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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 1
MIMO Zero-Forcing Receivers Part I:
Multivariate Statistical Analysis
Mario Kießling, Member, IEEE
Abstract
In this paper, we analyze the signal to noise ratio (SNR) statistics of a multiple input multiple output
(MIMO) zero-forcing (ZF) receiver in a correlated Rayleigh fading environment. We present a novel
mathematical approach based on multivariate complex Gaussian integrals that enables us for the first
time to calculate the moment generating function (MGF) and probability distribution function (PDF)
for arbitrary fading correlation at receive and transmit antenna arrays in closed form. It is demonstrated
that the MGF can be expressed in terms of the expected value of a ratio of determinants of complex
matrix Gaussian random quadratic forms. To the authors’ best knowledge, we calculate for the first
time closed form expressions for this expected value. Interestingly, we obtain concise formulas for
MGF and PDF in terms of certain elementary symmetric functions of the eigenvalues of the MIMO
channel correlation matrices. Based on the MGF and PDF, we calculate closed form SER expressions
for arbitrary quadrature amplitude modulation (QAM) constellations and present results on mean mutual
information. All results are exact and non-asymptotic. The new mathematical techniques presented in
this paper have a general scope and can be applied for solving other problems in information theory,
for example the performance analysis of MIMO minimum mean squared error receivers.
Index Terms
MIMO, ZF, zero-forcing receiver, multivariate statistics, quadratic forms, complex Gaussian
I. I NTRODUCTION
Research on the performance analysis of wireless MIMO systems in the majority of cases
focuses on Shannon capacity (in particular ergodic capacity) and pairwise error probability (PEP)
Manuscript received 2005
M. Kießling is with Bosch Blaupunkt.
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for maximum likelihood receivers. While ergodic capacity [1] [2] [3] [4] [5] [6] and PEP [7] [8]
are well understood, only little is known about the symbol error rate (SER) performance of low-
complexity linear MIMO receivers, especially in the presence of fading correlation at the receive
antenna array. For uncorrelated Rayleigh fading, it was shown in [9] in the context of smart
antenna systems that for zero-forcing (ZF) receivers, the subchannel signal to noise ratio (SNR)
(for each user) follows a simple gamma distribution. This result was extended for MIMO systems
to cover the case of fading correlation at the transmit antenna array in [10] and independently in
[11]. On the other hand, many results are available on the analysis of minimum mean squared
error (MMSE) processing (which is termed optimum combining in smart antenna literature) with
spatially uncorrelated fading. The exact subchannel SINR distribution for users with different
transmit powers was given in [12] based on a statistical result on certain matrix quadratic forms
in [13]. For equal-power interferers, an exact SER analysis was presented in [14], where the
eigenvalue probability density function of complex Wishart matrices was used for the derivation
[15]. However, to the authors’ best knowledge, no general exact analytical SER expressions
can be found in literature for the case of spatial fading correlation at the receive antenna array.
Available results for MMSE receivers are approximations or are semi-analytic [16], thus still
requiring lengthy Monte-Carlo simulations. For the special case of only two transmit and two
receive antennas, exact SER formulas were given in [11] for ZF receivers and in [17] for MMSE
receivers based on a random eigenvalue approach for systems with receive as well as transmit
correlation. However, these results could not be generalized for an arbitrary number of transmit
and receive antennas. In this paper, for the first time we present fully analytic SER expressions
for MIMO ZF receivers and an arbitrary finite number of transmit and receive antennas with
arbitrary fading correlation at the transmit as well as the receive antenna array. We emphasize that
correlation at the receiver (a practically relevant case also in multi-user beamforming scenarios)
can be taken into account, which is not possible with other mathematical approaches. In the
course of the derivation, we present expressions for the subchannel SNR moment generating
function (MGF) in terms of certain expected values of ratios of random determinants. As it
appears that there are no results available in literature for calculating these expected values, we
present closed form formulas that are derived by a novel mathematical approach. Specifically, we
make use of certain complex Gaussian integrals [6] [18] for the derivation. Based on the MGF,
we derive exact formulas for arbitrary moments as well as closed form expressions for PDF and
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CDF. We show that the SER of ZF receivers in the presence of correlated fading at transmit
and receive antenna array can be given in closed form for arbitrary square QAM constellations
by using a well-known integral representation of the Gaussian Q function [19]. Moreover, we
calculate exact formulas for the mean mutual information (MMI) of the subchannels. The details
of the SER and MMI derivations are given in part II of this paper, where we also present novel
asymptotical SER expressions for the high SNR regime, which allow for a simple assessment
of the influence of the various system parameters and especially fading correlation on the SER
performance. Finally, Monte-Carlo simulations for different propagation environments show that
the novel SER and MMI formulas exhibit a perfect match.
II. N OTATION AND SYSTEM MODEL
A. Notation
Vectors are denoted by bold lowercase lettersx, matrices by bold uppercase lettersX. Conjuga-
tion is indicated byX∗, transposition byXT and complex conjugate transpose (Hermitian) byXH.
An identity matrix of sizen×n is written asIn and diag(x1,x2, . . . ,xn) or diag(x), respectively,
returns a diagonal matrix with elementsxk on the diagonal. Equivalently, diag(X) returns the
vector of diagonal elements of square matrixX. The trace of a matrix is denoted by tr(X).
For brevity, we define etr(X) = exp(tr(X)). The matrix variate complex normal distribution
with meanM , m rows andn columns, covariance matrix of column vectorsΣΣΣ, and covariance
matrix of row vectorsΨΨΨ is written asNm,n(M ,ΣΣΣ,ΨΨΨ). By ∼ we denote ’is distributed as’ and
' means ’has the same distribution as’.X† is the pseudo-inverse, and the Kronecker product is
denoted by⊗. The expected value of a functionf (X) with respect toX has the representation
Ex [ f (X)]. We use the notationαk for index subsets of cardinality|αk|= k (the cardinality can
be omitted), complementary index subsets are written asγ = α\β . For exampleα3 = {1,3,5}with β2 = {1, . . . ,5}\α3 = {2,4}. By {X}α
βwe denote a matrix that results from selecting the
row subsetα and the column subsetβ from matrix X. Similarly, we let|X|αβ =∣∣∣{X}α
β
∣∣∣, where
|X| denotes the determinant of square matrixX. We make frequent use of elementary symmetric
functions of matrix argument with the definition
trk (X) =∑
αk
|X|αkαk
, (1)
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for squarem×m matrix X, where the sum is over all
(k
m
)different index subsets of
cardinalityk. Note that for vectord = (d1,d2, . . . ,dm)T and diagonalm×m matrix D = diag(d)
the elementary symmetric functions of matrix argument reduce to scalar elementary symmetric
functions (with indices{i1, . . . , ik})
trk (D) = trk (d) =∑
{i1,...,ik}di1 · · · · ·dik (2)
with the definition tr0(D) = 1 and trk (D) = 0 for all k < 0. Note that in (2) the sum is again
over all
(k
m
)index subsets of cardinalityk. For brevity we introduce the notation
tr(i)k (D) = trk (diag(d1,d2, . . . ,di−1,di+1, . . . ,dm)) . (3)
The complete symmetric function hk (x) with x = (x1 . . . ,xn)T is the sum of all monomials
mλ of total degreek in the variablesx1,x2, . . . ,xn so that [20]
hk (x) =∑
|λ |=k
mλ (4)
with h0(x) = 1, hk (x) = 0 for all k < 0, and h1(x) = tr1(x). For example, we have
h2((x1,x2)) = x21 +x1x2 +x2
2.
