulse (a simple sine wave) with a bandwidth BW is R = c2BW (1.3) 5 The bandwidth of a sine wave pulse is determined by . Mathematically, a sine wave pulse is modeled in time and frequency by the following Fourier transform pair: rect t cos(!0t) ! 12 sinc (! +! 0) 2 + sinc (! ! 0) 2 (1.4) The pulse width is modeled by the rectangle function. A rectangle function in time trans- forms to a sinc function in frequency. So, a simple sine wave pulse transforms to two sinc functions centered at !0 in the frequency domain. The bandwidth of this function is ap- proximately inversely proportional to the pulsewidth This means that we can replace BW in equation 1.3 with 1 . The range resolution equation for a sine wave pulse can now be rewritten as R = c 2 (1.5) To increase the average power, needs to be lengthened. To improve range resolution, needs to be shorter. These two diametric properties make using a simple sine wave pulse impractical. To solve the problem, the bandwidth of the pulse needs to be separated from its dependence on the pulsewidth, or the peak power needs to be increased. As previously mentioned, Pt is usually limited by hardware. Fortunately, there is a plethora of modulated waveforms that have a higher bandwidth than a sine wave. The technique of using a pulse with a modulated waveform with a higher bandwidth than an unmodulated pulse is pulse compression. 1.2.2 Pulse Compression Pulse compression waveforms can be separated into four di erent categories: linear frequency modulation (LFM), non-linear frequency modulation, time-frequency modulation, and phase-coded waveforms.Each type of modulation has advantages and disadvantages. Radar requirements determine the type of waveform that is selected. The radar system 6 analyzed in these pages implements LFM; therefore, the description of pulse compression waveforms is limited to LFM. LFM means that the frequency is modulated linearly with respect to time. Frequency can increase or decrease linearly. LFM is also commonly referred to as chirp. This comes from the chirping sound that the waveform makes in the audio range. LFM is de ned as x(t) = Arect t cos(2 f0 + t2) (1.6) where is the pulsewidth, f0 is the carrier frequency, and is the constant slope of the LFM. can be positive or negative. From this equation, the phase modulation term is (t) = t2 (1.7) The frequency modulation is the derivative of the phase modulation term with respect to time. It is f(t) = 12 d dt = t (1.8) Equation 1.8 shows that the frequency does indeed change linearly with respect to time. A plot that allows an easy way to visualize an LFM waveform is a frequency vs. time plot. Figure 1.3 is simply a straight line. This type of plot is important to introduce because it is used extensively in explaining how to process LFM returns. Deriving the Fourier transform of an LFM is not straightforward. A useful characteri- zation of LFM is the time-bandwidth product. The time-bandwidth product is the product of the pulsewidth and the frequency range swept during the pulse. For time-bandwidth products much greater than one, the spectrum of the LFM approaches a rectangle function, jX(f)j= rect f BW for BW >> 1 (1.9) 7 Figure 1.3: Frequency vs. time of an LFM waveform If the time-bandwidth product is not high enough, there is amplitude variation in the fre- quency spectrum. An example of an LFM waveform with a time-bandwidth product greater than 1000 in the time domain and frequency domain is given in Figure 1.4. The variation is negligible and is barely conceived at 0 Hz and the corners of the rectangle. 1.2.3 Stretch Processing Stretch processing is an alternative to using a matched lter. The analysis and imple- mentation of stretch processing is expounded in depth in later sections. Here, an introduction is given. \Stretch pulse compression is a technique for perfoming LFM pulse compression of wideband waveforms using a signal processor with bandwidth that is much smaller than the waveform bandwidth, without loss of SNR or range resolution [1]." Whereas a matched lter response is a spike in time corresponding to a target range, stretch processing produces a single-tone frequency corresponding to a target range. Alternatively, stretch processing can give another LFM waveform with a much smaller bandwidth. This is useful to reduce the bandwidth of the LFM waveform to a bandwidth that can be sampled by an ADC. It is 8 Figure 1.4: a) Time domain of LFM, b) Frequency domain of LFM then further processed in the digital domain. Stretch processing multiplies or mixes a return LFM with a reference LFM. The process can be implemented in the analog domain with a mixer that can support the bandwidth of the LFM waveforms. To understand how this works, recall the following two mathematical relations: cos(a) cos(b) = 12 [cos(a+b) + cos(a b)] (1.10) and time shifting a function in time f(t) delayed by seconds = f(t ) (1.11) Multiplying two cosine wave together results in the sum and di erence of the frequencies. When two LFM waveforms with the same are multiplied together the di erence is constant. The single-tone result is dependent only on the relative time di erence of the two waveforms. In stretch processing, an LFM waveform is transmitted. After waiting for a set amount of 9 Figure 1.5: Stetch Processing time after transmission, a reference or destretch waveform starts at one input to a mixer. The destretch waveform is mixed with any received waveforms. The result is a single-tone frequency that is proportional to the time di erence of the two waveforms. Figure 1.5 illustrates the process using a frequency vs. time plot. Not that the destretch waveform needs to be much longer than the transmitted waveform. Mathematically, stretch processing is modeled by 2 cos( (t R)2) cos( (t M)2) = cos( (t R)2 + (t M)2) + cos( (t R)2 (t M)2) (1.12) where R is the time delay of the received waveform and M is the time delay of the reference waveform. The time delays are relative to when the waveform was transmitted. In equation 1.12, the sum of the two terms can be ignored. In the frequency domain, it is \stretched" enough that its power is insigni cant. The subtraction term is the desired result. When the 10 two quadratic terms are expanded and combined, the result is cos(2 ( R m)t+ ) (1.13) where is all of the non-time-dependent terms combined into a constant phase term. The frequency of equation 1.13 is fM = ( R M) (1.14) The target range is calculated from this frequency. The only unknown in the equation is R. M, the reference waveform delay, represents the delay from a known relative range. = R M = fM (1.15) The equation for the relative range is r = (R R0) = c2 (1.16) where R0 = c2 M (1.17) The range window of the radar is limited by the pulsewidth of the destretch waveform. The pulsewidth is often limited by either the highest frequency that can be generated by the waveform generator or the bandwidth that can be handled by the stretch processing. Also, the Doppler frequency is coupled to the stretch frequency. It cannot be separated and is an error. The Doppler frequency can be determined and compensated for by processing multiple target returns in the digital domain. 11 Chapter 2 System 2.1 Simulation To ensure that the ROC works as expected when designed, a high-level simulation was designed. The main purpose is to model the signal development in the transmitter and the receiver. Once designed, the simulation can quickly simulate di erent values for the carrier frequency, IF frequency, LFM bandwidth, LFM slope, and stretch processing implementation. Filter speci cations can also be determined from the simulation. Figure 2.1 is a block diagram of the ROC. The bottom half of the diagram is the transmitter. The encircled numbers mark important points in the signal analysis. For the following equations and gures, the amplitude is neglected. Correct ampli cation is assumed. The waveform shape and frequency content along the transmitter and receiver is the focus. Also, to make the math easier to follow, the waveform windows or rectangle functions are ignored in the equations. These are very important in much of the radar analysis, but not much is lost by leaving them out in the equations to go along with the MATLAB simulation. 2.1.1 Transmitter Understanding the concept of quadrature mixing is important to understanding the radar operation. Quadrature mixing requires two mixers with the same local oscillator (LO) frequency, but one LO signal has a phase o set of 90 . In the radar diagram, there are three di erent occurrences of quadrature mixing. They appear as parallel mixers in the diagram. Notice that each quadrature pair has an I (inphase) signal at one LO port and a Q (quadrature phase) at the other LO port. Recall that cos(a) = sin(a+90 ); therefore, the I signal can be considered a cosine, and the Q signal is a sine. Quadrature mixing allows 12 Figure 2.1: ROC block diagram 13 canceling either the sum or di erence output of a mixer by either adding or subtracting the I and Q mixing results. [cos(a) cos(b)] [sin(a) sin(b)] = 12[cos(a+b)+cos(a b)] 12[cos(a b) cos(a+b)] (2.1) Quadrature mixing is especially useful for canceling a mixing result that is too close to the desired frequency to be ltered. Node 4 The LFM waveform generation begins with the DDS in the lower right corner Figure 2.1. The DDS has two outputs{ an in-phase and quadrature phase LFM waveform. These wave- forms need to be ltered with a low pass lter (LPF) to remove spurious frequencies inherent in the DDS output. Refer to Figure 1.4 for the visual representation of the baseband LFM waveform. I output: cos( t2) (2.2) Q ouput: sin( t2) (2.3) Node 3 The next block in the signal path is a quadrature mixing stage that mixes the baseband LFM waveform with the IF. The quadrature mixing is used to cancel the lower sideband of the LFM waveform in the next step. The only di erence in the outputs of the two waveforms is the phase of one of the terms. Figure 2.2 is the frequency spectrum of the output of one of the mixers. 14 Figure 2.2: LFM waveform at IF The output of the I mixer: cos( t2) 2 cos(!IFt) = cos(!IFt+ t2) + cos(!IFt t2) (2.4) The output of the Q mixer: sin( t2) 2 sin(!IFt) = cos(!IFt t2) cos(!IFt+ t2) (2.5) Node 2 After the quadrature mixing, the I and Q channels are combined with a summer. This cancels the lower sideband. Before the summation, the signal has two chirps. The chirps start at the IF, and one increases in frequency while the other decreases in frequency at the same slope. This waveform could be used for transmission. This system removes the lower sideband to reduce the power and bandwidth of the system.Figure 2.3 is the signal spectrum 15 Figure 2.3: Single-sideband LFM waveform at IF after the cancellation. It is exactly half the bandwidth of the bandwidth in Figure 2.2. cos(!IFt+ t2)+cos(!IFt t2) cos(!IFt t2)+cos(!IFt+ t2) = cos(!IFt+ t2) (2.6) Node 1 Once the signal is single-side band at IF, it is ampli ed with a variable gain ampli er. The signal is now ready to be up-converted to RF. The signal is mixed with the RF from the on-chip phase-lock loop (PLL). After mixing, the signal is ltered to get rid of the unwanted RF-IF frequency component. The last on-chip block that the signal goes through is the power ampli er driver. It prepares the signal to be input into an external power ampli er 16 Figure 2.4: Filtered RF waveform in the frequency domain before transmission. The nal on-chip signal shown in Figure 2.4 is cos(!IFt+ t2) 2 cos(!LOt) = cos(!Ct+ t2) + Filteredz }| { cos((!LO !IF)t t2) = cos(!Ct+ t2) (2.7) 2.1.2 Receiver Node 1 The received pulse is similar to the transmitted pulse. It has a time delay R, a Doppler frequency !D, phase noise , and noise vN added to the transmitted pulse. Equation 2.8 is the received signal at node 1: srec = rect t R T cos((!C +!dop)(t R) + (t R)2 + ) +vN (2.8) 17 The e ects of the noise, phase noise, Doppler frequency, and rectangle function will be ignored for the rest of the simulation results. For now, just the signal processing is shown. Each of these components will be examined in-depth in Chapter 3. After removing these components the signal looks like srec = cos(!C(t R) + (t R)2) (2.9) Node 2 After the received signal is ampli ed by either the wideband or narrowband LNA, it is mixed down to the IF frequency and ltered. The lter requirements were too stringent to be e ciently implemented on-chip. This IF SAW lter is one of the few components in the system that is not integrated. The bandwidth of this lter is as close to the bandwidth of the chirp as possible. The output of a received waveform at the SAW lter should be the same as Figure 2.3. The equation for the signal after the lter is sIF = cos(!IF(t R) + (t R)2) (2.10) Node 3 At node 3 the signal is down-converted from IF to baseband. This means that the signal is mixed with the IF. The 2!IF term is ltered out later by a baseband lter. The base band signal is sBB = cos(!IF(t R) + (t R)2) 2 cos(!IFt) = cos(2!IF(t R) + (t R)2) + cos( (t R)2) (2.11) Figure 2.5 shows the spectrum of the recieved baseband waveform. 