DESIGN OF DIRECT DIGITAL FREQUENCY SYNTHESIZER FOR WIRELESS
APPLICATIONS
Lakshmi Sri Jyothi Chimakurthy
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 8, 2005
DESIGN OF DIRECT DIGITAL FREQUENCY SYNTHESIZER FOR WIRELESS
APPLICATIONS
Lakshmi Sri Jyothi Chimakurthy
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
_________________________________
Signature of Author
__________________________________
Date
Copy sent to:
Name Date
iii
VITA
Lakshmi Sri Jyothi Chimakurthy, eldest daughter of Sri Rama Murthy and Sudha
Rani Chimakurthy, was born in Chirala, India. Her father is one of the leading
entrepreneurs in Hyderabad, India whose company manufactures Industrial transformers,
Inductors, UPS and a wide variety of electrical equipment.
Jyothi has received her Bachelor?s degree in Electronics and Communications
Engineering from Jawaharlal Nehru Technological University, Hyderabad, India in July,
2002. During her Bachelors she had done an internship with the Defense Research and
Development Laboratories, Hyderabad, India. She had joined Auburn University to
pursue her Masters degree in the area of RF Analog /Digital mixed signal circuit design
for wireless applications. She worked at AMSTC as a graduate research assistant under
the guidance of Dr. Foster Dai and Dr. Richard C. Jaeger.
iv
THESIS ABSTRACT
DESIGN OF DIRECT DIGITAL FREQUENCY SYNTHESIZER FOR WIRELESS
APPLICATIONS
Lakshmi Sri Jyothi Chimakurthy
Master of Science, August 8, 2005
(B-TECH, Jawaharlal Nehru Technological University, 2002)
94 Typed Pages
Directed by Foster Dai
Direct Digital Synthesis can be practically defined as a means of generating
highly accurate and harmonically pure digital representations of signals. High speed DDS
presents an attractive alternative to use of Phase locked loop approach in the design of
high bandwidth frequency synthesizers because of its features like sub-hertz frequency
resolution, fast settling time, continuous phase switching response and low phase noise.
Although the principle of the DDS has been known for many years, the DDS did not play
a dominant role until recent years. Earlier DDSs were limited to produce narrow bands of
closely spaced frequencies, due to limitations of digital logic and D/A converter
technologies. Recent advances in integrated circuit technologies have brought about
remarkable progress in this area. By programming the DDS, adaptive channel bandwidths
modulation formats, frequency hopping, and data rates are easily achieved. This is an
important step towards applications like software radio which can be used in various
systems. The DDS could be applied in the modulator or demodulator, in the
v
communication systems. The applications of DDS are restricted to the modulator in the
base station.
One of the important factors determining the spectral purity of DDS is the
resolution of the values stored in the sine look up table (ROM). Increasing the size of the
ROM for better spectral purity is not a good approach because of the higher power
consumption, lower speed and increased cost of a larger ROM. The first part of this thesis
discusses the design and implementation of phase accumulator in the high speed ROM
less DDS with sine-weighted DAC. The basic logic blocks are implemented in SiGe
technology. The second part of the thesis proposes a novel DDS architecture with a
compressed ROM without degradation of quantization noise. The ROM compression
algorithm proposed achieved a better compression ratio of 94.3:1 with little increase in
hardware when compared to many popular compression algorithms, giving a worst case
spur of -90.3 dBc.
vi
ACKNOWLEDGEMENTS
I would like to begin by thanking my parents Sri Rama Murthy and Sudharani,
my siblings Sri bala and Uday kiran. Over the years my family made numerous sacrifices
so that I could pursue my career goals. I would like to thank them for their support,
blessings and well wishes. Secondly, I would like to thank my close friends Malinky
Ghosh and Blessil George for their strong companionship and emotional support which
was very important for my fruitful research. I would also thank my cousins Subba Rao
and Mohan for their encouragement and confidence in me.
I am greatly indebted to Dr. Foster Dai, for being my advisor and teacher for the
last two years. He had constantly motivated me to work towards excellence. Despite my
repeated failures, his enthusiasm and strong support motivated me to finally succeed in
this project. I would like to express my deepest gratitude to Prof. Richard C. Jaeger for all
his support, valuable suggestions and making my graduate school life, a rich and
rewarding experience. He has constantly amazed me with his profound knowledge and
ability to simplify complicated research problems. I thank Dr. Niu for his complete
cooperation. I would like to thank my colleagues Dayu yang and Qi Xiu for their ideas
and helping me to successfully take the FPGA measurements. Life was never dull over
the last two years, thanks to the exciting and entertaining moments created by all the
friends I met at Auburn. The memorable times spent with Anand, Sailaja and Prasanthi
will be fondly remembered for years to come.
vii
Style manual or journal used IEEE Transactions on Circuits and Systems
Computer software used Microsoft Word 2000
viii
TABLE OF CONTENTS
LIST OF TABLES???????????????????????????.x
LIST OF FIGURES?.?????????????????????????..xi
1 INTRODUCTION ?????????????????????....?1
1.1 Direct Digital Frequency Synthesizer????????????..... .1
1.2 Overview of Work ?????????????????..?.........7
2 ROM-LESS HIGH SPEED DDS...???????????.?????...9
2.1 ROM-Less DDS Architecture ???????????????....10
2.2 Algorithm used for DAC Implementation??????????......10
2.3 Phase accumulator???????..???????????..?..12
2.3.1 SiGe Technology????.????????????..?.13
2.3.2 Background on CML and ECL circuit design????............14
2.3.2.1 DC Operation?????...????????..?..14
2.3.2.2 Transient Analysis?????...??????..?..15
2.4 Analysis of two input and three input NAND gate??????..?...18
2.5 Pipelined accumulator???????????..???.???......22
2.6 New Pipelined Accumulator Architecture??????..??............25
2.6.1 Full adder circuit???????.????.???..??.....25
2.7 Accumulator with Carry look ahead adders????????............29
2.7.1 Carry look ahead adder delay analysis???????..?......33
2.8 Modifications to the 4 bit CLA Adder???.????????...?34
2.8.1 Accumulator????????..?????????..........34
2.8.2 Carry look ahead adder delay analysis????..???..........37
2.9 Cosine Weighted Digital to Analog Converter?????????...37
3 NOISE ANALYSIS OF DDS OUTPUT SPECTRUM...???????....40
3.1 Reference clock????????????????.?????..41
3.2 Phase truncation??????????????????..............43
3.2.1 Mathematical Analysis of Phase truncation spurs?..???.. .44
3.3 Over sampling??????????????????????.50
3.4 Angle-to-amplitude mapping????????..???....................51
3.5 Quantization noise, DAC nonlinearities???????????.....55
4 A NOVEL DDS ARCHITECTURE WITH IMPROVED
COMPRESSION RATIO AND QUANTIZATION NOISE??????...57
4.1 Back ground of ROM Compression Algorithms???????..?..58
4.2 Algorithms commonly used in all ROM reduction techniques ???.59
4.2.1 Sine Quadrature Symmetry Algorithm??.??????..?59
4.2.2 Sine Phase Difference Algorithm?????????..?.?60
ix
4.3 Proposed Architecture???????????.?..?????.?.61
4.3.1 Generic Architecture???.???????..??????61
4.4 Quantization Error Analysis????????????????...65
4.5 Proposed Architecture with Non-linear Addressing???????...67
4.6 Simulation Results????????..??..??..............................69
4.7 FPGA Measurements?????.????????????...?..75
A MATLAB CODE???????????????????????.78
A.1 TRADITIONAL ROM DDS ARCHITECTURE????????...78
A.2 LINEAR ROM DDS ARCHITECTURE???????????...80
A.3 NON LINEAR ROM DDS ARCHITECTURE?????????..85
x
LIST OF TABLES
I. Dimensions of DDS with single ROM??????????????.....64
II. Memory word lengths versus worst case spur for non-linear addressing??.72
III. Memory lengths versus worst case spur for non-linear addressing???......73
IV. COMPARISION BETWEEN THE NICHOLAS? ARCHITECTURE
AND OUR PROPOSED ARCHITECTURE FOR P = 15 BITS?????..74
V. COMPARISON BETWEEN VARIOUS PROPOSED
ARCHITECTURES FOR P=12 BITS......................................................?...75
VI. FPGA Specifications????...?????????????????..76
xi
LIST OF FIGURES
1.1 A Conventional ROM Based DDFS Architecture?????????...2
2.1 High-speed DDS with a nonlinear cosine-weighted DAC??????10
2.2 ECL circuit diagram????????????????????...14
2.3 CML Circuit diagram????????????????????.15
2.4 Schematic showing DC analysis of NAND gate?????????...19
2.5 Schematic showing DC analysis of Level shifter?????????..19
2.6 The output of NAND gate showing a worst case propagation
delay of 26ps???????????????????????..20
2.7 Schematic showing DC analysis of 3 input NAND gate??????...21
2.8 The output of 3 input NAND gate showing a propagation
delay of 9ps????????????????????????21
2.9 Delay of 3 input AND gate is 50ps and 4 input AND gate is 79ps??...22
2.10 Generic Architecture of N x M Pipelined Accumulator?????????..23
2.11 Output waveform of an 8-bit fully pipelined adder????????...24
2.12 Test bench of new full adder circuit??????????????..26
2.13 Minimum propagation delay of traditional full adder is 35ps????...26
2.14 Worst case propagation delay of traditional full adder is 51ps????.27
2.15 Worst case propagation delay of the new full adder is 75ps.
It takes at least 90ps to reach either Vhigh or Vlow????????..28
2.16 Three level resetable latch??????????????????.29
2.17 Simulated output of an 8 bit pipelined accumulator
with CLA adders (N=4 and M=2) operating at
3.5 GHz clock frequency??????????????????...31
2.18 The Gate level 4 bit CLA adder used for implementation
of phase accumulator????????????????????.32
2.19 The comparison shows that 2D2 (delay of two 2-input AND gates)
is smaller than 1D3+D2 (delay of 3-input AND gate
and a 2-input AND gate)???????????????????35
2.20 Circuit diagram to implement logic function A+B.C????????36
2.21 Nonlinear DAC with current cells???????????????.38
2.22 Control logic of the nth DAC cell???????????????..39
3.1 DDS has four principle sources of spurs?????????????41
3.2 Spurs caused by modulation of the reference clock amplitude
are also reduced by 20log(N). The DDS?s input limiter, which
converts the amplitude modulation into a phase-modulation term,
provides additional suppression of this spur???????????..42
3.3 The DDS?s phase truncation spur mechanism models as a
noise source summed with an otherwise ideal synthesizer????????..43
3.4 A conventional phase accumulator representation with
xii
M bits truncated to L bits??????????????????...45
3.5 Spurs observed in the output spectrum due to phase
truncation, Spectrum for FCW = 2
9
+1?????????????.45
3.6 Spurs observed in the output spectrum due to phase
truncation, Spectrum for FCW = 1???????????????47
3.7 The phase truncation spurs are offset from the fundamental
by 390 kHz and its harmonics (a) The reference clock rate
and analyzing the 15 bit tuning word reveal this fact.
(b) The calculation shows how to predict the 390.625 KHz
spur spacing???????????????????????...49
3.8 Spectrum of the DDS with a traditional ROM showing the
carrier signal at about 762.93Hz and worst case spur for
FCW = 1, phase accumulator = 15 bits, DAC resolution = 8 bits,
clock freq. = 25MHz????????????????????..51
3.9 Limited phase resolution results in skipping DAC codes.
The dots at each phase increment show the calculated Va
i
?????...51
3.10 Spectrum showing the carrier signal and quantization
noise floor for FCW=1??????????????????.?..53
4.1 Block Diagram of Coarse-Fine ROM structure????????.??58
4.2 Architecture for constructing sine function using symmetry
around ?/2 and ??????????????????????...60
4.3 Block Diagram of Sine-phase difference algorithm????????..60
4.4 Graph showing
22
?
??
?
?