There is a close relation between complete and elementary symmetric functions [20], namely
for a n×1 vectorx
trk (x) =∣∣{h1−i+ j (x)}
∣∣1≤i, j≤k (5)
and equivalently
hk (x) =∣∣{tr1−i+ j (x)}
∣∣1≤i, j≤k . (6)
In (5) and (6)i and j are the row and column index, respectively, of thek×k matrices. For
brevity we introduce the following (normalized) complex matrix differential for complexM×N
matrix X
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DcX =M∏
m=1
N∏
n=1
ℜ{dx11}ℑ{dx11}· · ·ℜ{dxmn}ℑ{dxmn}π
(7)
and write for the multidimensional integral
∫f (X) DcX =
∫∫
ℜ{x11}ℑ{x11}
· · ·∫∫
ℜ{xmn}ℑ{xmn}
f (X) DcX, (8)
where each scalar integral is over the range−∞ to +∞. Throughout the paper we use the
definitions form×1 vectorx = (x1, . . . ,xm)T
Kx (l) =1∏m
n=1,n6=l (xl −xn)(9)
and
Kx (α1,α2) =1
xα1−xα2
(10)
with the relation
xα1 · Kx (α1,α2)+xα2 · Kx (α2,α1) = 1. (11)
B. System Model
We consider a flat fading MIMO link withT transmit andR receive antennas (see Fig. 1),
whereas theR×T channel matrix is given byH. There areL independent data channels and
the transmit symbols are arranged in aL×1 vectors. Furthermore, we introduce a linearT×L
transmit filter matrixF, which mapsL subchannels on theT transmit antennas. In general we
assumeL≤ T. On the receiver side we assume without loss of generality (w.l.o.g.) additive white
Gaussian noise (AWGN) modeled by theR×1 vectorn and theR×1 noisy received vector is
denoted byy. Colored noise can be taken into account via a modified receive correlation matrix
(see also below). The transmission over the MIMO channel with transmit prefiltering can than
be described by
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y = HFs+n = Ks+n (12)
with the R×L compound channel matrixK = HF. At the receiver side, the received vectory
is processed by the zero forcing (ZF) matrixG and theL×1 vectorz results
z = Gy. (13)
The zero-forcing receiver has the well known [10] pseudo inverse receiver matrix
G =(KHK
)−1KH = K†. (14)
Finally, we define the diversity of the system by
D = R−L+1. (15)
C. Statistics
In this paper, we investigate the transmission over a Rayleigh fading MIMO link, i.e. the
channel matrixH is complex Gaussian distributed
H ∼NR,T(0,RRX,RTX). (16)
Without loss of generality, we assume full rankRRX and RTX . Rank deficient correlation
matrices can be mapped on an equivalent system with full rank transmit and receive correlation
matrices with a smaller virtual number of transmit and receive antennas, respectively. We note
that (16) is the well known [21] [22] MIMO channel model with separable correlation matrices
at transmitterRTX and receiverRRX and
H ' AHHwB, (17)
where
RRX = AHA (18)
RTX = BHB. (19)
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The Rayleigh fading channel model can further be generalized by allowing for arbitrary
variances of the individual channel matrix elements. However, (17) is a good tradeoff between
complexity and accuracy. For later reference in subsequent derivations, we introduce the eigen-
value decomposition (EVD) of the receive correlation matrix with diagonal
O = diag(o) = diag(o1, . . . ,oR) , (20)
namely
RRX = VrOVHr (21)
with unitary matrixVr. Straightforward considerations lead to the distribution of the compound
channel
K ∼NR,L(0,RRX,C) (22)
with the equivalentL× L covariance matrixC, which comprises the effects of transmit
correlation as well as transmit prefiltering
C = FHRTXF. (23)
Throughout this paper we use the definition
(c11, . . . ,cLL)T
= diag(C−1) . (24)
The complex Gaussian pdf ofK is given by [23]
pK (K) =1
πRL|C|R|RRX|L ·etr(−C−1KHR−1
RXK). (25)
Without loss of generality we assume white transmit symbols with covariance
Rss= Es · IL, (26)
whereEs is the energy per transmit symbol. Other transmit covariance matricesRss can easily
be absorbed in a modified transmit correlation matrix. Equivalently, w.l.o.g. we consider AWGN
with covariance
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Rnn = N0 · IR, (27)
whereN0 is the noise variance per receive antenna. Colored noise can be taken into account by
straightforward absorption in the receive correlation matrix. Finally, in the following the mean
SNR is defined by
γ =Es
N0(28)
and will be used consistently throughout the paper. Furthermore, we use the scaled mean SNR
γk =γ
ckk (29)
and introduce the scaled vector of eigenvalues
o = (o1, . . . , oR)T = γk ·o. (30)
for brevity in later derivations. Throughout this paper we use the subchannel indexk.
III. SNR EXPRESSIONS
After splitting the vectorz at the output of the receive filterG in a signal componentzs and
a noise componentzn
z = GKs+Gn = zs+zn, (31)
it can be shown that by the zero forcing property
E[zszs
H]= Es · IL (32)
and for the noise component
E[znzH
n
]= N0 ·K†
(K†
)H. (33)
Therefore, the SNR on subchannel k after receive processing reads
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γSC,k =γ[
K†(K†)H]
kk
=γ[
(KHK)−1]
kk
. (34)
The subchannel SNR can be rewritten in terms of a random quadratic form that is later shown
to be well suited for a statistical analysis. For simplifying the notation, in the following we first
focus on the subchannel with indexk = 1. It is then a straightforward exercise to generalize the
results to an arbitrary subchannel.
We first partition the compound channel
K = [ k1 K ] , (35)
where k1 is a R× 1 column vector andK is a R× (L− 1) matrix. For rewriting the SNR
expression we can exploit the following result on partitioned inverses. Let
X =[ X11 X22
X21 X22
](36)
and
X−1 =[
X11 X22
X21 X22
]. (37)
It is then well-known that [24]
X11 =(X11−X12X−1
22 X21)−1≡ (X11·2)−1 . (38)
With the help of (38) it can be shown that the SNR on subchannel 1 is given by
γSC,1 = γ ·kH1
(IR− K
(KHK
)−1KH
)k1, (39)
which is a random quadratic form in complex Gaussian distributed vectors.
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IV. SUBCHANNEL SNR MGF
In this section we calculate the moment generating function (MGF) for the subchannel SNR,
which is the basis for derivations of the PDF and CDF. The MGF also serves as a basis for
mean mutual information (MMI) and SER calculations.
The MGF of the subchannel specific (index k) SNR is given by
Mk(s) = EK[exp(−s· γSC,k)
], (40)
where the expected value is with respect to the channel statistics. Note that in accordance
with common practice in the area of communication theory (e.g. [19]), we talk about the MGF
in (40), even though we use a minus sign in the exponent.
A. MIMO Channel Probability Distribution
For later integrations, it is convenient to reformulate the MIMO channel PDF. We partition
the covariance matrixC with scalarc11, (L−1)×1 vectorc21, and(L−1)× (L−1) matrix C22
as
C =
[c11 cH
21
c21 C22
]. (41)
Equivalently, we let
C−1 =
[c11 (c21)H
c21 C22
]. (42)
It is now possible to rewrite the exponential term of the channel PDF as
etr(−C−1KHR−1
RXK)
= etr(−(
c11kH1 R−1
RXk1 +C22KHR−1RXK
)) · . . . (43)
etr(−(
(c21)HKHR−1RXk1 +((c21)HKHR−1
RXk1)H)),
whereK andk1 are explicitly visible.
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B. Uncorrelated channel without prefilter
For the case of uncorrelated fading, the subchannel SNR statistics of MIMO ZF receivers
are well known [10] [25]. Basically, the subchannel SNR in this case can be expressed as
the marginal distribution of a complex Wishart matrix, which has been extensively studied
in multivariate statistical literature [26] [27] [28]. However, for an introduction of the novel
mathematical techniques deployed in this paper, we also consider this simple case. As expected
on the basis of symmetry considerations, the statistics are independent of the subchannel index.
Theorem 1:The MGF of the subchannel SNR in case of uncorrelated Rayleigh fading and
no prefilter at the transmitter sideF = IT is given by
Mu(s) =1
(1+s· γ)D(44)
with the obvious diversity of the systemD = R− L + 1. This is the MGF of a Gamma
distribution with D degrees of freedom. In case of no transmit correlation, the MGF is not
dependent on the subchannel indexk.
Proof: The channel PDF in case of uncorrelated fading is from (25) with the help of (43)
given by
pK ,u(K) =1
πRL ·etr(−(
kH1 k1 + KHK
)). (45)
The subchannel SNR MGF is then given by the integral (note again that due to the symmetry
of the problem, an arbitrary subchannel may be considered)
Mu(s) =∫
exp(−kH
1
((s· γ +1) · IR−s· γK
(KHK
)−1KH
)k1
)· . . . (46)
etr(−KHK
)Dck1DcK
Now carrying out the integral with respect tok1 using the well known vector variate Gaussian
integal (120) in Appendix I we find after some simple manipulations
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Mu(s) =1
(s· γ +1)R
∫1∣∣∣IR− s·γ
(s·γ+1)K(KHK
)−1 KH∣∣∣·etr
(−KHK)
DcK (47)
=1
(s· γ +1)R
∫1∣∣∣IL−1− s·γ
(s·γ+1)KHK
(KHK
)−1∣∣∣·etr
(−KHK)
DcK , (48)
where we have used (for matricesA andB of compatible size) [24]
|I +AB|= |I +BA|. (49)
After simplifying the determinant expression and carrying out the integral with respect toK
using (125) in Appendix I we finally have proven the theorem.
C. Channel with prefilter and transmit correlation
We present a first generalization of the results of the last subsection.
