Lattice-Reduction Aided Linear Equalization For Wireless
Communications Over Fading Channels
Except where reference is made to the work of others, the work described in this thesis
is my own or was done in collaboration with my advisory committee. This thesis does
not include proprietary or classifled information.
Wei Zhang
Certiflcate of Approval:
Jitendra Tugnait
James B. Davis and Alumni Professor
Department of Electrical and Computer
Engineering
Xiaoli Ma, Chair
Assistant Professor
Department of Electrical and Computer
Engineering
Min-Te Sun
Assistant Professor
Department of Computer Science and
Software Engineering
Stephen L. McFarland
Acting Dean
Graduate School
Lattice-Reduction Aided Linear Equalization For Wireless
Communications Over Fading Channels
Wei Zhang
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulflllment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 11, 2006
Lattice-Reduction Aided Linear Equalization For Wireless
Communications Over Fading Channels
Wei Zhang
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense. The author reserves
all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Wei Zhang was born on May 14, 1982 in Qingdao, a beautiful seaside city in China.
He graduated from Qingdao No. 2 Middle School in 2000. Then, he attended Zhejiang
University in Hangzhou, China. He got his Bachelor of Engineering degree with honors
from the Department of Information Science and Electronic Engineering in Chu Kechen
Honors College, Zhejiang University in 2004. After that, he joined Auburn University
to continue his graduate study in Department of Electrical and Computer Engineering.
iv
Thesis Abstract
Lattice-Reduction Aided Linear Equalization For Wireless
Communications Over Fading Channels
Wei Zhang
Master of Science, May 11, 2006
(B.S., Zhejiang University, 2004)
73 Typed Pages
Directed by Xiaoli Ma
Modern wireless communications ask for high data rate, high transmission perfor-
mance and low complexity. High data rate induces frequency-selective channels because
of the relatively shorter symbol duration than the delay spread. Wireless links intro-
duce fading which degrades performance and requires diversity techniques to combat.
Orthogonal Frequency Division Multiplexing (OFDM) is an efiective method to deal
with frequency-selective channels since it facilitates low complexity equalization and
decoding. To eliminate the efiects of the channel nulls and fading, linear precoded
OFDM is introduced to enable multipath diversity and the maximum likelihood decoder
is used to collect the diversity. But, the low complexity provided by OFDM is sacri-
flced. Multi-antenna techniques are shown to be able to boost the data rate and also
collect space diversity to combat fading. The V-BLAST (Vertical Bell Labs Layered
Space-Time) scheme enables higher data rate than single-antenna setup does, but it
also requires higher decoding complexity. As a combination, multi-input multi-output
(MIMO-) OFDM has been widely studied to boost the transmit-rate and performance
v
in terms of diversity. For MIMO-OFDM systems, many designs successfully exploit the
joint space-multipath diversity when maximum likelihood (ML) detector is adopted at
the receiver, which is well known for high complexity. To reduce the decoding complexity,
linear equalizers are favored in practical systems, but they usually induce performance
degradation. In this thesis, we flrst quantify the diversity of conventional linear equaliz-
ers for linear precoded OFDM, V-BLAST and MIMO-OFDM designs. Then, we propose
lattice reduction (LR-) aided equalizers to improve the performance, and show that LR-
aided linear equalizers achieve the same diversity order as that collected by ML detectors
for (MIMO-) OFDM systems and V-BLAST systems. Simulation results corroborate the
theoretical flndings.
vi
Acknowledgments
A journey is easier when you travel together. This thesis is the result of two years
of work during which I have been accompanied and supported by many people. It is a
pleasure that I now have the opportunity to express my gratitude for them.
First, I would like to express my sincere appreciation to my advisor Dr. Xiaoli Ma for
her guidance and support throughout this work. With her enthusiasm and inspiration,
this work has been carried out smoothly. During these two years? study, she provided
excellent mentoring, but what I beneflt most from her is her specially designed training
procedure which helped me to transmit from an undergraduate to graduate student
successfully. Her persistent encouragement and insightful advice make me believe in
my choice which results in this thesis. I owe her lots of gratitude for leading me to
the exciting world of wireless communications and showing me the way of performing
research. She could not even realize how much I have learned from her. Besides of being
an excellent supervisor, Dr. Ma is as close as a good friend to me. I am really glad that
I have known Dr. Ma in my life.
I am also grateful for the contributions of Dr. Jitendra Tugnait both as member
of my thesis committee and as a great teacher, and for his assistance throughout my
graduate studies. My appreciation also goes to Dr. Min-Te Sun from whom I broaden
knowledge to computer science from the weekly study group. Also, I thank my colleagues
Liying Song and Zhenqi Chen for their helpful discussions and support through this work.
It is not easy to start a new life in a difierent country. So I will give my appreciation
to my friends Wei Feng, Shuangchi He, Jin Li, Tao Li, Weidong Tang and Yuan Yao,
with whom my life becomes easier and happier.
vii
I would like to give my special thanks to my dear wife Xiaoming Li for her love and
support, and to my parents for their understanding, endless patience and encouragement.
As the only child of the family, I owe them a lot for studying abroad.
I would like to acknowledge the flnancial support provided by the U.S. Army Re-
search Laboratory and the U.S. Army Research O?ce under grant number W911NF-04-
1-0338 and through collaborative participation in the Collaborative Technology Alliance
for Communications & Networks sponsored by the U.S. Army Research Laboratory un-
der Cooperative Agreement DAAD19-01-2-0011.
viii
Style manual or journal used Journal of Approximation Theory (together with the
style known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (speciflcally
LATEX) together with the departmental style-flle aums.sty.
ix
Table of Contents
List of Figures xii
1 INTRODUCTION 1
2 PERFORMANCE ANALYSIS OF LLP-OFDM SYSTEMS 5
2.1 System Model of LLP-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Linear Equalization For LLP-OFDM . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 ZF Equalizer for LLP-OFDM . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 MMSE Equalizer for LLP-OFDM . . . . . . . . . . . . . . . . . . . 12
2.2.3 DFE Equalizer for LLP-OFDM . . . . . . . . . . . . . . . . . . . . 14
2.3 LR-aided Linear Equalization For LLP-OFDM . . . . . . . . . . . . . . . 15
2.3.1 LR-aided Linear Equalization . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Performance Analysis on LR-aided Linear Equalizers . . . . . . . . 18
2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 LINEAR EQUALIZATION FOR V-BLAST SYSTEMS 29
3.1 System Model of V-BLAST Systems . . . . . . . . . . . . . . . . . . . . . 29
3.2 Linear Equalizers and Performance Analysis . . . . . . . . . . . . . . . . . 30
3.2.1 ZF Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 MMSE Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 LR-Aided Linear Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 EXTENSION TO MIMO-OFDM SYSTEMS 40
4.1 FDFR-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 STF-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 CONCLUDING REMARKS AND FUTURE RESEARCH DIRECTIONS 49
Bibliography 51
Appendices 54
Appendix A 55
Appendix B 57
Appendix C 58
x
Appendix D 60
xi
List of Figures
2.1 Comparisons among difierent linear equalizers . . . . . . . . . . . . . . . . 24
2.2 Comparisons among difierent equalizers for LLP-OFDM . . . . . . . . . . 25
2.3 Comparisons among difierent group sizes using ML detector and complex
LR-aided equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Complexity comparison of difierent decoding methods . . . . . . . . . . . 27
3.1 BER of systems with Nt = 2 and Nr = 2,3,4 separately and BPSK
modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 SER of a system with (Nt,Nr) = (3,4) and QPSK modulation . . . . . . 37
3.3 Complexity comparison between complex LLL and real LLL algorithms . 38
3.4 Performance comparison between the complex and real LLL algorithms
with (Nt,Nr) = (4,4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Block Diagram of MIMO-OFDM . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Comparison among difierent equalizers for FDFR design . . . . . . . . . . 47
4.3 Comparison among difierent equalizers for STF design . . . . . . . . . . . 48
xii
Chapter 1
INTRODUCTION
Modern development of wireless communications requires reliable high-data-rate
services. To increase data rate, we can decrease the symbol period, but this will intro-
duce frequency-selectivity (and hence time-dispersive) channels. Orthogonal frequency
division multiplexing (OFDM) is known as an efiective method to deal with frequency-
selective channels since it facilitates low-comlexity equalization and decoding [22]. How-
ever, the original uncoded OFDM design neither guarantees symbol recovery, nor collects
the multipath diversity to combat fading. Several techniques were proposed to collect the
multipath diversity provided by the channel. One way to recover the symbol detectabil-
ity for single antenna OFDM systems, is linear complex-fleld coded (LCFC-) OFDM
(a.k.a. linear precoded OFDM) presented in [11, 22] to collect the multipath diversity
but at the cost of increased decoding complexity.
Another way to achieve high-data-rate is to adopt multi-antenna at the transmitter
and receiver. Space-time multiplexing of multi-antenna transmissions over multi-input
multi-output (MIMO) channels has well documented merits in combating fading, and
further enhancing data rates. The V-BLAST (Vertical Bell Labs Layered Space-Time)
architecture presented in [4, 25] is a well-known method for achieving high spectral
e?ciencies over a rich-scattered environment. Since high data rate is achieved through
multiple transmit- and receive-antennas, and high order signal constellation is usually
used, the high decoding complexity at the receiver for collecting the diversity provided
by the channels becomes the bottle-neck of the development of multi-antenna systems.
1
Frequency-selective MIMO fading channels provide space-multipath diversity to
combat fading. Thus, MIMO-OFDM becomes a strong candidate for next generation
wireless multi-antenna communications. Numerous space-time (ST) coding schemes have
been developed for MIMO-OFDM systems to collect space-multipath diversity (e.g., [11]
and [13]). Since all of them used maximum likelihood (ML) detection or near-ML schemes
such as sphere-decoding (SD) method, the decoding complexity is high especially when a
large number of transmit-antennas and/or high signal constellations are employed. Thus,
it is obvious that, no matter in single antenna LCFC-OFDM systems, V-BLAST systems
or in MIMO-OFDM systems, how to reduce the decoding complexity while exploiting
the diversity order is the problem we would like to study.
