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Performance Analysis of ZF and MMSE Equalizers for MIMO Systems: An In-Depth Study of the High SNR Regime

TL;DR: An in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas reveals several new and surprising analytical results.
Abstract: This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multiinput multioutput (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in terms of output signal-to-noise ratio (SNR), uncoded error and outage probabilities, diversity-multiplexing (D-M) gain tradeoff and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE equalizer (conditioned on the channel realization) is ρmmse = ρzf+η\ssrsnr, where ρzf is the output SNR of the ZF equalizer and that the gap η\ssrsnr is statistically independent of ρzf and is a nondecreasing function of input SNR. Furthermore, as \ssr snr\ura ∞, η\ssrsnr converges with probability one to a scaled F random variable. It is also shown that at the output of the MMSE equalizer, the interference-to-noise ratio (INR) is tightly upper bounded by [(η\ssrsnr)/(ρzf)]. Using the decomposition of the output SNR of MMSE, we can approximate its uncoded error, as well as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the MMSE equalizer relative to its ZF counterpart. The e-outage capacities of the two equalizers, however, coincide in the asymptotically high SNR regime. We also provide the solution to a long-standing open problem: applying optimal detection ordering does not improve the D-M tradeoff of the vertical Bell Labs layered Space-Time (V-BLAST) architecture. It is shown that optimal ordering yields a SNR gain of 10log10N dB in the ZF-V-BLAST architecture (where N is the number of transmit antennas) whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better and significantly so.

Summary (2 min read)

Introduction

  • This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multi-input multi-output (MIMO) systems with no fewer receive than transmit antennas.
  • Using the decomposition of the output SNR of MMSE, the authors can approximate its uncoded error as well as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the MMSE equalizer relative to its ZF counterpart.
  • As another example, the problem of obtaining the exact diversity-multiplexing (D-M) tradeoff [6] of V-BLAST with optimal detection ordering still remains open, and so does the quantification of the gain due to optimal detection ordering.
  • The lower bounds are shown to be asymptotically tight for high SNR.

A. Basics of ZF and MMSE Equalizers

  • Consider the MIMO channel model given in (1) where the N data substreams are mixed by the channel matrix.
  • Here [·]nn denotes the nth diagonal element.

B. Diversity-Multiplexing Gain Tradeoff

  • In [6], the authors established the framework of D-M gain tradeoff analysis in the asymptotically high SNR regime.
  • Denote R(snr) as the data rate of any communication scheme with input SNR snr.
  • The diversity gain and multiplexing gain are defined as follows [6].
  • A scheme is said to have multiplexing gain r and diversity gain d if the data rate R(snr) satisfies lim snr→∞ R(snr) log snr = r, (18) and the average error probability Pe(snr) satisfies lim snr→∞ log Pe(snr) log snr = −d. (19) Because Pe(snr) and R(snr) are related, so are d and r, also known as Definition II.1.
  • The authors denote d(r) the tradeoff between the diversity gain and multiplexing gain, which is always a non-increasing function.

C. Two Theorems

  • The following two theorems turn out to be very useful for the analysis in this paper.
  • Since the elements of the channel matrix H are i.i.d., the output SNRs of the N substreams are of identical (but not independent) marginal distributions.
  • It is readily seen from (26) that given According to the i.i.d.
  • The authors apply Theorem III.1, i.e. the relationship ρmmse = ρzf,n + ηsnr, to analyze the uncoded error probability, outage probability, and ²-outage capacity of the MMSE equalizer.

A. Uncoded Error Probability Analysis

  • The uncoded error probability of the ZF equalizer is well known but the authors state it here for the sake of completeness.
  • Because the output SNRs of all the N substreams are of identical distribution, the authors only need to focus on one substream.
  • Calculating the error probabilities of a general quadrature amplitude modulation (QAM) is straight- forward using the error probability expression in Q-function [27].
  • But their work focused on non-fading channel.
  • The Gaussian approximation is still quite accurate in terms of average error probability for channels that are full rank with probability one, a fact that is verified in a numerical example given in Section VII.

B. Outage Probability and ²-Outage Capacity

  • Consider employing independent codes of rate R each over the N antennas.
  • Moreover, as the authors can observe from (53), the gap would become smaller as R increases.
  • Since the cdfs of both ρzf,n and ρmmse,n are continuous.
  • The non-vanishing SNR gap between the outage probabilities of the zero-forcing and MMSE equalizers and the result in (59) may seem contradictory at first.
  • The authors obtain the exact D-M gain tradeoffs of the linear ZF and MMSE receivers when independent, equal rate (and equal power) SISO Gaussian codebooks are employed over the N antennas.