Theorem 2:In case of transmit correlation or the presence of a prefilter and uncorrelated
fading at the receive antenna array the subchannel SNR MGF is given by
Mk,TX(s) =1(
1+s· 1ckkγ
)D(50)
with the diversity of the systemD = R−L+1 and diag(C−1
)=
(c11, . . . ,cLL
)T.
Proof: We demonstrate two different proofs of the theorem. First, we consider the expected
value (with respect to the channel statistics) of an arbitrary functionf of the subchannel SNR
(again we consider exemplarily subchannelk = 1)
E[
f(γSC,1
)]=
∫f
γ[
(KHK)11] · 1
|C|R|RRX|L ·etr(−C−1KHR−1
RXK)
DcK . (51)
With the transformationX = C−1/2K (see e.g. [15] [26] [27] for an introduction to matrix
variate variable transformations) we obtain
E[
f(γSC,1
)]=
∫f
γ[(
C1/2KHKC1/2)11
] · 1
|RRX|L ·etr(−KHR−1
RXK)
DcK . (52)
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This can be written as
E[
f(γSC,1
)]=
∫f
γ ·
∣∣∣C1/2KHKC1/2∣∣∣
∣∣∣C1/222 KHKC1/2
22
∣∣∣
· 1
|RRX|L ·etr(−KHR−1
RXK)
DcK , (53)
where we have used [24] for square matrixX
x11 =|X22||X| . (54)
From that it can be seen that the expected value can be reformulated as
E[
f(γSC,1
)]=
∫f
(γ · 1
c11 ·∣∣KHK
∣∣∣∣KHK
∣∣
)· 1|RRX|L ·etr
(−KHR−1RXK
)DcK , (55)
and after generalizing for an arbitrary subchannel
E[
f(γSC,1
)]=
∫f
γk[
(KHK)kk] · 1
|RRX|L ·etr(−KHR−1
RXK)
DcK , (56)
with γk = γckk according to (29), i.e. the presence of fading correlation at the transmit antenna
array has just a scaling effect on the mean SNRγ .
We now present a second proof that makes extensive use of Gaussian integrals. The channel
PDF is from (25) with the help of (43) given by
pK ,TX(K) =1
πRL · |C|R ·etr
(−
(c11kH
1 k1 +(c21)H
KHk1 +((
c21)HKHk1
)H+C22KHK
)).
(57)
The subchannel SNR MGF for subchannel 1 is then given by the integral
M1,TX(s) =1|C|R ·
∫exp
(−s· γ ·kH
1
(IR− K
(KHK
)−1KH
)k1
)· (58)
etr
(−
(c11kH
1 k1 +(c21)H
KHk1 +((
c21)HKHk1
)H+C22KHK
))Dck1DcK .
Then note that from (123) in Appendix I, which is one of the key formulas for deriving the
novel results in this paper, we obtain the import relation
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exp(−s· γ ·kH
1
(IR− K
(KHK
)−1 KH)
k1
)= (59)
1(s·γ)L−1 ·
∣∣KHK∣∣ ·exp
(−s· γ ·kH1 k1
) ·∫ etr(−
(kH
1 Kx +xHKHk1 + 1s·γ ·xHKHKx
))Dcx ,
where we have removed the inverse in the exponent. Using (59) in (58) we get
M1,TX(s) =1
(s· γ)L−1 ·1|C|R ·
∫ ∣∣KHK∣∣ · (60)
etr
(−
((s· γ +c11) ·kH
1 k1 +(c21+x
)HKHk1 +
((c21+x
)HKHk1
)H))
·
etr
(−
(C22KHK +
1s· γ ·x
HKHKx))
Dck1 DcK Dcx.
Integrating with respect tok1 we find after rearranging the exponential
M1,TX(s) =1
(s· γ)L−1 · (s· γ +c11)R ·1|C|R ·
∫ ∣∣KHK∣∣ · (61)
etr
(−
(c11
s· γ ·ω xHKHKx − 1ω· (c21)HKHKx − 1
ω·xHKHKc21
))·
etr
(−
(C22KHK − 1
s· γ +c11(c21)HKHKc21))
DcK Dcx
with ω = s· γ +c11. Now carrying out the integral with respect tox we find after simplifying
M1,TX(s) =1
(s· γ +c11)R−L+1(c11)L−1· 1|C|R · (62)
∫etr
(−
(C22− c21(c21)
c11
H)
KHK
)DcK .
Finally integrating with respect toK we get
M1,TX(s) =(c11)R−L+1
(s· γ +c11)R−L+1 ·1
|C|R · (c11)R ·∣∣∣∣C22− c21(c21)
c11
H∣∣∣∣R. (63)
By using the relation for square matrixX [24]
∣∣X−1∣∣ =
∣∣X11∣∣∣∣∣X22−X21
(X11)−1
(X21)H∣∣∣ (64)
we get the final expression for the subchannel SNR MGF in case of transmit correlation.
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D. Channel with receive correlation only
Before we analyze the most general case with both receive and transmit correlation in the next
section, we first present a lemma for the case of receive correlation only, which is the starting
point for later derivations.
Lemma 1: In case of receive correlation only, the subchannel SNR MGF has the integral
representation
MRX(s) =1
|s· γ ·O+ IR| ·∫ ∣∣XHOX
∣∣∣∣∣XHO(s· γ ·O+ IR)−1X
∣∣∣·etr
(−XHX)
DcX. (65)
The integral can be interpreted as the expected value of a ratio of random determinants of complex
Gaussian matrix quadratic forms. Obviously the subchannel SNR and its MGF, respectively,
depends only on the eigenvalues of the receive correlation matrix.
Proof: If there is exclusively receive correlation present, the MIMO channel PDF is from
(25) with the help of (43) given by
pK ,RX(K) =1
πRL · |RRX|L·etr
(−(kH
1 R−1RXk1 + KHR−1
RXK))
. (66)
It is independent of the subchannel index. Using relation (59) we get for the subchannel SNR
MGF with the abbreviationϒϒϒ = s· γ · IR+R−1RX
MRX(s) =1
(s· γ)L−1 · |RRX|L·∫ ∣∣KHK
∣∣ ·etr(−(
kH1 ϒϒϒk1 +kH
1 Kx +xHKHk1)) · (67)
etr
(−
(1
s· γ ·xHKHKx + KHR−1
RXK))
Dck1 DcK Dcx
(68)
We first carry out the integral with respect tok1 and get
MRX(s) =1
(s· γ)L−1 · |RRX|L · |ϒϒϒ|·∫ ∣∣KHK
∣∣ · (69)
etr
(−
(xHKH
(−ϒϒϒ−1 +
1s· γ · IR
)Kx + KHR−1
RXK))
DcK Dcx
After integration with respect tox we find with the abbreviationΦΦΦ = s· γ ·RRX + IR
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MRX(s) =1
(s· γ)L−1 · |RRX|L−1 · |ΦΦΦ|·∫ ∣∣KHK
∣∣∣∣∣KH
(−ϒϒϒ−1 + 1
s·γ · IR
)K
∣∣∣· (70)
etr(−KHR−1
RXK)
DcK
By using the matrix inversion lemma for square matrixA [24]
I − (I +A−1)−1
= (I +A)−1 (71)
we find
MRX(s) =1|ΦΦΦ| ·
∫ ∣∣KHK∣∣
∣∣KHΦΦΦ−1K∣∣ · (72)
etr(−KHR−1
RXK) · 1
|RRX|L−1 DcK .
Now making the matrix variate transformationR−1/2RX K →X with JacobianJ
(R−1/2
RX K → X)
=
|RRX|L−1 we get
MRX(s) =1|ΦΦΦ| ·
∫ ∣∣XHRRXX∣∣
∣∣∣XHR1/2RX (ΦΦΦ)−1R1/2
RX X∣∣∣·etr
(−XHX)
DcX. (73)
By introducing the eigenvalue decomposition of the receive correlation matrix, we obtain the
lemma. We note that the JacobianJ(X ·V → X) = 1 for unitary matrixV.
Obviously, the MGF only depends on the eigenvalues of the receive correlation matrix and is
independent of the particular eigenvectors. Before we continue with the calculation of the MGF,
we further generalize the underlying channel model.
E. Transmit and receive correlation
Straightforward considerations lead to the following theorem.
Theorem 3:In case of Rayleigh fading with transmit and receive correlation we get similar
to (65) a matrix variate integral expression for the subchannel SNR MGF
Mk(s) =1
|s· γk ·O+ IR| ·∫ ∣∣XHOX
∣∣∣∣∣XHO(s· γk ·O+ IR)−1X
∣∣∣·etr
(−XHX)
DcX, (74)
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where(c11, . . . ,cLL
)T = diag(C−1
). The integral expression in (74) has again an interesting
interpretation as the expected value of a ratio of random determinants in generalized matrix
quadratic forms.
