The Performance of a Null-Steering Beamformer in Correlated Rayleigh Fading

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The Performance of a Null-Steering

Beamformer in Correlated Rayleigh Fading

Cornelius van Rensburg , Benjamin Friedlander

C van Rensburg is with Samsung Telecommunications America. E-mail: [email protected]

B. Friedlander is with The University of California Santa Cruz. E-mail: [email protected]. The work of

B. Friedlander was funded by the National Science Foundation under Information Technology Research Grant

CCR-0112508, and by the Office of Naval Research, under contract N00014-01-1-0075.

October 19, 2003 DRAFT

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Abstract

We consider the performance of a null-steering adaptive beamformer in a distributed Rayleigh fading

environment. To determine the beamformer coefficients, the signal and interference array response vectors

must be estimated. An estimator which makes use of the knowledge of the spatial distribution of the signal

is proposed, and compared with other estimators which do not use this information. As the angular spread

of the signal increases, the value of this information is reduced, until for the case of isotropic scattering

the estimate which uses the spatial distribution is equivalent to the estimate which does not. Performance

results are presented that compare the performance of the different estimators. We derive closed form

expressions for the SINRO and probability of error for the case where the training data are orthogonal.

Keywords

EDICS:1-ACOM,3-MCHA.

Array Processing, Communication, Fading, Estimation and Detection.


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I. Introduction

The need to separate co-channel signals by an antenna array arises in many situations,

and in cellular communications in particular. Separating signals from different users by

their spatial signatures can increase significantly system capacity beyond what can be

achieved by temporal processing alone.

Extensive work has been done on techniques for separating co-channel signals as de-

scribed in [1] and [2]. The overwhelming majority of this work assumes that the signals

are point sources (i.e., having zero angular spread), and that the propagation from trans-

mitter to receiver occurs along one path or a few distinct paths (multi path). A more

practical model used in the cellular communications environment represents the propaga-

tion as a continuum of different paths, where the signal arriving at the receiver antenna can

be characterized by a continuous spatial distribution of energy (sometimes called “angular

spread”) around the nominal direction of arrival.

Relatively little work has been done on co-channel signal separation for the case of

distributed signal sources. Previously [3] developed a performance bound for specifically

the isotropic scattering case. More recently, [4] proposed a structured estimate of the array

response, which performed better in a distributed fading environment to a system that used

an unstructured estimate. [5] looked at the effect that temporal and spatial correlation

has on the DOA estimation, and [6] investigated the performance of the general class of

semi-blind subspace-based estimators, like those that we are using here.

In this paper we study the effect of angular spread on the ability of an array to separate

co-channel signals. In particular we use a framework similar to that presented in [7] and

[8], where an isotropic scattering environment was assumed (essentially an angular spread

of 360 degrees ) which is characteristics of an indoor or pico cell environment. Here we

generalize the work of [7] for the case where the angular spread can take on any value

between zero (point source) and 360 degrees (isotropic scattering). Small to moderate

angular spread is characteristic of outdoor urban and rural communication where the

mobile is typically far from the base station and the fading happens in a relatively small

region around the mobile station.

The angular power distribution of the signal source induces the spatial covariance matrix


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of the received array response. In the case of a point source this covariance matrix is of

rank one, while for isotropic scattering it is the scaled identity matrix. In general the rank

of the covariance matrix increases with the angular spread.

In order to perform co-channel signal separation it is necessary to estimate the array

response vectors of the different users. As in [7] we assume that a training sequence is

available to do the estimation. In most TDMA cellular communication systems such as

GSM and IS-136, there are synchronization bursts at the start of every timeslot, which can

be used for this purpose. However, in addition to the training data it is possible to make

use of the statistical information embodied in the spatial covariance matrix associated

with each user. We consider two situations: (i) The case where the covariance matrix is

known a-priori, and (ii) The case where the covariance matrix is estimated from the data

itself. The first is useful as a reference point to show the best that can be done by using

this additional statistical information. The second is the more practical situation in which

the covariance matrix is not known a-priori. As is expected, using the covariance matrix

in addition to the training data can improve signal estimation performance.

We will consider four estimates of the array response, namely the perfect estimate A,

the estimate with unknown covariance R (UR), A, the estimate with known R (KR), AR,

and the estimate using an estimated covariance, AR, (ER).