A. The Linear ZF Equalizer

  • Consider the MIMO system that employs independent coding for each substream and the ZF equalizer at the receiver.
  • Each substream effectively experiences a scalar channel whose gain is of χ22(M−N+1) 17 distribution.
  • With the overall multiplexing gain r, each substream has a multiplexing gain rN .

B. The Linear MMSE Equalizer

  • The following theorem establishes that the equality holds in (67).
  • With the same but independent coding applied to all the substreams, the overall system outage probability would be dominated by that of the weakest substream, which is of order snr−dmin .

C. D-M Gain Tradeoff of V-BLAST with Channel-Dependent Ordered Detection

  • Based on Corollary V.3, the authors are ready to answer the long standing open question as to what really is the D-M tradeoff of V-BLAST with channel-dependent ordered decoding.
  • The authors see that the error probabilities of MMSE obtained via averaging over 105 Monte Carlo simulations match extremely well with the high SNR approximation of (46) for a moderate SNR (snr ≥ 10 dB).
  • Hence theorem VI.2 can be used to predict the SNR gain of ordered detection in V-BLAST architecture.
  • N. Jindal, “High SNR analysis of MIMO broadcast channels,” Proc. IEEE Int. Symp. Information Theory, Adelaide, Australia, Sept. 2005. [20].

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1
Performance Analysis of ZF and MMSE
Equalizers for MIMO Systems: An In-Depth
Study of the High SNR Regime
Yi Jiang Mahesh K. Varanasi Jian Li
Abstract
This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE)
equalizers applied to wireless multi-input multi-output (MIMO) systems with no fewer receive than transmit
antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in
terms of the well-known performance metrics of output signal-to-noise ratio (SNR), uncoded error and outage
probabilities, diversity-multiplexing (D-M) gain tradeoff, and coding gain. Contrary to the common perception
that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE
equalizer (conditioned on the channel realization) is ρ
mmse
= ρ
zf
+ η
snr
, where ρ
zf
is the output SNR of the
ZF equalizer, and that the gap η
snr
is statistically independent of ρ
zf
and is a non-decreasing function of input
SNR. Furthermore, as snr , η
snr
converges with probability one to a scaled F random variable. It is also
shown that at the output of the MMSE equalizer, the interference-to-noise ratio (INR) is tightly upper bounded
by
η
snr
ρ
zf
. Using the decomposition of the output SNR of MMSE, we can approximate its uncoded error as well
as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the
MMSE equalizer relative to its ZF counterpart. The ²-outage capacities of the two equalizers, however, coincide
in the asymptotically high SNR regime, despite the non-vanishing gap η
snr
. By analyzing a fictitious parallel
channel model with coding across the sub-channels in terms of the diversity-multiplexing (D-M) gain tradeoff,
we provide the solution to a long-standing open problem: applying optimal detection ordering does not improve
the D-M tradeoff of the V-BLAST (vertical Bell Labs layered Space-Time) architecture. However, by deriving
tight lower bounds to the outage probabilities of ZF and MMSE equalizers, we show that optimal ordering yields
a SNR gain of 10 log
10
N dB in the ZF-V-BLAST architecture (where N is the number of transmit antennas)
whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better, and
significantly so.
Keywords
This work was supported in part by the National Science Foundation Grant CCF-0423842 and CCF-0434410. This
pap er was presented in part at Globecom 2005.
Y. Jiang and M. Varanasi are with the Department of Electrical and Computer Engineering, University of Colorado,
Boulder CO 80309, USA. Email: yjiang.ee@gmail.com, varanasi@colorado.edu
J. Li is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130,
USA. Email: li@dsp.ufl.edu

2
Zero forcing, minimum mean squared error, MIMO, error probability, V-BLAST, diversity gain, spatial
multiplexing gain, tradeoff, outage capacity, outage probability.