Proof: The theorem follows from a combination of the results for the transmit correlated
only and receive correlated only cases above. Note that transmit correlation just leads to a scaling
of the effective SNR according to (56).
We now give a representation of the MGF in terms of a scalar integral only. It appears that
there are no comparable results available in literature on the expected value of ratios of random
determinants of complex Gaussian matrix quadratic forms. However, the formulas given in this
paper cover the well known vector variate case [29] [30] [31] [32] [33] [34].
Theorem 4:The subchannel SNR MGF has the following single scalar integral representation
(with matrix notation)
Mk(s) =∑
αL−1
|O|αL−1αL−1
·∞∫
0
tr(
U1,αU−12,α
)
|U2| · tL−2 dt, (75)
where the sum is over all index subsets of{1,2, . . . ,R} of cardinality L−1. For brevity, we
have introduced
U1,α = IL−1 +s· γckk ·Oα (76)
U2 = IL−1 + t ·O+s· γckk ·O
U2,α = IL−1 + t ·Oα +s· γckk ·Oα
with Oα = {O}αL−1αL−1
. We get the scalar representation
Mk(s) =∑
αL−1
|O|αL−1αL−1
·∑
αl∈αL−1
·∞∫
0
u1,αl[∏Rj=1u2, j
]·u2,αl
tL−2 dt, (77)
where we have introduced
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u1,αl = 1+s· oαl (78)
u2,l = 1+ t ·ol +s· ol (79)
u2,αl = 1+ t ·oαl +s· oαl . (80)
and ol = γk ·ol according to (30).
Proof: By using Theorem 11 in Appendix III we can derive (75) after simple manipulations
and (77) directly follows for diagonal matrices.
The complexity of the MGF expression can be reduced significantly. By deploying certain
elementary symmetric functions, the following theorem can be derived. It is the starting point
for later moment, SER, and MMI calculations.
Theorem 5:The MGF of the subchannel SNR in the presence if transmit and receive corre-
lation has the concise scalar integral representation
Mk(s) = 1−s· (L−1) ·∑
k
okR−1 · tr(k)L−1(O) ·Ko(k) ·
∞∫
0
tL−2
s+ 1ok
+ 1γk· t dt. (81)
Carrying out the integral yields the closed-form solution
Mk(s) = 1+(−1)L · γL−1k ·s· (L−1) ·
∑
l
ζl , (82)
where we have introduced the sum terms
ζl = olR−1 · tr(l)L−1(O) ·Ko(l) ·
(s+
1ol
)L−2
· log
(s+
1ol
)(83)
for brevity andol according to (30).
Proof: See Appendix IV.
F. Moments
Based on the closed form MGF expressions in Theorem 5, arbitrary moments of the subchannel
SNR can be calculated.
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Theorem 6:Let ν be the order of the moments. For the caseL > ν the moments of the
subchannel SNR are given by
mk (ν) = (−1)L−ν−1 · γνk ·
ν ·Γ(L)Γ(L−ν)
·∑
l
olR−L+ν · tr(l)L−1(O) ·Ko(l) · log
1ol
. (84)
For the caseL≤ ν the moments have the representation
mk (ν) = γνk ·ν ·Γ(L) ·Γ(ν−L+1) ·
χ∑
j=1
(−1) j+1 ·hν−L+1− j (O) · trL−1+ j (O) (85)
with χ = min(R−L+1,ν−L+1).
Proof: If we want to calculate theν th moment, we get from the MGF by exchanging
the sequence of differentiation and integration (this can be justified by Lebesgue’s dominated
convergence theorem; details are omitted here for brevity)
mk (ν) = (−1)ν+1 · (L−1) ·∑
l
olR−1 · tr(l)L−1(O) ·Ko(l) ·
∞∫
0
∂ ν
∂sνtL−2 ·s
s+ 1ol
+ 1γk· t
∣∣∣∣∣s=0
dt. (86)
Making use of the fact that
∂ ν
∂xνx
x+a
∣∣∣∣x=0
= (−1)ν+1 ·Γ(ν +1) · 1aν (87)
we arrive at
mk (ν) = Γ(ν +1) · (L−1) ·∑
l
olR−1 · tr(l)L−1(O) ·Ko(l) ·
∞∫
0
tL−2
(1ol
+ 1γk· t
)ν dt. (88)
Integration by parts yields the following formula for integerm> 0,n > 1 and constantsa,b
∞∫
0
xm
(a+bx)n dx = − xm
n−1· 1
b· (a+bx)n−1
∣∣∣∣∣∞
0
+1b
mn−1
∫ ∞
0
xm−1
(a+bx)n−1 dx. (89)
After application of (89) to (88) it can be readily seen by virtue of Lemma 5 in Appendix V
that the first term resulting from (89) vanishes at the integration boundaries. Therefore we can
find after iteratively applying (89) the following simplified integral formula for the caseL > ν
after simple modifications
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mk (ν) = γν−1k · ν ·Γ(L)
Γ(L−ν)·∑
l
olR−1 · tr(l)L−1(O) ·Ko(l) ·
∞∫
0
tL−ν−1
1ol
+ 1γk· t dt. (90)
Using (187) in Appendix IV for reformulating the fraction under the integral as a power series
we find with the help of Lemma 5 in Appendix V
mk (ν) = (−1)L−ν−1 · γL−1k · ν ·Γ(L)
Γ(L−ν)·∑
k
okR−1 · tr(k)L−1(O) ·Ko(k) · o1+ν−L
k · log1ok
. (91)
This proves the first part of the theorem.
For the caseL≤ ν we can apply the following integration formula valid for integerm< n+1
and constantsa,b
∞∫
0
xm
(a+bx)n dx =Γ(n−m−1) ·Γ(m+1)
an−m−1 ·bm+1 ·Γ(n)(92)
and find for the moments
mk (ν) = γνk ·ν ·Γ(L) ·Γ(ν−L+1) ·
∑
l
olR−L+ν · tr(l)L−1(O) ·Ko(l) . (93)
This yields with the help of Lemma 6 in Appendix V the second part of the theorem.
V. PDF AND CDF OF SUBCHANNEL SNR
Starting with the MGF expression in Theorem 5, the subchannel SNR CDF and PDF can be
calculated by inverse Laplace transforms.
Theorem 7:The CDF of the subchannel SNR is given by
qk(γSC,k) = 1−Γ(L) ·∑
l
µo,l ·(
γk
γSC,k
)L−1
·exp
(− 1
olγSC,k
)(94)
with
µo,l = olR−1 · tr(l)L−1(O) ·Ko(l) . (95)
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By expanding the exponential term in a power series inγSC,k, it can be shown by Lemma 5
in the appendix thatq(0) = 0. The corresponding subchannel SNR PDF has the representation
pk(γSC,k) = Γ(L) ·∑
l
µo,l ·(
γk
γSC,k
)L−1
·exp
(− 1
olγSC,k
)·(
1ol
+L−1γSC,k
). (96)
Again, by an expansion of the exponential term it can be shown that forR> L it is pk(0) = 0.
For completeness we note that in case of a transmit correlated or uncorrelated MIMO channel
the PDF of the subchannel SNR is a well-known Gamma PDF withD degrees of freedom
pTX,k(γSC,k) =1
Γ(D) · γk·(
γSC,k
γk
)D−1
·exp
(−γSC,k
γk
). (97)
Accordingly, we obtain for the CDF
qTX,k(γSC,k) = 1−exp
(γSC,k
γk
)·D−1∑
j=0
1Γ( j +1)
·(
γSC,k
γk
) j
. (98)
Proof: The CDFqk(γSC,k) =∫ γSC,k
0 p(t)dt has from Theorem 5 the Laplace transform
Qk(s) =Mk(s)
s=
1s− (L−1) ·
∑
l
µo,l ·∞∫
0
tL−2
s+ 1ol
+ 1γk· t dt. (99)
We can now make use of the Laplace transform pairs
1s+a
← e−ax (100)
1
(s+a)2 ← xe−ax.
With an inverse Laplace transform we obtain
qk(γSC,k) = 1− (L−1) ·∑
l
µo,l ·exp
(− 1
olγSC,k
)·
∞∫
0
tL−2 ·exp
(− 1
γk· γSC,k · t
)dt.(101)
Carrying out the integral we find the first part of the theorem. Finally, by differentiating with
respect toγSC,k we obtain the PDFpk(γSC,k).
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VI. SER CALCULATION
The conditional symbol error rate (conditioned on the subchannel SNR) in the presence of
Gaussian noise for square M-QAM constellations is given by [19]
Ps,k,cond = b·[Q
(√2c· γSC,k
)− b4·Q2(√
2c· γSC,k)]
(102)
with constants
b = 4·(
1− 1√M
)
c = 32·(M−1) .