The flrst straightforward thought to solve this problem is to use linear equalizers
such as zero-forcing (ZF) and minimum mean square error (MMSE) equalizers. It is
well-known that linear detection methods have much lower complexity than ML and
SD methods but introducing an inferior performance. Interestingly, it has been shown
that even linear equalizers guarantee maximum multipath diversity for certain precoded
OFDM systems [21] (e.g., the system in [22]). However, the decoding complexity of linear
equalizers in [21] depends on the number of subcarriers which usually is large. Grouped
LCFC-OFDM design has been proposed in [11] to reduce the decoding complexity by
performing smaller size of ML. The major difierence for the LCF coder design of [22,
21] and the one in [11] is that LCF coder in [11] depends on the lattice structure of
the transmitted symbols. Therefore, to difierentiate these two, we call the grouped
LCFC-OFDM scheme as linear lattice-based precoded (LLP)-OFDM. In general, the
performance of linear equalization has not been studied for LLP-OFDM, V-BLAST and
MIMO-OFDM systems in the literature. In this thesis, we analyze the performance
2
of the linear equalizers for LLP-OFDM systems and extend the results to V-BLAST
systems and MIMO-OFDM systems.
Recently LR technique has been used to improve the performance of linear equaliz-
ers over MIMO systems (e.g., [8] and [24]). A class of real LR-aided linear equalizers was
presented in [26, 27], to transform the system model into an equivalent one with better
conditioned channel matrix while maintaining the low complexity. It is shown that LR
technique can help to collect the maximum diversity order while does not increase the
complexity much. Therefore, in this thesis, we develop complex LR-aided linear equal-
izers to decode LLP-OFDM systems and analyze the performance in terms of diversity.
The results can also be extended to V-BLAST and MIMO-OFDM designs to show the
diversity order explicitly.
This thesis is organized as follows. In Chapter 2, we study the performance of
LLP-OFDM systems: flrst, the system model of LLP-OFDM is presented; then the
performance of LLP-OFDM systems with three kinds of low-complexity equalizers (ZF,
MMSE and decision-feedback equalizer (DFE)) is analyzed separately; LR-aided linear
detection methods for LLP-OFDM systems are developed and analyzed. We show that
LR-aided linear equalizers exploit multipath diversity. The simulation results will cor-
roborate our theoretical claims. Chapter 3 and Chapter 4 will follow similar structure
while Chapter 3 studies the V-BLAST systems and Chapter 4 copes with MIMO-OFDM
systems, where two kinds of multi-antenna OFDM designs are analyzed as examples. The
last chapter presents concluding remarks and future research directions.
Notation: Upper (lower) bold face letters will be used for matrices (column vectors).
Superscript H denotes Hermitian, ? conjugate, and T transpose. We will reserve ? for
the Kronecker product, d?e for integer ceiling, and E[?] for expectation; diag[x] will stand
3
for a diagonal matrix with x on its main diagonal. IN will denote the N ?N identity
matrix. Z is the integer set and C stands for the complex fleld. Z[p?1] denotes the
Gaussian integer ring whose elements have the form Z+p?1Z.
4
Chapter 2
PERFORMANCE ANALYSIS OF LLP-OFDM SYSTEMS
In this chapter, we consider single-antenna LLP-OFDM systems. First, the system
model of grouped LLP-OFDM is introduced. With this model, the performance of
linear equalizers is studied. Then, we develop the complex LR-aided linear equalizers for
LLP-OFDM systems and analyze the performance to show it can collect the multi-path
diversity.
2.1 System Model of LLP-OFDM
For high-rate transmissions, when the maximum delay spread (?d) of the channel
exceeds the symbol period (Ts), inter-symbol interference (ISI) cannot be ignored and
the channel exhibits frequency-selectivity. Suppose the channel is random, with flnite
impulse response, consisting of L+1 taps, where L is deflned as b?d/Tsc. Channel taps
are denoted as a vector h = [h0,h1,...,hL]T and modelled as complex Gaussian random
variables. Here, the conventional uncoded OFDM system we consider includes cyclic
preflx (CP) insertion and inverse FFT (IFFT) operations at the transmitter, and CP
removal and FFT operations at the receiver. It has been shown that plain OFDM enables
symbol-by-symbol low complexity decoding. However, as stated in [3] and [22], the
performance of uncoded OFDM sufiers from loss of diversity. LLP-OFDM [11, 12, 22] has
been proposed to combat frequency-selective fading by collecting the multipath diversity
provided by the channel.
5
In order to exploit the multipath diversity in OFDM, the LCFC-OFDM designs
flrst linearly encode the ith information block s(i) = [s(iN),...,s(iN + N ? 1)]T 2
Z[p?1]N?1 (N is much greater than L, but less than the normalized channel coherence
time) by a time-invariant matrix ? 2 CN?N; and then multiplexes the coded symbols
u(i) = ?s(i) 2 CN?1 using conventional OFDM. Collecting the ith received OFDM
block after CP removal and FFT operations as: y(i) = [y(iN),...,y(iN +N ?1)]T, the
input-output (I/O) relationship of the overall system model can be expressed as:
y(i) = DHu(i) +w(i) = DH?s(i) +w(i), (2.1)
where DH = diag[H(0),H(1),...,H(N ?1)], with H(n) = PLl=0 hle?j2?ln/N. w(i) is
the ith white complex Gaussian noise vector observed at the receiver with zero mean
and covariance matrix 2wIN. Since we consider block-by-block decoding, for notational
simplicity, we will drop our OFDM block index i from now on.
There are several ways to design ? to achieve maximum diversity without any chan-
nel knowledge at the transmitter. One way is to design ? as a tall Vandermonde matrix
with properly chosen generators [22]. For this case, though the bandwidth e?ciency is
sacriflced, it has been proved that even linear equalizers (ZF and MMSE) can also achieve
the full diversity [21]. But since the complexity of linear equalizers in [21, 22] depends on
the block size N with a order of O(N3), when N is large, even linear equalizers require
high decoding complexity.
Another way to design the LCF encoder uses algebraic number theory and grouping
method [11]. Suppose the N subcarriers are split into Ng groups and each group has
size K. To achieve the maximum coding gain, we select the gth group as [11] and [12]:
6
sg = [s(g),s(g + Ng),...,s(g + (K ?1)Ng)]T, with s(n) denoting the nth symbol in s.
The K equi-spaced symbols are precoded (or linear block coded) by a K ? K unitary
matrix ?, and the precoded symbol ug = ?sg is mapped into K equi-spaced carriers. It
is readily seen that analogous to (2.1), the I/O relationship for the gth group becomes:
yg = DH,g?sg +wg = Hequsg +wg, (2.2)
where Hequ := DH,g? is the equivalent channel matrix with DH,g = diag[H(g),H(g +
Ng),...,H(g + (K ? 1)Ng)], and wg is the corresponding noise of the gth group. We
observe that the I/O relationship of the gth group in (2.2) has the same form as (2.1).
Thus, ML or SD method can be used to collect full multipath diversity. As proposed in
[12, 11], the maximum achievable multipath diversity order is Gd = min(K,Rh), where
Rh denotes the rank of the channel coe?cients correlation matrix E(hhH).
Compared with the ungrouped version in [22], grouped LLP-OFDM has lower com-
plexity since each group only has K < N symbols. However, ML or near-ML decoder is
required to collect multipath diversity. One natural question now is what if one wants
to further reduce the complexity and just uses linear equalizer to (2.2) and what the
diversity is in this case. In the next section, we analyze the performance of the linear
equalizers for LLP-OFDM systems and answer this question. For brevity, we will drop
the group index g in (2.2).
2.2 Linear Equalization For LLP-OFDM
Linear equalizers are favored in practical systems because they have the lowest
complexity among all kinds of detection methods. However, at the same time, linear
7
equalizers are ?blamed? because they usually have inferior and unpredicted performance
(e.g., unknown diversity order). In the literature, the performance in terms of diversity
with optimal decoders (e.g., ML) has been well-documented (see e.g., [18, 12]). However,
the performance of linear equalizers is not well studied. Recently, it has been shown
that even linear equalizers can collect the multipath diversity for LLP-OFDM with tall
Vandermonde LCF encoders [21], while it is unclear on the performance of LLP-OFDM
with linear equalizers. In this section, we study the performance of LLP-OFDM systems
with ZF equalizer, MMSE equalizer and decision-feedback equalizer (DFE). We will keep
our proofs general so that they can be applied to other linear systems.
2.2.1 ZF Equalizer for LLP-OFDM
Suppose that the receiver has perfect channel knowledge. Based on the model in
(2.2), the output of ZF equalizer is given as:
x = (Hequ)?1y = s+(Hequ)?1w = s+n, (2.3)
where n := H?1equw is the noise after equalization. The channel matrix Hequ = DH?
has full rank with probability one (wp1) because ? is a unitary matrix and the diagonal
matrix DH has full rank wp1. Note that the noise vector n is no longer white and its
covariance matrix depends on the equalization matrix H?1equ. The next step is to map
the output to signal constellation:
?si = Q(xi) = arg
s2S
minjxi ?sj, (2.4)
where xi denotes the ith element of x and Q(?) means the quantization of the symbol.
8
Starting from (2.3), we now analyze the diversity collected by ZF equalizer. Suppose
that the ith transmitted symbol is si, and at the receiver it is erroneously decoded as
?si 6= si. The error probability is given as:
P(si ! ?sijHequ) = P(jxi ? ?sij2 < jxi ?sij2 j Hequ),
where ni is the ith element of n. If we deflne ei = si ??si, then the error probability can
be further simplifled as:
P(si ! ?si j Hequ) = P(jei +nij2 < jnij2 j Hequ)
= P
?e
in?i ?e?ini
2 >
jeij2
2
flfl
flflHequ
?
. (2.5)
Though as we stated, the ZF equalized noise vector n is no longer white, it is not di?cult
to verify that for each channel realization, n is still complex Gaussian distributed with
zero mean and covariance matrix
E[nnH] = 2w(HHequHequ)?1 = 2wC, (2.6)
where C := (HHequHequ)?1. Deflne a random variable vi = (?ein?i ?e?ini)/2. Given the
error symbolei, vi is real Gaussian distributed with zero mean and variancejeij2E[jnij2]/2 =
jeij2 2wCii/2, where Cii is the (i,i)th element of C in (2.6). Thus, the error probability
in (2.5) can be re-written as:
P(si ! ?si j Hequ) = Q
?s
jeij2
2 2wCii
!