18 Figure 2.5: Received BB waveform in the frequency domain Node 4 After the signal is down-converted to baseband, it is ampli ed with a variable gain baseband ampli er. Next, the stretch processing occurs. To understand why this particular implementation of stretch processing, two conditions need to be described. Firstly, only one ADC is used on the chip. This is for space and power savings. Many systems use two ADCs to convert both the I and Q channels to the digital domain. Having both I and Q channels allows for phase information to be retained. Secondly, the scenario where a return arrives at the ROC before the destretch process begins. For example, a 10 s chirp arrives at the receiver 1 s before the destretching begins. 1 s is lost, but the remaining 9 s are destretched. This results in a negative frequency corresponding to a negative range. Some power is lost in returns coming in early, but they may still have signi cant power. Figure 2.6 demonstrates returns coming in 3 s from the destretch waveform. The results are frequencies at 27MHz. The concept of a negative range or negative frequency may seem odd at rst glance. Think of the destretch waveform as a reference range. For example, if the destretch waveform starts 30 s after transmission, it represents a range of c 30 s2 = 4:5km. The frequencies from 19 Figure 2.6: Destretching with a return at 3 s the destretching represent ranges relative to the relative range of the destrecth waveform. In this example, any range less than 4.5 km will show up as a negative frequency. Ideally, the distinguishing of positive and negative ranges is done in the digital domain, but this requires I and Q channels that are not available for this system. With a single ADC, positive and negative frequencies are practically the same. This range ambiguity has to be solved in the analog domain. Quadrature destretch waveforms are used for the stretch processing. Eventually, this allows for the canceling of a sideband that allows the distinguishing of negative and positive frequencies. At node 4, the output of the two mixers are as follows. I Mixer: sI = 2 cos( (t M)2) cos( (t R)2) = cos(2 ( R M)t ) + cos( (t R)2 + (t M)2) (2.12) 20 Figure 2.7: Example of a received waveform after destretching Q Mixer: sI = 2 sin( (t M)2) cos( (t R)2) = sin(2 ( R M)t ) + sin( (t R)2 + (t M)2) (2.13) M is the time delay of the destretch signal and is a phase term introduced by the stretch processing. The di erence term from the two mixers is the single-tone frequency that is wanted. The sum term overlaps the single-tone frequency. This adds to the noise, but its impact is nearly negligible because its power is so spread out. Figure 2.7 is an example of a received waveform after stretch processing. Note that it is impossible to know if the relative range is negative or positive. Node 5 Both channels have an LPF after the stretch processing. These lters set the range of frequencies that the system can process. For example, a lter with a cuto of 10 MHz means 21 that only ranges correspond to a frequency range of 10 MHz are acceptable. The purpose of the lters will become more clear. After the lters, each channel is mixed with an o set frequency. The o set frequency needs to be at least the frequency of the cuto of the baseband lter. This ensures that any negative frequency is now positive. For example, a frequency of -9 MHz becomes a frequency of 1 MHz. This is why the lters are needed. Any frequency that is still negative after mixing will give a false range reading. A frequency of -11 MHz is mixed with 10 MHz to get to -1 MHZ. A single ADC will sample this frequency as 1 MHz{ the same as 1 MHz in the previous example that resulted from mixing up -9 MHz. The gures demonstrating this part of the system use a lter with a cuto of 10 MHz and an o set frequency of 10 MHz. To keep the math easy to follow, the sum terms are dropped from the equations 2.14 and 2.15. The results of the mixing are as follows: I Mixer: sI = cos(2 ( R M)t ) 2 cos(!offsett) = cos(!offsett+ 2 ( R M)t ) + cos(!offsett 2 ( R M)t+ ) (2.14) Q Mixer: sQ = sin(2 ( R M)t ) 2 sin(!offsett) = cos(!offsett+ 2 ( R M)t ) + cos(!offsett 2 ( R M)t+ ) (2.15) Now any negative frequencies have been mixed to a positive frequency. After mixing with the o set frequency, there is still an unwanted frequency term{ the di erence of the o set and the stretch processing frequency. Figure 2.8 is Figure 2.7 mixed with the o set frequency. 22 Figure 2.8: Destretched waveform return mixed with the o set frequency Node 6 Due to the quadrature stretch processing and o set mixing, the unwanted frequency can be canceled by subtracting Equation 2.14 and Equation 2.15. Figure 2.9 is the result of subtracting the two channels. This is the nal frequency output that is sampled by the ADC where = R M. s = sI sQ s = cos(!offsett+ 2 ( t )) (2.16) 2.2 Cadence Simulation The ROC was designed using Cadence IC 6.1. The system parts were designed by multiple designers. Each system part was tested individually by its designer. Testing the system as a whole is still very important though. As with any complex system, testing is also complex and non-trivial. Some system parts cannot be easily included in a full system test. For example, the PLL and the ADC are better tested separately. The important 23 Figure 2.9: Result of subtracting the two channels consideration for any part left out of the system simulation is the interconnect between it and the rest of the system. 2.2.1 TX Simulation Ideally, the a full length transient simulation of a chirp could be performed on the system from the DAC at the output of the DDS to the power ampli er driver. This type of simulation is not practical though. The problem is that changing time domains from baseband to X-band make simulating a signi cant length of time at baseband prohibitive. The time step for simulating a waveform at X-band is very small relative to the length of time required to simulate a baseband waveform. Instead of simulating a chirp, a series of single tone frequencies that span the chirp bandwidth is simulated. To test the basic operation of the transmitter, baseband frequencies of 45 MHz, 120 MHz and 195 MHz were simulated with a transient simulation for 130 ns. The IF is 1.76 GHz and the PLL is 1.76 GHz 4 = 7.04 GHz. All three tests were successful. The amplitude of the output frequencies are similar and all spurs are about 40 dB below the output. Also there are no spurs within the LFM bandwidth. The output spectrum has a frequency resolution 24 of 10 MHz; therefore, the 45 MHz and 195 MHz simulations have the energy split between 40 MHz and 50 MHz, and 190 MHz and 200 MHz respectively. Figure 2.10, Figure 2.11, Figure 2.12 give the results of the simulation. 2.2.2 RX Simulation Unlike the transmitter, the receiver needs to be tested with an LFM waveform. It does not need to be tested from RF to the ADC with an LFM waveform though. This creates the aforementioned problem of simulation resources. The important part of the receiver that needs to be tested is the stretch processing. This spans from IF to the ADC input. A transient simulation of a full chirp from IF to baseband requires a lot of resources. A six core computer with 8 GB of ram requires several days for each simulation. Since each simulations requires so many resources, multiple received chirps were simulated at once. As input to the simulation, a series of chirps spanning the whole width of the 20 MHz range was generated with a MATLAB script. There are several things to watch for in the simulation results. The processing gain of the stretch processing needs to be close to what is expected. The destretched frequencies need to be narrowband. The image frequencies that may appear because of quadrature stretch processing need to be well below the destretched frequencies. Figure 2.13 is the received LFM waveforms at IF. The individual echoes are not distinguishable at this point. The simulation was performed twice. Once with low gain and again with high gain in the baseband VGA. The results are in Figure 2.14 and Figure 2.15. In the high gain results, image frequencies are starting to emerge in the spectrum. They are still more than 20 dB below the desired frequency though. Table2.1 shows the delays and destretched frequency values of the simulation. 25 Figure 2.10: 45 MHz transmitter simulation result 26 Figure 2.11: 120 MHz transmitter simulation result 27 Figure 2.12: 195 MHz transmitter simulation result 28 Figure 2.13: RX IF VGA output 29 Figure 2.14: RX simulation results for low gain 30 Figure 2.15: RX simulation results for low gain 31 Table 2.1: Delay and frequency values for RX simulation Delay (s) Frequency (Hz) -1.66E-06 -4.94E+06 -1.11E-06 1.00E+03 -1.00E-06 1.00E+06 -6.67E-07 4.00E+06 0.00E+00 1.00E+07 1.11E-07 1.10E+07 5.56E-07 1.50E+07 1.00E-06 1.90E+07 1.66E-06 2.49E+07 32 Chapter 3 Stretch Processing 3.1 Pulsewidth and LFM Start Frequency There are many factors that determine the performance and characteristics of the ROC. The maximum system bandwidth, maximum slope of the LFM, the baseband lter, and the performance of the ADC can all be limiting factors for the radar performance. For example, if the LFM bandwidth is 150 MHz, each component in the transmitter and the receiver until the stretch processing is complete must be able to handle this bandwidth without distortion. The DDS limits the slope of the LFM waveform; therefore, it sets the minimum pulsewidth required for a given bandwidth. Discovering the limiting factors of the system and optimizing system performance is very important. Remember that range resolution and pulse length can be set independently. If pulse length and bandwidth are de ned by system speci cations such as average power and range resolution, simply divide the bandwidth by the pulsewidth to calculate the slope of the LFM waveform. Make sure that the time-bandwidth product is much greater than one to avoid amplitude ripple in the frequency domain. 3.1.1 Pulsewidth Setting the delay M and the length of the destretch waveform is required. If the pulsewidth is too long, power and time are wasted. There is no need to process returns that will promptly be ltered by the baseband lter. If the pulsewidth is too short, power is lost in the stretch processing. For the way stretch processing is implemented on the ROC, the destretch waveform needs a di erent start frequency to make sure that all of the power is captured from returns in the range window. The minimum delay of tauM is determined 33 by the length of the transmitted pulse. The ROC cannot start receiving until it is done transmitting. The acceptable frequency range for the ROC is fcutoff of the baseband lter. The optimum pulsewidth to cover this frequency range is T = time of transmitted pulse + 2fcutoff (3.1) 3.1.2 O set Frequency Equation 3.1 only holds true for the way stretch processing is implemented on the ROC. To use the minimum pulsewidths and make sure all returns in the acceptable range are fully processed a frequency o set between the two pulses is required. The o set is equal to the maximum possible frequency value. fstart = fmaxM (3.2) The o set frequency has to be taken into account when calculating the relative range. r = c2 (fM fstart) (3.3) Without any o set frequency, a negative relative range implies that the return arrives at the receiver before the destretch waveform starts. The o set makes sure that the earliest processed return occurs at the beginning of the stretch processing. Examine Figure 3.1 to see why power is wasted without an o set frequency. 