(103)
Based on a well known finite scalar integral representation of theQ function, we can use the
closed form subchannel SNR MGF expressions for calculating exact SER formulas for square
M-QAM constellations. An extension of the results to other QAM modulations is straightforward.
Theorem 8:The average SER of subchannelk of a MIMO ZF receiver in correlated Rayleigh
fading with receive correlation is given by
Ps,k = b·[
1−b/4− γL−1k · (−1)L−1 ·
∑
l
ςl ·[
Λ1,l +bπ· (Λ2,l +Λ3,l
)]]
(104)
with the auxiliary terms
Λ1,l =
√c·
(1ol
+c
)(105)
Λ2,l = −√
c·(
1ol
+c
)·arctan
√1+
1c· ol
,
ςl = olR−1 · tr(l)L−1(O) ·Ko(l) · 2
L−1 · (L−1)!(2L−3)!!
·(
1ol
+c
)L−2
(106)
and
Λ3,l = c
1
2+
L−3∑
l=0
(2L−2l −5)!! ·(
1ol
+2c)L−2−l
2L−1−l · (L−2− l)! ·(
1ol
+c)L−2−l
· log
(2+
1c· ol
)(107)
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for readability. In case of transmit correlation only or uncorrelated fading, the SER is given by
Ps,k = b·[
Ψ1−b/4·[
14− µc
π(Ψ2−Ψ3)
]](108)
with the following terms for brevity
µc =
√cγk
1+cγk, (109)
Ψ1 =(
1−µc
2
)L
·L−1∑
l=0
(L−1+ l
l
)(1+ µc
2
)l
, (110)
Ψ2 =(π
2−arctanµc
)·
L−1∑
l=0
(2l
l
)
[4(1+cγk)]l , (111)
Ψ3 = sin(arctanµc) ·L−1∑
l=1
l∑
i=1
Til
(1+cγk)l · [cos(arctanµc)]
2(l−i)+1 , (112)
and finally
Til =
(2l
l
)
(2(l − i)
l − i
)·4i · [2(l − i)+1]
. (113)
Proof: Due to space limitations, in this paper we omit a proof of the first SER formula of
this theorem. It will be presented in the second part of this paper. The SER expression in (108)
is a well known result from [19].
A. Calculation of Mean Mutual Information
The mean mutual information (MMI) of MIMO subchannel k in nat per channel use is given
by
Ik = E[log
(1+ γSC,k
)](114)
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The expected value in (114) can be calculated using the CDF expression in (94) in closed
form.
Theorem 9:The MMI of subchannel k of a MIMO link with ZF receiver in correlated Rayleigh
fading is given by
Ik = (−1)L−1 ·Γ(L) · γL−1k ·
∑
l
µo,l ·[
E1
(1ol
)·exp(− 1
ol)−
L−1∑
m=1
1
Γ(m) · om−1l
], (115)
whereE1 is the exponential integral [35],γk from (29), o from (30), and
µo,l = olR−1 · tr(l)L−1(O) ·Ko(l) . (116)
Proof: A proof is omitted in this paper due to the space limitation. It will be presented in
part II of this paper, where we also consider the MGF of mutual information.
VII. N UMERICAL RESULTS
In this section we study systems with white input signals of powerEs and additive white
Gaussian noise with varianceN0 per receive antenna
Rss= Es · IT
Rnn = N0 · IR. (117)
Furthermore, due to their simple structure, in the following we consider exponential correlation
matrices [36] at the transmitter and the receiver with
RRX =[r |i− j|RX
]
RTX =[r |i− j|TX
], (118)
where i and j are the row and column indices, respectively. The correlation coefficient at
the receiver (transmitter)rRX (rTX) ranges from0 to 1 and models the correlation between two
neighboring receive (transmit) antennas. With the given channel model, correlation between two
antenna elements decreases exponentially with their distance. Finally, the SNR in dB is defined
by
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γdB≡ 10· log10ρ ·Es
N0= 10· log10(ρ · γ) [dB], (119)
whereρ is the transmit power constraint and we assume in the following numerical results
ρ = T in accordance with (117).
In Fig. 2 we have plotted the empirical PDF of a MIMO system with ZF receiver andT = 4
transmit antennas,L = 4 independent subchannels, andR= 6 receive antennas. For the given
scenario, we assume a MIMO channel with receive correlation only withrRX = 0.9 andrTX = 0.
It is demonstrated that there is an exact match with the analytical PDF given in Theorem 7.
For the same channel correlation properties, the CDF is plotted according to Theorem 7 in
Fig. 3.
The influence of receive correlation on the PDF can be seen in Fig. 4. With increasing
correlation, as expected the PDF gets more peaky and the maximum of the PDF moves closer to
zero. A considerable change of the PDF can be observed when the receive correlation coefficient
rRX is increased from0.7 to 0.9.
In Fig. 5 we have plotted SER curves for a system with 16 QAM modulation and varying
receive correlation. Theoretical results according to the closed form SER expressions in Theo-
rem 8 and numerical results of a Monte Carlo simulation perfectly match. Again, the negative
effect of receive correlation, especially for valuesrRX > 0.7 can be observed.
SER curves for a system with again 16 QAM modulation are depicted in Fig. 5. We show
curves for an uncorrelated channel as well as a receive correlated channel withrRX = 0.7, whereas
we note that due to symmetry considerations all subchannels for these two scenarios have the
same SER. On the other hand, if there is additionally transmit correlation present withrTX = 0.7,
again due to symmetry there are two different SER on the subchannels.
In Fig. 7 SER curves for two systems withR= {4,8} receive antennas are depicted. Curves
are shown for weakly and strongly correlated receive antennasrRX = {0.3,0.9}. Obviously, the
full diversity of the systems withD = {1,5} is achieved, independently of the strength of the
receive correlation, for higher SNR. However, receive correlation leads to a considerable shift
of the SER curves.
In Fig. 8 analytical curves of the MMI according to Theorem 9 and Monte Carlo simulation
results perfectly agree for different scenarios with correlation at the receive antenna array.
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VIII. C ONCLUSION
For the first time we have determined the exact probability distribution of the statistics of
MIMO ZF receivers in correlated Rayleigh fading with transmit as well as receive correlation.
We have derived a novel probability distribution function that can be expressed in terms of
certain elementary symmetric functions of the eigenvalues of the receive correlation matrix.
Based on the closed form probability expressions, which are valid for an arbitrary finite number
of transmit antennas, we have calculated exact formulas for the symbol error rate of square QAM
constellations and presented results on mean mutual information. A new mathematical approach
based on complex Gaussian integrals has been introduced for the derivation of the statistics. The
authors expect that this approach will find numerous applications in other fields of information
theory, particularly in the analysis of linear MIMO receivers like minimum mean squared error
(MMSE) receivers.
APPENDIX I
COMPLEX GAUSSIAN INTEGRALS
Basic material on real vector variate Gaussian integrals can be found in [37] and [38]. The
straightforward extension to the complex case is e.g. given in [18]. For complexm×1 column
vectorsx,a,b and real positive definete matrixA the basic complex Gaussian integral is given
by
∫exp
(−xHAx +aHx+xHb)
Dcx =1|A| exp
(aHA−1b
). (120)
It can furthermore be shown that
∫xHAx ·exp
(−xHBx)
Dcx =1|B| · tr
(AB−1) . (121)
Due to its importance in the derivations in this paper, we emphasize that from (120) we obtain
the following integral representation of an inverse determinant
1|C| =
∫exp
(−xHCx)
Dcx (122)
Furthermore, (120) can be reformulated as
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exp(aHA−1b
)= |A| ·
∫exp
(−xHAx +aHx+xHb)
Dcx, (123)
i.e. we can get rid of the inverse in the exponent via an integral representation. In the matrix
variate case we get similar to (120) forM×N matricesX,A,B, M×M matrix M , andN×N
matrix N (see e.g. [6])
∫etr
(−NXHMX +AHX +XHB)
DcX =1
|M ⊗N|etr(N−1AHM−1B
)(124)
with the special case
∫etr
(−XHMX)
DcX =1
|M |N . (125)
APPENDIX II
MATRIX VARIATE DISTRIBUTIONS AND RELATED INTEGRALS
We base our derivations on certain expected values of random determinants for establishing
some important integral equalities. In this context, we derive an exact closed form solution of
the expected value of a noncentral matrix quadratic form and the corresponding matrix variate
integral. It appears that this result until now was not available in literature in this explicit form.