. (2.7)
9
Based on (2.7), the diversity order collected by the ZF equalizer is established in
the following:
Proposition 1 Given the model in (2.2), if the channel taps are complex Gaussian
distributed with zero mean, then the ZF equalizer in (2.3) exists wp1 and collects diversity
order 1 over LLP-OFDM systems with frequency-selective channels.
Proof: According to the design in [11] or [12], there exists at least one unitary matrix ?
that enables maximum diversity for any block size K and constellations that belong to
Gaussian integer ring. The diagonal matrix DH has full rank wp1. Thus, both of them
are invertible wp1. This shows the ZF equalizer exists wp1.
Because ? is unitary and recalling the Vandermonde structure of ?, we notice that
the elements of ? have amplitude 1/pK. Thus, matrix C in (2.6) can be written as
C = (HHequHequ)?1 = (?HDHHDH?)?1 = ?HD?1H (DHH)?1?,
and the (i,i)th entry of C, Cii becomes
Cii = Hi D?1H (DHH)?1 i = 1K
K?1X
k=0
1
jH(k)j2 (2.8)
where i is the ith column of the LCF encoder ?. Based on (2.8), we can bound Cii as :
1
KjH(c)j2 ? Cii ?
1
min
0?k?K?1
jH(k)j2 (2.9)
where c is any integer, c 2 [0,K ? 1]. The left inequality holds because Cii is the
summation of K nonnegative numbers and is surely larger than or equal to any one of
10
them. The right inequality holds because 1jH(k)j2 ? 1min
0?i?K?1
jH(i)j2 for any 0 ? k ? K ?1.
According to (2.7), the post-processing signal-to-noise ratio (SNR) of ZF equalizer is
= jeij22 2
wCii
. Plugging the inequality (2.9) into (2.7), we will have:
Kjeij2jH(c)j2
2 2w ? =
jeij2
2 2wCii ?
jeij2 min
0?k?K?1
jH(k)j2
2 2w . (2.10)
Thus, the outage probability [23] can be bounded as:
P
jH(c)j2 ? 2
2w th
Kjeij2
?
? P( < th) ? P
min
0?k?K?1
jH(k)j2 ? 2
2w th
jeij2
?
. (2.11)
Next, to show the diversity order collected by ZF equalizer, we need the following
lemma:
Lemma 1 Given N random variables X1,X2,...,XN (either dependent or indepen-
dent), they are all central Chi-square distributed with degrees of freedom 2M. Let Xmin
denote the minimum of them, and then we have P(xmin < ?) ? cu?M, where cu is a
constant depending on N and M.
The proof is given in Appendix A. Since jH(k)j2 is exponentially distributed (or Chi-
square distributed with degrees of freedom 2), according to Lemma 1, we can obtain:
P
min
0?k?K?1
jH(k)j2 ? 2
2w th
jeij2
?
? cu th
je
ij2
2 2w
??1
So the performance upper-bound in (2.11) shows diversity one. Since H(c) is complex
Gaussian distributed, jH(c)j2 is Chi-square distributed with degrees of freedom 2. Then,
11
we can obtain that:
P
jH(c)j2 ? 2
2w th
Kjeij2
?
= exp
?
2w th
Kjeij2
?
,
which shows that the performance lower-bound also has diversity one (see also [23] for
similar argument).
So the diversity order of the outage probability is just 1. According to [23], the
diversity of outage probability is the same as that of average error probability. Thus,
the diversity collected by the ZF equalizer for LLP-OFDM system is just 1. ?
Interestingly, difierent from the claim in [21], Proposition 1 shows that if we use
ZF linear equalizer for LLP-OFDM systems, the diversity order for the performance is
only 1. This is because of the difierent structures of the precoders in [11, 22]. For LLP-
OFDM, although the ZF equalizer has low complexity, it cannot collect any multipath
diversity.
2.2.2 MMSE Equalizer for LLP-OFDM
Another often used linear equalizer is minimum mean square error (MMSE) equal-
izer. Based on the model in (2.2), the linear MMSE equalizer for LLP-OFDM systems
is given as:
x = (HHequHequ + 2wIK)?1HHequy. (2.12)
12
It is easy to verify that the MSE of the symbols after MMSE equalization is
E[ks? ?sk2] = 2w ?HHequHequ + 2wIK??1 . (2.13)
Deflning C = ?HHequHequ + 2wIK??1 and plugging in the fact that Hequ = DH? into
C, we have the (i,i)th entry of C
Cii = Hi (DHHDH + 2wIK)?1 i = 1K
KX
k=1
1
jH(k)j2 + 2w. (2.14)
The approximate BER after MMSE equalization is [16]:
Pe ?
X
m
fimQ
?
flm
s
1
2wCii ?1
!
, (2.15)
where fim and flm depend on the symbol constellation. Based on (2.14), we flnd that
the SNR for the error probability is bounded by:
min
0?k?K?1
jH(k)j2
2w ?
1
2wCii ?1 ?
K
2w(jH(c)j
2 + 2
w). (2.16)
Similar to the ZF equalizer case, based on (2.16), the diversity order collected by
the MMSE equalizer is established in the following:
Proposition 2 Given the model in (2.2), if the channel taps are complex Gaussian dis-
tributed with zero mean, then the MMSE equalizer in (2.12) for the LLP-OFDM system
collects diversity order 1.
13
2.2.3 DFE Equalizer for LLP-OFDM
Generalized decision feedback equalizer (GDFE) is proposed and compared with the
Nulling-Cancelling (NC) equalization ([6]) in [5]. It is shown that the operation of GDFE
is equivalent to the NC processing. There are two types of GDFE: ZF-GDFE and MMSE-
GDFE. Here, we consider the performance of ZF-GDFE while it is straightforward to
see the performance of MMSE-GDFE following the similar analysis. For ZF-GDFE (or
ZF-NC), the output of the forward equalizer is:
x = QHy, (2.17)
whereQR is the QR-decomposition of Hequ with unitary matrixQ and upper triangular
matrix R. Then the decision process becomes
for n = 0 : K ?1
?sK?n = Q((xK?n ?Ppi=1 RK?n,K?n+i?sK?n+i)/RK?n,K?n)
end
where Rp,q is the (p,q)th entry of R and xi is the ith entry of x. According to [5], the
diversity of LLP-OFDM systems with ZF-GDFE equalizer is determined by the lowest
degrees of freedom of jRk,kj2,k 2 [1,K], where R is the QR decomposition of DH?
and Rk,k is the (k,k)th entry of R. Considering the QR decomposition process, RK,K
ofiers the lowest degrees of freedom. Because C = (HHequHequ)?1 = (RHR)?1, it can
be verifled that jRK,Kj2 = C?1KK satisfles:
min
0?k?K?1
jH(k)j2 ? jRKKj2 ? KjH(c)j2. (2.18)
14
Thus, following the analysis for ZF equalizer, we can see that the maximum diversity
order that ZF-GDFE can collect is just 1. Similarly, the diversity of the LLP-OFDM
systems with MMSE-GDFE is also 1. Proof is omitted here.
Remark 1 From our analysis, we notice that for LLP-OFDM systems, the linear equal-
izers cannot exploit any multipath diversity, though they have much lower decoding
complexity compared with ML or SD methods. Furthermore, the performance of ZF
equalizer is the worst among these three linear equalizers while MMSE-GDFE equalizer
has the best performance of them.
2.3 LR-aided Linear Equalization For LLP-OFDM
As we know, for linear systems, if Hequ in (2.2) is diagonal, ZF equalizer has the
same performance as ML decoder (e.g., for plan OFDM case). However, in general Hequ
is not diagonal, and thus ZF equalizer has inferior performance (e.g., in Section III we
have shown that ZF equalizer cannot collect multipath diversity). This motivates us to
flnd a way to make Hequ close to a diagonal matrix.
If the symbols s are drawn from Gaussian integer ring (QAM, PAM constellations),
then Hequs belongs to a lattice generated by the columns of Hequ. Thus, to estimate the
information symbols is equivalent to searching for the closet point in the lattice [1]. The
motivation to use lattice reduction (LR) technique in equalization is to make the decision
region of the linear equalizers more like that of ML detector by flnding a more orthogonal
basis for the lattice, thus increase the performance. Recently, Lenstra-Lenstra-Lov?asz
(LLL) algorithm has been adopted for LR-aided linear equalizers to flnd more orthogonal
basis for communication MIMO channels, because it guarantees polynomial complexity
15
to flnd a set of bases within a factor of the shortest vectors (see e.g., [8, 24, 27]). The
LLL reduction was flrst posted in [10]. The LR-aided linear equalizers were flrst applied
to MIMO systems (see e.g., [8, 24, 27]). However, the methods in [24, 26, 27] are based
on real lattice reduction methods, which increase the decoding complexity by splitting
the complex channels into real and imaginary parts. [8] uses complex lattice reduction
method, but only for two-transmit-antenna case. In [14], it is mentioned that complex
LLL algorithm can be extended to any number of transmit-antennas. Unfortunately,
detailed complex LLL algorithm is not provided in [14]. In general, these existing re-
sults show that LR performed on the channel matrix can improve the linear detectors?
performance while does not increase the complexity much.
2.3.1 LR-aided Linear Equalization
We flrst extend the LLL algorithm to complex fleld for any number of transmit-
antenna. Our proposed complex LLL algorithm can be found in Appendix B, where
we use the conventional Matlab notation (e.g., R(k,k) denotes the (k,k)th element of
matrix R). Compared with the real LLL algorithm in [27], the major difierence of the
complex LLL algorithm exists at Steps (8) and (16) in Table 1. Later by simulations,
we show that the complex LLL algorithm enables lower computational complexity than
the real LLL algorithm without sacriflcing any performance.
Given the system model in (2.2), we adopt CLLL algorithm to reduce the lattice
basis of Hequ and obtain ?H = HequT, where T is a unimodular matrix which means all
the entries of T and T?1 are Gaussian integers and the determinant of T is ?1 or ?j.