3.1.3 Range Window One comparative disadvantage of stretch processing compared to matched ltering is the small range window. Several factors a ect the size of the range window such as and 34 Figure 3.1: LFM with frequency o set the bandwidth of the baseband lter. The equation for the range window is range window = c2frange (3.4) where frange is the total bandwidth of frequencies corresponding to a stretch processed fre- quency. For the ROC, frange = 2fcutoff. can be varied by the DDS. When is varied, the pulsewidth of the transmitted waveform also has to be varied to maintain a constant range resolution. If the frequency is increasing slower (smaller slope), more time is needed to cover the whole bandwidth of the LFM waveform. If the slope is decreased by a factor of 10, the range window and pulsewidth are increased by a factor of 10. If the range resolution is not set, the pulsewidths can also be varied with the changing slope to achieve a higher PRF. 35 Table 3.1: Varying to adjust the range window with constant range resolution (Hzs ) Pulsewidth ( s) Destretch Pulsewidth ( s) Range Window (m) Minimum PRF (kHz) 9.00E+12 16.7 18.9 3.33E+02 28.1 9.00E+11 167 189 3.33E+03 2.81 4.50E+12 33.3 37.8 6.67E+02 14.1 4.50E+11 333 378 6.67E+03 1.41 Bandwidth of TX pulse = 150 MHz Bandwidth of destretched frequencies = 20 MHz 3.2 Doppler The Doppler frequency is inseparable from the destretched frequency; therefore, it adds directly to the error. Fortunately this error is generally negligible. A Doppler shift occurs when there is a relative radial velocity between the radar and the target. When the radiating source is moving towards the point of observation, each wave crest (or any other periodic point in the wave) is emitted closer to the point of observation. This causes an increase in the frequency. If the radiating source is moving away from the observation point, the distance between wave crests is increased, causing a decrease in frequency. Detecting Doppler shifts is very important for many radar applications. For the scenario of a moving target, consider the receiver as the point of observation and the target as the source. The target is the source because it re-radiates the transmitted waveform. The Doppler frequency shift can be accurately approximated as fdop = 2vr (3.5) When the Doppler frequency is added to the nal stretch processed waveform in Equa- tion 2.16, it becomes cos((!offset +!dop)t+ 2 t ) (3.6) where !dop = 2 fdop. Consider the following example to illustrate how a Doppler shift a ects the range measurement of a return. For an RF carrier of 9.4 GHz and an LFM slope of 9e12 36 Hz/s, an object moving at 343 m/s (speed of sound) creates a range measurement error as follows fdop = 2 343 9:4e9c (3.7) fdop = 21:5 kHz (3.8) range error = 0:4m (3.9) The Doppler frequency range error for stretch processing can be derived as follows. Remem- ber that the frequency from stretch processing corresponds to the delay of the target echo. Recall these Equations 1.14 and 1.17 from Chapter 1: = fM R = c2 Substituting the Doppler frequency for fM in the time delay equation and plugging the result into the range equation gives the range error from Doppler shifts. dop = 2vrfCc Rdop = c2 2vrfCc Rdop = vrfC (3.10) where fC is the RF carrier of the transmitted waveform. The error is inversely proportional to the slope of the LFM waveform. This is intuitive because as the slope increases, the each added hertz to the destretched frequency corresponds to a smaller increase in range. 37 The Doppler frequency can be extracted using digital processing and multiple pulses. This is possible because the ROC uses coherent processing. It is coherent because the same PLL is used to generate IF and RF for transmit and receive. If a target is stationary, the phase di erence between the return and the LO should stay the same for each pulse. As the target moves, the phase di erence will change also. There is ambiguity introduced if the target moves more than a wavelength between pulses. 3.3 Group Delay Group delay, which is an indicator of phase distortion, is a possible source of error for stretch processing. It has the e ect of smearing the target frequency in the frequency domain. Group delay (GD) is the rate of change of the phase with respect to the frequency. It has the unit of seconds. Wide bandwidth radar waveforms are especially susceptible to phase distortion. The equation for GD is GD = d d! (3.11) GD is used to measure the delay of a linear system block such as a lter. If the GD is constant, the system has a constant delay. This means that all frequencies of concern are delayed an equal amount. A constant GD implies a linear phase response. Do not mistake GD with transient analysis or the settling time of a system block. GD is a measure of how frequencies are delayed, not how long it takes a system block to settle after turning on. Consider an ideal transmission line as an example to understand the concept of GD. e j!T (3.12) The phase for the ideal transmission line is !T, so the GD is a constant T. The GD is determined by the length of the transmission line. As the length of the transmission line 38 increases, the GD also increases. Non-constant GD means that some frequencies are delayed di erently. This is common in lters for frequencies around the cuto frequency. With the preceding explanation of GD, there should be some intuition about how GD can a ect stretch processing. Constant GD is not much of a concern. If the constant GD is large enough, it will add a range error. If the GD is not a constant, it will spread the power of the destretched frequency in the frequency domain, therefore, reducing the range resolution. Quantifying how GD a ects stretch processing is possible because of the relationship between frequency and time. Consider a destretched frequency with frequency dependent phase distortion (!) destretched with it. How the phase distortion a ected the destretching needs to be determined. cos(2 t+ (!)) (3.13) The e ect of the phase distortion is not immediately obvious since it is a function of frequency. The instantaneous frequency of this signal is fM = + 12 d (!)dt (3.14) The frequency dependent distortion has the following relationship to time during stretch processing ! = 2 t (3.15) To nd the e ect of the GD on the destretched frequency, time can be replaced with its relationship to frequency in Equation 3.14. fM = + d (!)d! (3.16) fM = ( + GD) (3.17) 39 If GD is not constant, it will vary with time for the duration of the pulsewidth. If the frequency varies, the range resolution decreases. The range equation can be written with the GD e ects included. r = fM c2 r = ( + GD)c2 (3.18) 3.3.1 GD Variation From the previous analysis, constant group delay is not detrimental to the radar system performance. If the ROC introduces a signi cant amount of constant group delay, it can be corrected through digital processing. Group delay variation is a much more detrimental issue. Group delay variation is non-linear phase distortion. Consider the following second order phase distortion : = a!2 +b! (3.19) The constant coe cient a determines the e ect of group delay variation on the signal. Trans- lating the e ect of a on the signal in the time domain is done as follows: GD = d d! = 2a! b Because of the relationship between time and frequency, ! can be rewritten in terms of time. After the substitution, the equation is in terms of instantaneous frequency. GD = 2a(2 t) b = 4 a t b (3.20) 40 The instantaneous frequency of the GD variation looks very similar to the instantaneous frequency of a LFM waveform. Indeed, it is a LFM waveform with a much smaller slope. The original is scaled down by multiplication with a. This equation can be substituted for the GD in Equation 3.17 to get the result for GD variation. fM = ( 4 a t b) (3.21) The previous derivation of the e ect phase distortion through GD may be a slightly convo- luted route. The second order phase distortion can be directly included in the destretched waveform. The same result is obtained either way. s(t) = cos(2 t+ (!)) = cos(2 t+a!2 +b!) = cos(2 t+a(2 t)2 +b(2 t)) = cos(2 ( +a2 t+b)t) (3.22) 3.3.2 GD Simulation The previously derived e ect of constant and linear GD can be veri ed through simu- lation. The phase distortion can be added to the stretch processing in both the time and frequency domain. The results should be the same and match the equations. Stretch pro- cessing can be achieved in the frequency domain by convolving the received and reference waveforms. Computationally, this is not nearly as e cient as multiplying in the time domain. The phase distortion is added to the signal in the frequency domain multiplying the Fourier transform of the time signal with a complex exponential. The Fourier transform properties and pairs needed for simulating GD in the frequency domain are ( is the symbol 41 for convolution) cos(!0t), [ (! +!0) + (! !0)] (3.23) cos(!0t+ ), e j (! +!0) +ej (! !0) (3.24) x(t)v(t),X(!) V(!) (3.25) Before simulation, knowing the bandwidth of the signal with GD will be helpful. Also note that the simulation assumes that the linear GD is constant across the whole waveform bandwidth. In reality, the linear GD will be worse around the edges of the signal BW. The bandwidth of the GD signal can be estimated by the non-constant part of Equation 3.21. Where Tchirp is the pulsewidth of the LFM, is the slope of the LFM, and a is the constant coe cient from Equation 3.19, the bandwidth is BW = 4 a 2Tchirp (3.26) The parameters for the GD simulation are given in Table 3.2. Using the parameters in the table with Equation 3.26, the expected BW result of the simulation is 2.44 MHz. The undistorted result is a frequency at 9 MHz. The constant GD, b, causes an o set of 900 kHz. Table 3.2: GD Simulation Parameters Parameter Value Description Tchirp 16.666 s pulsewidth 9E12 Hzs slope ofLFM 1E-6 s time delay of received signal a 1.44E-16 2nd order phase distortion coe cient b 0.1 E-6 1st order phase distortion coe cient The simulation results show that the GD added in the frequency domain and the GD added in the time domain are nearly the same. There is only a barely perceptible di erence in the ripple of the destretched signals. Figure 3.2 shows a plot of the results. The plot on the left is the results found in the time domain, and the left plot is the results from the 42 Figure 3.2: GD Veri cation frequency domain simulation. The green plot on the right graph is the destretched signal with no GD. The 3-dB bandwidth of the GD signal is about 2.34 MHz. This is slightly less than predicted. There are a few reasons for this. Firstly, the actual bandwidth of an LFM waveform is about 0.98 of the frequency range covered by the LFM waveform. Secondly, the time bandwidth product of the LFM waveform produced by the linear GD is not >> 1. 3.3.3 GD Power Loss Figure 3.2 illustrates another problem that GD creates. The peak power in the waveform with GD is much lower than the signal with no GD. The di erence in the gure is about 16 dB. The total energy of the signals is the same. The energy in the signal with no GD is concentrated at a single frequency; whereas, the energy in the waveform with linear GD is spread across a BW. This can make a signi cant di erence when trying to detect low SNR target returns. 43 Sine wave pulses and LFM pulses have approximately the same average power. The equation to nd the average power of a pulse of length T is Pavg = 1T Z T 0 s(t)2dt (3.27) The average power of sine wave can be calculated with Equation 3.27 or found in any number of electrical engineering text books. The average power of a sine wave with amplitude A is A2 2 . The average power for an LFM waveform does not have a closed form solution, but it can be approximated Pavg = 1T Z T 0 (Acos( t2))2dt = 1T Z T 0 A2(1 + cos(2 t 2)) 2 dt = 1T Z T 0 A2 2 dt+ Z T 0 A2 cos(2 t2) 2 dt = A 2 2 + A2C(2p t) 4p T 0 C(x) is a Fresnel integral. The integral converges to 12 as x approaches in nity. For x > 0 All of the values of C(x) are between 0 and 0.8, and for x > 3, the values stay between 0.4 and 0.6. For GD variation and LFM waveforms, >> 1. This means that the Fresnel integral can be approximated as 12 and dividing it by thep makes it a very small number. This means that the average power in a sine wave and an LFM wave of equal amplitude is equal. PLFMavg = Psineavg = A 2 2 (3.28) Since the total energy is the same for a waveform with and without GD variation, the gain-bandwidth product should be the same for each waveform as long as the amplitude is equal. For an ideal sine wave the bandwidth is in nitely small. In, practice this is not the case. The sine wave pulse is a sine wave multiplied by a rectangle function. In the frequency 44 domain, the sine wave pulse, is a delta function convolved with a sinc function. The sinc function determines the bandwidth of the pulse. If the main lobe of the sinc is approximated as a rectangle, and the pulse a ected by group delay is approximated as a rectangle, the following relationship is true: BWsinePsinepeak = BWGDPGDpeak (3.29) Refer to Figure 3.2 as an example. The 3-dB bandwidth of the sinc function is 55 kHz and the peak power is -3 dB. The bandwidth of the GD pulse is about 2.34 MHz. Using Equation 3.29, the peak power of the GD pulse should be about -19 dB{this is con rmed by examining the plot. 3.3.4 GD Summary The e ect that GD has on both power level and signal bandwidth has been established. The best way to compile the analysis into useful design equations is to look at how much it varies from the start to the nish of an LFM bandwidth. If the variation is not linear, use the di erence between the highest and lowest GD values. This di erence is referred to as GD. This strategy is useful because design software such as Cadence has the ability to calculate and plot GD. If the GD is not directly available, export the phase of the system and perform a numerical derivative using software such as MATLAB. GD can be found using Equation 3.20. Substitute Tchirp for t. Once GD is found, the bandwidth and peak power loss can be calculated for a given LFM slope, : BWGD = GD (3.30) Ploss = 10 log 1 TchirpBWGD (3.31) 45 3.3.5 GD Range Resolution E ect The range resolution is reduced by the GD. As a target return is spread out in the frequency domain, it may overlap with a nearby return, making the targets indistinguish- able. Recall the original range resolution equation, which is inversely proportional to the bandwidth of the pulse: R = c2BW The bandwidth of an LFM waveform is the slope multiplied with the pulsewidth of the waveform. Substitute the bandwidth of the LFM waveform with the bandwidth in the range resolution equation to get the following: R = c2 T chirp (3.32) The relative range equation for stretch processing can be written in terms of the destretched frequency as range = c2fM where fM is the destretched frequency. If the equation is divided by the received frequency on each side of the equal sign, it represents meters per Hz. In other words, it represents how many meters correspond to a 1 Hz change in the destretched frequency. It is the relationship or mapping between range and frequency in stretch processing. range fM = c 2 (3.33) Notice that the right hand side of Equation 3.33 is part of Equation 3.32, the range resolution equation for stretch processing. Equation 3.32 is comprised of two parts: the meters per Hz multiplied by 1T chirp . 