First, we note that a noncentrally distributed complex Gaussian matrixG of dimensionm×n
with i.i.d. elements of unity variance and meanC has the PDF
pG(G) =1
πmn ·etr(−(
G−C)H (
G−C))
. (126)
It was conjectured in [39] and finally proven in [40] that
EG
[∣∣GHG∣∣] =
Γ(m+1)Γ(m−n+1)
+n−1∑
i=0
Γ(m− i)Γ(m−n+1)
· tri+1(Q) (127)
whereQ = CCH for brevity andGHG has a so-called complex noncentral Wishart distribution.
Now note that from (127) together with (126) we can derive the important integral identity for
rank 1 matrixC
∫ ∣∣GHG∣∣ ·etr
(−(
G+C)H (
G+C))
DcG =1
Γ(m−n+1)· [Γ(m+1)+Γ(m) · tr(Q)] . (128)
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In the central case we directly get from (128)
∫ ∣∣GHG∣∣ ·etr
(−GHG))
DcG =Γ(m+1)
Γ(m−n+1). (129)
We now generalize the result in (128). First, we note that then×n matrix GHMG with m×m
matrix M is a complex noncentrally distributed generalized random matrix quadratic form. In
the following, we calculate the expected value of a special random determinant.
Theorem 10:Consider the expected value of the random determinant
g = EG
[∣∣GHMG∣∣] =
∫ ∣∣GHMG∣∣ ·etr
(−(
G+C)H (
G+C))
DcG, (130)
whereG is noncentrally complex Gaussian distributed. It can be calculated in closed form
g =∑
αn
|M |αnαn·[
Γ(n+1)+n−1∑
i=0
Γ(n− i) · tri+1(C1,αCH
1,α)]
, (131)
where the sum is over all
( n
m
)subsets of cardinalityn and then×n matrix
C1,α = {C}αn1,...,n. (132)
In case of a rank 1 matrixC the result simplifies to
g1 = Γ(n) ·∑
αn
|M |αnαn· [n+ tr
(C1,αCH
1,α)]
. (133)
Proof: We first expand the determinant expression in the integral of (130). To this end,
we can make use of the general formula fork×k matrix K = C ·D · · · · ·R ·S (where matrices
C,D, . . . ,R,S are of compatible sizes)
|K |=∑
αk
∑
βk
· · ·∑
δk
∑
σk
|C|{1,2,...,k}αk
· |D|αk
βk· · · · |R|δk
σk· |C|σk
{1,2,...,k} . (134)
The sums in (134) are over all partitionsαk, βk, δk, σk of cardinality k. Direct application yields
∣∣GHMG∣∣ =
∑
αn
|M |αnαn· ∣∣G∣∣{1,2,...,n}
αn· ∣∣G∣∣αn
{1,2,...,n} . (135)
Now we define a complementary index subset of cardinalitym−n
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βm−n = {1,2, . . . ,m}\αn (136)
and for brevity we introduce the auxiliary matrices
G1,α = {G}αn1,...,n (137)
G2,α = {G}βm−n1,...,n
and equivalently
C1,α = {C}αn1,...,n (138)
C2,α = {C}βm−n1,...,n.
With the help of (135) and the partitionings (137)(138) we can rewrite (130) as
g =∑
αn
|M |αnαn
∫ ∣∣GH1,αG1,α
∣∣ ·etr(−(
G+C)H (
G+C))
DcG. (139)
We further focus on the integral in (139), which can be split into the product of two independent
integrals
Iα =∫ ∣∣GH
1,αG1,α∣∣ ·etr
(−(
G1,α +C1,α)H (
G1,α +C1,α))
DcG1,α · (140)∫
etr(−(
G2,α +C2,α)H (
G2,α +C2,α))
DcG2,α .
Using the matrix integral (127) we get
∫ ∣∣∣GH1,αG1,α
∣∣∣ ·etr(−(
G1,α +C1,α)H (
G1,α +C1,α))
DcG1,α = (141)
Γ(n+1)+∑n−1
i=0 Γ(n− i) · tri+1
(C1,αCH
1,α
)
and by straightforward considerations we find
∫etr
(−(
G2,α +C2,α)H (
G2,α +C2,α))
DcG2,α = 1 . (142)
After combining the partial results we obtain the important theorem.
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APPENDIX III
A RATIO OF RANDOM DETERMINANTS
In this Appendix, we explicitly calculate the expected value of a ratio of random determinants
in complex generalized matrix quadratic forms. It appears that until now there were no results
available in literature for this general matrix variate case, which also comprises the well-known
vector variate case. In this paper, we give a scalar integral representation that is useful for the
derivations of this paper. However, we note that the remaining integral can be calculated in
closed form with the help of the residue theorem.
Theorem 11:Assume thatX is a m× n complex Gaussian distributed matrix with i.i.d.
elements and PDF
pX(X) =1
πmn ·etr(−XHX
). (143)
The following expected value of random determinants
r = EX
[∣∣XHCX∣∣
|XHDX|
](144)
with respect toX with diagonalm×mmatricesC = diag(c1,c2, . . . ,cm) andD = diag(d1,d2, . . . ,dm)
can be calculated by the single scalar integral expression
r =∑
αn
|C|αnαn·
∞∫
0
tr((In + t ·D1,α)−1
)
|Im+ t ·D| · tn−1 dt (145)
with the auxiliaryn×n matrix
D1,α = {D}αn1,...,n. (146)
Proof: We first express the expected value as an integral
r =∫ ∣∣XHCX
∣∣|XHDX| ·etr
(−XHX)
DcX. (147)
For carrying out the integral, we can use (122) and rewrite withn×1 vectorx
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r =∫ ∫ ∣∣XHCX
∣∣ ·etr(−xHXHDXx
) ·etr(−XHX
)DcX Dcx. (148)
First, we focus on solving the integral
r =∫ ∫ ∣∣XHCX
∣∣ ·etr
(1κ·xHXHDXx
)·etr
(−XHX)
DcX Dcx, (149)
where we have introduced a variableκ that is chosen such that the integrals that appear in the
following derivations are formally correct and convergent. In the final result, it can be shown
that the solution is valid for allκ and we letκ =−1 such that we can establish a solution for
the problem in (148). In (149) we can introduce another integral expression withm×1 vector
y according to (123) in Appendix I
etr
(1κ·xHXHDXx
)=
1κm ·
∫etr
(−
(κ ·yHy+xHXHD1/2y+yHD1/2Xx
))Dcy (150)
=1
κm ·∫
etr(−κ ·yHy
) ·etr(−(
UHX +XHU))
Dcy
with the auxiliarym×n matrix U for brevity
U = D1/2yxH. (151)
From (149) we obtain after straighforward manipulations and completing the square in the
exponent
r =1
κm ·∫ ∫ ∫
Ix ·etr(−κ ·yHy
) ·etr(UHU
)DcX Dcx Dcy (152)
with the auxiliary term for brevity
Ix =∣∣XHCX
∣∣ ·etr(−(X +U)H (X +U)
). (153)
The integral with respect toX can directly be solved (note again that auxiliary matrixU is
of rank 1) via integral identity (133)
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r =1
κm ·Γ(n) ·∑
αn
|C|αnαn·∫ ∫
Ixy Dcx Dcy (154)
with
Ixy =[n+ tr
(U1,αUH
1,α)] ·etr
(−κ ·yHy) ·etr
(UHU
). (155)
After introducing the partitionings of the matrixD
D1,α = {D}αnαn
(156)
D2,α = {D}βm−n
βm−n
and the vectory
y1,α = {y}αn{1} (157)
y2,α = {y}βm−n{1}
we get for the two parts of auxiliary matrixU
U1,α = D1/21,αy1,αxH (158)
U2,α = D1/22,αy2,αxH .
Using (158) in the integral expression of (154) we get
rα =∫ ∫ [
n+yH1,αD1,αy1,αxHx
]etr
(−yH1,α
(κ · In−xHxD1,α
)y1,α
)Dcy1,α · (159)
∫etr
(−yH
2,α(κ · Im−n−xHxD2,α
)Hy2,α
)Dcy2,α Dcx.
We can make use of (120) and (121) for calculating the integrals and obtain
rα =∫
1|κ · Im−xHx ·D|
[n+ tr
(xHx ·D1,α
(κ · In−xHx ·D1,α
)−1)]
Dcx. (160)
This can be further simplified via application of the general formula forn×1 vectorx
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∫f (xHx) Dcx =
1Γ(n)
∞∫
0
f (t) · tn−1 dt, (161)
which can be derived via a variable transformation to polar coordinates. We obtain
rα =(−1)m
Γ(n)
∞∫
0
1|−κ · Im+ t ·D| tr
(In− t ·D1,α (−κ · In + t ·D1,α)−1
)· tn−1 dt, (162)
which can be simplified with the matrix inversion lemma to
rα =(−1)m
Γ(n)
∞∫
0
1|−κ · Im+ t ·D| · tr
((In− t
κ·D1,α
)−1)· tn−1 dt. (163)
As an important result, the expression in (163) becomes forκ =−1
rα |κ=−1 =(−1)m
Γ(n)
∞∫
0
1|Im+ t ·D| · tr
((In + t ·D1,α)−1
)· tn−1 dt. (164)
The integral is convergent, as the integrand has no poles in the integration interval and behaves
like 1tm−n+2 for larget. Substituting (164) in (154) we arrive at the single scalar integral expression
given in the theorem.