Then, we apply the LR-aided ZF equalizer ?H?1 instead of H?1equ, and the output can be
16
S1: Map the information bits to symbols s whose constellation belongs to Gaussian
integer ring;
S2: Obtain (2.2) after LLP-OFDM transceiver operations;
S3: Perform the Complex LLL algorithm to reduce the lattice basis of the equivalent
channel matrix: [ ?H,T] = CLLL(Hequ);
S4: Rewrite the system as y = HequT(T?1s)+w = ?Hz +w;
S5: Apply the ZF equalization based on the new system to obtain ?z = Q( ?H?1y);
S6: Use ?z and T to recover the original information: ?s = Q(T?z).
Table 2.1: Lattice-Reduction Aided ZF Equalization
written as [c.f. (2.2)]:
x = T?1s+ ?H?1w = z +n. (2.19)
Since all the entries of T?1 and the signal constellation belong to Gaussian integer ring,
the entries of z are also Gaussian integers. One can estimate z from x by quantization.
After obtaining z, one can recover s by using T and mapping to the appropriate constel-
lation. We summarize the main steps of LR-aided ZF equalizer for LLP-OFDM systems
in Table .
Note that the equivalent channel matrix Hequ can be other communication channels
(e.g. MIMO channel matrix later in Chapter 3, MIMO-OFDM matrices in Chapter 4).
To perform the LR-aided MMSE equalizer, we cannot just apply the conventional MMSE
equalizer to the new system in Step S4, because the average power of z is not easy to
determine. In [27], it shows that LR-aided MMSE equalizer agrees to the LR-aided ZF
equalizer with respect to an extended system. Compared with the conventional linear
equalizers, LR-aided linear equalizers increase the complexity only in the CLLL reduction
step, we will compare the complexity over difierent systems by simulations.
17
Remark 2 Here, we notice that in Step S5, the quantization of ?H?1y is simply round-
ing to the nearest integer in order to reduce the complexity. Thus, we must make sure
the constellation of symbols s belongs to Gaussian integer ring, and both the real and the
imaginary parts are drawn from an integer set whose elements are consecutive integers,
or can be transferred to consecutive integers by shifting and scaling. Otherwise, for each
realization of channel Hequ, one needs to calculate T and then flnd the set of possible ?z.
Typically, M-QAM constellations satisfy this prerequisite, e.g., 4-QAM symbols s whose
constellation is f?1?jg, can be transferred to f1(0)+1(0)jg by performing 12(s+1+j).
Difierent from the simple rounding in Step S5, the quantization in Step S6 is to map
the product of T and ?z to the original information constellation.
2.3.2 Performance Analysis on LR-aided Linear Equalizers
In this section, we prove the diversity order collected by LR-aided linear equalizers.
To make our proof compact, we introduce some important deflnition and lemmas flrst.
Deflnition 1 An orthogonality deflciency (od) of an M?N matrix B = [b1,b2,...,bN]
as:
od(B) = 1? det(B
HB)
QN
n=1 jbnj2
(2.20)
where jbnj,1 ? n ? N is the norm of the nth column of B.
Note that 0 ? od(B) ? 1, 8B and if B is singular, od(B) = 1; and if the columns of B
are orthogonal, od(B) = 0. It has been shown that LLL algorithm tries to reduce the
18
od of the studied matrix [10]. The quantitative result on od reduction is given in the
following lemma:
Lemma 2 Given a matrix H 2 CM?N with rank N, ?H is obtained after applying
complex LLL (CLLL) algorithm with parameter ? on H [30]. Then, the orthogonality
deflciency of ?H satisfles:
q
1?od( ?H) ?
4
4? ?1
??N(N?1)/4
:= c?, (2.21)
where c? is determined by ? and N, and ? can be any flxed real number in (1/4,1).
For real H, Lemma 2 is consistent with the result in [10, Proposition 1.8]. Here, we
extend it to complex fleld according to the CLLL algorithm in [30]. Given ? and any
integer N ? 1, c? is always less than 1. Therefore, the od(H) is bounded by 1 ?c2?. If
H is singular, i.e., rank(H) < N, then Lemma 2 does not hold true. In this case, we
need to reduce the size of H and then apply CLLL algorithm.
Since information symbols s belong to Gaussian integer ring, then Hs generates a
lattice L 2 CM?1 with a set of basis vectors H = [h1,h2,...,hN]. The following lemma
shows an important statistical property of the minimum distance of the lattice L, which
will be useful for our proof on diversity order.
Lemma 3 Let H = [h1,h2,...,hN] be a set of bases for a lattice L in CM?1. Deflne
hmin as the vector in L which has the minimum non-zero norm among all the vectors. If
all entries of H are complex Gaussian distributed with zero mean, the following inequality
19
holds:
Pfjhminj2 ? ?g ? ch?D, (2.22)
where ch is a constant determined by N and M, and
D = min
8p6=0 rank(E(pp
H)), (2.23)
where p 2 L and p 6= 0.
The proof can be found in Appendix C. From Lemma 3 we notice that the degrees of
freedom D ofhmin is determined by the minimum rank of all possible covariance matrices
generated by the vectors L. Apparently, it depends on the covariance matrices of each
column hn and the cross-correlation among the columns.
To facilitate the use of Lemma 3, we give a corollary as follows:
Corollary 1 Let H = [h1,h2,...,hN] be a set of bases for a lattice L in CM?1. If 1) all
the entries of H are complex Gaussian distributed with zero mean; 2) rank(E[hnhHn ]) =
D, 8n 2 [1,N]; and 3) all the columns are linear independent with each other on the
Gaussian integer ring, then we have Pfjhminj2 ? ?g ? ch?D.
Now we are ready to analyze the diversity order collected by the LR-aided ZF
equalizer for the LLP-OFDM systems which is quantifled in the following proposition:
Proposition 3 Considering an LLP-OFDM system with group size of K and frequency-
selective channel order of L, given the model in (2.2), the diversity order collected by an
20
LR-aided ZF equalizer is min(K,Rh) which is the same as that obtained by ML detector,
where Rh = E[hhH] and h = [h(0),...,h(L)]T.
Proof: The output of the LR-aided ZF equalizer is stated in (2.19). Because the entries
of z are integers, if the real and imaginary parts of each entry of n = ?H?1w are in
the interval (?12, 12), one is able to decode z correctly and thus obtain s correctly. Let
us denote ?H?1 as [a0,a1,...,aK?1]T, where aTi ,i 2 [0,K ? 1] is the ith row of ?H?1.
Hence, if jaTi wj is less than 12, we will deflnitely decode the ith symbol correctly. Thus,
PejHequ, the error probability for a given Hequ is upper-bounded by
PejHequ ? P
jaTi wj ? 12
flfl
flflHequ
?
.
From [19, Lemma 1], we obtain the following inequality:
jaTi j ? 1q
1?od( ?H)j?hij
(2.24)
where ?hi,i 2 [0,K ?1] represents the ith column of ?H. Because
jaTi wj ? jaTi jjwj ? jwjq
1?od( ?H)j?hij
,
if jwj is less than 12
q
1?od( ?H)j?hij, we will have jaTi wj ? 12. Furthermore, since ?H is
reduced from Hequ using CLLL algorithm,
q
1?od( ?H) ? ( 44??1)?K(K?1)/4 according
to Lemma 2 wp1. Deflne Hequ := [h0,h1,...,hK?1], where hi,i 2 [0,K ?1] is the ith
column of Hequ. Let hmin represent the vector with minimum non-zero norm of all the
vectors in the lattice generated by Hequ. Since T is unimodular, ?H spans the same
21
lattice as Hequ. It is easy to verify that jhminj is less than or equal to min
1?i?K
j?hij. Thus,
we can see that if jwj is less than 12c?jhminj, then jwj is less than 12
q
1?od( ?H)j?hij for
any i 2 [1,K]. Here we notice that though c? is independent on Hequ, hmin depends on
Hequ. In summary, we have:
PejHequ ? P
jaTi wj ? 12
flfl
flflHequ
?
? P
jwj ? 12c?jhminj
flfl
flflHequ
?
. (2.25)
Since w is complex Gaussian white noise, jwj2 is a central Chi-square random vari-
able with degrees of freedom 2K and mean K 2w. Thus, by averaging (2.25) with respect
to the random matrix Hequ (or hmin, the error probability can be further simplifled as:
Pe ? EH
?
P
jwj ? 12c?jhminj
flfl
flflHequ
??
? P '(c?jhminj)2 ? 4K 2w?P 'jwj2 ? K 2w?
+P '4K 2w ? (c?jhminj)2 ? 4t2K 2w?P 'jwj2 ? t2K 2w?
+P '4t2K 2w ? (c?jhminj)2 ? 4t3K 2w?P 'jwj2 ? t3K 2w?+... (2.26)
where t is a positive constant that satisfles t > 1. Here, we notice that all the entries
of Hequ are complex Gaussian distributed with zero mean and all the columns are
linear independent with each other with probability one. Furthermore, the rank of the
covariance matrices of hi,i 2 [0,K ?1] is min(K,Rh) [11]. With these three conditions
satisfled, according to Corollary 1, we have Pfjhminj2 ? ?g ? ch?min(K,Rh). Thus, we
can get the probability that:
P '(c?jhminj)2 ? aK 2w? ? ch ?
aK 2
w
c2?
?min(K,Rh)
.
22
Because jwj2 is Chi-square distributed, we obtain that [9, p. 25]
P 'jwj2 ? aK 2w? = e?aK
K?1X
k=0
(2aK)k
k! ? cKe
?aKaK,
where cK = PK?1k=0 (2K)kk! is a constant that only depends on K. By using Gd to represent
min(K,Rh), Eq. (2.26) is simplifled as :
Pe ? chcK
4K
c2?
?Gd 1
2w
??Gd " 1X
n=0
tn(K+Gd)e?Ktn
#
. (2.27)
It is not di?cult to show that the summation (2.27) converges to a constant which only
depends on K,t, and Gd when t > 1. Therefore, the diversity order of the LR-aided ZF
equalizer is greater than or equal to Gd = min(K,Rh). However, as we know, the maxi-
mum diversity order for each group is min(K,Rh). Thus for LLP-OFDM, the LR-aided
ZF equalizer collects diversity order min(K,Rh). ?