1T chirp is the bandwidth of the destretched frequency. Think of the pulsewidth of the chirp as a rectangle function multiplying the destretched frequency. This rectangle function transforms to a sinc in the frequency domain with a bandwidth of 1T chirp . 46 The range resolution for stretch processing is just the bandwidth dependent on the pulsewidth multiplied with a constant that maps the bandwidth to meters. Group delay variation spreads the bandwidth of the destretched frequency. Once the bandwidth of the linear GD variation is larger than 1T chirp , it can be used in the range resolution equation. Equation 3.26 or Equation 3.31 can be used to calculate the bandwidth of the GD a ected signal. The modi ed range resolution equation is R = 8> >< >>: c 2 1 Tchirp if 1 Tchirp > BWGD c 2 1 BWGD if 1 Tchirp < BWGD (3.34) Equation 3.34 gives some idea of what may be an acceptable amount of GD variation for a system. For BWGD less than 1T chirp , the e ects may be tolerable. Small amounts of GD variation can still cause a frequency shift. For example, 1T chirp = BWGD = 60kHz. The bandwidth of the signal a ected with GD is not much larger than an undistorted signal. The center of the signal is now shifted from the ideal received frequency. fM to fM + BWGD2 . Figure 3.3 is an example of GD variation causing two targets to appear as one. The plot on the left shows the two returns with no GD variation. The frequencies are separated by 540 kHz. The plot on the right shows the same two retruns with BWGD = 600 kHz. The two returns are indistinguishable. 3.4 IQ Imbalance The stretch processing for the ROC is implemented with two channels. The received baseband signal is split into two channels before stretch processing. Each channel destretches the baseband signal. The destretch signal for one channel is cos( t2) and is labeled the in- phase or I channel. The other destretch signal is o set by 90 and is labeled the quadrature- phase or Q channel. The quadrature signal can be written as sin( t2). 47 Figure 3.3: The e ect of GD on range resolution After the stretch processing, each channel has a baseband variable gain ampli er, a lowpass lter, and a mixer before being combined. Ideally, each channel is identical with the exception of the phase o set in the initial stretch processing. Unfortunately, due to process variations and circuit layout inequalities, the channels are not equal. The channel inequalities are modeled as an amplitude and phase di erence. Channel inequalities have two e ects on the processed signal: The expected frequency loses some power, and most importantly, an image frequency appears in the output spectrum of the processed signal. To begin to analyze IQ channel imbalances, start with the ideal channel outputs for a single target. I Mixer: sI = cos(!offsett+ 2 ( )t) + cos(!offsett 2 ( )t) (3.35) Q Mixer: sQ = cos(!offsett+ 2 ( )t) + cos(!offsett 2 ( )t) (3.36) 48 Ideally, the two channels are subtracted to entirely cancel the foffset frequency term. Because of IQ imbalances, this frequency is not entirely canceled. If the image frequency is strong enough, it will appear as a false return in the digital processing. As Equations 3.36 and 3.35 show, the outputs of the two channels are not actually in quadrature phase when they are combined. The channels are more accurately described as quadrature stretch processing used to cancel an image frequency. The term \IQ imbalance" is used to refer to the phase and amplitude inequalites that occur because of the two signal paths. 3.4.1 Phasor Representation The easiest way to derive the e ects of IQ imbalances on the nal frequency spectrum is to treat the frequency terms as phasors. Using Euler?s formula, a cosine wave can be represented with complex exponentials: Acos(!t+ ) = Ae j(!t+ ) +e j(!t+ ) 2 (3.37) = RefAej(!t+ )g (3.38) When dealing with linear combinations of sinusoids of the same frequency, all of the terms share ej!t. This term can be dropped during analysis, and each sinusoid can be represented with a magnitude and phase term. This is called phasor representation. The magnitude is always positive. A negative number is represented by changing the phase of the phasor. 3.4.2 Linear Combination Consider the summation of two sinusoids with the same frequency but di erent phases and amplitudes. Acos(!t+ ) +Bcos(!t+ ) = RefAej(!t+ )g+ RefBej(!t+ )g (3.39) = C cos(!t+ ) (3.40) 49 Equation 3.40 reduces to a cosine wave of the same frequency but di erent amplitude and phase. The magnitude of the wave can be found using phasor representation and the following property: The square root of a complex number multiplied by its complex conjugate is the magnitude. C = p (Aej +Bej ) (Ae j +Be j ) = p A2 +ABej( ) +ABe j( ) +B2 = p (A2 +B2 + 2ABcos( )) (3.41) If the phases are equal, the magnitude C is simply A + B. If the phases are not equal, the magnitude will always be less than A+B. If the phases are 180 out of phase, the magnitude reduces to C = A B; the cosines are subtracted. The phase of Equation 3.40 is a function of both the magnitudes and phases of the cosines being combined. For any complex number x + jy, the phase is arctan(yx). The sum of the cosines can be separated into the real and imaginary parts to nd the phase: Aej +Bej = A(cos( ) +jsin( )) +B(cos( ) +jsin( )) = Acos( ) +Bcos( ) +j(Asin( ) +Bsin( )) The phase can now be found as = arctan Asin( ) +Bsin( ) Acos( ) +Bcos( ) (3.42) The cosine terms that are supposed to constructively add when combining the channels from stretch processing are not as susceptible to IQ imbalances as the terms that are supposed to cancel. The !offsett+2 ( )t may lose some power or gain power because of amplitude variations. Phase variations will always cause a reduction in power. If the amplitudes are 50 the same, the e ect of a phase di erence on the amplitude can be calculated as follows: C = 2Acos 2 (3.43) The terms that are supposed to cancel at the combination of the two channels are the ones that need to be balanced. Discrepancies between the channels will cause an unwanted image frequency at foffset . The location of the image frequency is always the correct frequency mirrored around the o set frequency. For example, an o set frequency of 10 MHz and a destretched frequency of 1 MHz creates an image frequency at 19 MHz. To simplify the analysis, let one of the channels be normal or ideal. The other channel has the phase and amplitude variation. To accomplish this, set B = 1 and = 0 in Equation 3.40. The equation for the linear combination of two cosines can now be written as Acos(!t+ )+cos(!t) = p A2 + 2Acos( + ) + 1 cos !t+ arctan Asin( + ) Acos( + ) + 1 (3.44) Since the two cosines are going to be subtracted, needs to be added to . This accounts for the negative amplitude of A. For determining how much power the image frequency has, the phase of the cosine on the right hand side of Equation 3.44 is not important. The power of the image frequency is Pimage = A 2 2Acos( ) + 1 2 (3.45) Figure 3.4 is a contour plot showing the power of the image frequency for phase and amplitude variation. The y-axis is the amplitude variation in dB. The x-axis is the phase variation in degrees. The contours are also in dB. The power of the image frequency is approximately equal to the amplitude variation for small values of phase variation. 51 Figure 3.4: Contour plot for the power of the image frequency 3.5 Processing Gain Processing gain is a term that describes a system?s ability to reject interference. In spread spectrum systems, processing gain is the ratio of the spread bandwidth to the base- band bandwidth. This immediately gives an idea of what processing gain means for stretch processing. To make it analogous to spread spectrum, the LFM waveform is the spread signal. The energy of the signal is spread across a wide bandwidth. After stretch processing, this energy is compressed into a single frequency. Intuitively, the processing gain for stretch processing is going to be related to the ratio of signal bandwidth before stretch processing to after stretch processing. Assuming a rectangular window, the bandwidth after stretch processing is 1T chirp . From this reasoning, processing gain should be proportional or equal to BWTchirp. This number has already been seen in Chapter 1 as the time-bandwidth product. More generally, the processing gain Gp can be de ned as the ratio of the output SNR to the input SNR: Gp = SNRoSNR i (3.46) 52 3.5.1 Matched Filter Matched lters are relevant to the discussion of processing gain. A matched lter is designed to have a maximum output SNR. A matched lter convolves a noisy received signal with the conjugated, time-reversed version of the expected signal. The SNR at the output of a matched lter is SNRmf = EN 0 (3.47) where E is the energy of the signal and N0 is the one-sided noise power spectrum. Notice that the signal power is only dependent on the energy in the signal . Since a matched lter is ideal, its output SNR can serve as a benchmark to other lters and processing techniques. The Radar Handbook [1] de nes a lter matching loss as Lm = SNRmfSNR 0 (3.48) This equation will be useful for creating a gure of merit for the stretch processing imple- mentation of the ROC. According to [3], the processing gain for an LFM waveform with a matched lter is the time-bandwidth product. This matches up with the previous speculation that the processing gain of stretch processing is equal or proportional to the time-bandwidth product. The authors of [2] claim that the lter loss from stretch processing is equal to or less than matched ltering depending on the noise bandwidth. 3.5.2 Processing Gain Derivation Although the authors of [2] claim to have derived the lter loss for stretch processing, a thorough explanation and derivation of the processing gain for stretch processing seems to be lacking in available literature. The goal of this section is to provide an intuitive, easy-to-follow explanation as well as the derivation. 53 Consider a received signal in the form of s(t) = rect t Tchirp e(j (t )2) (3.49) The complex exponential is used instead of a cosine merely out of convenience. The math is much easier to follow. Input noise n(t) is added to the signal. Assume that the noise band-limited white Gaussian noise. The noise is described by its power spectrum density N0 and the system noise bandwidth Bn. The system noise bandwidth should be Bn Bs where Bs is the bandwidth of the received LFM waveform. Most systems should have a noise bandwidth very close to the bandwidth of the received signal. For the sake of completeness, they are assumed to be di erent. The noise power Pn = N0Bn. The signal plus the noise is the input into the correlation mixer. The average power of an LFM waveform is derived in Section 3.3.3 and Equation 3.28 as A22 . Since a complex exponential is used here instead of a cosine, the average power is simply A2. The SNR of the input waveform can now be calculated as SNRi = A 2 BsN0 (3.50) The reference signal is also in the form of a complex exponential: r(t) = rect t Tr e( j t2) (3.51) The reference signal has a time delay of 0, and a bandwidth Br Bs+2 max. The reference signal and the received signal are multiplied together to implement stretch processing. y(t) = (s(t) +n(t))r(t) = rect t Tchirp ej( 2 (t 2) +n(t)r(t) (3.52) 54 The rst part of Equation 3.52 is simply the result of stretch processing. The noise multiplied by the reference is of interest though. First the Fourier transform of y(t) can be found. Y(f) = ATchirpe(j 2)sinc((f + )Tchirp) +nout(f) (3.53) The Fourier transform is a sinc function centered around the destretched frequency. This is exactly what is expected from earlier discussions of stretch processing. The important thing to realize here is the power at the frequency : Pout = (ATchirp)2 (3.54) The last piece missing from the processing gain equation is the output noise power. Since nout(f) is the Fourier transform of n(t)r(t), the frequency domain solution is the convolution of the reference signal with the noise. The expected value of the noise power at a given frequency is the convolution of the noise power spectrum with the power spectrum of the reference signal. The noise power spectrum is given simply by Pn(f) = N0rect( fB n ) (3.55) The magnitude squared or power of the reference signal spectrum is approximated with a scaled rectangle function. jR(f)j2 = 1 rect( fB r ) (3.56) Notice that the convolution is going to be of two scaled rectangle functions. This is an interesting result. Recall that the convolution of two equal rectangle functions is a triangle. If the rectangle are of di erent sizes, the triangle becomes an isosceles trapezoid. In other words, the triangle gets a at top with a size dependent on how big the di erence between the rectangles is. This means that in some situations the output SNR can be dependent on the destretched frequency. The convolution needs to be solved before the rami cations are 55 discussed any further. E[nout(f)] =jR(f)j2 Pn(f) = Z 1 1 jR(f0 f)j2Pn(f0)df0 = N0 Z 1 1 rect f0 f Br rect f0 Bn df0 Before the solution to the convolution is presented, some limits on the problem can simplify it. The only frequencies of interest are the ones that can correspond to a received time delay. Equation 3.1 can be multiplied by and solved for fcutoff to set the upper limit of the convolution solution as Br Bs2 . E[nout(f)] = 8 >>>> >>< >>>> >>: N0 Br if jfj< Bn Br 2 N0 Bn if jfj< Br Bn 2 N0 Br+Bn 2 f jBn Brj 2 jfj Br Bs 2 (3.57) Now that the noise power is calculated, the output SNR can be calculated. The output SNR is dependent on the noise bandwidth. Using Equations 3.56 and 3.57, the output SNR is SNRout = 8 >>> >>>< >>> >>>: A2TchirpBs N0Br if jfj< Bn Br 2 A2TchirpBs N0Bn if jfj< Br Bn 2 2A2TchirpBs N0(Bn+Br j2fj jBn Brj 2 jfj Br Bs 2 (3.58) This is an interesting result. In some instances, the output SNR is dependent on the de- stretched frequency. In other words, the output SNR is dependent on the time delay of the received signal. The important observation, which was clear from the beginning, is that the noise bandwidth of the system needs to be as close to the bandwidth of the received LFM waveform as possible. 