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].
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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