A
A
A
E
for memory length of 14 bits?????66
4.5 Phase to Sine conversion architecture with non-linear addressing???69
4.6 Error versus samples for P = 15 bit, ROM is 2
8
x 12 bit,
FCW = 69, linear addressing, error of the order of 10
-5
???????70
4.7 Error versus samples for P = 15 bit, nonlinear addressing,
FCW = 33????????????????????????..71
4.8 Memory word lengths versus worst case spur for linear addressing??.72
4.9 Different word-lengths and memory lengths of ROM1
and ROM2 with same spurious response????????????...73
4.10 Spectrum of DDS with linear architecture for FCW = 1,
phase accumulator = 15 bits, DAC resolution = 8 bits,
clock freq. = 25MHz????????????????????..76
4.11 Spectrum of DDS with non-linear architecture for FCW = 1,
phase accumulator = 15 bits, DAC resolution = 8 bits,
clock freq. = 25MHz????????????????????..76
xiii
1
CHAPTER 1
INTRODUCTION
Modern communication systems, especially spread spectrum systems, are placing
increasing demands on the resolution and bandwidth requirements of frequency
synthesizer sub systems in order to gain improved performance. Today?s spread spectrum
applications require a frequency synthesizer that is capable of tuning to different output
frequencies with extremely fine frequency resolution with a switching speed of the order
of nanoseconds. The resolution requirements of many systems are so severe that they are
surpassing the performance capabilities of conventional analog phase locked loop.
Although limited by Nyquist criteria, DDS allows frequency resolution control on the
order of milli-hertz or even nano-hertz of phase resolution control. The increasing
availability of high speed DACs make a direct digital approach to frequency synthesis an
enticing alternative to conventional analog synthesizers.
1.1 Direct Digital Frequency Synthesizer
The simplest architecture of DDFS uses an N-bit accumulator, a read only
memory (ROM) and a digital to analog converter, implemented using the same reference
clock f
clk
as is presented by Tierney Rader, Gold et al [1]. A conventional ROM-based
DDFS is show in Figure 1.1. The output frequency of the DDFS depends on the W bit
input word called Frequency control word (FCW). At each positive edge of the reference
2
clock cycle the FCW is added to the value in the W-bit accumulator. At any instant the
value in the accumulator represents the phase of the sinusoid, so it is referred as phase
accumulator register. The value in the phase accumulator register is used to address the
ROM storing the values of the sinusoid. And its resolution is shown in Figure 1.1. The
output frequency of the DDS depends on f
clk
as
2
N
clkout
FCW
ff = (1.)
The finest resolution possible is
2
N
clk
res
f
f =
(1.2)
Figure 1.1 - A Conventional ROM Based DDFS Architecture
DDS is a digital technique for generating an output waveform (sine, square or
triangular) or clocking signal from a fixed-frequency clock source.
Some DDS advantages:
? Micro Hertz frequency resolution and sub-degree phase offset resolution.
3
? Extremely fast frequency transition or ?hopping speed? : <5 ns to change frequency.
? All digital control eliminates need for manual system tweaking and allows output
frequency to be conveniently adjusted.
? Easily synchronized allowing quadrature and other exact signal phase relationships to
be obtained.
? Unparalleled matching of I and Q outputs.
? Provides accurate, high speed PSK (phase shift keying) and FSK (frequency shift
keying).
DDSs are found in a wide range of applications:
-waveform generation
-impedance measurement
- sensor excitation
- digital modulation/demodulation
- test and measurement equipment
- clock recovery
- programmable clock generator
- liquid and gas flow measurement
- sensory applications
- medical equipment
- HFC data, telephony, and video modem
- FM chirp source for radar and scanning systems
- agile, LO frequency synthesis
- commercial and amateur RF exciter
4
- wireless and satellite communications
- cellular base station hopping synthesizer
- broadband communications
- VHF/UHF-LO synthesis
- tuners
- military radar
- automotive radar
- SONET/SDH clock synthesis
- acousto-optic device driver
- PSK/FSK/Ramped FSK modulation
Many companies like Analog devices, Qualcomm etc. have integrated Complete-
DDS products [2] which present an attractive alternative to analog PLLs for agile
frequency synthesis applications. Direct digital synthesis (DDS) has long been
recognized as a superior technology for generating highly accurate, and frequency-agile
(rapidly changeable frequency over a wide range), low-distortion output waveforms.
A major advantage of a DDS system is that its output frequency and phase can be
precisely and rapidly manipulated under digital processor control. Other inherent DDS
attributes include the ability to tune with extremely fine frequency- and phase resolution
(frequency control in the millihertz (mHz) range and phase control < 0.09?, and to rapidly
"hop" in frequency (up to 23 million output frequency changes per second). These
characteristics have combined to make the technology extremely popular in military radar
and communications systems. In fact, DDS technology was previously relegated almost
exclusively to high-end and military applications: it was costly, power-hungry
5
(dissipations specified in watts), difficult to implement, required a discrete high-speed
DAC, and had a set of user-hostile system interface requirements.
DDS should be uniquely attractive for local oscillator (LO) and up/down
frequency conversion stages-which were until now the exclusive domain of PLL-based
analog synthesizers. The Complete-DDS architecture by Analog devices holds distinct
advantages over an equivalent PLL-based agile analog synthesizer. For example:
? Output frequency resolution: The AD98x0 C-DDS products have 32-bit phase
accumulators, which enable output frequency tuning resolutions much finer than a PLL-
based synthesizer can enjoy. The AD9850 has a tunable output resolution of 0.06 Hz,
with a clock frequency of 125 MHz; the AD9830 has a tuning resolution of 0.012 Hz,
with a reference clock of 50 MHz. Furthermore, the output of these devices is phase-
continuous during the transition to the new frequency. In contrast, the basic PLL-based
analog synthesizer typically has an output tuning resolution of 1 kilohertz; it lacks the
inherent resolution afforded by the digital signal processing.
? Output-frequency switching time: The analog PLL frequency switching time is a
function of its feedback loop settling time and VCO response time, typically > 1 ms. C-
DDS-based synthesizer switching time is limited only by DDS digital processing delay;
the AD9850's minimum output frequency switching time is 43 ns.
? Tuning range: A critical feedback loop bandwidth and input reference frequency
relationship determines the stable (usable) frequency range of the typical analog PLL
circuit. C-DDS-based synthesizers are immune to such loop filter stability issues and are
tunable over the full Nyquist range (< 1/2 the clock rate).
6
? Phase noise: Because of the frequency division, C-DDS-based solutions have a clear
advantage over analog PLL synthesizers in output phase-noise. The output phase noise of
a C-DDS synthesizer is actually better than that of its reference clock source, while
analog PLL-based synthesizers have the disadvantage of actually multiplying the phase
noise present in their frequency reference.
? Implementation complexity: DDS which include the signal DAC, translate to ease of
system design. There is no longer an element of RF design expertise required to
implement a DDS solution. A simple digital instruction set for control minimizes the
complexity of support hardware. Digital system design replaces the analog-intensive
system design required for PLL-based analog synthesizer solutions to similar problems.
This basic DDFS architecture however incorporates a huge ROM, which restricts
high speed applications and requires huge power consumption. To reduce the power
dissipation and the die area there are two possible options:
? Implement ROM compression techniques.
? Replace the ROM and linear DAC with non linear DAC.
This work suggests a new algorithm for ROM compression with an improved
compression ratio at the same quantization noise levels. The second part of the work
involves study of ROM-less DDS with nonlinear DAC, design and implementation of
high-speed accumulator in SiGe technology for High speed DDS.
1.2 Overview of the work
In chapter 1 the operation of conventional DDS as presented by Tierney Rader,
Gold et al [1] has been discussed. DDS has been described as an attractive alternative to
analog PLL for agile frequency synthesis. Its applications and advantages and
7
comparison to PLL have been provided. Two alternatives have been suggested to
improve the speed and reduce the power consumption of the ROM based DDS. The first
one is the design of ROM-less high speed DDS with non-linear sine weighted DAC and a
novel ROM compression technique without degradation in spurious response has been
presented as a second alternative.
In chapter 2 the architecture for the ROM-less DDS has been discussed. CML
logic has been used to implement the basic logic blocks of the high speed DDS. Two
types of logic gates with levels of three transistors stacked vertically i.e. two input gates
and with levels of four transistors stacked vertically i.e. three input gates have been
proposed. A novel resettable flip-flop architecture has also been discussed. Merging logic
has been used to combine the gates in a full adder. Finally a full adder circuit with a
single current source to implement the carry logic and a single current source to
implement the sum logic has been presented. Discussion on different accumulator
architectures like the pipelined architecture and the carry look ahead adder accumulator
architecture has been made. These architectures have been implemented in CADENCE
and simulation results have been presented. An 8 bit fully pipelined phase accumulator at
6.5 GHz has been presented.
Chapter 3 discusses the noise analysis of the output of DDS, which is very
important for the frequency planning. Brief discussion about the effect of reference clock
spurs, phase truncation, angle to amplitude mapping (quantization noise due to finite
resolution of values stored in ROM), DAC?s quantization noise, and nonlinearities on the
output spectrum of DDFS has been presented.
8
In chapter 4, a novel ROM compression algorithm with improved compression
ratio, without degradation of quantization noise at the output of the DDFS has been
presented. The code for the proposed DDS architecture has been written in VERILOG
and the results are verified on Xilinx Spartan II FPGA at 25 MHz. The results of this
proposed architecture have been compared to the architecture proposed by Nicholas
which is the most popular DDS architecture with compressed ROM. Analog devices
manufactured a DDS chip based on Nicholas?s architecture.
9
CHAPTER 2
ROM-LESS HIGH SPEED DDS DESIGN
Direct digital synthesis (DDS) provides precise frequency resolution and direct
modulation capability. However, the majority of the DDS designed so far is limited to
low frequency applications with clock frequencies less than a few hundred MHz.
Digitally generating highly complex wide bandwidth waveforms at the highest possible
frequency instead of down near base band would considerably reduce the transmitter
architecture in terms of size, weight and power requirements as well as cost. This chapter
presents a low power, high-speed direct digital synthesizer (DDS) designed at Auburn
University in a 47GHz SiGe technology. The ROM-less DDS includes an 8-bit
accumulator, column/row decoders and an 8 bit cosine-weighted digital-to-analog
converter (DAC) operating at maximum 6GHz clock frequency to synthesize and
modulate the inter-median frequency (IF) of up to L-band. The DDS could achieve better
than -45dB spectral purity. The DDS core occupies an area of 2 mm
2
and consumes less
than 2W power with a 3.3V supply voltage. The 5GHz MMIC provides a frequency
synthesis and modulation means for L-band applications. The cosine weighted DAC,
eliminate the sine look up table, which is the bottleneck of speed and area for high-speed
DDS implementations. The low power consumption we achieved using SiGe technology
is much lower than that of an equivalent DDS implemented in a 137GHz InP technology
[3], where the DDS consumes about 15 W, operating at 9.2GHz clock frequency with an
10
8-bit accumulator and an 8-bit nonlinear DAC.
2.1 ROM-Less DDS Architecture
The conventional DDS architecture utilizes a ROM look-up table to convert the
accumulated phase word into sine/cosine words that are further used to drive a linear
binary weighted DAC. The speed bottleneck of a conventional DDS architecture lies
upon the large look-up table with multi-level decoders. The huge look-up table not only
restricts high speed operation, but also occupies large area and consumes large amount of
power. To reduce the power dissipation and the die size, a nonlinear DAC with cosine-
weighted current cells is employed in this design as shown in Figure 2.1. The nonlinear
DAC converts the digital phase information directly into an analog output. The proposed
architecture exploits the quadrant symmetry property of the sine function around ? /2
and ? . Thus the 2 MSB?s of the accumulator output signify the different quadrants of the
sinusoid. The remaining w-2 bits from the complementor are used to generate the
waveform of sinusoid in the first quadrant. The complete sinusoid can be generated with
the 2 MSB?s.
FCW
Phase
Accumlator
Complementor
Output
Sine
Non-Linear
DAC
W
W-2
2MSBs
2
nd
MSB
MSB
W-2
i+1
f
clk
f
out
Figure 2.1 - High-speed DDS with a nonlinear cosine-weighted DAC.
2.2 Algorithm used for DAC Implementation [4]
The phase output ? of the accumulator is divided into three parts ? (most
significant part), ? (middle part), ? (least significant part) such that number of bits in
11
? , ? and ? are a, b and c respectively.