As we have shown, linear equalizers cannot exploit the multipath diversity for LLP-
OFDM systems, however, after introducing the LR technique into the linear equalization
process, multipath diversity is collected. Similar to the proof for Proposition 3, one
can show that LR-aided MMSE estimator also collects multipath diversity. Note that
LR-aided linear equalizers have some unique properties: i) the decoding complexity
is much lower than ML and quite close to linear ones; ii) unlike SD, the complexity
does not depend on SNR; iii) the complexity of CLLL part does not change along with
constellation size.
23
0 5 10 15 20 25 3010?5
10?4
10?3
10?2
10?1
100
SNR in dB
BER
LLP?OFDM w/ ZFLLP?OFDM w/ MMSE
LLP?OFDM w/ ZF?GDFELLP?OFDM w/ MMSE?GDFE
Plain OFDM w/ ML
Figure 2.1: Comparisons among difierent linear equalizers
2.4 Simulation Results
In this section, we use computer simulations to verify our theoretical claims on the
diversity order of linear equalizers and the performance of LR-aided linear equalizers.
Example 1 (Performance comparison of difierent equalizers): We flrst compare
the performance of the LLP-OFDM with difierent equalizers with the conventional
(plain) OFDM. We select L = 3, and the total subcarriers N = 8. The channel taps are
i.i.d complex Gaussian random variables with zero mean and variance 2 = 1/(L + 1).
To enable the multipath diversity Gd = L + 1 = 4, we adopt the LLP-OFDM. The
subcarriers are split into Ng = 2 groups with group size K = 4. QPSK modulation is
used. The bit-error-rate (BER) versus SNR for linear equalizers is shown in Figure 2.1.
24
0 5 10 15 20 25 3010?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR in dB
BER
LLP?OFDM w/ LR?aided ZFLLP?OFDM w/ LR?aided ZF?GDFE
LLP?OFDM w/ LR?aided MMSELLP?OFDM w/ ML
Figure 2.2: Comparisons among difierent equalizers for LLP-OFDM
For plain OFDM, we only consider symbol-by-symbol ML detection. For LLP-OFDM,
we consider ZF, ZF-GDFE, MMSE, MMSE-GDFE. From Figure 2.1, we notice that: i)
plain OFDM only achieves diversity one, and so do ZF, ZF-GDFE, MMSE and MMSE-
GDFE detectors for LLP-OFDM; ii) for LLP-OFDM, the performance of ZF equalizer
has the worst performance, while MMSE(-GDFE) has better performance than others.
Figure 2.2 shows the BER curves of LR-aided ZF, LR-aided MMSE, LR-aided ZF-GDFE
and ML detectors for LLP-OFDM. From this flgure, we observe LR-aided ZF, MMSE or
ZF-GDFE detectors of LLP-OFDM collect diversity order L+1 and so does ML detector.
The performance of LR-aided MMSE equalizer is better than that of LR-aided ZF and
ZF-GDFE, but there still exists a gap between the performance of LR-aided equalizers
and ML detector. This is one of our future topics.
25
0 5 10 15 20 25 3010?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR in dB
BER
LR?aided ZF with K=2LR?aided ZF with K=3
LR?aided ZF with K=4SD with K=2
SD with K=3SD with K=4
Figure 2.3: Comparisons among difierent group sizes using ML detector and complex
LR-aided equalizer
Example 2 (Performance comparison of difierent group sizes):In this example,
we flx the total number of subcarriers N = 12, channel order L = 2 and the number
of total channel taps is L + 1 = 3. We use difierent group sizes to simulate the LLP-
OFDM, and then compare the performance of them. The group size is chosen to be
K = 2,3,4 respectively, and precoder ? is designed according to [29]. At the receiver,
both SD decoder and complex LR-aided ZF equalizer are employed and compared. Fig-
ure 2.3 shows the performance for difierent cases. Theoretically, the diversity order is
Gd = min(K,L+1) [12]. Based on the simulation results, we observe that the diversity
order collected by both SD and complex LR-aided ZF equalizer is 2,3 respectively cor-
responding to group size K = 2,3. When K = 4, the channel taps in each group are
26
2 2.5 3 3.5 4 4.5 5 5.5 60
0.5
1
1.5
2
2.5
3x 104
Group Size K
Number of Arithmetic Operations
ZF DetectionGeneral ZF
LR?aided ZFSD method
Figure 2.4: Complexity comparison of difierent decoding methods
correlated. We notice that our LR-aided linear equalizer still collects full diversity.
Example 3 (Complexity comparison of decoding schemes):After comparing the
performance of difierent decoding methods with difierent group size, we verify here the
difierence of the complexity. Here, we choose the number of total sub-carriers of OFDM
as N = 12 and the order of channel L = 5, then the multipath diversity order is 6.
To compare the complexity of difierent decoding methods, we flx SNR = 30dB and
count the number of arithmetic operations (real additions and real multiplications). In
Figure 2.4, we plot four curves to represent: SD method [7], complex LR-aided ZF
equalizer, general ZF equalizer and ZF detection for LLP-OFDM. Here, general ZF
detection means using pseudo-inverse equalizer ((DH?)?1) and the simple ZF detection
27
is ?HD?1H . From Figure 2.4, we notice that, the curve of LR-aided ZF equalizer is
much colser to general ZF detection than to SD method. This means that the decoding
complexity of LR-aided ZF is near that of general linear equalizers, and much lower than
SD method. Furthermore, the ratio of the gap between LR-aided ZF and SD to the
gap between LR-aided ZF and general ZF increases as group size K increases. Thus,
LR-aided ZF equalizer becomes computational preferable as K increases. All of these
four curves increase as K increases, which means the complexity increases as the group
size increases while the performance is getting better. From Figure 2.4, we can see that
the simplifled ZF equalizer for LLP-OFDM is quite low but it can only collect diversity
1. However, with a complexity that is a little bit higher than ZF equalizer, LR-aided ZF
equalizer can gurantee the same diversity as ML detector.
28
Chapter 3
LINEAR EQUALIZATION FOR V-BLAST SYSTEMS
In this chapter, we study the performance of V-BLAST multi-antenna architecture
which is presented in [4] and [25]. Similar to the LLP-OFDM systems, through the
analysis, we show that LR-aided linear equalizers can exploit the same diversity as ML
detection while linear equalizers lose diversity which means performance degradation.
3.1 System Model of V-BLAST Systems
Consider a multi-antenna system withNt transmit-antennas andNr receive-antennas,
where Nr ? Nt. In V-BLAST model ([4, 25]), the data stream is divided into Nt
sub-streams and transmitted through Nt antennas. These sub-streams consist of M-
QAM symbols (or in general, symbols on Gaussian integer ring). For notation sim-
plicity, we assume that the power of each transmit-antenna is normalized to one. Let
s = [s1,s2,...,sNt]T represents the Nt ? 1 transmitted data vector at one time slot,
while w = [w1,w2,...,wNr]T denotes the white Gaussian noise vector observed at the
Nr receive-antennas with zero mean and variance 2w. Without any coding, we consider
that E'ssH? = INt and E'wwH? = 2wINr.
The received signal at one time slot is denoted as: y = [y1,y2,...,yNr]T which is
represented by:
y = Hs+w, (3.1)
29
where H is the channel matrix, which consists of Nr ?Nt uncorrelated complex Gaus-
sian channel coe?cients with zero mean and unit variance. We assume a at-fading
environment that the channel coe?cients are constant during a frame and change inde-
pendently from frame to frame. Also we assume that the channel matrix H is known at
the receiver.
Theoretically, at the receiver, we can perform maximum-likelihood (ML) detection
to obtain the optimal performance of the system. Recall that the ML estimate is given
as:
?sML = arg
s2SNt
min k y?Hs k2, (3.2)
where k ? k denotes the 2-norm. The diversity order collected by an ML decoder for
the system in (3.1) is Nr, the number of receive-antennas (see e.g., [28]). However, the
cardinality of the searching space is jSNtj = MNt, thus the exponential complexity pro-
hibits the use of ML detection in practical systems (especially for systems that have large
constellations and numbers of antennas). In the next section, we will introduce the linear
detectors, whose decoding complexity is much lower than ML and thus computationally
preferable in certain situation.
3.2 Linear Equalizers and Performance Analysis
In this section, we brie y describe the linear equalizers for V-BLAST systems and
then focus on analyzing their performance.
30
3.2.1 ZF Equalizers
The zero-forcing (ZF) detector for the input-output relationship in (3.1) is given as:
xZF = Hyy = s+Hyw = s+n, (3.3)
where Hy denotes the Moore-Penrose Pseudo-inverse of the channel matrix H and n :=
Hyw is the noise after equalization. The pseudo-inverse matrix can be written as :
Hy = (HHH)?1HH (3.4)
where the channel matrix H has full column rank with probability one. Note that the
noise vector n is no longer white and its covariance matrix depends on the equalizer
matrix Hy. The quantization step is then used to map each entry of x into the symbol
alphabet S :
?sZFn = Q(xn) = arg
s2S
minjxn ?snj, (3.5)
where xn denotes the nth element of xZF and Q(?) means the quantizer of the symbol.
Starting from (3.3), we flrst study the performance of the ZF equalizer. Following
the proof for Proposition 1 with Hequ = H, we can get the error probability given H
as:
P(si ! ?si j H) = Q
?s
j ei j2
2 2wCii
!
, (3.6)
31
where Cii is the (i,i)th element of C and C can be expressed as:
C = Hy(Hy)H = (HHH)?1. (3.7)
Based on (3.6), the diversity order collected by ZF equalizer is established in the
following:
Proposition 4 Given the model in (3.1), if the channels are i.i.d. complex Gaussian
distributed, then the ZF equalizer in (3.3) exists with probability one and the diversity
order for the system is Nr ?Nt + 1.
Proof: See Appendix D.