56 Calculating the processing gain is now a straightforward step. The output SNR divided by the input SNR is the processing gain. Gp = 8 >>> >>>< >>> >>>: TchirpBsBs Br if jfj< Bn Br 2 TchirpBsBs Bn if jfj< Br Bn 2 2TchirpBsBs (Bn+Br j2fj jBn Brj 2 jfj Br Bs 2 (3.59) The processing gain is the same as a matched lter for Bs = Bn. If the noise bandwidth is larger than the reference signal bandwidth, the degradation in the processing gain is proportional to Br=Bs. If the noise bandwidth is smaller than the reference signal bandwidth, the degradation in the processing gain is proportional to Bn=Bs. 3.5.3 ROC Processing Gain The processing gain of the ROC is not ideal. The system is sub-par not because of hardware and process variations. It is not ideal because of the design itself. The problem is rooted in the image frequency that is the result of stretch processing. Recall that stretch processing produces the single-tone di erence frequency and the sum frequency, which will be termed the chirp-squared image. The chirp-squared image is another LFM waveform with a bandwidth twice as large as the original. If the reference signal delay is set to zero, the chirp squared term from the ROC can be written as chirp2 = cos(2 (t2 t+ 2 2 )) (3.60) Its presence in the spectrum limits the output SNR of any received signal to the processing gain. For example, consider an input signal with an SNR of 5 dB and a processing gain of 34 dB. The output SNR should be 39 dB, but if the chirp-squared term is present in the spectrum, the output frequency is only 34 dB (the processing gain) above the chirp-squared term. This may not seem like a problem at rst. All it does to an input signal is limit the 57 Figure 3.5: Plot showing chirp-squared term SNR to the processing gain. Input signals that are below the noise oor are not a ected at all. Unfortunately, the chirp-squared term causes another major problem. It shields other signal returns with less power. With the previous example, any signal with an SNR below 5 dB after stretch processing will be hidden by the chirp-squared term. As an example, Figure 3.5 shows the spectrum of a destretched return with the single tone and the chirp-squared term. The received signal has an SNR of 10 dB. The processing gain is ideally around 34 dB. The output SNR should be 44 dB, but it is 34 dB because of the chirp-squared term. Notice that at 300 MHz the chirp-squared term ends. This is because the chirp-squared bandwidth is twice the original bandwidth of 150 MHz. As an example of the chirp squared term hiding a smaller signal, a signal with -40 dB of power is added to the signal in Figure 3.5. The stretch processing is calculated for both 58 Figure 3.6: a)Chirp-squared term hiding a smaller tone b)Ideal stretch Processing chirp-squared interference and ideal stretch processing. The left plot in Figure 3.6 is the chirp-squared term. There is no second tone visible. The plot on the right is ideal stretch processing. The low power frequency is visible, and the stronger frequency has a higher SNR than the plot on the left. 3.6 Correct Stretch Processing on ROC The interference from the chirp-squared term is a serious aw. In the initial design of the ROC, focus was centered on di erentiating positive and negative ranges and using one ADC. The chirp-squared term was not forgotten about, but it was given secondary importance. After the rst fabrication of the ROC, the system design was improved to remove the chirp-squared term. Recall the building blocks that are the most costly for RFIC design. In any changes made to the system, extra lters, frequencies not easily available, and ADCs are to be avoided if possible. Adding mixers and summers are much more desirable. 59 There are two possibilities to get rid of the chirp-squared term. First, quadrature mixing can be used to cancel the image. Second, the stretch processing can be done at an intermediate frequency. This moves the chirp squared term to fIF +chirp2. As long as the IF is high enough to move the chirp-squared term completely out of the possible range of received frequencies, it can be ltered out. Stretch processing at IF is the best option for the ROC. Only one extra mixer and summer is added to the system. No new frequencies are needed. In the original implementation of the ROC, the quadrature stretch processing is per- formed in the baseband. The quadrature stretch processing could not be performed at IF because no quadrature IF+LFM was available. The quadrature phase of the IF signal can be obtained from the available signals. Figure 3.7 is the updated stretch processing dia- gram. The diagram is not the whole transceiver. It only shows the blocks necessary for the stretch processing. The input signal that has already been down-converted to IF is injected at mixers M5 and M6. 3.6.1 Analysis Using Figure 3.7, the system analysis for the stretch processing that avoids the chirp- squared interference can easily shown. For the analysis, the input and output of each relevant system block will be shown. The blocks will have an equation with the block input on the left side and the block output on the right side. The ampli ers and bu er is not shown because the signal frequency does not change. Following the analysis of stretch processing in Chapter 2, the rectangle functions are ignored throughout the analysis. They can be added to the nal result without loss of accuracy as long as the received signal is assumed to be within the range-window. Mixers M1-M4 and summers S1 and S2 are used to generate the quadrature LFM waveforms at the IF. These waveforms are used for the stretch processing. M1 cos(!IFt) cos( t2) = 12 cos(!IFt t2) + 12 cos(!IFt t2) (3.61) 60 Figure 3.7: Updated Stretch Processing 61 M2 sin(!IFt) cos( t2) = 12 sin(!IFt+ t2) + 12 sin(!IFt t2) (3.62) M3 cos(!IFt) sin( t2) = 12 sin(!IFt+ t2) 12 sin(!IFt t2) (3.63) M4 sin(!IFt) sin( t2) = 12 cos(!IFt t2) 12 cos(!IFt+ t2) (3.64) S1 is the di erence between M1 and M4. The in-phase IF+LFM is generated at its output. S1 1 2(cos(!IFt+ t 2) + cos(!IFt t2)) +12(cos(!IFt t2) cos(!IFt+ t2)) = cos(!IFt t2) (3.65) S2 is the sum of M2 and M3. The quadrature-phase IF+LFM is generated at its output. S2 1 2(sin(!IFt+ t 2) + sin(!IFt t2)) +12(sin(!IFt+ t2) sin(!IFt t2)) = sin(!IFt t2) (3.66) Next the quadrature stretch processing is performed at M5 and M6. Remember that the received signal has already been down-converted to the IF. This is the step that o ers improvement over the orignal stretch processing implementation in the ROC. The chirp- squared term is mixed up to 2!IF. 62 M5 cos(!IFt+ t2) Acos(!IFt+ (t )2) = A2 (cos(2!IFt+ 2 (t2 t+ 2 2 )) + cos( (2 t 2))) (3.67) M6 sin(!IFt+ t2) Acos(!IFt+ (t )2) = A2 (sin(2!IFt+ 2 (t2 t+ 2 2 )) + sin( (2 t 2))) (3.68) After the stretch processing the stretch processing, each channel is ampli ed and ltered. The lter cut-o frequency only needs to be as large enough to pass the frequency corresponding to the maximum time delay. The LPFs also remove the 2!IF terms. After ltering each channel is mixed with an o set frequency. The o set frequency has to be at least the same as the lter cuto frequency to move any negative frequencies into the positive range. The o set frequency has an I and Q output. This allows for the image to be canceled when the two channels are summed. M7 A 2 cos( (2 t 2))2 cos(!offt) = A2 (cos( (2 t 2) +!offt) + cos( (2 t 2) !offt)) (3.69) M8 A 2 sin( (2 t 2)) 2 sin(!offt) = A2 (cos( (2 t 2) !offt) cos( (2 t 2) +!offt)) (3.70) 63 After the o set mixing, the two channels are subtracted to cancel the image frequency. Can- celing this term through quadrature mixing is very important. It allows for the distinction between positive and negative relative ranges. S3 A 2 (cos( (2 t 2) +!offt) + cos( (2 t 2) !offt)) A2 (cos( (2 t 2) !offt) cos( (2 t 2) +!offt)) = Acos( (2 t 2) +!offt) (3.71) After the two channels are combined, the result is passed through an anti-aliasing lter before being bu ered and sampled by the ADC. The anti-aliasing lter is necessary to remove any haromonics introduced during the mixing. 3.7 Interference Much like matched ltering, stretch processing is tolerant to some interference. Whereas as the received LFM waveform is compressed in frequency, uncorrelated waveforms are stretched. Interference of a waveform with a similar slope will cause a problem though. The ideal processing gain for stretch processing is the time-bandwidth product. A received pulse has energy evenly spread across the bandwidth of the waveform. When the pulse is destretched the energy is compressed into a bandwidth equal to the inverse of the pulsewidth, 1 Tchirp. Conversely, when a single-tone pulse is multiplied with the correlation mixer the en- ergy is spread from a bandwidth of 1T chirp to the bandwidth of the reference LFM waveform. The ratio of input bandwidth to output bandwidth of the interfering tone is 1TrBr where Tr is the pulsewidth of the reference waveform and Br is the bandwidth. The ratio of the output 64 signal to interference to the input signal to interference for single interference is SNIout SNIin = TchirpTrBr (3.72) = GpBsTrBr (3.73) More generally, the signal to interference gain for stretch processing can be described as the processing gain multiplied with the ratio of the output of bandwidth of the interfering signal divided by the input bandwidth of the interfering signal. 65 Chapter 4 Circuits 4.1 Group Delay Compensation Nonlinear phase response in the receiver adversely a ects stretch processing by smearing the received signal across a wider bandwidth than expected. One of the biggest sources of phase nonlinearity is at the edge of the passband in an ampli er. One solution to this kind of phase nonlinearity is to slightly extend the passband. There are two possible negative a ects to increasing the bandwidth. First, the gain of the ampli er may be reduced. Second, increasing the bandwidth will also increase the noise bandwidth. As a concrete example of group delay, Figure 4.1 is a fth order Butterworth LPF. The GD slope gets steeper as it nears the cuto frequency. Remember group delay variation, not constant group delay, corresponds to phase nonlinearity. Figure 4.1: Fifth order Butterworth LPF with GD response 66 Figure 4.2: Equalizer 4.1.1 Equalizer If the only circuit parameter of concern is group delay, maintaining constant group delay would be easy. For example, a simple notch lter circuit has a positive phase and therefore a negative group delay. Using a notch lter to compensate for group delay would lter the frequencies that need to be corrected! Some lter types have better group delay responses than others. The Bessel-Thomson lter has an optimally at group delay response. This lter generally has unacceptable stop-band attenuation and a wide transition band. The group delay variation occurs when the amplitude of an ampli er is starting to fall o . A circuit with a positive slope of equal magnitude where the ampli er magnitude starts decreasing will smooth the group delay. A circuit that meets this description can be described as an equalizer [11]. It?s called an equalizer because it attens or \equalizes" the gain. A common equalizer is based on the pi attenuator. This design does not work well for ICs though. Passives, especially inductors, take up too much space to be used without discretion. A simpler equalizer is given in Figure 4.2. This equalizer only requires one inductor. 