Based on trigonometric identities, the first quadrant of the sinusoid
]
)1(2
sin
)1(2
)(
cos
)1(2
cos
)1(2
)(
[sin
)1(2
)(
sin
22222
??
+
+
??
+
=
?
++
++++++++++ cbacbacbacbacba
??????????????
(2.1)
]
)1(2
sin
)1(2
)(
cos
)1(2
)(
[sin
)1(2
)(
sin
2222
??
+
+
?
+
=
?
++
++++++++ cbacbacbacba
????????????
(2.2)
The Equation (2.2) is implemented using a nonlinear DAC divided into coarse DAC and
fine DAC. The first term on the right hand side is realized using a coarse DAC and the
second term is realized using a fine DAC. The first term is monotonic and can be realized
as a coarse nonlinear DAC using the full thermometer code non linear DAC technique. If
the nonlinear DAC has i+1 bits of amplitude resolution, each coarse DAC cell output (o
k
)
is
for 0=k (2.3)
for 11
2
???
+ba
k
where k is
2
c
?? +
, a and b are the number of bits in segments ? and ? respectively. As
the number of phase bits increase the number of DAC cells increase and therefore the
power requirements, die area also increase. Since the number of bits in ? and ? are less
than the total number of input phase bits without segmentation the number of coarse
DAC cells are much less than that required without segmentation. The second term in
Equation (2.2) forms the fine nonlinear DAC to interpolate the amplitudes between
adjacent coarse DAC outputs. In Equation 2.2 the value of ? has been approximated
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
=
+
+
=
1
012
)5.0(
sin)1(int
12
)5.0(
sin)1(int
2
2
2
2
k
n
nba
i
ba
i
k
o
o
k?
?
12
using the average value, ?
avg
so that the fine DAC term depends only on ? and ? to
reduce the number of fine non linear DAC cells. The fine DAC output is not monotonic
in ? and ? . So a fine DAC is constructed using 1
2
?
a
nonlinear sub DACs, where a is the
number of bits in segment ? . The m
th
DAC cell output in the th? sub DAC,
o m,?
, can be
approximated as
for m=0
for 1? m? 2
c
-1 (2.4)
The total output is the sum of the current outputs of the coarse DAC and the fine sub
DACs if current steering technique is used. Different segmentations give different
performance due to the different amplitude errors in the approximation. An architecture
based on this algorithm has been discussed in [5].
2.3 Phase accumulator
Phase accumulator is the first module in the DDS design. The phase accumulator
adds the N-bit input frequency control word to itself once every clock cycle. The speed of
the phase accumulator depends upon the N bit adder and the flip-flop design. In the
proposed architecture, even if the nonlinear DAC operates at 10 GHz, the speed of the
DDS is limited by the speed of accumulator. This thesis discusses two different
architectures for the 8-bit phase accumulator, the pipelined accumulator and the pipelined
accumulator with Carry Look Ahead (CLA) adders.
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
+
?
?
?
?
?
?
?
?
?
??
+
?
?
?
=
++++
++++
=
1
0
,
,
12
)5.0(
sin.
)1(2
)(
cos)1(int
12
)5.0(
sin.
)1(2
)(
cos)1(int
22
2
22
2
m
n
n
cbacba
avg
i
cbacba
avg
i
k
o
o
m
?
?
?
??
?
??
?
?
13
2.3.1 SiGe Technology
Silicon-germanium (SiGe) semiconductor technology has been used to implement
the phase accumulator. SiGe technology has long held the promise of high-frequency
operation with high levels of integration. This technology claims device transition
frequencies that could surpass expensive gallium-arsenide (GaAs) technology while
using standard silicon wafers. While the technology is now making its way into a variety
of integrated circuits (ICs), mainly for cellular handsets and wireless-local-area-network
(WLAN) cards, its integration potential has been largely untapped. Companies like
Centellax, Broadcom have worked on high speed fractional N-synthesizers that take
advantage of the excellent high-frequency performance of SiGe as well as the potential
for integration when using silicon CMOS processing.
Advantages
? high levels of integration
? excellent high-frequency performance
? potential for integration when using silicon CMOS processing.
? Traditionally, HBT devices have been fabricated on GaAs or InP substrates, requiring
complex and expensive process technologies. One of the benefits of SiGe HBT devices is
that they can be fabricated with conventional silicon CMOS technology with only a few
additional process steps.
SiGe made it possible for integration of VCO, synthesizer, and digital signal
processing on a single chip offering a significant savings in size, cost, and power
consumption compared to existing solutions for a wide range of broadband applications
in commercial, industrial, and military systems. CML logic with SiGe Transistors has
14
been used for implementation of all the basic logic blocks in the accumulator.
2.3.2 Background on CML and ECL circuit design
2.3.2.1 DC Operation
ECL and CML circuits are based on the non saturated emitter-coupled pair. The
CML circuit (Figure 2.3, [6]) is a simplified version of an ECL circuit (Figure 2.2, [6]) in
which the emitter follower is omitted for a higher integration density. This emitter
follower acts as a level shifter and as an impedance adapter (it supplies current to the
output and as a result the circuit speed and fan-out are higher). A direct consequence of
using the emitter follower is that the Q1 collector voltage is isolated against any output
load. Taking the proper values for R
L
and Io, the circuit can be designed so that the
emitter-coupled pair transistors will never operate in the saturation region. This is the
main reason for the low propagation delay time of the ECL and CML circuits. In most
cases, both, fan-out and fan-in are not determinant factors in this kind of circuits.
However, this is true only in DC design. In transient design, if a high circuit operation
speed is desired, the propagation
Figure 2.2 ? ECL circuit diagram
15
Figure 2.3 ? CML Circuit diagram
delay has to be taken into account and the number of gates loading a given gate has to be
limited. The study of the problems related to transient design is the object of the next
section.
2.3.2.2 Transient Analysis [6]
There are several factors that contribute to a logic circuit propagation delay. One
of them is the wiring capacitance. This capacitance appears mainly due to the coupling
between the interconnections and the surface material. At this point, it should be stated
that, GaAs and its alloys show smaller capacitances than Si for the same structure. This
leads to a higher operation speed in III-V HBTs as compared with silicon BJTs. Another
source of capacitance is fan-out. Gates inputs are outputs capacitances to the preceding
gates associated with the active components. This capacitance varies with the voltage,
making difficult the estimation of the propagation delay. Another effect is the intrinsic
16
delay associated with the electron transit time across the transistor. This transit time is
characterized by the intrinsic frequency fT, ?=1/2?fT. The transit time can be minimized
using materials or structures with high electron mobilities. The propagation delay t
p
and
rising and falling times (t
pLH
and t
pHL
) degrade with fan-out. For a given fan-out, ECL
gates demonstrate a faster response than CML ones. Besides, ECL gates have a higher
fan-out. However, as it will be discussed later, the power consumption of an ECL gate is
greater than that of a CML circuit. Furthermore a CML circuit has fewer components,
hence a higher integration density for a given package. The linear wiring capacitance has
a decisive influence on the transient response. In logic circuitry this capacitance ranges
over a wide range depending on the interconnection length. The circuit propagation delay
can be estimated by the well known expression,
?
?
?
?
?
?
=
dt
dv
Ci . It is necessary to increase the
available current for charging and discharging this load capacitance when the gate is
strongly loaded. The propagation delay variation with the load capacitance is smaller than
with the fan-out. Besides, CML gates are more sensitive to a load capacitance variation
than ECL gates.
The average power consumption of a logic circuit can be expressed as a linear
combination of the average static power consumption and the average dynamic power
consumption. We calculate the static power consumption, i.e. the power consumption
when there is a high level at the output (P
OH
) and the power consumption when there is a
low level at the output (P
OL
). For the ECL circuit P
OH
can be calculated from,
()
()
IV
R
VV
ee
VIP ref
ref
ef
eeOOH
.
1
2
+
+
+= (2.5)
where V1 is the high level voltage. Similarly, P
OL
can be expressed as,
17
()
()
IV
R
V
o
V
ee
VIP ref
ref
ef
eeOOL
.
2
+
+
+= (2.6)
where V
0
is the low level voltage. These expressions can also be used for the CML circuit
shown in Figure 2.3 removing the term related to the emitter follower. Because of this,
the power consumption of CML gates is smaller than that of ECL gates. Furthermore, if a
little current I
ref
is supposed for both gates (this is valid because it is a base current), it
can be seen that, for the ECL gate, the power consumption varies with the input (Vin),
while for the CML gate the power consumption does not depend on the input. As a
consequence, the P
OH
and P
OL
difference for the ECL circuit is greater than that of a CML
circuit.
The static power values have to be added to the dynamic power consumption. The
dynamic power consumption is a direct consequence of the required energy to charge and
discharge the circuit capacitances in a unit time. For ECL and CML circuits, these
capacitances are concentrated in a unique load capacitance C
L
. This capacitor models the
wiring capacitance between two consecutive stages. The required energy to charge and
discharge this capacitor is
2
2
VCL
?
, ?V being the voltage swing. If the input signal is a
periodic signal and its frequency is f, the capacitor charges and discharges once a cycle.
Consequently, the dynamic power consumption can be expressed as,
VC
f
P LD
2
?= . In
typical designs, the contribution of P
D
to the overall power consumption at the working
frequency can be neglected. For example, for a C
L
of 100 fF and a frequency of 1GHz the
dynamic power consumption is 16?W. This dynamic power consumption is three orders
of magnitude less than the static power. As a result, the total average power consumption
can be expressed approximately as the average static power consumption. The
18
propagation delay of ECL and CML circuits are mainly affected by two elements: the
wiring capacitance (C
L
), and the fan-out (FO). The behavior of the propagation delay is
quite linear over a wide range of C
L
values. Propagation delay dependence with fan-out is
not linear. tp is more sensitive to fan-out variations than to C
L
variations. For this reason,
in a design using ECL or CML circuits, the designer has to put more attention to fan-out
requirements than to the wiring capacitances requirements. ECL and CML power
consumption is more than the power consumed by other circuits like MESFETs. For this
reason, ECL and CML gates are only useful for high performance circuits where the
power consumption is not so critical or specific applications where the logic gates are
strongly loaded. An example of this kind of application is a circuit with high wiring
capacitance or with high fan-out. A figure of merit used in logic circuits evaluation is the
power-delay product. MESFETS circuits are better in this regard than the other circuits,
even HBTs ones. Another figure where the fan-out is included is the power-delay product
divided by the maximum fan-out of the logic family. This figure provides evidence for
using HBTs in digital circuits with high loads in its nodes.
2.4 Analysis of two input and three input NAND gate
A NAND gate has been implemented in differential mode. The bias current is set
at 400uA. The resistor values have been taken as 530 ohms which gives an output swing
of about 430 mV. The peak ft current for 1um npn transistor is about 600um. It is biased
at 400 um for optimal speed and minimum current consumption. The bottom transistor
when biased properly acts as a current source. The current source should be always out of
saturation.
19
Figure 2.4 ? Schematic showing DC analysis of NAND gate
Figure 2.5 ? Schematic showing DC analysis of Level shifter.
Qp
VCC = 3.3 V
530 Ohm
Bias at 2.9 V
Ap
Am
Qm
i=96.61 u i= 291.2 u
3.248 V 3.144 V
3.1 V
3.1 V
2.249 V 2.249 V
Bias at 1.85 V
Vref = 1 V
722/3 = 240.7 Ohm
Bp
Bm
Vbe= 865.9
mV
Vbe= 847.1 mV
3.144 V
3.144 V
Vbe= 847.1 mV
Vbe= 865.2 mV Vbe= 865.2 mV
1.825 V 1.825 V
850.6 mV
1 V
86.31 mV
2 i/p NAND
400 uA norm, 1 um npn
2.27 V
VEE
20
The B pair of transistors in the NAND gate has to be biased as at 1.85 V. The level shifter
used is shown in the Figure 2.5. Schottky diodes provide an extra voltage drop of about
0.3-0.4 V. The worst case output delay of this NAND gate is about 26 ps.
Figure 2.6 ? The output of NAND gate showing a worst case propagation delay of 26ps.