APPENDIX IV
PROOF OFTHEOREM 5
With qk =−1+t·okok
=− 1γk·(
1ok
+ t)
we get from Theorem 4 the equivalent MGF representation
Mk(s) =1
|γkO|·∑
αL−1
|O|αL−1αL−1
·∑
αm∈αL−1
∞∫
0
s+ 1oαm[∏R
l=1,l 6=αm(s−ql )
]· (s−qαm)2
· tL−2 dt. (165)
We can now decompose the integrand into partial fractions with respect to s
s+ 1oαm[∏R
l=1,l 6=αm(s−ql )
]· (s−qαm)2
tL−2 =R∑
l=1,l 6=αm
Xl (αm)+Y1(αm)+Y2(αm) . (166)
With the short-hand notations
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Fl = s+1ol
+1γl· t (167)
and
Zo(l) =1
∏Rn=1,n6=l
(1ol− 1
on
) (168)
we get
Xl 6=αm (αm) = (−γk)R−1 ·Zo(l) · 1
1ol− 1
oαm
· tL−1
Fl+ (169)
(−γk)R−1 ·Zo(l) · t
L−2
Fl
= Xl 6=αm,1(αm)+Xl 6=αm,2(αm) .
Y1(αm) = (−γk)R−1 ·Zo(αm) · t
L−2
Fαm
+ (170)
(−γk)R ·Zo(αm) ·
R∑
n=1,n6=αm
11
oαm− 1
on
· tL−1
Fαm
= Y11(αm)+Y12(αm) .
Y2(αm) = (−γk)R−2 ·Zo(αm) · tL−1
(Fαm)2 . (171)
Using integration by parts we obtain
∞∫
0
Y2(αm) dt = (−γk)R−1 ·Zo(αm) · t
L−1
Fαm
∣∣∣∣∞
0− (172)
(L−1) ·∞∫
0
(−γk)R−1 ·Zo(αm) · t
L−2
Fαm
dt (173)
= Y21(αm)− (L−1) ·∞∫
0
Y22(αm)dt.
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In the original equation (165) the termsY11 and Y22 cancel and we find after some tedious
algebra
Mk(s) =1γk·∑
αL−1
|O|αL−1αL−1
·∑
αm∈αL−1
(U (αm)+ Iαm) . (174)
The main terms in (174) are
Iαm =
∞∫
0
R∑
l=1,l 6=αm
Vl (αm)+W (αm)− (L−2) ·Q(αm)
dt, (175)
Vl 6=αm (αm) = olR−1 ·oαm · Ko(αm, l) · Ko(l)
Fl· tL−1 + (176)
olR−2 · Ko(l)
Fl· tL−2,
W (αm) = oαmR−1 ·
R∑
n=1,n6=αm
on · Ko(αm,n)
· Ko(αm)
Fαm
· tL−1, (177)
U (αm) = oαmR−2 · Ko(αm) · t
L−1
Fαm
∣∣∣∣∞
t=0, (178)
and finally
Q(αm) = oαmR−2 ·Ko(αm) · t
L−2
Fαm
. (179)
For the reformulation we have used
1|O| ·Zo(l) = Ko(l) ·oR−2
l · (−1)R−1 . (180)
The MGF can be further simplified. First, we do a resummation
Mk(s) =1γk·∑
m
om · tr(m)L−2(O)
U (m)+
∞∫
0
R∑
l=1,l 6=m
Vl (m)+W (m)− (L−2) ·Q(m) dt
. (181)
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After a rearrangement of the terms, we obtain
Mk(s) =1γk·∑
m
tr(m)L−2(O) ·oR−1
m · Ko(m) · tL−1
Fm
∣∣∣∣∞
t=0(182)
+omR−1 ·
∞∫
0
R∑
l=1,l 6=m
ol · tr(l)L−2(O) · 1om· Ko(m)
Fm· tL−2
+
tr(m)
L−2(O) ·om ·R∑
l=1,l 6=m
ol · Ko(m, l)+R∑
l=1,l 6=m
o2l · tr(l)L−2(O) · Ko(l ,m)
· Ko(m)
Fm· tL−1
−(L−2) · tr(m)L−2(O) ·Ko(m) · t
L−2
Fmdt
].
A first simplification with the help of Lemma 8 in Appendix V yields
Mk(s) =1γk·∑
m
tr(m)L−2(O) ·oR−1
m · Ko(m) · tL−1
Fm
∣∣∣∣∞
t=0(183)
+omR−1 ·
∞∫
0
R∑
l=1,l 6=m
ol · tr(l)L−2(O) · 1om· Ko(m)
Fm· tL−2
+(L−1) · tr(m)L−1(O) · Ko(m)
Fm· tL−1
−(L−2) · tr(m)L−2(O) ·Ko(m) · t
L−2
Fmdt
].
Application of Lemma 4 in Appendix V yields the simplification
Mk(s) =1γk·∑
m
tr(m)L−2(O) ·oR−1
m · Ko(m) · tL−1
Fm
∣∣∣∣∞
t=0(184)
+omR−1 ·
∞∫
0
[(L−1) · tr(m)
L−1(O) · 1om· Ko(k)
Fm· tL−2
+ (L−1) · tr(m)L−1(O) · Ko(m)
Fm· tL−1 dt
].
Then note that
1ol ·Fl
+tFl
= γk ·(
1− sFl
). (185)
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We thus find from (184)
Mk(s) =1γk·∑
m
tr(m)L−2(O) ·oR−1
m · Ko(m) · tL−1
Fm
∣∣∣∣∞
t=0(186)
−γk ·s·omR−1 · (L−1) · tr(m)
L−1(O) ·Ko(m) ·∞∫
0
tL−2
Fmdt.
We can make use of the formula
xn
a+bx=
(−1)n ·an−1
bn ·[
1
1+ bax−
n−1∑
i=0
(−1)i(
ba
x
)i]
(187)
for rewriting the termtL−1
Fm.
Using Lemma 5 in Appendix V for simplifying the sum resulting from application of (187)
we can finally prove the first part of the theorem.
APPENDIX V
ELEMENTARY SYMMETRIC FUNCTIONS
A powerful tool for deriving identities for elementary symmetric functions is the generating
function (GF) approach. For the elementary symmetric functions (ESF) of them×1 vectorx it
reads
E (x, t) =m∏
l=1
(1+xl · t) =m∑
l=0
trl (x) · t l . (188)
We use (188) to derive a number of important ESF identities.
Lemma 2:For 0≤ n≤m−1 and1≤ k≤m
trn+1(x) = tr(k)n+1(x)+xk · tr(k)n (x) . (189)
Proof: We can rewrite the GF as
E (x, t) =m∏
l=1,l 6=k
(1+xl · t)+xk · t ·m∏
l=1,l 6=k
(1+xl · t) . (190)
Using (188) we obtain the equationm∑
l=0
trl (x) · t l =m−1∑
l1=0
tr(k)l1(x) · t l1 +xk · t ·
m−1∑
l2=0
tr(k)l2(x) · t l2. (191)
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By equating coefficients of like power int we can establish the lemma.
Lemma 3:For m×1 vectorx the following relation holdsm∑
l=1
xl · tr(l)n (x) = (n+1) · trn+1(x) . (192)
Proof: Differentiating the GF we can derive
∂∂ t
E (x, t) =m∑
k=1
xk ·m∏
l=1,l 6=k
(1+xl · t) =m∑
r=1
r · trr (x) · tr−1. (193)
By equating coefficients of like power int we can establish the lemma.
Lemma 4:For m×1 vectorx the following relation holdsm∑
l=1
xl · tr(l)n (x) = (n+1) ·[tr(k)n+1(x)+xk · tr(k)n (x)
]. (194)
Proof: The lemma directly follows from application of Lemma 2 and Lemma 3.