This proposition shows that not surprisingly, the ZF linear equalizer enables the
same diversity as nulling-cancelling (NC) method for V-BLAST system [28], since NC
method agrees with ZF equalizer when detecting the flrst symbol ( the Ntth symbol of
s). However, the diversity order is lower than the ML equalizer.
3.2.2 MMSE Equalizers
Another type of linear equalizer is minimum mean square error (MMSE) detector
which takes into account the noise variance. It minimizes the mean-square error between
the actually transmitted symbols and the output of the linear detector and thus improves
the performance. Given the model in (3.1), the MMSE equalizer is :
xMMSE = ?HHH + 2wINt??1HHy. (3.8)
32
Again, the quantization step is the same as the one in (3.5). As shown in [26], with the
deflnition of an extended system, the MMSE detector agrees with the ZF detector.
For the MMSE equalizer, we start from (3.8) and rewrite the system model as:
xMMSE = s+?HHH + 2wINt??1 (HHw? 2ws). (3.9)
Deflne the equivalent noise vector n = (HHH+ 2w INt)?1(HHw? 2ws). Given a signal
vector s, the noise vector n has mean and covariance matrix, respectively as:
?n = ? 2w(HHH + 2wINt)?1s
? = 2w(HHH + 2wINt)?1 ? 4w(HHH + 2wINt)?2. (3.10)
Similar to the analysis for the ZF equalizer, we can verify that error probability is given
as:
P(si ! ?si j H) = Q
?s
(j ei j2 +e?i ?ni +ei?n?i)2
2jeij2?ii
!
, (3.11)
where ?ni is the ith element of noise mean ?n and ?ii is the (i,i)th element of noise
covariance matrix ? in (3.10). At high signa-to-noise ratio (SNR), i.e., when 2w is much
smaller than 1, Eq. (3.11) can be approximated as:
P(si ! ?si j H) ? Q
?s
j ei j2
2 2w ?Cii
!
, (3.12)
33
where ?Cii is the (i,i)th element of (HHH + 2wINt)?1. It is ready to verify that ?Cii
has the same degrees of freedom as Cii in (3.6). This shows that MMSE detector can
achieve the same diversity as ZF detector:
Proposition 5 Given the model in (3.1), if the channels are i.i.d. complex Gaussian
distributed, then MMSE equalizer in (3.8) exists with probability one and the diversity
order for the system is Nr ?Nt + 1.
In general, the MMSE equalizer outperforms the ZF equalizer with larger coding ad-
vantage because ?Cii is always less than Cii. Furthermore, if we spell out ?Cii and Cii,
for the same SNR, as the number of receive-antennas (Nr) increases, the performance
gap between MMSE and ZF detectors decreases. In [21], it has been shown that certain
linear precoded OFDM systems can also achieve maximum diversity by linear equaliz-
ers. However, here we have shown that linear equalizers for V-BLAST systems can only
achieve diversity order Nr?Nt+1, which is less than the maximum diversity Nr. In the
following, we show that using lattice reduction methods, we can restore the maximum
diversity order Nr.
3.3 LR-Aided Linear Equalizers
The performance of LR-aided linear detectors for V-BLAST system is established
in the following proposition:
Proposition 6 The diversity order collected by a Lattice-Reduction aided linear detector
(LR-aided ZF or LR-aided MMSE) for an MIMO V-BLAST system with Nt transmit-
antennas and Nr receive-antennas is Nr, which is the same as that obtained by maximum-
likelihood detector.
34
The proof is similar to that of Proposition 3. What we need to prove is Pfjhminj ? ?g ?
ch?2Nr, where hmin is the vector with the minimum norm of all the vectors in the lattice
spanned by H. Since all the entries of H are i.i.d. complex Gaussian random variables
with zero-mean, the rank of E[hnhHn ] is Nr for n 2 [1,Nt]. Furthermore, the Nt columns
are linear independent with each other on the Gaussian integer ring with probability
one. Thus, according to Corollary 1, we can obtain that Pfjhminj ? ?g ? ch?2Nr. Then
following the proof of Proposition 3, we can see, for V-BLAST systems LR-aided ZF
equalizer can collect diversity Nr, which is the same as that exploited by ML detection.
3.4 Simulation Results
In this section, we use computer simulations to verify our theoretical claims on the
diversity order of linear equalizers and the performance of LR-aided linear equalizers for
V-BLAST systems. The channels are generated as i.i.d. complex Gaussian variables
with zero mean and unit variance. The SNR is deflned as symbol energy per transmit-
antenna versus noise power.
Example 1 (Diversity collected by linear equalizers): The ZF and MMSE equal-
izers in (3.3) and (3.8) are considered for V-BLAST systems in this example. We consider
Nt = 2 transmit-antennas, and difierent numbers of receive-antennas Nr = 2,3,4. BPSK
is used as the modulation scheme. The bit-error rate (BER) versus SNR is depicted in
Figure 3.1. Reading the slopes of the curves in Figure 3.1, we observe that the diversity
orders enabled by either ZF or MMSE equalizer are indeed Nr ? Nt + 1, which in this
35
0 5 10 15 20 25 3010?5
10?4
10?3
10?2
10?1
100
SNR in dB
SER
ZF equalizer(Nr=2)ZF equalizer(Nr=3)
ZF equalizer(Nr=4)MMSE equalizer(Nr=2)
MMSE equalizer(Nr=3)MMSE equalizer(Nr=4)
Figure 3.1: BER of systems with Nt = 2 and Nr = 2,3,4 separately and BPSK modu-
lation
example are 1,2, and 3. Furthermore, we observe that the performance of MMSE detec-
tion is better than ZF detection and the gap between them decreases as Nr increases.
Example 2 (LR-aided linear equalizers): In this example, we flx the number of
transmit- and receive-antennas as Nt = 3,Nr = 4, and use QPSK as the modulation
scheme. Five detection methods are applied to the system: ZF detection, MMSE de-
tection, complex LR-aided ZF detection, complex LR-aided MMSE detection and ML
detection. Observing Figure 3.2, we notice that the linear detectors can only achieve the
diversity order Nr ? Nt + 1 which is 2 in this case. The ML detector enables diversity
order Nr = 4. As expected, the LR-aided linear detectors achieve the same diversity
order as the ML does. The performance gap between LR-aided linear detectors and ML
36
0 5 10 15 20 25 3010?5
10?4
10?3
10?2
10?1
100
SNR in dB
SER
ZF DetectionLR?aided ZF Detection
MMSE DetectionLR?aided MMSE Detection
ML Detection
Figure 3.2: SER of a system with (Nt,Nr) = (3,4) and QPSK modulation
detector is because the LLL algorithm cannot perfectly diagonalize the channel matrix.
Example 3 (Complex LLL algorithm): As shown in Chapter 2, we have extended
the LLL algorithm to complex fleld for any number of transmit-antenna. The detailed
CLLL algorithm can be found in Appendix B. In this example, we compare the complex
LLL algorithm with the real LLL algorithm in terms of complexity and performance. The
arithmetic operations we count are the number of real additions and real multiplications.
In Figure 3.3, we count the average number of arithmetic operations needed by the
complex and real LLL algorithm separately as Nr = Nt = n increases. We can see
that the complexity of the real LLL algorithm is about O(n4) which is consistent with
the result in [10]. The number of arithmetic operations that the real LLL algorithm
37
2 3 4 5 6 7 8 9 100
5000
10000
15000
Nr=Nt
Number of Arithmetic Operations
Complex?LR?aided ZFReal?LR?aided ZF
Figure 3.3: Complexity comparison between complex LLL and real LLL algorithms
needs is about 1.5 times of that of the complex LLL algorithm. Therefore, the complex
LLL algorithm is more e?cient. In Figure 3.4, we also compare the performance of
complex LR-aided and real LR-aided schemes. It can be seen that complex LR-aided
linear equalizers have the same performance as the real ones. So we can see, with
lower complexity and the same performance, complex LLL algorithm is computational
preferable to real LLL algorithm.
38
0 5 10 15 20 25 3010?5
10?4
10?3
10?2
10?1
100
SNR in dB
SER
ZFComplex?LR?aided ZF
Real?LR?aided ZFML
Figure 3.4: Performance comparison between the complex and real LLL algorithms with
(Nt,Nr) = (4,4)
39
Chapter 4
EXTENSION TO MIMO-OFDM SYSTEMS
As we have shown, linear equalizers can not exploit the multipath diversity for LLP-
OFDM systems, however, after introducing the LR technique into the linear equalization
process, multipath diversity is collected. In this chapter, we see similar situation happens
to the MIMO-OFDM designs. The block diagram of MIMO-OFDM is depicted in Fig.4.1.
There are Nr receive-antennas and Nt transmit-antennas, and the frequency-selective
wireless links between transmit- and receive-antennas have the same channel order L.
The channel taps of the link between the ?th transmit-antenna and the ?th receive-
antenna are denoted as h(?,?)l for l 2 [0,L] and ? 2 [1,Nr],? 2 [1,Nt]. x?n(p) represents
the symbol transmitted on the pth subcarrier from the ?th transmit-antenna in the nth
OFDM time slot while y?n(p) is received at the ?th receive-antenna on the pth subcarrier
in the nth OFDM slot.
There are many coding schemes designed for MIMO-OFDM and difierent designs
will present difierent performance even with the same equalizers. In this section, we will
use two coding schemes: full-diversity-full-rate (FDFR) design in [13] and space-time-
frequency (STF) design in [11] as examples to show the diversity that linear equalizers
and LR-aided linear detection methods can collect.
4.1 FDFR-OFDM
Multi-antenna systems not only provide space diversity, but also boost transmission
rate. FDFR design has been introduced to enjoy both advantages [13]. When channels
40
Figure 4.1: Block Diagram of MIMO-OFDM
are frequency-selective, FDFR design can be combined with OFDM to reduce the equal-
ization complexity and collect diversity to combat fading. However, the main price paid
here is decoding complexity (see [13] for details). In the following, we brie y introduce
the FDFR-OFDM design and then give the performance analysis when linear equalizers
or LR-aided equalizers are employed.