A three input NAND gate has been implemented to reduce the power consumption of the
adders. The current source is implemented using an nFET and is biased in saturation. The
current source is always in saturation as V
ds
> V
gs
- V
t
over the entire range of the input
swing. If a bipolar transistor had been used in the place of nFET the current source could
be in deep saturation with the voltage levels handled.
21
Figure 2.7 ? Schematic showing DC analysis of 3 input NAND gate.
.
Figure 2.8 ? The output of 3 input NAND gate showing a propagation delay of 9ps.
22
Figure 2.9 ? Delay of 3 input AND gate is 50ps and 4 input AND gate is 79ps.
2.5 Pipelined accumulator
The proposed architecture uses an 8 bit pipelined accumulator. Figure 2.10 shows
generic architecture of an NxM pipelined accumulator. To realize an 8 bit accumulator
we have taken N=1 and M=8. Pipelining enables us to run the NxM bit accumulator at
the speed of N bit accumulator. Therefore the 8 bit accumulator runs at the speed of a 1
bit accumulator consisting of a full-adder and a reset flip-flop. Maximum frequency
attained by a 1-bit accumulator implemented using CML logic with 400uA tail bias
current is about 6.5 GHz. The gate delay to obtain the output includes a delay of the flip-
flop and 3-gate delay of the full adder circuit. The accumulator flip-flops are rising-edge
triggered master slave flip-flops with reset. The reset is active low asynchronous reset.
23
ACC (N:2N-1)
ACC (0:N-1)
M-1 FLIP FLOPS
N
A(0:N-1)
B(0:N-1)
SUM
C_IN
C_OUT
N BIT-ADDER
N
N
1-FLIP
FLOP
CLK RESET
D
OUT
A(0:N-1)
B(0:N-1)
SUM
C_IN
C_OUT
N N
N
N BIT-ADDER
4
N-FLIP
FLOPS
CLK RESET
D
OUT
N
4
N-FLIP
FLOPS
CLK RESET
D
OUT
N
4
N-FLIP
FLOPS
CLK RESET
D
OUT
N
FCW (0:N-1)
4
N-FLIP
FLOPS
CLK RESET
D
OUT
4
4-FLIP
FLOPS
CLK RESET
D
OUT
FCW (N:2N-1)
4
4-FLIP
FLOPS
CLK RESET
D
OUT
4
4-FLIP
FLOPS
CLK RESET
D
OUT
A(0:N-1)
B(0:N-1)
SUM
C_IN
C_OUT
N BIT-ADDER
N
N
1-FLIP
FLOP
CLK RESET
D
OUT
4
N-FLIP
FLOPS
CLK RESET
D
OUT
N
4
N-FLIP
FLOPS
CLK RESET
D
OUT
N
4
N-FLIP
FLOPS
CLK RESET
D
OUT
FCW ((M-1)*N:M*N-1) ACC ((M-1)*N:M*N-1)
MS
T
A
G
E
S
M FLIP FLOPS
M-2 FLIP FLOPS
Figure 2.10 - Generic Architecture of N x M Pipelined Accumulator.
The accumulator flip-flops are rising edge triggered master slave flip-flops with
active low asynchronous reset. For low supply voltage, the reset transistors cannot be
vertically stacked at the bottom of the CML circuitry without saturating the latch
transistors. It is rather critical for high-speed circuit to keep all the transistors from
saturation. We have used a novel reset CML latch circuitry that implements the reset in a
parallel structure with only three levels of transistors including the current source
transistor. The proposed reset CML flip-flop circuitry is very suitable for low supply
24
voltage applications although the delay is little higher. This architecture has a latency
period of about 7 clock cycles.
The full adder used in the accumulator architecture is realized using the following
equations:
ACCBBACARRY ?+?+?=
CBASUM ??= (2.7)
Maximum frequency attained by a 1-bit full adder is about 10GHz. All the gates
used to realize the full adder are two input gates. Hence the maximum delay to obtain the
carry in a full adder is 3 gate delays. A fully pipelined 8-bit accumulator contains about
210 gates and consumes about 150mA. The power consumption is about 0.5W and the
area occupied is approximately 210 x (2025um)
2
= 425250 um
2
. The simulated output of
the 8-bit accumulator operating at 6.5GHz clock frequency is shown in Figure 2.11. The
number of clock cycles till the vertical line in the Figure 2.11 indicates the latency period.
Figure 2.11 - Output waveform of an 8-bit fully pipelined adder.
25
2.6 New Pipelined Accumulator Architecture
An 8-bit fully pipelined accumulator has been redesigned using a new full adder
circuit and resettable flip flops to reduce the power consumption. It is expected to run at
almost the same speed as the old accumulator with lower power requirements. Also the
area requirements are reduced.
2.6.1 Full adder circuit
A new full adder has been designed by merging the gates to use a single current
tail for calculating sum output and a single current tail for calculating the carry output.
The equations used are:
()
CBASUM
CBACBACARRY
??
)(
=
?++=
(2.8)
The circuits have four levels of transistors stacked vertically including the current
source transistor. The fastest changing input is assigned to upper level of transistors and
the slowest changing inputs are assigned to lower level transistors to gain speed and
reduce glitches in the output. This new architecture helps in reducing the power
consumption to a great extent without suffering much loss in terms of speed. The current
source transistor can be a FET instead of a bipolar to use Vcc = 3.3.V and maintain the
current source to be out of saturation.
26
Figure 2.12 ? Test bench of new full adder circuit.
Figure 2.13 - Minimum propagation delay of traditional full adder is 35ps.
27
Figure 2.14 - Worst case propagation delay of traditional full adder is 51ps.
28
Figure 2.15 ? Worst case propagation delay of the new full adder is 75ps. It takes at least 90ps to
reach either Vhigh or Vlow.
In the Figure 2.15, Vhigh and Vlow indicate the maximum output swing.
29
VCC = 3.3 V
530 Ohm
Dp
Dm
Qp
Qm
Latch, active low reset
400 uA norm, 1 um npn
CLKp
CLKm
Bias at 1.3 V
Rstp
Rstm
Vref = 1 V
722/3 = 240.7 Ohm
VEE
Figure 2.16 - Three level resetable latch.
The maximum speed attained by this full adder is about 10 GHz. The 8bit accumulator
built with this full adder is expected to run at 6 GHz.
2.7 Accumulator with Carry look ahead adders
The proposed DDS architecture can incorporate various modulation schemes. The
DDS modulation waveform configurations include chirp, step frequency, FM, MSK, PM,
AM, QAM, and other hybrid modulations. To allow DDS to operate with modulations,
the developed pipelined accumulator needs to be modified to allow variable frequency
control words (FCW) as its inputs. In chirp or step frequency modulation the FCW input
of the DDS changes continuously. To incorporate frequency modulation techniques in
DDS, the latency period has to be reduced.
30
The 8 bit pipelined accumulator architecture has a large latency period of 7 clock
cycles. This latency can be reduced to 1 clock cycle using an 8 bit accumulator with two
4-bit CLA adders (N=4 and M=2 in Figure 2.10). The latency can be reduced to zero
using an 8-bit CLA adder with (N=8 and M=1). However, the architectures suffer a
considerable reduction in speed. The maximum speed attained by N=4 and M=2
pipelined accumulator with CLA adders is same as the maximum speed attained by a 4-
bit accumulator, which runs at maximum speed of 3.5 GHz. The speed of the 4-bit
accumulator depends upon the speed of the 4-bit CLA adder and the reset flip-flops. All
the gates used are two input gates. The basic building blocks in the adder circuit are
NAND gate and EXOR gate. All other logic gates like NOR gate, AND gate and OR gate
can be realized from NAND and XOR gates. Differential topology has been used to build
the basic building blocks for the improvement of noise characteristics. SiGe npn
transistors with an emitter length of 1um have been used. The power dissipation per gate
is about 1.32mw. The total delay to obtain the output sum and carry is 6-gate delay for a
4-bit CLA adder, while the delay in a fully pipelined architecture is 3-gate delay. The
generation of carries in the CLA can be summarized as:
c
pgC
?
+=
0
00
1 (2.9)
c
ppgpgC
0
01012
2 ??+?+= (2.10)
c
pppgppgpgC
0
012012122
3 ???+??+?+= (2.1)
c
ppppgpppgppgpgC
0
01230123123233
4 ????+???+??+?+= (2.12)
31
Figure 2.17 - Simulated output of an 8 bit pipelined accumulator with CLA adders (N=4 and
M=2) operating at 3.5 GHz clock frequency.
where
ba
g
nn
n
?= is the carry generator and
ba
p
nn
n
?= is the carry propagate to
calculate (n+1)th carry. The schematic of the 4 bit CLA adder has been shown in Figure
2.18.
32
s3
CARRY
INPUT
A0
B0
A1
B1
A2
B2
A3
B3
s0
s1
s2
CARRY
Figure 2.18 - The Gate level 4 bit CLA adder used for implementation of phase accumulator
33
2.7.1 Carry look ahead adder delay analysis
A CLA adder is constructed according to the carry propagate and carry generate
logic. The basic gates have to be wired such that the delays due to pairs of gates are
properly pipelined to achieve maximum speed. For the ith input bits of an adder the
propagate term is denoted by pi and generate term by gi where,
yixigi ?= has one gate delay denoted by 1D
yixipi ?= has one gate delay denoted by 1D
c1=g0+p0.c0
(1D+2D)
= 3D
c2 = (g1+p1.g0) + (p1.p0.c0)
The delay according to the Equation is (1D+2D+3D),
Pipelining the delays will give (1D+2D) +3D
=3D+3D=4D
c3=g2+p2.g1+p2.p1.g0+p2.p1.p0.c0
= (1D+2D+3D+2D.2D)
Pipelining the delays will give
(1D+2D)+3D+ (2D.2D)
= (3D+3D+3D)
=5D
c4= g3+p3.g2+p3.p2.g1+p3.p2.p1.g0+p3.p2.p1.p0.c0
According the equation the delays are (1D+2D+3D+2D.2D+2D.2D.0D) (Considering all
the gates to be two input gates).
34
= (1D+2D) +3D+ (2D.2D) + (2D.3D)
=3D+3D+3D+4D=3D+ (3D+3D) +4D
=3D+4D+4D= (3D+4D) +4D=5D+4D
=6D.
The equation can now be represented with brackets of precedence as
c4= ((g3+p3.g2) + (p3.p2.g1+ ((p3.p2).(p1.g0))))+((p3.p2).(p1.p0.c0))
2.8 Modifications to the 4 bit CLA Adder
2.8.1 Accumulator
In the new architecture of carry look ahead adder, two 3 input building blocks
have been used. The two building blocks are A+B.C and A.B.C, where A, B, C are input
signals to the gates. Apart from these, 2 input EXOR, OR and AND gates are also used.
The 3-input gates are designed with the same power supply VCC=3.3 V, to reduce the
total power consumption and area. Two different level shifters, shifting voltage levels
from 3.3V to 2.2V and from 2.2V to 1.3V are required. Using three input gates the
number of level shifters required is also reduced.
The generation of carry in the CLA can be summarized as:
()()
()()()
()()()()()
c
ppppgpppgppgpgc
c
pppgppgpgc
c
ppgpgc
c
pgc
0
01230123123233
0
012012122
0
01011
0
00
4
3
2
1
???+???+??+?+=
???+??+?+=
??+?+=
?+=
The brackets in the expression provide the order of precedence in which the operations
are performed. This specific order has been chosen to obtain minimum power
consumption and gate delay. To decide this order, comparisons have been made between
35
Figure 2.19 - The comparision shows that 2D2 (delay of two 2-input AND gates) is smaller than
1D3+D2 (delay of 3-input AND gate and a 2-input AND gate).
36
the delay of 4-input AND gate using a 3-input AND gate and a 2-input AND gate and
that using two 2-input AND gates as shown in Figure 2.19.
For the three input gates the slowest changing input is assigned to the last stage of
transistors for better speed performance.
Bias at 3.1 V
Bias at 2.2 V
VCC = 3.3 V
530 Ohm
Cp
Cm
Qp
Qm
A+B.C
400 uA norm, 1 um npn
Bp
Bm
Bias at 1.3 V
Am
Ap
Vref = 1 V
722/3 = 240.7 Ohm
96.64 mV
i= 400 u
VEE
Figure 2.20 ? Circuit diagram to implement logic function A+B.C.