67 Figure 4.3: Equalizer plots a) impedance b) phase c) group delay The impedance of the equalizer circuit is y = R + j!L1 !LC (4.1) Assuming that it is in parallel with a 50 load, ZL, the impedance is Z = ZLYZ L +Y (4.2) The values of the components are chosen to meet system speci cations.L and C are chosen to adjust the resonant frequency and Q-factor. While maintaining the same resonant frequency higher inductor values give lower Q-factors. Higher Q-factors provide a steeper slope; lower Q-factors provide wider bandwidth. Lower resistance values give a less steep and more linear slope than high resistance values. The resistance values are limited by the return loss of the equalizer though. Away from resonance, the the impedance of the circuit approaches R for low values. This decreases the matching of the circuit. Figure 4.3 shows plots of the impedance, phase and GD of the circuit. The range of R values is from 25 to 150 . 68 One reason that the circuit is so convenient is that it can be inserted into existing designs. This may be easier than re-designing an ampli er to meet group delay speci cations. Also, standard ampli er design processes can be followed, and the equalizer can be added at the end if it is needed. Before an application is given, a summary is useful. Wide bandwidth signal can su er from phase nonlinearity. The phase nonlinearity can be characterized by the group delay which is the negative rate of change of the phase versus frequency. Constant group delay is good and represents linear phase. Varying group delay represents phase distortion. Stretch processing can be vulnerable to varying group delay. The group delay shows up in the stretch processed spectrum as spreading the bandwidth of the single-tone frequency. Phase nonlinearity is usually the worst at pass band edges of ampli ers and lters. An equalizer can be inserted into a design to alleviate the group delay variations at a pass band edge. Note It could be a good idea to design a DDS to pre-compensate for both group delay and amplitude variations. Correcting issues digitally may be preferable to changing a hardware design. 4.1.2 LNA with Equalizer Narrowband LNA design techniques for SiGe technology are widely available. Particu- larly, the design techniques presented in [5] result in well matched, high performance LNAs. As the previous references suggest, the cascode is a good con guration for LNA design. The cascode con guration is a common-emitter ampli er with a common-base ampli er connected to the output. This design has several advantages: 1. The Miller capacitance of the common-emitter ampli er is reduced by connecting it to the common-base ampli er. This increases the high frequency performance. 2. The isolation S12 between the input and the output is increased relative to just a common emitter ampli er. 69 Figure 4.4: Cascode LNA 3. Because of the high frequency performance of the cascode, there is a plethora of liter- ature discussing design techniques. A disadvantage of the cascode is that it requires a higher voltage because of the two transis- tors are stacked. A simpli ed schematic of the cascode is shown in Figure 4.4. The design process discussed in [5] is loosely followed in the following steps to determine the parameters of the LNA. With any LNA design power consumption, noise gure, gain, linearity, stability, and maybe the bandwidth have to meet certain speci cation. Usually, there are trade o s for increasing any one performance metric. For example, reducing power consumption may reduce gain and linearity. 1. The st step is to determine the optimum current density for the cascode. This can be done in simulation by setting the emitter lengths at a mid-range value and sweeping the collector current. Find the collector current that gives the lowest NFmin and calculate the current density at this collector current. Since the optimum noise gure is not very sensitive to Ic changes, it may be increased from the optimum value to increase gain and linearity. The current density is the ratio Ic to emitter area. 70 2. A NFmin has an optimum source resistance associated with it. This is the value that the source resistance needs to be matched to for achieving the minimum noise gure. Assuming that the LNA is designed for a 50 system, it is desirable for the optimum source resistance Rs;opt to also be 50 . The emitter length can be scaled while maintaining the same current density to set Rs;opt. This will increase the power consumption. If power consumption is a priority, choose Ic rst. Then select emitter length to balance noise gure and gain. 3. The next step is to set LB and LE. LE provides inductive degeneration and matches the real part of the input impedance. Inductive degeneration increases linearity but decreases gain and increases noise gure. The equation for the emitter inductor is LE = 502 f T (4.3) where fT is the unity gain frequency for the particular current density. The base in- ductor cancels the base-emitter capacitance. This capacitance increases as the emitter length increases. Correctly setting LB will match noise and the input. The equation for the base inductor is LB = 1!2C BE LE (4.4) These equation are approximations. It is recommended to use the values as starting points and sweep sets of inductor values in simulation to nd the best values. 4. All that is left is the output matching. RC is used to decrease the Q-factor and increase stability. It may not be needed. The LC network can be determined using a Smith chart [13]. The preceding design steps are for a narrowband design. Recall that group delay is also a concern for this LNA. Group delay is worst around band edges. As an example, an X-band LNA is designed. After it is designed, an equalizer is added to its output. First look at 71 Figure 4.5: LNA gain with no equalizer the gain before and after the equalizer is added. Figure 4.6 shows that the LNA with the equalizer has a much atter gain. Without the equalizer the gain drops from 20.4 dB at 9 GHz to 18.3 dB at 11 GHz. The equalizer attens the gain. It varies only 0.8 dB from 9-11 GHz. The peak gain is dropped to 18.6 dB. It is important to note that the noise gure is una ected by the equalizer. The noise gure for both circuits is shown in Figure 4.7. As expected, the group delay decreases quickly at the band edge. The group delay changes by about 40 picoseconds in the circuit with no equalizer. The equalizer group delay drops by 15 picoseconds between 9 and 10 GHz before it starts increasing to the starting level at 9 GHz. There are two major drawbacks to using the equalizer. First is the added space that the circuit takes up. The inductor is especially large. Second the output matching is de- creased. The value of the equalizer resistance controls the output matching. The value of the resistor in the equalizer was selected to keep S22 below -10 dB for the whole band of interest. Figure 4.8 demonstrates how the equalizer changes the circuit from narrowband output matching to worse but wideband output matching. 72 Figure 4.6: LNA gain with equalizer 73 Figure 4.7: LNA NF Figure 4.8: a) S22 with no equalizer b) S22 with equalizer 74 Also, the resonant frequency of the equalizer can be moved lower to decrease both the bandwidth and the group delay variation. There is no need to have a wider bandwidth than needed. That will only add to the noise. In conclusion, a cascode is arguably the best con guration for high frequency SiGe LNAs. Due to their widespread use, optimized design processes are available. Adding an equalizer to a circuit allows for the use of narrowband designs while still taking into account bandwidth and group delay. 4.2 Common-Base Ampli er The common-base (CB) ampli er also has good high frequency performance because there is no Miller e ect. The input impedance for the CB ampli er is approximately 1gm. This can be much less than the typical 50 system and creates a challenge for achiev- ing a good input match. The voltage-voltage feedback solves the problem of low input impedance.Figure 4.9 shows a simpli ed schematic of a CB ampli er with voltage-voltage feedback. The idea of using voltage-voltage feedback to control input matching for a high frequency SiGe CB LNA is presented in [6]. The voltage gain of a CB ampli er is approxi- mately gmRL. The current gain is approximately 1. In a 50 system, a gain of 50gm may be too low and current gain may be needed to drive the next stage. The most obvious solution to this issue is to follow the CB ampli er with a common-collector (CC) ampli er. This resolves several issues. First, the current gain of a CC ampli er is 0 + 1. Also, the input impedance of a CC ampli er is Zin = r + ( 0 + 1)ZL (4.5) The load impedance is magni ed by 0 + 1. Since this is the impedance that the CB stage sees, a much larger gain is achieved than by just a CB stage with a 50 load. After deciding on a circuit con guration, the next steps in designing an LNA are setting the bias current and sizing the transistors. The rst step is to choose a current density Jc 75 Figure 4.9: Common-base ampli er with voltage-voltage feedback that gives a low minimum noise gure for the CB ampli er. To determine the optimum Jc, Ic is swept for a xed emitter length. Figure 4.10 shows the results of this sweep. The minimum noise gure is somewhere between 4 mA and 5 mA. 4.5 mA is chosen which corresponds to a Jc of 2.5 mA/ m2. The emitter length is determined next. Unlike the cascode, varying the emitter length is not used to set the input impedance. This allows for more freedom in setting the emitter length. Figure 4.11 is a graph of the minimum noise gure (NFmin) versus emitter length. The NFmin drops slightly as the emitter length is increased. A much smaller emitter length can be used compared to a cascode LNA at the same frequency, since the base resistance is not as important. This allows for a transistor with a high fT to be chosen. An emitter length of 20 m with a corresponding Ic of 6 mA is chosen to compromise between noise gure, power consumption, and linearity. As already described, the ampli er is a two-stage CB-CC ampli er. The implementation uses a current sharing scheme to save power. The CC ampli er is stacked on top of the CB 76 Figure 4.10: NFmin vs. IEE for determining Jc Figure 4.11: NFmin vs. emitter length 77 Figure 4.12: The interface between CB and CC ampli er. There is an LC tank circuit connected to the collector of the CB and the emitter of the CC. This tank circuit allows the DC current to pass through the inductor, but represents high impedance to the frequency band of interest. Figure 4.12 shows a zoomed in picture of the connection. The capacitor C11 in Figure 4.12 represents an ac short; thereby, letting the output signal of the CB ampli er pass to the input of the CC ampli er. Although this may look like some kind of oscillator, the ampli er is unconditionally stable in the frequency band of interest. The emitter length of the CC ampli er is 15 m. It is smaller than the CB ampli er to increase the input impedance, 0, and linearity of the CC ampli er. The calculation of the input impedance can be done using feedback ampli er theory. Zin = Zopen loopin (1 + Gopen loop) (4.6) For this circuit, Equation 4.6 reduces to Zin = 1g m + 0ZL (4.7) The 0 is present because of the magni cation of the load impedance by the CC ampli er. The input impedance is set by the output impedance. Because ZL is a function of frequency, 78 the input impedance can be tuned by tuning the output load. Ideally if is correctly chosen, the input and output impedance can be simultaneously matched for a range of frequencies by changing a single reactive element in ZL. For implementing the design, the output is matched with a lumped element matching network. The output matching is done before adding any feedback. A Smith chart is used to design the matching network by moving the output load to S 22. Next, the feedback is added. Figure 4.13 shows the schematic of the whole ampli er. The biasing circuitry is not shown. The feedback network is formed by the capacitive divider consisting of C25 and C17. = C25C25+C17 . C25 is chosen as small as possible so that it will be high impedance to the operating frequency. This is needed to keep it from a ecting the output matching network. For this reason, C25 is set at 100 fF which is the lower limits of the technology. Unfortunately, there is still a noticeable impact on the output matching. In the current implementation, C17 = 1.1 pF. This results in = 0:08. This feedback factor is small enough that the gain is not a ected very much, but it is the required factor to match the input. The two capacitors are also theoretically noiseless elements, so the noise gure should not be a ected by them. The supply voltage is 2.2 V and the current consumption is 6 mA resulting in a power consumption of 13.2 mW. One issue to consider is that when the bias circuitry is inserted, the ideal current source will be gone. The signal injection still needs to see high impedance here. This could pose an issue for an on-chip inductor. A similar implementation has used a high-quality surface mount inductor o chip. An improvement is the elimination of a large o -chip inductor at the signal injection. A tank circuit with a much smaller inductor realizes the high impedance that needs to be seen by the input signal. 