Let y(p) 2 CNr?1 represent the symbol vector received through Nr receive-antennas
on the pth subcarrier. By stacking y(p),p 2 [0,Nc ?1] (Nc is the number of total sub-
carriers which we assume to be Nt(L+1) for simplicity) into one symbol vector, the I/O
relationship of FDFR design is represented as:
2
66
66
64
y(0)
...
y(Nc ?1)
3
77
77
75 =
2
66
66
64
H(0)(P1Dfl)? T1
...
H(Nc ?1)(PNtDfl)? TNc
3
77
77
75s+
2
66
66
64
w(0)
...
w(Nc ?1)
3
77
77
75 = HFDFRs+w,(4.1)
where the permutation matrix Pn and the diagonal matrix Dfl are deflned as:
Pn :=
2
64 0 In?1
INt?n+1 0
3
75, D
fl := diag[1,fl,...,flNt?1], (4.2)
41
and scalar fl can be found in [13], H(p),p 2 [0,Nc ?1] is the MIMO channel matrix on
the (p+1)st sub-carrier with the (?,?)th entry H(?,?)(p) = PLl=0 h(?,?)l e?j2?plNc , Tn is the
nth row of Nc?Nc LCF encoding matrix ? and s comprises N2t (L+1) symbols. Based
on this system model, we flrst summarize the results for FDFR design as follows:
Proposition 7 Considering an FDFR-OFDM system with Nt transmit-antennas and
Nr receive-antennas, and the frequency-selective channel order of L, given the model in
(4.1), if the channel taps are independently complex Gaussian distributed with zero mean,
then the ZF equalizer exists wp1 and collects diversity order Nr ?Nt + 1. An LR-aided
ZF equalizer exists wp1 and collects full diversity NrNt(L+1) which is the same as that
exploited by ML detector.
According to the design of fl and ? in [13] and the structure of HFDFR, it is not
di?cult to verify that HFDFR has full rank wp1, which means ZF equalizer exists wp1
for the model given in (4.1). Following the general proof of Proposition 1, we can express
the error probability of the FDFR design using ZF equalizer as:
P(si ! ?si j HFDFR) = Q
?s
jeij2
2 2wCii
!
. (4.3)
where Cii is the (i,i)th entry of the matrix C = (HHFDFRHFDFR)?1. Given HFDFR
as in (4.1) we can get the speciflc form of Cii as:
Cii = 1N
c
Nc?1X
p=0
Ckk(p) = 1N
c
Nc?1X
p=0
1
C?1kk (p), k = (m+p)?
?m+p
Nt
?
?1
?
Nt (4.4)
42
where m =
l
i
Nc
m
and Ckk(p) is the (k,k)th entry of the matrix C(p) = (HH(p)H(p))?1.
Thus, we can bound Cii as:
1
NcC?1kckc(c) ? Cii ?
1
min
0?p?Nc?1
C?1kk (p) (4.5)
Here c is a constant number, c 2 [0,Nc ?1] and kc is determined by c and i. As stated
in Appendix. D, C?1kk (p) is a Chi-square random variable with degrees of freedom of
2(Nr ? Nt + 1). With the error probability expressed in (4.3), by using Lemma 1, the
outage probability can be bounded as:
(Nr ?Nt + 1)Nr?Nt
?(Nr ?Nt +1)
2 2
w th
Ncjeij2
?Nr?Nt+1
? P( < th) ? cu
2 2
w th
jeij2
?Nr?Nt+1
. (4.6)
So we can obtain that the diversity order of ZF equalizer for FDFR design is Nr?Nt+1.
Regarding the performance of LR-aided ZF equalizer for FDFR designs, the proof is
similar to that for Proposition 3. We only need to prove Pfjhminj2 ? ?g ? ch?NrNt(L+1),
where hmin is the vector with minimum non-zero norm of all the vectors in the lattice
spanned by HFDFR. Given the system in (4.1), we can write the speciflc form of the
ith column of HFDFR as :
hi =
2
66
66
66
66
66
66
4
H(0)mfl(m?1)?1, i?(m?1)Nc
...
H(p)(p+m)?(n?1)Ntfl(m?1)?p+1, i?(m?1)Nc
...
H(Nc ?1)(Nc+m)?(n?1)Ntfl(m?1)?Nc, i?(m?1)Nc
3
77
77
77
77
77
77
5
, (4.7)
43
where m = d iNce and n = dm+pNt e. H(p)(p+m)?(n?1)Nt is the ((p+m)?(n?1)Nt)th
column of the matrix H(p), and ?p+1, i?(m?1)Nc is the (p + 1, i?(m?1)Nc)th entry
of the LCF encoder ?. Since fl(m?1)?p+1, i?(m?1)Nc is deterministic given i and p, the
statistical property of the ith column is determined byH(p)(p+m)?(n?1)Nt,p 2 [0,Nc?1],
which is the frequency response of the pth subcarrier of the frequency-selective channels
between the ((p + m) ? (n ? 1)Nt)th transmit-antenna and Nr receive-antennas. Intu-
itively, the NrNt(L + 1) ? 1 vector hi selects L + 1 subcarriers from each of the total
NrNt frequency-selective channels. So the rank of the covariance matrix of each column
is NrNt(L + 1). The N2t (L + 1) columns are linear independent with each other which
is guaranteed by the LCF encoder ? and the choice of fl in [13]. Thus, similar to LLP-
OFDM, following the proof of Proposition 3, we can get that the diversity order of the
LR-aided linear equalizer for FDFR design is NrNt(L + 1) which is also the maximum
diversity order that the system can achieve.
4.2 STF-OFDM
STF design is also composed of two stages: LCF encoding across subcarriers and ST
multiplexing using ST orthogonal code [20]. For example, when Nt = 2, the 2NgK ?1
symbol vector s is split into 2Ng groups, sn 2 CK?1,n 2 [1,2Ng]. Then, we transmit
every two groups through two transmit-antennas using Alamouti scheme [2] after encod-
ing the symbol blocks sn with the K?K LCF encoding matrix ?. The I/O relationship
44
for each group is:
2
66
66
64
yn(0)
...
yn(K ?1)
3
77
77
75 =
2
66
66
64
jH(0)j
...
jH(K ?1)j
3
77
77
75?sn + ?w = HSTFsn + ?w. (4.8)
where ?w is the equivalent white Gaussian noise vector, and
jH(p)j2 =
NrX
?=1
NtX
?=1
jH(?,?)(p)j2 =
NrX
?=1
NtX
?=1
flfl
flfl
fl
LX
l=0
h(?,?)l e?j2? p lK
flfl
flfl
fl
2
. (4.9)
Based on this system model in (4.8), we apply the ML detection or SD method to get
the estimation of information symbols ?sn. According to [11], when the group length K
is greater than or equal to L + 1, these optimal and near optimal detectors exploit the
full diversity NrNt(L+1).
Given the model in (4.8), we can see that the ZF equalizer for the STF design
exits with probability one. Since (4.8) has the same form as (2.2), we can get the
error probability as in (2.7), where Cii now is the (i,i)th entry of the matrix C =
(HHSTFHSTF)?1. Similarly, we can bound Cii as:
1
K
1
jH(c)j2 ? Cii =
1
K
K?1X
k=0
1
jH(k)j2 ?
1
min
0?k?K?1
jH(k)j2, (4.10)
where c is a constant in [0,K?1]. Since jH(p)j2 is the summation of NrNt independent
Chi-square random variable with degrees of freedom 2 as shown in (4.9), according to
[15], jH(p)j2 still performs like a Chi-square random variable with degrees of freedom
45
2NrNt. Following the proof of Proposition 3, we can bound the outage probability as:
(NrNt)NrNt?1
?(NrNt)
2 2
w th
Kjeij2
?NrNt
? P( < th) ? cu
2 2
w th
jeij2
?NrNt
(4.11)
So, the diversity order that the ZF equalizer can exploit for STF design is NrNt.
To show that the diversity order collected by LR-aided ZF equalizer for STF design
is NrNt(L + 1), we need to show for the lattice spanned by HSTF, Pfjhminj2 ? ?g ?
ch?NrNt(L+1), where hmin is the vector with minimum non-zero norm of all the vectors in
the lattice. In other words we need to make sure the three conditions in Corollary 1 are
satisfled. Obviously, all the entries in HSTF are complex Gaussian distributed with zero
mean. From (4.8), we can see that the kth column of the equivalent channel matrixHSTF
of the system is hk = diag[jH(0)j,...,jH(K ?1)j] k,k 2 [1,K], the square norm of
which is a Chi-square random variable with degrees of freedom 2NrNt(L+1), where k
is the kth column of ?. Further, because of the linear independence of LCF encoding
matrix ?, all the columns of the equivalent channel matrix HSTF are linear independent
with each other. So according to Corollary 1, we have Pfjhminj ? ?g ? ch?2NrNt(L+1).
Thus, following the proof of Proposition 3, we can get that the diversity order of LR-aided
ZF equalizer for STF design is NrNt(L+1). The results for STF design are summarized
in the following proposition:
Proposition 8 Consider an STF-OFDM system with Nt transmit-antennas and Nr
receive-antennas, and the frequency-selective channel order of L. Given the model in
(4.8), if the channel taps are independently complex Gaussian distributed with zero mean,
then ZF equalizer in (2.3) exists wp1 and exploits diversity order NrNt. For STF design,
46
0 5 10 15 20 25 3010?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR in dB
BER
ZF equalizerLR?aided ZF
MMSE equalizerLR?aided MMSE
SD method
Figure 4.2: Comparison among difierent equalizers for FDFR design
an LR-aided ZF equalizer collects diversity order NrNt(L+1) which is the same as that
obtained by ML detector.
4.3 Simulation Results
Example 1 (Performance comparison of difierent equalizers for FDFR):Con-
sider a multiantenna system with Nt = Nr = 2 and frequency-selective channel order
L = 1. Five detectors are employed respectively on model (4.1): ZF, MMSE, LR-aided
ZF, LR-aided MMSE detectors and sphere decoder (SD). QPSK modulation is used for
modulation. The BER versus SNR performance is depicted in Figure 4.2. It shows that
the linear detectors (ZF, MMSE) can only achieve diversity Nr ? Nt + 1 = 1. How-
ever, the LR-aided linear detectors achieve maximum diversity NrNt(L + 1) = 8 with
47
0 5 10 15 20 2510?8
10?6
10?4
10?2
100
SNR in dB
BER
ZF equalizerLR?aided ZF
SD method
Figure 4.3: Comparison among difierent equalizers for STF design
much lower complexity than the SD detector though there exists a gap between SD and
LR-aided linear detectors.