37
2.8.2 Carry look ahead adder delay analysis
Consider a 3 input gate, for example A+B.C, from the figure the fastest changing
input should be connected to C and the slowest changing input should be connected to A,
for optimum speed. We can assign numbers to the amount of delays, the delay from input
A to output can be assigned number 2. The delay from B to output as 1 and from C to
output as 0. The connections have to be made according to the inputs indicated by the
delay number on each gate as shown below.
() ()(1) 0 2
p
0
00
1 C
g
C
?+=
()()
() ()() () () ()2 0 1 1 0 2
C
p p g p
C 0
01011
2
??+?+= g
()( ) ( )( )
() ()()( ) ( )()( ) () ( ) ( )
() () ()
0 1 2
2 0 1 2 1 0 1 0 2
C
p p p g p p g p g
0
012012122
3
???+??+?+=
C
2.9 Cosine Weighted Digital to Analog Converter [7]
The high-speed DDS utilizes a cosine-weighted DAC operating in current mode,
which does not require op-amp buffer at the output and its speed would not be limited by
the bandwidth of the op-amp. The DAC contains a current-cell matrix [4] as shown in
Figure 2.21. Each DAC cell outputs a current proportional to cosine value of the
corresponding phase indicated by the a+b bits. The sinusoidal output is obtained by
summing the output currents from all the cells through an external pull-up resistor. An
npn transistors with minimum emitter length of 1um can be used to switch the current
cells. The bias current should be carefully chosen considering the speed, power
38
consumption and DAC output full scale voltage swings. The accuracy of the bias current
is ensured using an optimized band-gap reference design with calibrating capability.
Dynamic performance of the DDS rapidly degrades with frequency due to transient
glitches in the DAC. These glitches can be minimized by using thermometer decoding
scheme that ensures the minimum number of cells switching simultaneously. In addition,
all switching control signals should be buffered to ensure differential synchronous
switching for all cells.
a+b
b
VCC
Thermometer column decoder logic
a
Co
n
t
r
o
l
l
o
g
i
c
CELL 0
CELL 2^(a+b)
Co
n
t
r
o
l
l
o
g
i
c
R[0]
R[(2^a)-1]
T
her
m
o
m
e
t
e
r
r
o
w
d
e
c
o
d
e
r
l
o
gi
c
R[1]
R[(2^a)-2]
C[0] C[(2^b)-1]
MSB
MSB
Figure 2.21 - Nonlinear DAC with current cells.
The DAC control logic is shown in Figure 2.22, where the signal MSB represents
the MSB bit of the phase accumulator output. Signals A and B are differential pair signals
which control the current switches in the DAC cell.
39
Figure 2.22 - Control logic of the nth DAC cell.
40
CHAPTER 3
NOISE ANALYSIS OF DDS OUTPUT SPECTRUM
DDS (direct digital synthesizer) has its applications in radios, instrumentation,
and radar systems etc. Though large and unpredictable spurious responses have troubled
old designs, innovations have improved DDS performance, and the worst-case spurs are
now smaller and predictable. Careful frequency planning allows us to place the worst-
case spurs outside the bandwidth of interest, so that they can be easily filtered. Most DDS
applications use only a fraction of their output spectrum and attenuate the remainder with
external filters. The bandwidth of interest is typically from 0 Hz to about 40% of the
sampling frequency. The sub-Nyquist limitation is due to the transition band of the
external image-rejection filter. Some applications can use the image band and eliminate
an upconversion stage, but the reduced power in the image lowers the SNR. Image use
also requires bandpass filtering rather than a lowpass filter. The DAC's zero-order
sample-and-hold imparts a Sinc (sin(x)/x) attenuation envelope to the fundamental,
images, and harmonics in the DDS spectrum.
A DDS has four principal spur sources: the reference clock, truncation in the
phase accumulator, angle-to-amplitude mapping errors, and DAC error terms, including
nonlinearities and quantization noise as shown in Figure 3.1. The spur frequencies'
predictability allows you to develop an effective frequency plan.
41
ANGLE TO
AMPLITUDE
CONVERTOR
1/z
DAC
cos
P
Tuning Word
Reference Clock
Phase Truncation Spurs
Sources of Predictable Spurs
Reference Clock Spurs Noise
Angle to Amplitude Mapping Spurs
Quantization noise and Non ideal DAC Spurs
D
N
Figure 3.1 - DDS has four principle sources of spurs.
3.1 Reference clock
A DDS functions like a high-resolution frequency divider with the reference clock
as its input and the DAC as its output. The spectral characteristics of the reference clock
directly impact those of the output. Though phase noise and spurs on the reference clock
also appear at the DAC output, they do so at a reduced magnitude due to the frequency
division. The improvement, expressed in decibels, is 20 log(N), where N is the ratio of
input to output frequencies. For example, two trials dividing a 300-MHz clock down to
80 and 5 MHz result in a difference in their phase-noise plots of 20 log(16)=24 dB. The
DDS's internal reference-clock path is the dominant contributor of phase noise from the
DDS. Modulating the clock amplitude generates spurs in its output spectrum. If a 400-
MHz RF carrier with 10% AM (amplitude modulation) by a 100-kHz sine-wave signal is
observed at reference clock and 10.119-MHz DDS outputs, this effect can be
demonstrated. The Figure 3.2 [2] which superimposes the reference-clock and DAC-
42
output spectra, shows the amplitude reduction in AM clock spurs at the DDS output. The
attenuation calculation, 20 log (400 MHz/10.119 MHz), predicts a 32-dB improvement,
although the plot shows more. The additional spur attenuation is due to the fact that the
modulated sine wave of the reference-clock signal encounters a limiter or squaring circuit
at the DDS input. The limiter stage converts the sine wave to a square wave, and the AM
spurs are thus converted to PM (phase modulation) spurs. The AM-to-PM conversion
results in an additional attenuation of spurs that depends on the characteristics of the
limiter circuit but is typically on the order of ?6 dB.
Figure 3.2 - Spurs caused by modulation of the reference clock amplitude are also reduced by
20log(N). The DDS?s input limiter, which converts the amplitude modulation into a phase-
modulation term, provides additional suppression of this spur.
The reference clock imposes limits on DDS performance in ways that are often
recognizable. The reference clock usually causes those DDS spurs that maintain their
relationship to the carrier as you change the output frequency. Also, there is some degree
43
of noise at the input of any circuit. A high-slew-rate reference clock spends less time
traversing the region where noise can cause jitter.
3.2 Phase truncation
Consider a DDS with a 32-bit phase accumulator. If the design maintained all 32
bits throughout, the DDS core would occupy a large die area and dissipate significant
power. Truncating the value from the phase accumulator i.e., passing only the
accumulator's most significant bits to the angle-to-amplitude mapper reduces the power
dissipation and die area as well as the phase resolution of the angle-to-amplitude mapper.
The DDS's phase-truncation spur mechanism models as a noise source summed with an
ANGLE TO
AMPLITUDE
CONVERTOR
1/z
DAC
DAC
ANGLE TO
AMPLITUDE
CONVERTOR
1/z
ANGLE TO
AMPLITUDE
CONVERTOR
1/z
cos
sin
cos
sin
cos
20
20
12
8
20
20
20
12
12
8
FULL RESOLUTION ANGLE MAPPING
PHASE TRUNCATION NOISE SOURCE
0
Figure 3.3 - The DDS?s phase truncation spur mechanism models as a noise source summed with
an otherwise ideal synthesizer.
44
otherwise ideal synthesizer as shown in Figure 3.3. The example truncates a 20-bit phase
accumulator to 8 bits. M is the phase accumulator's tuning-word width. Each update of
the reference clock adds the value of M to the accumulator output. The output is divided
into two sections, the truncated phase word, P, which is sent to the mapper, and the
discarded bits, D=M?L. As the value in the discarded section accumulates, it eventually
overflows into the truncated phase word. One effect of this overflow is production of
phase-modulation spurs. A second effect is that those overflows maintain the full-
frequency resolution of T. Note, if no bits in the discarded portion are set to logic one,
then no phase-truncation spurs occur.
3.2.1 Mathematical Analysis of Phase truncation spurs [8]
When the input frequency control word, FCW, is represented in M-b, the M-b
phase value of the phase accumulator is updated as
[] []()
2
mod1
M
FCWnn +?=+? (3.1)
and truncated to an L-b phase value. Then, the truncation error of can be obtained in a
recursive equation given by
[] []()
2
mod1
LM
Rnn
?
+?=+? (3.2)
where R is a least significant (M-L)-b value of FCW given by
2
2
LM
LM
FCW
FCWR
?
?
?
?
?
?
?
?
?
?= (3.)
Finally, the cosine output of the DDFS becomes
[]
[] []()
?
?
?
?
?
? ???
=
2
2
cos
M
nn
nx
?
(3.4)
45
[] []
?
?
?
?
?
? ?
?
?
?
?
?
? ?
=
2
2
cos
2
2
cos
MM
nn ??
(3.5)
[] []
?
?
?
?
?
? ?
?
?
?
?
?
? ?
+
2
2
sin
2
2
sin
MM
nn ??
(3.6)
In this case, the cosine corresponding to []n? only has a desired frequency
component. []n? is periodic as well as []n? , and the period of []n? is
2
/
LM
R
?
. In the
case of 2>L , the period of []()
2
/2cos
M
n?? and []()
2
/2sin
M
n?? is determined by []n? .
Thereby, the periodic truncation error creates spurs at the harmonic frequencies of
2
/
LM
R
?
. If the input frequency control word is decomposed as RNFCW
LM
+?=
?
2
, the
L-b truncated phase value is increased by N or N+1, and the rate of the increment by N+1
is given by
2
/
LM
R
?
. The mean value of the increment observed at the L-b phase value is
given by
2
/
LM
RN
?
+ . Accordingly, the conventional M-b phase accumulator with the L-
b truncation can be equivalently decomposed into two accumulators for L-b and (M-L)-b.
The update of the (M-L)-b accumulator corresponding to the truncation error is equal to
Figure 3.4 - A conventional phase accumulator representation with M bits truncated to L bits.
1/z
1/z
M
L
M-L
CARRY OUT
(M-L) b reg
L b reg
M-L
L
{ 0,1 } Sequence Generator
Phase out
L
46
Equation 3.2. The L-b accumulator is increased by N+1 when the (M-L)-b accumulator is
overflowed. Otherwise, it is increased by N. Thus, the (M-L)-b accumulator can be
simply considered as a {0, 1} sequence generator controlled by the (M-L)-b control word
of R. In the conventional phase accumulator, the {0, 1} sequence from the (M-L)-b
accumulator is periodic, because the truncation error is periodic. Figure 3.5 and Figure
3.6 shows the spectrum of the {0, 1} sequence generated by (M-L)-b accumulation of the
conventional phase accumulator. These harmonic tones contribute to the spurs in the final
output. If the periodicity of the {0, 1} sequence is eliminated, the power of the spurs can
be reduced significantly.
Figure 3.5 ? Spurs observed in the output spectrum due to phase truncation, Spectrum for FCW =
2
9
+1,
47
Figure 3.6 ? Spurs observed in the output spectrum due to phase truncation, Spectrum for FCW =
1.
The phase truncation spurs for a phase accumulator of 15 bits truncated to 9 bits,
DAC resolution of 8 bits at 25 MHz and FCW = 2
9
+1 is shown in Figure 3.6. Figure 3.7
shows phase truncation spurs for FCW=1. It can be observed that the decibel level of
phase truncation spurs do not change with FCW. Phase-truncation spurs are proportional
to the LSB weight in the phase word and, therefore, are not typically issues. Conversely,
in applications in which a DDS drives a PLL, phase-truncation spurs within the loop
bandwidth of the PLL will be amplified by 20 log(N) dB, where N is the PLL-
multiplication factor. The accumulator's phase-word output should be 3 or more bits
wider than the DAC resolution. It can be calculated that the worst-case spurious
magnitude attributable to the phase-word width is ?6.02P dBc, where P is the number of
bits in the phase word. So, a 12- to 19-bit phase resolution produces a ?72- to ?114-dBc
spurious magnitude. But the 8 bit DAC that has been used to observe the results has
48
about 5~6 effective number of bits that result in about 30~36 dB quantization noise floor
which has been observed during the test. Thus the worst case spurious response is 30~36
dBc instead of the expected 48 dBc. The tuning-word width, its fraction disposed of
through truncation, and the reference-clock frequency combine to reveal the frequency
offset of the truncation-spur phase-modulated sidebands.