3
I. Introduction
Consider the complex baseband model for the wireless multi-input multi-output (MIMO) channel
with N transmit antennas and M receiver antennas
y = Hx + z, (1)
where y C
M×1
is the received signal and H C
M×N
is a Rayleigh fading channel with independent,
identically distributed (i.i.d.), circularly symmetric standard complex Gaussian entries, denoted as
h
ij
N(0, 1) for 1 i M, 1 j N . We assume that the number of receive antennas is no less
than the number of transmit antennas (M N). We also assume that the N data substreams have
uniform power, i.e., x C
N×1
has covariance matrix E[xx
] = σ
2
x
I
N
, where E[·] stands for the expected
value, (·)
is the conjugate transpose, and I
N
is an N ×N identity matrix. The white Gaussian noise
z N(0, σ
2
z
I) is also circularly symmetric. The input signal-to-noise ratio (SNR) is defined as
snr =
σ
2
x
σ
2
z
. (2)
In this paper, we present an in-depth analysis of the performance of the zero forcing (ZF) and
minimum mean squared error (MMSE) equalizers applied to the channel given in (1). The linear
ZF and MMSE equalizers are classic functional blocks and are ubiquitous in digital communications
[1]. They are also the building blocks of more advanced communication schemes such as the decision
feedback equalizer (DFE), or equivalently, the V-BLAST (vertical Bell Labs layered Space-Time)
architecture [2][3], and various other MIMO transceiver designs (see, e.g., [4][5] and the references
therein). Despite their fundamental importance, however, the existing performance analyses of the ZF
and MMSE equalizers
1
are far from complete. For instance, it is commonly understood that ZF is a
limiting form of MMSE as snr . But when the ZF and MMSE are applied to the MIMO fading
channel given in (1), one may observe through simulations that the error probabilities of MMSE and
ZF do not coincide even as snr . To the best of our knowledge, no rigorous account of such a
phenomenon is available in the literature. As another example, the problem of obtaining the exact
diversity-multiplexing (D-M) tradeoff [6] of V-BLAST with optimal detection ordering still remains
open, and so does the quantification of the gain due to optimal detection ordering. In this paper,
we attempt to provide an in-depth look at the classical ZF and MMSE equalizers with respect to the
well-known performance metrics of output SNR, uncoded error and outage probabilities, diversity-
1
In the sequel we refer to the ZF and MMSE equalizers as ZF and MMSE for simplicity.

4
multiplexing (D-M) gain tradeoff, and SNR gain.
The major findings of this paper are summarized in the following.
R1 A common perception about ZF and MMSE is that ZF is the limiting form of MMSE as snr .
Therefore, it is presumed that the two equalizers would share the same output SNRs, and consequently,
the same uncoded error or outage probability in the high SNR regime. We show, however, that the
output SNRs of the N data substreams using MMSE and ZF are related by
ρ
mmse,n
= ρ
zf,n
+ η
snr,n
, 1 n N, (3)
where ρ
zf,n
and η
snr,n
are statistically independent and η
snr,n
is a nondecreasing function of snr. More-
over,
η
snr,n
η
,n
with probability one (w.p.1), as snr , (4)
where
M N + 2
N 1
η
,n
F
2(N1),2(MN+2)
is of F-distribution.
2
Further, the interference-to-noise
ratio (INR) of the nth substream at the output of MMSE (denoted as inr
n
), is approximately upper
bounded as
inr
n
.
η
snr,n
ρ
zf,n
. (5)
with the approximate upper bound being asymptotically tight for high SNR. Since
η
snr,n
ρ
zf,n
is inversely
proportional to the input SNR, (5) implies that the higher the input SNR, the smaller the leakage
from the interfering substreams.
R2 Using R1, we obtain tight approximations of the uncoded error and outage probabilities of MMSE
which can be evaluated via numerical integration rather than Monte-Carlo simulations. This analysis
also confirms that there is a non-vanishing SNR gain of MMSE over ZF as snr . Interestingly,
however, the ²-outage capacities of MMSE and ZF coincide in the asymptotically high SNR regime in
spite of the SNR gap between their outage probabilities.
R3 We obtain the following upper bounds of the output SNRs for the ZF and MMSE equalizers:
ρ
mmse,n
λ
2
N
snr + 1
u
1 and ρ
zf,n
λ
2
N
snr
u
, (6)
where λ
N
is the smallest singular value of H and u is a Beta random variable that is independent of
λ
N
with a probability density function (pdf)
f
u
(x) = (N 1)(1 x)
N2
, 0 x 1. (7)
2
Given two independent Chi-square random variables a χ
2
m
and b χ
2
n
. The ratio c =
a/m
b/n
is a random variable
with distribution f
c
(x) =
Γ
(
m+n
2
)
n
n
2
m
m
2
x
m
2
1
Γ
(
m
2
)
Γ
(
n
2
)
(n+mx)
m+n
2
, where Γ(z) =
R
0
t
z1
e
t
dt. We denote c F
m,n
[7].

