# 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)

Jump to: [Introduction] – [A. Basics of ZF and MMSE Equalizers] – [B. Diversity-Multiplexing Gain Tradeoff] – [C. Two Theorems] – [A. Uncoded Error Probability Analysis] – [B. Outage Probability and ²-Outage Capacity] – [A. The Linear ZF Equalizer] – [B. The Linear MMSE Equalizer] and [C. D-M Gain Tradeoff of V-BLAST with Channel-Dependent Ordered Detection]

### 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(N−1),2(M−N+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)

N−2

, 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

z−1

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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14 Sep 1984

TL;DR: In this article, the distribution of the Mean Vector and the Covariance Matrix and the Generalized T2-Statistic is analyzed. But the distribution is not shown to be independent of sets of Variates.

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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