2
M
f=f
N
REFs
(3.7)
where f
S
is the spur offset frequency, f
REF
is the reference-clock output frequency, M is
the decimal value of the discarded bits, and N is the number of discarded bits. These
sidebands appear on both sides of the fundamental. If the offset frequency is greater than
the output frequency or greater than the difference between the output and the Nyquist
frequencies, the sidebands fold around dc, Nyquist, or both. Figure 3.7 shows phase-
truncation spurs that were generated from a DDS with 9-bit phase resolution and a 15-bit
tuning word. The first phase-truncation sideband spurs are approximately ?30 dBc as it
would be expected from the resolution of DAC.
49
(a)
DISCARDED BITS
15- BIT TUNING WORD
TRUNCATED BITS
DAC WIDTH
8 BITS
6 BITS
0000 01 00
9 BIT PHASE WORD
04
100001000
9
PHASE TRUNCATION SPUR SPACING =
DECIMAL EQUIVALENT OF DISCARDED BITS
2
NUMBER OF DISCARDED BITS
X f
REF
b)
Figure 3.7 - The phase truncation spurs are offset from the fundamental by 390 kHz and its
harmonics (a) The reference clock rate and analyzing the 15 bit tuning word reveal this fact. (b)
The calculation shows how to predict the 390.625 KHz spur spacing.
50
3.3 Over sampling
The carrier to noise spectral density varies with the FCW selected by the user.
The worst case spurious magnitude is given by -6.02N where N is the number of bits in
the phase register used to address the ROM. An improvement in the output spectral
density can be observed by reducing the FCW. For a DDS with N=15 bit, FCW=1 and
f
clk
=25 MHz
Hz
M
f
out
93.762)1(
2
25
15
=?=
The Nyquist frequency is given by
out
f2
. The over sampling ration is defined as
FCWFCWf
f
OSR
NN
out
s
1
2
2
2
2
?
===
For FCW = 1, the over sampling ratio is 2
14
. For every doubling of over sampling ratio
over worst case (OSR =1, FCW = 2
14
) an improvement of 3dB can be observed in the in-
band quantization noise spectrum. In this case an improvement of 14 x 3 = 42 dB will be
observed. The quantization noise floor due to the 8bit DAC/ADC (effective 5~6 bits) is at
-30dBc, thus the quantization noise floor with OSR = 2
14
is below the carrier by
dBcdBdBc 7214330 ?????
51
Figure 3.8 - Spectrum of the DDS with a traditional ROM showing the carrier signal at about
762.93Hz and worst case spur for FCW = 1, phase accumulator = 15 bits, DAC resolution = 8
bits, clock freq. = 25MHz.
3.4 Angle-to-amplitude mapping
DDS designs can implement the phase-to-amplitude block algorithmically, which
reduces die area and power consumption or as a ROM look-up table. The algorithmic
approach enhances hardware efficiency, but its approximations may generate higher spur
levels.
The phase-to-amplitude conversion of a time-sampled sine wave is
( )?
i
pi
sinV=Va (3.8)
where Va
i
is the sample amplitude, Vp is half of the DAC's full-scale voltage, and
i
is
the value of the sample's phase word.
52
Phase Index
Amplitude
(LSBs)
0
1
2
3
-1
-2
-3
-4
3b DAC- 3b Phase Word
P2A(n,N,D)
Ideal(n,N,D)
n
Figure 3.9 - Limited phase resolution results in skipping DAC codes. The dots at each phase
increment show the calculated Va
i
.
It's unlikely that the Va
i
that the system calculates corresponds exactly to a DAC
code, so the DDS selects the nearest code, resulting in a residual error. If the phase word
has too few bits, the Va
i
calculation may skip over DAC codes (Figure 3.9). Conversely,
retaining more bits in the phase word reduces these errors. To guarantee that all DAC
codes are available to the phase-to-amplitude converter, a good rule of thumb is to set the
phase word to a minimum of 3 bits wider than the DAC. A Fourier transform of the time-
domain sine plot would display a corresponding spectral plot with discernible frequency
spurs. The error can be interpreted as modulating signal acting on the sine wave. The
resulting spurs' frequency locations can be determined and their amplitudes can be
approximated, although the amplitudes are subject to some architectural dependencies. In
some DDSs, the amplitude of the worst-case spurs ranges from ?12 to ?24 dB below the
DAC-quantization-noise level. The quantization noise (SNR) is proportional to the DAC
resolution:
)(76.102.6 dBNSNR +=
(3.9)
where N is the DAC resolution in bits.
53
Figure 3.10 - Spectrum showing the carrier signal and quantization noise floor for FCW=1.
A normalized spectral plot of a simulated look-up-table based DDS displays the spurious
response that the amplitude-error signal causes (Figure 3.10). The DDS tuning word is 15
bits, the phase word is 9 bits, the reference clock is 25M samples/sec, tuning word,
FCW=1 and the DAC resolution is 8 bits(effective 5 ? 6 bits due to board errors). The
location of spurs due to quantization noise depends on the tuning word and the DDS
architecture. Their power level is below the DAC's expected SNR, which is a goal of the
design. A DAC with greater resolution would decrease the magnitudes of these spurs.
Instead of performing a Fourier transform of the amplitude-error signal, a
relatively simple method determines the frequency of the most pROMinent spurs for a
given tuning word. This method uses a test tuning word with only one bit set. The
resulting spectrum consists of the test carrier and its spurs, the offsets and spacing of
which harmonically relate to the carrier frequency. The spectral region where these spurs
reside is
2
f=f
nb
REFs
?
(3.10)
54
where b represents the location of the single bit asserted in the tuning word, and n is the
DAC resolution in bits. This analysis method proceeds through nine steps:
Accurately measure and record the reference-clock frequency, f
REF
, to a ?1 Hz tolerance.
Counting from the MSB of the tuning word, assert only the bth bit. The bth bit is defined
as bth?n+8; that is, if the DAC width is 10 bits, then set 18th bit or higher counting from
the MSB.
Calculate the tuning-word frequency, f
C
, from f
REF
and the tuning word. Use
Equation 3.10 to locate the frequency region of the pROMinent spur set to measure. Note
that the spacing between these sets of spurs is two times the tuning-word frequency.
Measure and record the frequency of each individual spur in the worst-case set.
Individually divide the frequencies of the worst-case spur set by the tuning word. Round
the results to the nearest whole number. The results will probably be consecutive odd
numbers. Let this be step 1.
These results in step 1 are harmonically related to the tuning word. When you
change the tuning word, predict the locations of the corresponding spurs by multiplying
the new tuning word by each value in step 1. Let this be step 2. Terms from step 2 greater
than Nyquist, f
REF
/2 will alias. To locate the alias if the product is above Nyquist but
below f
REF
, subtract the f
REF
from the product; the difference is where the alias resides.
To locate the alias if the product is above f
REF
, divide by f
REF
and analyze the remainder
of the quotient. If the remainder is below 0.5, multiply it by f
REF
Otherwise, subtract it
from 1 and then multiply by f
REF
. let this be step 3. Repeat steps 2 and 3 for every value
found in step 1.
55
3.5 Quantization noise, DAC nonlinearities
A DAC's quantization noise and distortion determine its SNR. You can calculate a
first-order approximation of SNR by taking the ratio between the quantization-noise
power, integrated over the Nyquist bandwidth, and the power in the fundamental. As a
result, SNR is proportional to the DAC resolution in bits, as given in Equation 3.9. This
SNR calculation describes an ideal DAC. Real DACs also have nonlinearities due to
process mismatches and imperfect bit-weight scaling. Nonideal switching characteristics
also add distortion and nonlinearity. The most prominent DAC spurs are usually due to
nonideal switching characteristics, which, along with any nonlinearity in the transfer
function, appear as lower order harmonics of the fundamental. Both quantization noise
and the nonideal DAC properties produce a response that consists of harmonically related
spurs of the fundamental. This relationship is the key to understanding how to predict the
frequency location of the prominent spurs.
Harmonics alias because the DAC is a time-sampled system. As a result, the
carrier's harmonics, the reference clock, and the reference clock's harmonics create
numerous sum- and difference-mixing products. The well-defined mathematical
relationship of these products makes predicting the spur locations possible. Harmonics
beyond the first Nyquist zone are mapped back to the first Nyquist zone.
For example, a DDS tuned to 25.153 MHz with a reference clock of 100M
samples/sec generates low-order odd harmonics close to the fundamental. Once the
harmonic series exceeds the Nyquist frequency, they alias back into the first Nyquist zone
in a predictable way (difference product). This DDS has a 14-bit DAC. The SFDR
(spurious-free dynamic range) within the 4-MHz bandwidth is better than ?73 dBc.
56
Increased over-sampling, by raising f
REF
to 400M samples/sec eliminates the alias
products of the third, fifth, and seventh harmonics within the first Nyquist zone.
Significant benefits arise in DDS applications from running the DDS and
comparator at a simple subharmonic of the reference clock. These benefits include
reduced jitter and a simpler reconstruction filter. Because f
REF
and f
C
are related by an
integer ratio, the DAC quantization noise and spurs caused by other error sources fall
exactly on top of the harmonics of f
C
. In such cases, harmonics don't produce jitter,
because they are phase coherent with the carrier.
57
CHAPTER 4
A NOVEL DDS ARCHITECTURE WITH IMPROVED COMPRESSION RATIO AND
QUANTIZATION NOISE
In this section a novel DDS ROM compression technique is presented. The
technique is based on two basic properties of the sine function, (a) piecewise linear
characteristic of the sinusoid for infinitesimal difference in phase angle (b) variation in
the slope of the sinusoid with the phase angle. Storing only some of the values of the
sinusoid and constructing the entire sinusoid using compression techniques reduce the
memory requirements in DDS. In the proposed architecture, the first property of sinusoid
is used to interpolate the values of sinusoid that have not been stored in the ROM,
thereby greatly reducing the memory requirement. The second property has been used to
increase the number of values that can be interpolated as the slope of the sinusoid
decreases, without degradation in the performance. The variation in the number of values
interpolated has been achieved through a nonlinear addressing scheme. The ROM
incorporated in the proposed architecture for a phase resolution of 15 bits is 1216 bits
resulting in a spectral purity of -90.6 dBc. The proposed architecture has better
compression ratio than the Nicholas architecture [9] with the same hardware efficiency
and spurious response. The linear, nonlinear and traditional architectures have been
implemented in a Xilinx Spartan II FPGA. Measured spectra show that the compressed
and uncompressed ROMs end up with the same noise floor and thus the proposed ROM
59
compression algorithms are verified. In the proposed technique the sine wave has been
assumed to be piece-wise linear between the values stored in the ROM. The technique
requires a small multiplier, which does not present a bottleneck to low power and high-
speed applications. The results show that the technique can achieve a considerable
reduction in ROM size with reduced circuit complexity without degradation in the
spurious response. The technique has been implemented on Xilinx Spartan II FPGA.
4.1 Back ground of ROM Compression Algorithms
Different algorithms have been proposed for compression of look up tables. A
simple architecture for ROM compression was first proposed by Hutchison [10], which
was improved by Sunderland and Nicholas [9]. The architecture proposed by Nicholas
[9] has been considered to be most efficient due to its simplicity and minimum hardware
requirements. Nicholas architecture provides reduction in the look up table storage
requirements by replacing the storage requirements of one large ROM of size 2
A+B+C
words with two smaller ROMs of sizes 2
A+B
words and 2
A+C
words, whose outputs are
added together to reconstruct the sine function.
Figure 4.1 ? Block Diagram of Coarse-Fine ROM structure.