Lemma 5:For k = 0. . .m−1 we havem∑
l=1
xkl · tr(l)n (x) ·Kx (l) = (−1)n ·δ (k− (m−n−1)) . (195)
For m> k > m−n−1 we therefore find the important special case
m∑
l=1
xkl · tr(l)n (x) ·Kx (l) = 0 ∀ m> k > m−n−1. (196)
Proof: We begin the proof with the expansion in partial fractions in Lemma 9 in Ap-
pendix VI for 0≤ µ ≤m
(−1)µ · tµ∏m
i=1(1+xi · t) =m∑
l=1
Kx (l) · xm−µ−1l
1+xl · t(197)
=m∑
l=1
Kx (l) ·xm−µ−1l ·∏m
j=1, j 6=µ(1+x j · t
)∏m
ν=1(1+xν · t)
=∑m
l=1Kx (l) ·xm−µ−1l ·∑m−1
n=0 tr(l)n (x) · tn∏m
ν=1(1+xν · t)By comparing like powers oft we find
(−1)µ ·δ (µ−n) =m∑
l=1
Kx (l) ·xm−µ−1l · tr(l)n (x) . (198)
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Then settingk = m−µ−1 proves the lemma.
Lemma 6:For k≥m we have withτ = min(m−n,k+1−m)
m∑
l=1
xkl · tr(l)n (x) ·Kx (l) =
τ∑
j=0
(−1) j+1 ·hk+1−m(x) · trn+ j (x) . (199)
Proof: We begin the proof with the expansion in partial fractions in Lemma 10 in Ap-
pendix VI
(−1)µ
tµ ·∏mi=1(1+xi · t) =
m∑
l=1
Kx (l) · xm+µ−1l
1+xl · t+
µ∑
j=1
(−1) j · hµ− j (x)t j (200)
=γ1 + γ2
tµ ·∏mν=1(1+xν · t)
=δ1 +δ2
tµ ·∏mν=1(1+xν · t)
with the auxiliary terms
γ1 =m∑
l=1
Kx (l) ·xm+µ−1l · tµ ·
m∏
i=1,i 6=µ
(1+xi · t) (201)
γ2 =µ∑
j=1
(−1) j ·hµ− j (x) · tµ− j ·m∏
n=1
(1+xn · t)
and
δ1 =m∑
l=1
Kx (l) ·xm+µ−1l ·
m−1∑
i=1
tr(l)i (x) · t i+µ (202)
δ2 =µ∑
j=1
(−1) j ·hµ− j (x) ·m∑
n=1
trn(x) · tµ+n− j
for brevity. By comparing both sides forµ ≥ 1 we find
m∑
l=1
Kx (l) ·xm+µ−1l ·
m−1∑
i=1
tr(l)i (x) · t i+µ +µ∑
j=1
(−1) j ·hµ− j (x) ·m∑
n=1
trn(x) · tµ+n− j = 0. (203)
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Now we compare like powers oft on both sides. To this end, we consider (203) for fixedi
m∑
l=1
Kx (l) ·xm+µ−1l · tr(l)i (x) · t i+µ =
min(m−i,µ)∑
j=1
(−1) j+1 ·hµ− j (x) · tri+ j (x) · t i+µ . (204)
Finally settingk = m+ µ−1 proves the lemma.
Lemma 7:For the two distinct indicesk1 andk2
Kx (k2,k1) ·[tr(k1)
n (x)− tr(k2)n (x)
]= tr(k1,k2)
n−1 (x) . (205)
Proof: The lemma can be derived via a generating function approach. To this end we show
∏
l1 6=k1
(1+xl1 · t)−∏
l2 6=k2
(1+xl2 · t) =(
11+xk1t
− 11+xk2t
)·∏
l
(1+xl · t) (206)
= (xk2−xk1) · t ·∏
l 6={k1,k2}(1+xl · t)
Comparing like powers oft proves the lemma.
Lemma 8:For 1≤ k≤m and0≤ n≤m−1
xk · tr(k)n (x) ·m∑
l=1,l 6=k
xl · Kx (k, l)+m∑
l=1,l 6=k
x2l · tr(l)n (x) · Kx (l ,k) = (n+1) · tr(k)n+1(x) . (207)
Proof: From Lemma 4 we obtain
(n+1) · tr(k)n+1(x) =m∑
l=1,l 6=k
xl · tr(l)n (x)+xk · tr(k)n (x)− (n+1) ·xk · tr(k)n (x) . (208)
Now using (11) we can write
(n+1) · tr(k)n+1(x) =m∑
l=1,l 6=k
xl2Kx (l ,k) · tr(l)n (x)+ (209)
m∑
l=1,l 6=k
xl xkKx (k, l) · tr(l)n (x)−n·xk · tr(k)n (x) .
Comparing (207) and (209), in order to prove the lemma we have to show that
xk · tr(k)n (x) ·m∑
l=1,l 6=k
xl · Kx (k, l) =m∑
l=1,l 6=k
xl xkKx (k, l) · tr(l)n (x)−n·xk · tr(k)n (x) . (210)
We rewrite (210) as
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n· tr(k)n (x) =m∑
l=1,l 6=k
xl · Kx (k, l) ·[tr(l)n (x)− tr(k)n (x)
]. (211)
Now using Lemma 7 we get
n· tr(k)n (x) =m∑
l=1,l 6=k
xl · tr(l ,k)n−1 (x) (212)
and finally by Lemma 3 we can prove the lemma.
APPENDIX VI
EXPANSIONS IN PARTIAL FRACTIONS
The two lemmas of this section are given without proof.
Lemma 9:For integerk and0≤ k≤m
(−1)k · tk∏m
l=1(1+xl · t)=
m∑
l=1
Kx (l) · xm−k−1l
1+xl · t. (213)
Lemma 10:For integerk and0≤ k≤m
(−1)k
tk ·∏ml=1(1+xl · t)
=m∑
l=1
Kx (l) · xm+k−1l
1+xl · t+
k∑
j=1
(−1) j · hk− j (x)t j . (214)
ACKNOWLEDGMENT
The authors would like to thank...
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Mario Kießling studied at the University of Stuttgart, Germany, where he received his Dipl. Ing. degree in
Electrical Engineering in 2000. From 2001 to 2004 he has been with the Institute of Telecommunications
at the University of Stuttgart and Siemens Information and Communication Mobile in Ulm, Germany. In
2004 he received a Dr.-Ing. degree in Electrical Engineering. Since then he has worked at Bosch Blaupunkt
as a member of the international management trainee program with focus on research and development.
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 44
T
H Gs z
n R R
y
LLF
Fig. 1. System Model
0 1 2 3 4 50
0.5
1
1.5
x
p(x)
T=4, R=6, L=4
Empirical PDFAnalytical PDF
rRX
=0.9
Fig. 2. Probability Density Function,rRX = 0.9, T = L = 4, R= 6
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 45
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
q(x)
T=4, R=6, L=4
Empirical CDF
Analytical CDF
rRX
=0.9
Fig. 3. Cumulative Distribution Function,rRX = 0.9, T = L = 4, R= 6
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 46
0 1 2 3 4 50
0.5
1
1.5
x
p(x)
T=4, R=6, L=4
rRX
=0.9
rRX
=0.7
rRX
=0.5
rRX
=0.3
rRX
=0.9
rRX
=0.7
rRX
=0.5
rRX
=0.3
Fig. 4. Probability Distribution Function,rRX = {0.3,0.5,0.7,0.9}, T = L = 4, R= 6
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 47
0 5 10 15 20 25 3010
−4
10−3
10−2
10−1
100
SNR (dB)
SE
R
T=4, R=6, L=4
Analytical Results
Monte Carlo Sim.
rRX
=0.3
rRX
=0.9
rRX
=0.7rRX
=0.5
Fig. 5. Symbol Error Rate,rRX = {0.3,0.5,0.7,0.9}, T = L = 4, R= 6, M = 16 QAM
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 48
0 5 10 15 20 25 3010
−4
10−3
10−2
10−1
100
T=4, R=6, L=4
SNR (dB)
SE
R
Uncorrelated
rRX
=0.7, rTX
=0
rRX
=0.7, rTX
=0.7
Fig. 6. Symbol Error Rate,T = L = 4, R= 6, M = 16 QAM
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 49
0 5 10 15 20 25 30 35 4010
−4
10−3
10−2
10−1
100
T=4, R={4,8}, L=4
SNR (dB)
SE
R
rRX
=0.9
rRX
=0.3
R=4
R=8
Fig. 7. Symbol Error Rate,rRX = {0.3,0.9}, T = L = 4, R= {4,8}, M = 16
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TRANSACTIONS ON INFORMATION THEORY, VOL. 1,NO. 1, JANUAR 1111 50
−10 −5 0 5 10 15 20 25 300
1
2
3
4
5
6
7
SNR (dB)
MM
I (na
ts p
er c
hann
el u
se)
T=4, R=6, L=4
Analytical Result
Monte Carlo Sim.
rRX
=0.3
rRX
=0.9
rRX
=0.7
rRX
=0.5
Fig. 8. Mean Mutual Information,rRX = {0.3,0.5,0.7,0.9}, T = L = 4, R= 6
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