4.2.1 CB-CC Simulation Results The simulation was done in Cadence IC Design 6.1. S-parameter, periodic steady state (for linearity analysis), and dc simulation where required to get all the results. Figure 4.14 79 Figure 4.13: Schematic of CB-CC LNA 80 Figure 4.14: S11 and S22 are simultaneously matched shows the simultaneously matched input S11 and S22. Recall that this is with no standard input matching network, only the capacitive divider feedback. Figure 4.15 is the real part of the input impedance. It reaches 50 right at 8 GHz. Without the feedback, the input impedance would be much less than 20 . A possible use for this LNA circuit is for use in non- fty Ohm systems since the input impedance is easily set. The noise gure is 1.7 dB at 8 GHz. This is with ideal passive components, but it should remain reasonable when 0.13 m technology library parts are exchanged for ideal parts. Figure 4.16 is the NF and the NFmin. There is only 0.5 dB di erence between the NF and NFmin. Figure 4.17 is the gain of the LNA. Although the peak of the gain is not exactly where S11 and S22 are matched, there is plenty of gain at 8 GHz. The gain variation at 8 GHz is not signi cant for narrow bandwidths. The gain at 8 GHz is 18.6 dB. The ampli er simulated above has a novel architecture and excels in some performance areas. The gain, matching, power consumption, and stability are all exceptional. This ampli er may be a viable alternative to the cascode structure. Setting S11 only by using 81 Figure 4.15: Input impedance of CB-CC Figure 4.16: NF and NFmin comparison 82 Figure 4.17: CB-CC Gain the output load is especially desirable. It requires three inductors which is the same as the standard cascode structure. 4.1 is a performance summary of the LNA. Table 4.1: CB-CC performance summary Frequency 8 GHz Gain 18.6 dB Power 13.2 mW NF 1.7 dB S11 -25 dB S22 -23 dB P1dB -19 dBm IIP3 -9 dBm 4.2.2 Integrated Duplexer The authors of [8] and [9] have suggested the possibility of integrating a duplexer using a hybrid transformer [10] and electrical balance. A hybrid transformer has several useful characteristics: 83 Figure 4.18: Autotransformer used as hybrid transformer All ports can be simultaneously matched. The ports are bi-conjugate. This means that TX and RX can be electrically isolated. The incoming power from one port can be split into any ratio of power to the receiving ports. The duplexer is realized with an autotransformer. An autotransformer is a simple imple- mentation of a hybrid transformer. It has only one winding and a center tap. The au- totransformer can be fabricated on an IC with a di erential inductor. This simple hybrid transformer is shown in Figure 4.18. The autotransformer can be matched with the following set of equations: Rc = Rb Ra = 2Rb Rd = Rb=2 84 Figure 4.19: Duplexer Some port terminations in the duplexer cannot be 50 . If the ports are matched, Ra and Rd are isolated; and Rb and Rc are isolated. [2] selected the con guration in Figure 4.19 as the optimum con guration of the autotransformer. The power ampli er cannot be integrated, so it is o -chip. A major concern for the duplexer is that it cannot handle the power from the power ampli er. The TX to RX isolation is very good. It is also very wide bandwidth. The bandwidth of the isolation is limited by parasitics. Rbal in Figure 4.19 is a tunable resistor. The power ampli er output appears as common-mode voltage to the LNA. The LNA and the power ampli er do not have 50 matched states. This is why the CB-CC ampli er already described in this section is viable for the LNA. The LNA has to be modi ed to be di erential. The input impedance should be 100 if the antenna impedance is 50 . The duplexer and LNA were simulated with modest results. The S-parameters are in Figure 4.20. S22 is the only concern with slightly more than -10 dB in the frequency band. The insertion loss for the PA is about 5 dB.A higher tuning resolution should reduce this. The isolation from the PA to the LNA is abot -50 dB and very wideband. The reverse 85 Figure 4.20: S-parameters for LNA with duplexer 86 Figure 4.21: Duplexer isolation isolation from the LNA output to the PA is greater than 100 dB. Figure 4.21 is the isolation. The noise gure is made much worse by adding the duplexer. The insertion loss of the RX adds to the noise. It is slightly less than 5 dB as shown in Figure 4.22. 87 Figure 4.22: Duplexer and LNA NF 88 Chapter 5 Software 5.1 Test Setup The ROC package has about 100 pins. Since the ROC had to be carefully tested, dozens of testing pins were used to monitor the system at di erent points during transmitting and receiving. Over thirty pins are used for VDD, GND, and voltage references. There are also pins for clocks, connecting lters to the IC, and the ADC outputs. The point is that pin real estate on the IC is scarce. This makes serial communication necessary for controlling the dozens of control bits for the ROC. A serial peripheral interface bus or SPI was designed to control the ROC. A SPI has a master and a slave. It only requires four lines for communication: SPI DI: serial data input SPI DO: serial data output SPI CLK serial clock SPI CS chip select Figure 5.1 is a simple diagram of how the spi communication is wired. The ROC is controlled with a Xilinx Spartan-6 FPGA on an SP601 evaluation board. The SP601 is the master Figure 5.1: SPI diagram 89 Figure 5.2: ROC communication diagram device in the SPI. The ROC is the slave. The ROC itself is on custom designed PCB for testing. The SP601 communicates with a computer through a USB-to-UART device. The USB-to-UART device converts serial data transmission from the Spartan-6 to comply with USB transmission. This allows the SP601 to communicate with a computer through a standard USB connection. UART stands for universal asynchronous receiver/transmitter. It translates the parallel data form internal to the Spartan-6 to serial form. The pins for UART transmission are provided on the Spartan-6 and connected to the USB-to-UART chip on the SP601. The UART itself has to be custom designed by the user. The USB-to-UART device manufacturer provides a free Virtual COM Port device driver which allows the USB-to-UART device to appear as a COM port on a computer. The block diagram for how the ROC communicates with a computer during testing is shown in Figure 5.2. The tester needs an intuitive, quick way to communicate with the ROC. Designing good software will save time and reduce errors in testing. 5.2 Software Needs To begin with, the features and functionality of the software need to be de ned. The most important function of the software is to send updates to the ROC. This most basic functionality could be provided with communication software such as HyperTerm. This would require sending a string of ascii characters such as \asdf." There is no information about what is being sent to the ROC. Ideally, information about the ROC control bits is available to the user. The ROC control bits are divided into di erent 90 registers such as the PLL register, ADC register, or DDS register. The software should be able to select a register or individual bits of a register to update. The current state of each register should also be available. With hundreds of di erent control bits in a dozen registers, con guring the ROC to a particular state can be cumbersome. Saving and loading ROC states will save time and enable consistent test setups. The saved format of test states should also allow editing or the creation of new test states. Modular software design permits the software to be dynamic and easily updated. New needs may arise during testing. Communication protocol with the ROC may change if the SP601 is changed or replaced. Changing as little of the software as possible as conditions change ensures that testing time will not be lost. The software needs to be platform independent. For obvious reasons, the software does not need to be tied to any one computer. The software has to be able to read and store information from the ADC. Viewing the time and frequency data from ADC reads is another good feature. The user interface has to be intuitive and easy to learn. One option is a command line interface. A command line interface is inherently not as easy to use as a well-designed GUI. A command line interface requires a user to navigate tiers of menus for each action. A GUI, while taking more development time, is a better option for ease of use. 5.3 Platform Choice Python is the language chosen for software development. The Python v2.7 help docu- mentation describes Python as follows [12]: Python is an easy to learn, powerful programming language. It has e cient high-level data structures and a simple but e ective approach to object-oriented 91 programming. Python?s elegant syntax and dynamic typing, together with it?s interpreted nature, make it an ideal language for scripting and rapid application development in may areas on most platforms. The Python interpreter and exten- sive library are freely available in source or binary form for all major platforms form the Python website... and may be freely distributed. The same site also contains distributions of and pointers to may free third party Python modules, programs and tools, and additional documentation. What is not to like about Python!? One key thing about Python is the plethora of third party packages that address the needs of the ROC control software. Python is easy to learn. It is also free. There are third party packages that address COM port communication, GUI development, and Excel spreadsheet creation and editing. The ease of use of any pro- gramming language is highly dependent on the integrated development environment (IDE). Eclipse with the pydev extension was chosen as the IDE. Like Python, Eclipse is free. It provides real time parsing, code completion, debugging, a le explorer, a code outline, and a command line interface. 5.4 Code Structure 5.4.1 GUI Python has several options for GUI development. PyQt was chosen for this project. PyQt is advantageous because it has a low learning curve before a useful GUI can be cre- ated. From PyQt?s website, \PyQt is a set of Python bindings for Nokia?s Qt application framework... The bindings are implemented as as a set of Python modules and contain over 300 classes and over 6,000 functions and methods." The advantage of being able to use Qt for GUI development is using Qt Designer. Qt Designer allows for the graphical design of a GUI. A GUI can be built by dragging and dropping objects such as menus, tables, or buttons onto a window. Once the GUI is 92 built and all of the objects are labeled intuitively, it is converted to a python class through a command line tool provided with PyQt. An instance of the class is then created in the main Python program. This process allows the GUI to be updated seamlessly as long as object names do not change. Once the user interface is added to the main program, it has to be connected to other functions to correctly handle all events. An event is any input or action to the GUI that elicits a response i.e., a button press, a menu selected, text entered, or a tab changed. This is done using commands that connect an event to a speci c function. An event can have multiple functions connected to it. A function may have multiple events connected to it. 5.4.2 ROCDriver The core of the software is a class representing the ROC driver. This class contains all information about the ROC state, the default values, the registers, the register addresses, and functions required for updating the ROC. The ROC SPI has twelve registers. Each register has an address that is prepended to to the updates of the control bits. The registers are de ned as follows: PLL SPI is a 64-bit register containing all of the control bits for the PLL. ADC SPI is a 20-bit register containing all of the control bits for the ADC. CTRL SPI1 is a 20-bit register controlling the overall operation of the ROC. CTRL SPI2 is a 64-bit register that controls the wait time after transmitting and after receiving. DCDO SPI1 is a 64-bit register that controls the start and stop frequency of the trans- mitted LFM waveform. DCDO SPI2 is a 64-bit register that controls the start and stop frequency of the reference LFM waveform during receiving. 