Example 2 (Performance comparison of difierent equalizers for STF):STF de-
sign is simulated and 16-QAM is chosen as symbol modulation scheme. We use the chan-
nel conflguration in previous example. ZF, LR-aided ZF and SD equalizers are applied
to the system model in (4.8). Performance of difierent equalizers is plotted in Figure
4.3. Reading the slopes of the curves in Figure 4.3, we observe that the diversity orders
for ZF equalizer is indeed NrNt = 4 while LR-aided ZF equalizer can achieve maximum
diversity NrNt(L+ 1) = 8, the same as SD. Note that Figure 4.3 also shows that if the
spatial diversity order (NtNr) is high, linear equalizers achieve similar performance as
SD for a large SNR range.
48
Chapter 5
CONCLUDING REMARKS AND FUTURE RESEARCH DIRECTIONS
In this thesis, we have discussed the performance in terms of diversity of linear
detectors for LLP-OFDM systems, V-BLAST systems and two MIMO-OFDM designs.
It shows that conventional linear equalizers can only collect diversity 1 for LLP-OFDM
systems and Nr ?Nt+1 for V-BLAST systems though they have the lowest complexity.
For MIMO-OFDM designs, it depends on the ST coding schemes. However, with slightly
increased complexity, LR-aided linear equalizers achieve maximum diversity for LLP-
OFDM, V-BLAST and MIMO-OFDM designs, which is the same as that collected by
ML detector. The complexity of LR-aided linear equalizers is much lower than those of
ML and near-ML detectors.
Based on this thesis, future research directions are listed but not limited as follows:
First, the performance analysis in terms of coding gain will be studied. Based on
our simulation results, it has been shown that although LR-aided equalizers achieve the
same diversity as ML does, there still exists a performance gap which is mainly due to
coding gain loss. In this research thrust, we will flrst quantify the coding gain of LR-
aided linear equalizers and then try to modify the equalizers to reduce the performance
gap with ML. Furthermore, we will study the performance of coded linear systems with
LR-aided decoding and compare the performance and complexity with other alternatives
in the literature.
Another future research direction is the study of the orthogonality deflciency of the
equivalent channel matrix for wireless communications. As we know, for linear systems,
49
when the equivalent channel matrix is diagonal, the performance of ZF equalizer is the
same as that of ML detector. However, in general the channel matrix Hequ is not
diagonal, and thus linear equalizers have inferior performance. Orthogonality deflciency
describes how ?diagonal? a matrix is. So the study of orthogonality deflciency provides a
useful tool to quantify the diversity and coding gains of linear equalizers. Furthermore,
how it will afiect the performance of linear equalizations may guide us to construct
the coding schemes to reach full diversity and high coding gain when linear equalizers
are adopted at the receiver. Hybrid equalizers will also be designed to trade-ofi the
performance with complexity.
50
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53
APPENDICES
54
APPENDIX A: Proof of Lemma 1
Suppose the joint probability density function (pdf) of Xn?s is f(x1,x2,...,xN).
The cumulative density function (cdf) of Xmin is:
F(v) = P(xmin < v) = 1?P(xmin ? v)
= 1?
Z 1
v
dx1
Z 1
v
dx2 ???
Z 1
v
f(x1,x2,...,xN)dxN. (1)
Then, we can obtain the pdf of Xmin by taking the derivative of the cdf:
f(v) = ddvF(v) = ?
Z 1
v
dx1 ddv
Z 1
v
dx2 ???
Z 1
v
f(x1,x2,...,xN)dxN
?
+
Z 1
v
dx2 ???
Z 1
v
f(v,x2,...,xN)dxN
=
NX
n=1
Z 1
v
dx1 ???
Z 1
v
dxn?1
Z 1
v
dxn+1 ???
Z 1
v
f(x1,??? ,xn?1,v,xn+1 ...,xN)dxN
?
NX
n=1
fXn(v), (2)
where fXn(x) is the pdf of Xn. According to [9, p. 25], we know that for a central
Chi-square random variable Xn with degrees of freedom 2M, we have
P(xn < ?) = 1?e??/2
M?1X
k=0
??
2
?k
k! = e
??/2
1X
k=M
??
2
?k
k! ?
??
2
?M
= cM?M,
55
where cM := 2?M is a constant only dependent on M. With this inequality, we can
obtain:
P(xmin < ?) =
Z ?
0
f(v)dv ?
Z ?
0
NX
n=1
fXn(v)dv ?
NX
n=1
cM?M = cu?M, (3)
where cu := NcM. Thus, Lemma 1 is proved. We notice that, Lemma 1 can be general-
ized to Xn?s with other pdfs. ?
56
APPENDIX B: Complex LLL algorithm
Here, we give the details of complex LLL algorithm in conventional Matlab nota-
tion in the following table.
INPUT: H
OUTPUT: Q, R, T
(1) [Q,R] = QR Decomposition(H);
(2) ? = 0.75;
(3) m = size(H,2);
(4) T = Im;
(5) k = 2;
(6) while k ? m
(7) for n = k ?1 : ?1 : 1
(8) u = round((R(n,k)/R(n,n)));
(9) if u ?= 0
(10) R(1 : n,k) = R(1 : n,k)?uR(1 : n,n);
(11) T(:,k) = T(:,k)?uT(:,n);
(12) end
(13) end
(14) if ?(R(k ?1,k ?1)2) > kR(k ?1,k)?Q(:,k ?1) +R(k,k)?Q(:,k)k2
(15) Swap the (k-1) th and k th columns in R and T
(16) ? =
?fi? fl
?fl fi
?
where fi = R(k?1,k?1)kR(k?1:k,k?1)k; fl = R(k,k?1)kR(k?1:k,k?1)k;
(17) R(k ?1 : k,k ?1 : m) = ?R(k ?1 : k,k ?1 : m);
(18) Q(:,k ?1 : k) = Q(:,k ?1 : k)?H;
(19) k = max(k ?1,2);
(20) else
(21) k = k +2;
(22) end
(23) end
Table 1: The complex LLL algorithm
57
APPENDIX C: Proof of Lemma 3
Let x be a vector in the lattice L. Since L is spanned by the columns of H, then x
can be expressed as Ha, where a is an N ?1 column vector with all entries belonging
to Gaussian integer ring. So based on the deflnition of hmin, we can obtain:
hmin = arg minx
2L, x6=0jxj
2 = HaH, (4)
where aH 2 Z[p?1]N?1. By deflning an MN ?1 vector h = [hT1 ,...,hTN]T, we have:
jhminj2 = jHaHj2 = hH(IM ??(aTH)HaTH?)h = hHAHh, (5)
where AH = IM ? ?(aTH)HaTH?. Suppose that the correlation matrix of the channel
vector is E[hhH] = Rh. The singular-value decomposition (SVD) of Rh is Uh?hUHh
where Uh is a unitary matrix and ?h is a diagonal matrix with all singular values.
Suppose rank(Rh) = Rh and deflne an Rh ?1 vector ?h with i.i.d. entries. Then h and
Uh?
1
2
h?h have identical distribution and thus the same statistical properties. Similar to
pairwise error probability (PEP) analysis in [18] and [22], we can get an upper bound
for the probability that jhminj is less than ? by averaging (5) with respect to the random
basis H:
P(jhminj2 ? ?) = P(?hH?
1
2
hU
H
h AHUh?
1
2
h?h ? ?) ? ch?
D, (6)
58
where
D = min
8aH rank(RhAH).
Since H are complex Gaussian distributed, the set of possible aH can be the whole
Gaussian integer ring. Thus, we can represent D as
D = min
8p6=0 rank(E(pp
H)). (7)
?
59
APPENDIX D: Proof of Proposition 4
In (3.7), we have deflned that C = (HHH)?1. By properly permutating H to
[hi,Hi], where hi is the ith column of H and Hi denotes the rest columns, we obtain
that
C = (HHH)?1 = P
2
64 hHi hi hHi Hi
HHi hi HHi Hi
3
75
?1
PT,
where P is a permutation matrix such that HP = [hi Hi]. Based on the matrix
inversion lemma, we know that the (i,i)th element of C is
Cii =
?
hHi hi ?hHi Hi?HHi Hi??1HHi hi
??1
. (8)
Suppose the singular value decomposition (SVD) of Hi is Hi = VDUH, where V
is an Nr?(Nt?1) unitary matrix, D is an (Nt?1)?(Nt?1) diagonal matrix and U is
an (Nt ?1)?(Nt ?1) unitary matrix. Plugging this SVD result into (8), we are ready
to verify that:
Hi(HHi Hi)?1HHi = VV H.
It is straightforward to see that the rank of VV H is Nt ?1. More speciflcally, VV H is
an Nr ?Nr matrix whose eigenvalue decomposition can be written as:
VV H = ?V
2
64 INt?1 0Nt?1,Nr?Nt+1
0Nr?Nt+1,Nt?1 0Nr?Nt+1,Nr?Nt+1
3
75 ?V H,
60
where ?V is an Nr ?Nr unitary matrix with the flrst Nt ? 1 columns same as V . This
is because the matrix VV H has the same non-zero eigenvalues as V HV . Therefore, we
can verify
C?1ii = hHi (INr ?VV H)hi
= hHi ?V
2
64 0Nt?1 0Nt?1,Nr?Nt+1
0Nr?Nt+1,Nt?1 INr?Nt+1,Nr?Nt+1
3
75 ?V Hh
i.
It can be seen that the number of degrees of freedom in C?1ii is 2(Nr ? Nt + 1). Fur-
thermore, C?1ii is a chi-squared random variable with 2(Nr ?Nt +1) degrees of freedom,
because the channel coe?cients are complex Gaussian distributed. As shown in [22], if
we integrate the right side of (2.7) with respect to this random variable, we obtain that
the diversity order is equal to Nr ? Nt + 1. This means the diversity order of the ZF
detection is Nr ?Nt + 1. ?
61