These are called the coarse ROM and fine ROM. Certain values of the sine function have
60
been stored in the course ROM which is addressed by the MSB bits of the phase
accumulator?s output. The values of sine function between the coarse ROM values have
to be interpolated. These values are obtained using the values stored in a second ROM
called fine ROM. In the Nicholas architecture, none of the values of the sine function are
calculated. All the values of the sine function are indirectly stored. As the speed of
technology is increasing, calculation of values is faster as well as easier; hence the
memory requirements can be reduced by calculating the values that have to be
interpolated rather than storing them as adopted by Nicholas architecture. This is the
basis for the proposed architecture where all the values that have to be interpolated are
calculated thereby greatly reducing the memory requirements. Further, the number of
values to be calculated can be increased, which reduces the number of values that have to
be stored, by exploiting the properties of sinusoid. The architecture proposed by
Bellaouar [11] and [12] also uses the course and fine ROM structure long with a
multiplier to calculate the values that have to be interpolated.
4.2 Some of the algorithms commonly used in all ROM reduction techniques
4.2.1 Sine Function Symmetry
The simplest way to reduce the ROM size to ? is to exploit the sine function
symmetry [9] about ?/2 and ? . By incorporating this technique into the architecture, only
values between 0 and 90
0
are needed to be stored. The architecture to construct the entire
sine function using the values between 0 - 90
0
is as shown in Figure 4.2. All the values
are represented in 2?s complement format. A ? LSB offset has been introduced in to the
value that has to be complemented so that a 1?s complementor can be used in place of 2?s
61
complementor.
1
st
MSB
P Phase
Accumulator
Proposed
Architecture
ROM
LOOK UP
TABLE
m
m-1k-2
k
2
2
nd
MSB
1's comple-
mentor
1's comple-
mentor
Figure 4.2 - Architecture for constructing sine function using symmetry around ?/2 and ?.
4.2.2 Sine Phase Difference Algorithm
The sine phase difference algorithm [9] is one of the popular techniques used in
almost all the compression algorithms proposed. According to this algorithm the ROM
stores the values obtained from the function
22
sin)(
xx
xf
??
?
?
?
?
?
?
?
=
(4.1)
instead of the value of function sin (?x/2). Two bits of amplitude are saved in the ROM
by incorporating this algorithm without degradation in the spurious response,
because
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
sinmax21.0
2
sinmax
P
P
P ??
, but this algorithm requires an
additional adder at the output of the ROM to calculate the function
xxf +)( (4.2)
DELAY
( )
TABLE
LOOKUP
P
P
?
2
sin
?
m-3
m-1
( )
2
sin
P?
Figure 4.3 - Block Diagram of Sine-phase difference algorithm.
62
4.3 Proposed Algorithm
4.3.1 Generic Architecture
The proposed technique is based on two basic properties of sine function, (a) the
piecewise linear characteristic of sinusoids for infinitesimal differences in phase angle (b)
variation in the slope of the sinusoid with phase angle. In the architectures of DDFS using
ROM compression techniques, certain values of the sinusoid are stored in the ROM, and
entire sinusoid has to be constructed using these values. For these the values, which are
not stored in the ROM, have to be interpolated. In our architecture, the two most popular
techniques of storage reduction (i) sine phase difference algorithm and (ii) sine function
quadrature symmetry have been used. The sine phase difference algorithm helps to
reduce the width of the word which has to be stored in the ROM by 2 bits. Incorporating
this algorithm in to our architecture requires 2 additional adders. In the proposed
architecture, sine phase difference algorithm has been used to reduce the ROM from 1408
bits to 1216 bits. Exploiting the sine function quadrature symmetry reduces the memory
requirements by 4 times. At any clock cycle the value in the phase register indicates the
phase of the sine function. As all the values of the sine function are not stored in the
ROM, out of the N bits of the phase address register only the first m bits are used for
addressing. The remaining d bits (d = N-m) indicate the relative position of the sine value
that has to be interpolated from the value that has been stored in the ROM. The value in
the phase registers R at any instant reads,
)1( ??= iFCWR
(4.3)
63
where FCW is the input frequency control word to the phase accumulator and i is the
number of clock cycles. The word length of R is N bits. The phase of the sine value that
has to be calculated is given by
2
2
N
R
B
??
=
?
(4.)
The equation used for calculating the value that has to be interpolated is
())sin(]0:1[
)2(
)sin(sin
)sin( Ads
AC
B
d
+??
?
= (4.5)
where )sin(B is the sine function value that has to be calculated and )sin(A and )sin(C
are sine values stored in the ROM where )sin(C > )sin(A . d = N - m gives the number of
values that have to be interpolated if FCW = 1. The LSB bits s[d-1:0] of the phase
register, indicate the relative position of the sine value that has to be interpolated from the
sine value that has been stored in the ROM. In other words, it represents the number of
clock cycles required for the value in the phase register to change from A to B for
FCW=1. The output of the ROM has a word length of k bits.
In the proposed architecture the sine function is assumed to be piecewise linear.
The division in Equation 4.5 causes increase in the quantization noise due to the rounding
effect. In order to avoid the division in Equation 4.5 which involves hardware, the
formula has been modified to
)()sin(])0:1[())sin()(sin()sin( dAdsACB <<+???= (4.6)
This change makes the output word length n = k + d bits. This increases the precision of
representation and does not increase the quantization noise levels. The k + d bits are later
truncated to k bits to meet the specifications of the DAC. The spurious response has been
64
analyzed after truncating the word length to k bits. The memory requirements here are
k
m
?2 bits.
The expression ])0:1[())sin()(sin( ??? dsAC of Equation 4.6 is calculated by
using a Multiplier. This architecture is termed as linear architecture and has been
implemented in MATLAB for the dimensions provided in the Table I. The size of the
subtractor was chosen by taking into account the maximum difference between
successive ROM samples. The values in the ROM have been optimized to give minimum
error. These values are the averaged values that have been obtained after a number of
simulation trials.
TABLE I: Dimensions of DDS with single ROM
Phase accumulator
size s[14:0]
15 bits
Addressing lines to
ROM s[12:5]
8 bits
No. of LSB bits for
interpolation s[4:0]
or d
5 bits
Width of the ROM
with sine phase
difference
algorithm
12 bits
output word length
n = k + d
14 + 5 = 19 bits
Truncated output
word length
14 bits
Subtractor size 7 bits
Multiplier size 7 x 5
Memory requirements have further been reduced by exploiting the second
property of sine function. The slope of sine ranges from 1 to 0 as the angle changes from
0
0
-90
0
. So the slope of the sine wave changes gradually between successive ROM values
at the beginning and reduces to almost 0 at 90
0
. Hence more ROM values can be stored in
65
the region of the sinusoid having higher slope and a smaller number of ROM values can
be stored in region of sinusoid with lesser slope. Therefore, the number of values that
have to be interpolated are less in the high-sloped region than the number of values that
have to be interpolated in the low-sloped region of the sinusoid. For the architecture
shown in Figure 4.5, the sinusoid is partitioned in to two parts; 0-45
0
and 45
0
-90
0
. The
sine values for phases between 0
0
-45
0
are stored in ROM1, and those for 45
0
-90
0
in
ROM2. As the change in the slope of the sinusoid, is less between 45
0
-90
0
,
the number of
ROM values stored to calculate sine value after 45
0
can be reduced to almost half without
significant drop in the spurious response. So the number of addressing lines to ROM2
changes accordingly to m-1, where m is the number of addressing lines to ROM1. The
number of values that have to be interpolated between 45
0
-90
0
will increase (twice the
number that have to be interpolated in the single ROM architecture) and the relative
position of the value that has to be interpolated is indicated by LSB d
2
(d
2
= N-m+1) bits.
Equation 4.6 gets modified to
)()sin(])0:1[())sin()(sin()sin(
222222
dAdsACB <<+???= (4.7)
)()sin(])0:1[())sin()(sin()sin(
111111
dAdsACB <<+???= (4.8)
Subscripts 2 and 1 indicate that the values under discussion are from ROM2 and ROM1
respectively. d
2
is one bit greater than d
1
(d
1
= d) so shifting the )sin(
2
A value by d2 will
make the output word length k+d+1 bits. In order to have uniform output the width of
ROM2 is made k-1 instead of k. The memory requirements for this architecture are
)1(22
21
??+?
??
kk
mm
bits.
66
4.4 Quantization Error Analysis
The number of bits in each word of the ROM will determine the amplitude
quantization error. The finite amplitude quantization in the sine ROM values leads to
degradation of the output spectrum. So the quantization noise is discussed in detail in this
section. Without phase truncation, the output of the DDS is
)(
2
)(2
sin n
E
nP
A
N
??
?
?
?
?
? ?
(4.9)
where P (n) is the N-bit value in the phase accumulator register at the n
th
clock cycle and
E
A
(n) is the quantization error due to finite ROM values at the n
th
clock cycle. The
amplitude quantization errors with an uncompressed ROM can be assumed to be
uniformly distributed and uncorrelated within each quantization step if the step size is
small, we have
22
?
??
?
?
A
A
A
E
(4.10)
with the quantization step size
2
1
k
A
=
?
where k is word length of values stored in the
ROM.
The sinusoid signal power is
2
2
A
PA
= (4.1)
and the error power is
{}
12
1
2
2
2
22
?
=
?
?
??
?
=
dEEE
E
AA
A
A
A
A
(4.12)
Assuming that the amplitude quantization error gets its maximum absolute value ?
A
/2 at
67
Figure 4.4 - Graph showing
22
?
??
?
?
A
A
A
E
for memory length of 14 bits.
every sampling instance, and all the energy is in one spur, the carrier to spur ratio is
dBck
P
S
C
A
)02.601.3(
4
log10
2
10
+?=?
?
?
?
?
?
?
= (4.13)
However, in the worst case the sum of the discrete spurs is approximately equal to
{}
dBck
EE
P
S
C
A
A
)02.676.1(log10
2
10
+=
?
?
?
?
?
?
?
?
= (4.14)
We have used the last formula to analyze the error due to finite amplitude quantization in
the proposed architecture. E{E
2
A
} is the mean of the variance of the error. Therefore,
?
=
=
=
Nj
j
jA
e
N
EE
1
22
1
}{ (4.15)
where j represents at the j
th
clock cycle. N is the total number of clock cycles to
construct one period of the sine function and
j
e
is
)sin()sin( BB
e
cal
j
?= at the j
th
clock cycle (4.16)
68
Figures 4.6 and 4.7 show
j
e , which is the error between the actual sinusoid function and
that obtained from the proposed architecture for Linear and Nonlinear addressing
respectively, at each clock cycle. The values of error shown in the graphs are used for
calculating the mean of the variance of the error.
4.5 Proposed Architecture with Non-linear Addressing (Two ROMs)
This section discusses the implementation of the generic algorithm with two
ROMs. For the architecture with two ROMs, the sinusoid is partitioned in two parts; 0-
45
0
and 45
0
-90
0
.The sine values for phase between, 0-45
0
being stored in ROM1, and
45
0
-90
0
in ROM2. This architecture is simulated for dimensions of ROM1 equal to 2
6
x
12 bits and ROM2 equal to 2
5
x 11 bits. The width of ROMs has been reduced from 14
bits to 12 bits in ROM1 and 13 to 11 bits in ROM2 by using the sine-phase difference
algorithm. Let the 15 bit output of the phase accumulator register be represented by
variable s[14:0] where 14:0 indicate bit 14 is MSB and bit 0 is LSB. The first 2 bits out of
the 15 bits of the phase accumulator register indicate the quadrant of the sine wave.
Hence the remaining 13 bits are used to address the sine wave between 0-90
o
. FROM the
ROM sizes mentioned, the number of values that have to be interpolated are
approximately 64 between 0-45
0
and 128 in between 45
0
-90
0
if FCW = 1. The address
lines required to address ROM1 are 7 i.e., s [12:6] and that required for ROM2 are 6 i.e.,
s [12:7]. According to Equations 4.7 and 4.8, the values of ROM1 are shifted by 6 and
the values of ROM2 are shifted by 7 before being added to the output of the multiplier to
obtain the output sine value. The position of sine value that has to be interpolated
according to the value in the phase register is given by 6 LSBs s [5:0] for sine phase
between 0 to 45
o
and by 7 LSBs s [6:0] for sine phase between 45
o
to 90
o
. The whole sine
69
value is represented with a precision of 15 + 6 = 21 bits. This value is truncated to 14 bits
before sending to the DAC. This technique requires a 7x8 bit multiplier. The increase in
the hardware overhead is the trade off for the great reduction in the memory requirement.