5
Based on these upper bounds, we prove that for both ZF and MMSE, the D-M gain tradeoff of a
fictitious parallel channel (with N independent sub-channels) with coding across the N substreams is
the same as that for the ZF and MMSE equalizers applied to the MIMO channel with independent
coding over each individual substream, and this trade-off is given as
d(r) = (M N + 1)
³
1
r
N
´
. (8)
That is, the SNR gain gap between the MMSE and ZF equalizers cannot be captured by the D-M
gain tradeoff analysis.
R4 As an important corollary of R3, we solve the well-known open problem on the diversity gain of
the V-BLAST architecture with optimal detection ordering [2]. Note that the V-BLAST architecture
can be regarded as employing ZF or MMSE equalizers combined with decision feedback [3], which in
the sequel are referred to simply as ZF-VB and MMSE-VB, resp ectively. We prove that with equal
rate for each substream and for any order of decoding, both ZF-VB and MMSE-VB have the D-M
gain tradeoff
d
vb
(r) = (M N + 1)
³
1
r
N
´
, (9)
which means that the so-called V-BLAST order [2] does not yield an improvement in the D-M gain
tradeoff relative to unordered decoding.
R5 We also derive lower bounds on the outage probabilities of MIMO systems that use ZF and MMSE
(without decision feedback). The lower bounds are shown to be asymptotically tight for high SNR.
Based on these bounds, we prove that for ZF the strongest substream has a SNR gain of as much as
10 log
10
N dB over an average one at high SNR. For MMSE, the SNR gain is even higher, and that too
by a significant margin. When applied to systems with decision feedback, as in V-BLAST, because
the overall outage probability is dominated by that of the first detected substream, this result also
quantifies the coding advantage of optimally ordered decoding over fixed order decoding.
The results R1 and R2 are on the distribution of the output SNR of the MMSE equalizer, the
asymptotic normality of interference-plus-noise at its output, and the coded (outage) and uncoded
error probability p erformance. Such problems are also investigated in [8][9] for the asymptotic property
of linear multiuser receivers. While their work focuses on large systems, we study finite systems with
asymptotically high SNR. The influence of non-Gaussian interference upon error probability in finite
CDMA systems is studied in [10] which shows that the larger an interfering user’s amplitude, the
smaller its effect on bit-error rate [10]. The (tight) upper bound of INR given in (5) yields more

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Abstract: Preface to the Third Edition.Preface to the Second Edition.Preface to the First Edition.1. Introduction.2. The Multivariate Normal Distribution.3. Estimation of the Mean Vector and the Covariance Matrix.4. The Distributions and Uses of Sample Correlation Coefficients.5. The Generalized T2-Statistic.6. Classification of Observations.7. The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance.8. Testing the General Linear Hypothesis: Multivariate Analysis of Variance9. Testing Independence of Sets of Variates.10. Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices.11. Principal Components.12. Cononical Correlations and Cononical Variables.13. The Distributions of Characteristic Roots and Vectors.14. Factor Analysis.15. Pattern of Dependence Graphical Models.Appendix A: Matrix Theory.Appendix B: Tables.References.Index.

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Q1. What are the contributions mentioned in the paper "Performance analysis of zf and mmse equalizers for mimo systems: an in-depth study of the high snr regime" ?

This paper presents an in-depth analysis of the zero forcing ( ZF ) and minimum mean squared error ( MMSE ) equalizers applied to wireless multi-input multi-output ( MIMO ) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, the authors reveal several new and surprising analytical results in terms of the well-known performance metrics of output signal-to-noise ratio ( SNR ), uncoded error and outage probabilities, diversity-multiplexing ( D-M ) gain tradeoff, and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, the authors show that the output SNR of the MMSE equalizer ( conditioned on the channel realization ) is ρmmse = ρzf + ηsnr, where ρzf is the output SNR of the ZF equalizer, and that the gap ηsnr is statistically independent of ρzf and is a non-decreasing function of input SNR. By analyzing a fictitious parallel channel model with coding across the sub-channels in terms of the diversity-multiplexing ( D-M ) gain tradeoff, the authors provide the solution to a long-standing open problem: applying optimal detection ordering does not improve the D-M tradeoff of the V-BLAST ( vertical Bell Labs layered Space-Time ) architecture. However, by deriving tight lower bounds to the outage probabilities of ZF and MMSE equalizers, the authors show that optimal ordering yields a SNR gain of 10 log10 N dB in the ZF-V-BLAST architecture ( where N is the number of transmit antennas ) whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better, and significantly so. Furthermore, as snr → ∞, ηsnr converges with probability one to a scaled F random variable. Using the decomposition of the output SNR of MMSE, the authors can approximate its uncoded error as well as outage probabilities through a numerical integral which accurately reflects the respective SNR gains of the MMSE equalizer relative to its ZF counterpart.