93 DCDO SPI3 is a 64-bit register controlling the initial phase and the frequency step of the LFM waveform. DCDO SPI4 is a 15-bit register controlling miscellaneous DDS functions such as DAC gain control and inverse sinc lter enable. PA SPI is a 3-bit register enabling the power ampli er driver and setting the power ampli- er driver current. PHASE SPI is not implemented in the current ROC fabrication TX SPI is a 26-bit register that controls transmitter options such as lter tuning, VGA control, and transmitter powerdown. RX SPI is a 26 bit register that controls receiver options such as LNA select, mixer current controls, and VGA control. The ROCDriver class contains all relevant information about the ROC. The necessary infor- mation is a map relating each control bit to its register, register addresses, default control bit values, current control bit values, the location of the control bits within a register, and functions necessary for communication. One third party package is needed for the ROCDriver class. The package, bitstring allows for the easy storage and manipulation of arrays of bits. Each control bit can be stored separately and easily concatenated to form the register. The GUI operations are divided into ve di erent tabs. One tab is devoted to updating the control bits and showing the current values of the control bits. Individual control bits can be updated or entire registers can be updated. Figure 5.3 is a screenshot of the ROC interface software with the SPI tab active. The register is selected from the list on the left The control bits are shown in the table to the right. The columns are the bit numbers, bit name, current value, the value to be written on update, and a column to enter new bit values. Once the new values are entered, the \To Be Written" button is pressed. This updates the 94 Figure 5.3: ROC Interface SPI tab \Write on Update" column. Once all of the registers are updated, the large button at the top of the page labeled \Update SPI" is pressed to actually transmit the data to the ROC. 5.4.3 Serial Communication The third party package pySerial provides access to the windows COM port. It can create an object for the COM port that can be opened, closed, read from, and written to. As long as the correct COM port is chosen, communication to the ROC is trivial. The ROC Interface software has a tab devoted to opening a COM port. It scans the computer for open COM ports rst. The correct COM port is then selected and opened. A message appears stating the success or failure of the port opening. Figure 5.4 is a screenshot of the communication tab. 95 Figure 5.4: ROC Interface communication tab 96 5.4.4 Load and Save ROC States Loading and saving ROC test states is very valuable to the tester. This is accomplished in the ROC Interface software with the use of Excel spreadsheets. Three companion third party packages are needed to enable the use of Excel spreadsheets. xlrd is used to import and read spreadsheets. xlwt is used to edit and save spreadsheets, and xlutils is used to copy spreadsheets. When a test state is saved, all of the control bit names are put into a column sorted by registers and bit order. The bit values are placed in the column to the right. When a test state is loaded, each cell is checked to see if it matches a control bit name. If it matches a name, the cell to the right is assumed to have the values. The values are error checked to make sure it is a valid entry. This allows for test states to be created and edited in Excel. Other descriptive information can be added to the spreadsheet without causing error also. The Excel tab of the ROC Interface is divided into a \Load Test State" section and a \Save Test State" section. The \Load Test State" section opens a workbook rst. Once a workbook is loaded, the worksheet is selected. The \Load" button updates the \To Be Written" data for all of the control bits. The data is sent to the ROC once the \Update SPI" button at the top of the GUI is pressed. The \Save Test State" can save a test state to an existing workbook or create a new workbook. Worksheets can also be overwritten. There is a eld to enter a description. The description is displayed when the state is selected to load. Figure 5.5 is a screenshot of the Excel tab. The top is the load section, and the bottom is the save section. 5.4.5 Tools Tab The ROC Interface has a tab for miscellaneous operations. For now its main function is to calculate the correct 32-bit word for a given frequency. This function is necessary for the start and stop frequencies of the transmit and receive LFM waveform. 97 Figure 5.5: ROC Interface Excel tab 98 Figure 5.6: ROC Interface ADC tab Future versions will calculate control bits to specify certain radar parameters such as PRF or range resolution. 5.4.6 ADC Tab The ADC tab has all of the functions necessary for reading, saving, and viewing data from the ADC. When an ADC read command is sent to the ROC, the ADC data is rst saved to memory on-board the SP601. The memory size can be set by editing the Spartan-6 Verilog. Once the data is saved on the SP601, it is transmitted to the host computer. The ADC read may take up to a minute, and the ADC tab has a status bar showing the progress of the read. The save location and le name may be selected for each ADC read. The current data or any saved data can be plotted in from the ADC tab. A screenshot of the ADC tab is given in Figure 5.6. 99 5.4.7 Packaging Since Python and the third party packages are free, running the software on any com- puter can be accomplished by installing Python and the necessary packages. Although straightforward, installing Python and third party packages is time consuming. A list of the third party packages and their dependencies would also be required. Thankfully, there is a program named PyInstaller that converts Python programs to stand-alone executables. Once the program is thoroughly tested for crash scenarios, an executable is a very convenient way to package it. 100 Chapter 6 Summary This chapter provides a summary of the work in the preceding chapters. It will highlight the key conclusions and accomplishments. It should provide a condensed reference for the rest of the chapters. A suggestion of future work concludes the chapter. 6.1 Simulation The MATLAB simulation of the transceiver provides a quick analysis of the high-level details of the system. It can be easily modi ed to show changes in lters, frequencies, or stretch processing. Once the initial simulation is designed, the MATLAB simulation is faster and less error prone than hand calculations. The simulation is also much faster and easier to modify than a simulation in Cadence (the IC design software). The simulation can also model imperfections such as noise and interference. 6.2 Stretch Processing Stretch processing is an alternative to matched lters. A matched lter correlates a received signal with the expected signal. The result is a pulse in time when the waveforms \line up." A matched lter has optimum SNR. It is commonly implemented with SAW lters or in the digital domain. Every waveform has a unique matched lter. Multiple matched lters are required to vary the slope of an LFM waveform. A matched lter can be implemented digitally, but the bandwidth of a waveform is limited by the ADC. Stretch processing compresses LFM waveforms into a pulse in frequency. It multiplies the incoming signal with a reference LFM waveform. Stretch processing can be actualized with mixers on the IC. This allows the ADC to have a much smaller bandwidth than the 101 transmitted waveform. The reference waveform can come from the same waveform generator that supplies the transmitted waveform. Several constraints dictated how stretch processing could be designed for the ROC. One ADC was used on the ROC. This means that I-Q processing in the digital domain is not possible. The bandwidth of the ADC is also a limiting factor. The destretched or range frequency cannot exceed the bandwidth of the ADC. Since there is no digital I-Q processing, negative frequencies have to be distinguished from positive frequencies in the analog domain. Negative frequencies correspond to returns signals that arrive before the reference waveform starts. The ROC receiver resolves negative and positive ranges. The stretch processing on the ROC has one serious problem. The destretching is per- formed by mixing the received baseband signal with the reference LFM waveform at base- band. The di erence of the two waveforms is the single-tone signal. The sum of the signals is another LFM waveform at baseband. This sum or chirp-squared waveform usually does not a ect the corresponding di erence signal, but it can hide other signals that have lower power. The solution is to do the stretch processing at the IF. This requires quadrature IF+LFM waveforms. The in-phase waveform is already available. The quadrature phase signal is generated with a few extra mixer. The rest of the stretch processing remains the same. Figure 6.1 is the diagram for the improved implementation. 6.2.1 Analysis This section summarizes the key conclusions and equations from the ROC stretch pro- cessing analyses. To start with, the basic results for stretch processing are = fM (6.1) r = c2 (6.2) 102 Figure 6.1: Updated Stretch Processing 103 where is the delay beteween the reference waveform and the incoming signal and r is the relative range and is the slope of the LFM waveform. The pulsewidth of the reference waveform and o set frequency need to be set to ensure that possible ranges are completely processed and returns from outside of the range window are not processed. The equations for the reference pulsewidth is T = time of transmitted pulse + 2fcutoff (6.3) The o set frequency should be equal to the cuto frequency. The range window is one of the biggest disadvantages of stretch processing. Stretch processing can only occur as long as the reference waveform is \on." The reference waveform is limited either by the DDS or the system bandwidth capabilities. The equation for the range window is range window = c2frange (6.4) Doppler frequency directly adds to the range error of stretch processing. In most scenar- ios, the error is small though. The error cause by an object traveling with a radial velocity of vr is Rdop = vrfC (6.5) where fc is the carrier frequency. Group delay variation is a measure of phase distortion. Wideband LFM waveforms are susceptible to group delay variation. Group delay causes a reduction in SNR and worse range resolution. The group delay spreads the target frequency. The power loss caused by group delay is BWGD = GD (6.6) Ploss = 10 log 1 TchirpBWGD (6.7) 104 The range resolution because of group delay is R = c2 1BW GD (6.8) I-Q imbalance is an issue for any quadrature processing. In the ROC stretch processing, the channel inequalities cause an image frequency to occur with a power of Pimage = A 2 2ABcos( ) +B2 2 (6.9) where A and B are amplitudes and is the phase di erence. The processing gain is the ratio of output SNR to input SNR. Maximum processing gain is equal to the time-bandwidth product of the transmitted LFM waveform. It can be degraded if the noise bandwidth is larger than the LFM transmitted bandwidth. Where Bn is the noise bandwidth, Br is reference LFM bandwidth and Bs is the transmitted LFM bandwidth: Same as matched lter for Bs = Bn Loss is proportional to Br=Bs is the noise bandwidth is larger than the reference signal Loss proportional to Bn=Bs if the noise bandwidth is smaller than the reference band- width 6.3 Circuits Group delay variation is worst at the band edges of lters and ampli ers. Group delay variation compensation can be had by the use of an equalizer. An LNA can be designed using narrowband LNA design techniques, and an equalizer can be added at the output to smooth gain and group delay without a ecting noise gure. A common base{common collector with voltage-voltage feedback is a possible alternative to a traditional cascode. It has tunable input matching, good high frequency performance, 105 and input matching that does not depend on emitter lenght. The circuit tested used shared DC current by using a tank to make the connection appear as an open to the frequency operating range. 6.4 Software The ROC has hundreds of control bits that are controlled with a SPI. The ROC com- municates with a Xilinx Spartan-6 development board. The development board appears as a COM port to the computer. Testing software needs to provide a quick intuitive way to communicate with the ROC. Python is the platform used for software development. It is free along with useful third party packages. PyQt allows for streamlined GUI development. The software has the following features: Communicates with the ROC Can update bits individually or as a register Save and load test states using Excel spreadsheets Read and view data from the ADC Built into an executable 6.5 Future Work If the ROC is re-fabricated the improved stretch processing should be used. The stretch processing analysis needs to be compiled into a more useful form by mak- ing a software tool based on it. The user can input some system features such as range resolution and maximum bandwidth, and the other parameters are calculated accordingly. 106 The testing software can be updated to be able to control radar operation. Instead of a user entering control bits, radar parameters like PRF and range resolution are de ned. The integrated duplexer is an interesting circuit. Further research and maybe fabrica- tion needs to be done to see if it really is viable. 107 Bibliography [1] Skolnik, Merrill. Radar Handbook. 3rd ed. McGraw-Hill, 2008. [2] T. Long, Y. 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