The sinusoid can further be partitioned to 0-30
0
, 30
0
-60
0
, 60
0
-90
0
instead of 0-45
0
and
45
0
-90
0
, to obtain further reduction in the memory requirement. The values stored in the
ROM are optimized to minimize the mean square error by computer simulations. Worst-
case spur amplitude of about -88 dBc is achieved with this method. But the simulation
results show that a ROM of 2
6
x 13 + 2
5
x 12 bits provides a spurious rejection of -90.3
dB, which provides the best compression ratio.
Incorporating sine-phase difference algorithm in nonlinear addressing the word
length of ROM can be reduced by 2 bits. Introduction of this technique in to ROM
requires two extra adders. The values stored in the ROM are given by the expression
)()())sin()12(( mkciA
k
?<=32767
s=s-32767; %when s>2
15
the next cycle of sine wave starts
end
i=i+1; %indicates the number of clock cycles from the initial reset
sample(i,:)=ROM((s+1),:)/(16383); %gives the sine value between 0-1. width of ROM
%is 15 bits
error(i,:)= (sin ((pi/2)/8192*(i-1)*FCW)- sample(i,:));
%Eq to calculate error during each clk cycle
j=s;
end %phase accumulator loop ends
%error=sort(error(1:(i-1)/4));
plot(sample(1:100)); %plots the sine value on Y axis and no. of clock cycles
%on x-axis
%for loop to calculate the total signal power and error power
for N=0:1:(((i-1)/(4*FCW))-1) %checking output for only 1
st
quadrant of sine between
%0
o
-45
o
z=z+(error((N+1),:))^2; % gives the total error power during all the clock cycles
end
%Value of N after the for loop, gives total power in all samples during all the clk cycles
f=20*log10(sqrt(N/(z*2))); %f value gives the worst case spur value in dBc
nfft=(i-1);
fout=sample(1:nfft);
79
psd=20*log10(abs(fftshift(fft(fout(1:nfft))))+0.0000001);
%for N=0:1:i-2
% F=F+(psd((N+1),:))^2;
%end
%signal=max(psd);
%psd=psd-signal;
%stem(psd(1:nfft));
A.2 LINEAR ROM DDS ARCHITECTURE
FCW=input ('FCW:'); %FCW is the frequency control word that is provided by the user
s=0; %s is the variable storing the accumulator values
i=0; %i is variable indicating the number of clock cycles
q=0;
m=0;
sample=0; %Sample is the sine value corresponding to the phase word in the
%accumulator
ROM=load('6f.txt'); % load values from the ROM stored in the notepad file '6f.txt'
%y=load('2615.txt');
l=1; %l is variable to break the phase accumulator loop
step=0;
stepsize=0;
j=0;
k=0;
z=0;
80
qut=0;
F=0;
FC=0;
FC2=0;
a=0;
u=0;
s1=0;
while s <=(65536/2) & l~=0 %Phase accumulator s is 15 bit wide. When one complete
%sin wave is completed the value in s might not be 0 but
%less than 2
15
so j is used to break the loop
l=s;
si=s;
sb=dec2bin(s,15); %sb has the binary value of s packed in 15 bits, width of
%phase accumulator
u=sb([3 4 5 6 7 8 9 10 11 12 13 14 15 ]);
%u does not have first 2 bits as they only indicate
%quadrant
u=bin2dec(u);
MSB=sb([1 2]); %MSB indicates the quadrant
MSBd=bin2dec(MSB); %decimal value of the quadrant
switch MSBd
case 1
cc=bitcmp(u,13);
81
u=cc;
case 3
cc=bitcmp(u,13);
u=cc;
end
ub=dec2bin (u,15);
d=ub([ 11 12 13 14 15 ]); %d is the distance between the positions of the sine value
%that has to be calculated and that has been stored in the
%ROM.
c=ub([ 3 4 5 6 7 8 9 10 ]); %c is used to address the ROM values
ci=bin2dec(c);
di=bin2dec(d);
i=i+1;
m=ROM(ci+1); %m is assigned the sine value addressed by c
if(k~=m)
j=k;
k=m;
end
diff=ROM(ci+2)-ROM(ci+1); % diff has difference between successive ROM samples
step=di*(diff/32); %calculates the step size
if i==248*4 %for FCW=33, i=248*4 gives one complete cycle of sine
break
end
82
switch (MSBd) %each case indicates one of the four quadrants
case 0
sample(i,:)=((ROM(ci+1))+ step)+16; %value 16 is added to reduce the error and
%has been decided after many iterations of
%simulations
samp=dec2bin(sample(i,:),19);
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]); % only 14 MSBs of the sine value
%calculated are considered
sample(i,:)=bin2dec(sampl);
sample(i,:)=(sample(i,:))/(2^14-1); % sample has sine value between 0-1.
error(i,:)=(sin ((2*pi)*FCW*(i-1)/(2^15)+ 2*pi/2^16 )- sample(i,:));
%2*pi/2^16 is due to ? LSB shift.
case 1 %for the second quadrant
sample(i,:)= ROM(ci+1)+ step+16;
samp=dec2bin(sample(i,:),19);
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]);
sample(i,:)=bin2dec(sampl);
sample(i,:)=sample(i,:)/(2^14-1);
error(i,:)= (sin ((2*pi)*FCW*(i-1)/(2^15) + 2*pi/2^16)- sample(i,:));
case 2 % for III quadrant
sample(i,:)=(ROM(ci+1)+ step)+16;
samp=dec2bin(sample(i,:),19);
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]);
83
sample(i,:)=bin2dec(sampl);
sample(i,:)=-1*(sample(i,:))/(2^14-1); %sine is negative in III quadrant
error(i,:)=(sin ((2*pi)*FCW*(i-1)/(2^15)+ 2*pi/2^16 )- sample(i,:));
case 3 % for IV quadrant
sample(i,:)=(ROM(ci+1) + step)+16;
samp=dec2bin(sample(i,:),19);
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]);
sample(i,:)=bin2dec(sampl);
sample(i,:)=-1*(sample(i,:))/(2^14-1);
error(i,:)=(sin ((2*pi)*FCW*(i-1)/(2^15)+ 2*pi/2^16 )- sample(i,:));
end
s=s+FCW;
if s>=65536/2;
s=s-65536/2;
end
l=s;
end
plot(sample(1:100)); %Plots the sine wave
title('ROM samples')
xlabel('no of samples')
%title('scaled sine value 2^12*sin(90*2^6*(N)/2^(12))')
%xlabel('Number of samples')
nfft=i/4;
84
fout=sample(1:nfft);
psd=20*log10(abs(fftshift(fft(fout(1:nfft))))+0.0000001);
%% ERROR CALCULATION
stem(error(1:247*4)); %shows the error plot
for N=0:1:246*4
F=F+(error((N+1),:))^2; %F indicates the total error power
end
f=20*log10(sqrt(N/(F*2))); %Gives the value of worst case spur, N is the value
%of the total signal power as the amplitude of
%signal is 1.
A.3 NON LINEAR ROM DDS ARCHITECTURE
MATLAB CODE
FCW=input ('FCW:'); %FCW is the frequency control word provided by the user
s=0; %s is the variable storing the accumulator values
i=0; %i is variable indicating the number of clock cycles
q=0;
m=0;
sample=0; %Sample is the sine value corresponding to the phase word in the
%accumulator
ROM1=load('6f1.txt'); % load values from the ROM1 stored in the notepad file '6f1.txt'
ROM2=load('6f2.txt'); % load values from the ROM2 stored in the notepad file '6f2.txt'
l=1; %l is variable to break the phase accumulator loop
step=0;
85
stepsize=0;
j=0;
t=0;
k=0;
z=0;
qut=0;
F=0;
FC=0;
FC2=0;
a=0;
u=0;
while s <=(65536/2) & l~=0 %Phase accumulator s is 15 bit wide. When one
%complete sin wave is completed the value in s
%might not be 0 but less than 2
15
so j is used to
%break the loop
l=s;
si=s;
sb=dec2bin(s,15);
%sb2=dec2bin(s,14);
FCL=dec2bin(FCW,15); %FCL has the binary form of FCW
if s<=4160 %if accumulator phase is less than 45
o
c=sb([ 3 4 5 6 7 8 9 10]); %8 MSBs are used to address the ROM1
ci=bin2dec(c);
86
else
c=sb([3 4 5 6 7 8 9 ]); %7 MSBs are used to address the ROM 2 as it is
%half the size of %ROM1
ci=bin2dec(c)-63; % to start the addressing from 0. the first values in
%ROM2 should have an address 0.
end
MSB=sb([1 2]);
MSBd=bin2dec(MSB);
if s<=4160 %if phase <=45
o
d=sb([ 11 12 13 14 15 ]); %last 5 bits indicate the distance between the positions
%of the sine %value that has to be calculated and that
%has been stored in the ROM1.
else
d=sb([10 11 12 13 14 15]); %last 6 bits indicate the distance between the positions
%of the sine value that has to be calculated and that
%has been stored in the ROM2
end
di=bin2dec(d); %di had decimal value of the distance
i=i+1;
if s<=4160 %if phase<=45
o
take values from ROM1
m=ROM1((ci+1));
diff=ROM1(ci+2)-ROM1(ci+1);
else
87
m=ROM2((ci+1)); %if phase>45
o
take values from ROM2
diff=ROM2(ci+2)-ROM2(ci+1);
end
if(k~=m)
j=k;
k=m;
end
t(i,:)=ci;
if s<=4160
step=di*(diff/32);
else
step=di*diff/64;
end
if i==252 %for FCW=33, i=252 shows more than complete I quadrant of sine
end
switch (MSBd)
case 0
if s<=4160
sample(i,:)=(ROM1(ci+1)+ step);
else
sample(i,:)=(ROM2(ci+1)+ step);
end
samp=dec2bin(sample(i,:),19);
88
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]);
sample(i,:)=bin2dec(sampl);
sample(i,:)=sample(i,:)/(2^14-1);
error(i,:)=(sin ((2*pi)*FCW*(i-1)/(16384*2))- sample(i,:));
case 1
if s<=4160
sample(i,:)=(ROM1(ci+1)+ step);
else
sample(i,:)=(ROM2(ci+1)+ step);
end
samp=dec2bin(sample(i,:),20);
sampl= samp([1 2 3 4 5 6 7 8 9 10 11 12 13 14 ]);
sample(i,:)=bin2dec(sampl);
sample(i,:)=sample(i,:)/(2^14-1);
error(i,:)= (sin ((2*pi)*FCW*(i-1)/(16384*2))- sample(i,:));
case 2
if s<=4160
sample(i,:)=(ROM1(ci+1)+ step);
else
sample(i,:)=(ROM2(ci+1)+ step);
end
sample(i,:)=(-1)*sample(i,:)/(2^20-1);
error(i,:)= (sin ((2*pi)*FCW*(i-1)/(16384*2))- sample(i,:));
89
case 3
if s<=4160
sample(i,:)=(-1)*(ROM1(ci+1)+ step);
else
sample(i,:)=(-1)*(ROM2(ci+1)+ step);
end
sample(i,:)=sample(i,:)/(2^20-1);
error(i,:)= (sin ((2*pi)*FCW*(i-1)/(16384*2))- sample(i,:));
end
s=s+FCW;
if s>=65536/2;
s=s-65536/2;
end
l=s;
end
stem(sample(1:(i-1)));
title('ROM samples')
xlabel('no of samples')
%title('scaled sine value 2^12*sin(90*2^6*(N)/2^(12))')
% xlabel('Number of samples')
nfft=i/4;
fout=sample(1:nfft);
psd=20*log10(abs(fftshift(fft(fout(1:nfft))))+0.0000001);
90
%% ERROR CALCULATION
stem(error(1:251));
for N=0:1:250
F=F+(error((N+1),:))^2;
end
f=20*log10(sqrt(N/(F*2)));
91
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94