II. Problem Statement

A. Notation

We use lower case boldface letters to denote vectors and upper case boldface letters to

denote matrices. In addition we use the following conventions throughout the paper,

X⊗Y Kronecker product

Av The vec() operator, which is the concatenation

of the columns of A

()′ The conjugate transpose

()T Transpose

()∗ Conjugate

<(A) The real part of A


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=(A) The imaginary part of A

E[] Expected value

I Identity matrix

S† Moore-Penrose pseudo inverse

∼ Nc A complex Gaussian distribution

B. The Signal Model

Consider an M-element antenna array with an array response ak, corresponding to a

signal sk(tn) at time tn. The total of K transmitted signals are corrupted by an AWGN

term u(tn), so we can write the received signal (the M-vector) as

x(tn) =K∑

k=1

aksk(tn) + u(tn) (1)

where u(tn) ∼ Nc(0, ηI). The k’th signal has a power of pk = |sk(tn)|2. Defining the total

array response as

A = [a1 . . . aK ], (2)

and the total signal at time tn as

s(tn) = [s1(tn) . . . sk(tn)]T , (3)

we can write

x(tn) = As(tn) + u(tn). (4)

If we now collect Nd data points and define the matrices

X = [x(t1) . . .x(tNd)] (5)

S = [s(t1) . . . s(tNd)] (6)

U = [u(t1) . . .u(tNd)], (7)

then we can finally write the model as

X = AS + U. (8)

Using the estimate for A, namely A, we will reconstruct the signal at the receiver using

a null-steering beamformer that would attempt to null-out the interferers completely by


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implementing this equation:

s(tn) = (A′A)−1A′x(tn), (9)

which is the least squares (LS) minimizer for the cost function ||x(tn)− As(tn)||2F .

C. Channel Model

In a faded multi path environment the array response vector is

ak =∫ π

−πψk(θ)w(θ)dθ, (10)

where w(θ) is the steering vector for the particular array and ψk(θ) is the complex fading

from the signal sk. Note that ψk is independent of ψl for l 6= k, and ψk(θ1) is independent

of ψk(θ2) for θ1 6= θ2, such that

E[ψk(θ1)ψ∗l (θ2)] = Ψ(θ1 − θk)δ(θ1 − θ2)δk−l (11)

E[ψk(θ1)ψl(θ2)] = 0. (12)

The covariance of the array response can be expressed as

Rk = E[aka′k] =

∫ π

−πΨk(θ)w(θ)w′(θ)dθ, (13)

where

Ψk(θ) = E[|ψk(θ)|2], (14)

is the spatial energy distribution of the k’th signal. Therefore, the columns of A will be

random with

ak ∼ Nc(0,Rk), ∀ k = 1, . . . , K (15)

but the fading in the different columns are independent, thus

E[aka′i] = 0 ∀ k 6= i. (16)

The problem now is how to estimate the channel matrix A, using the covariance.

III. Channel Estimation

All the estimators considered here assume that there is a known training sequence St,

of length N available to the receiver to estimate the channel.


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A. The Estimate if R is Known (AR)

In this section, we develop an array response estimate AR in which we use the a priori

knowledge of the covariances of the columns of A,

ak = R12k yk, (17)

yk ∼ Nc(0, ILk), (18)

E[yky′i] = 0, ∀ k 6= i (19)

where Lk is the rank of Rk. Note that when Rk is low rank, some of the eigenvalues will

be zero. Equation (2) can be vectorized and expressed as

Av =

R121 0 0

0. . .

...

0 0 R12K

y1

...

yK

= R12y. (20)

We define the matrix square root as the M × Lk matrix

R12k = ΓkΛ

12k (21)

which means that Rk = R12k R′ 12

k , Γk is the matrix of eigenvectors of Rk and Λ12k is the

diagonal matrix of the square roots of the eigenvalues. This property will be used in the

performance analysis section. Vectorizing (8)

Xv = (STt ⊗ IM)Av + Uv = QR

12y + Uv, (22)

where Q = STt ⊗ IM, is a reformatted version of the training symbols. The LS estimator

can be written as

AvR = R

12 (QR

12 )†Xv = R

12y + R

12 (QR

12 )†Uv (23)

or alternatively

ak = R12k [(QR

12 )†Xv]k = R

12k yk + R

12k [(QR

12 )†Uv]k, (24)

where []k is the k’th M ×Lk block sub matrix. This estimator is the LS minimizer for the

cost function

||Xv −QR12y||2F .


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TABLE I

The Estimators AR, AR, A.

Case Estimator

Known R R12k [(QR

12 )†Xv]k

Estimated R R12k [(QR

12 )†Xv]k

Unknown R [(Q)†Xv]k

B. Estimate using no a-priori Knowledge (A)

This case is just a special case of the estimator presented in Section III.A, where the

Rk = I, thus

Av = (Q)†Xv = y + (Q)†Uv. (25)

C. The Estimate where R is Estimated (AR)

In this section, the covariance is assumed to be unknown and first needs to be estimated.

To do this we first make Na estimates of the channel using the method presented in Section

III.B, which assumes no-prior knowledge of the covariance. The estimated covariance is

then the sample covariance of these estimates :

Rk =1

Na

Na∑

n=1

ak(n)a′k(n), (26)

Once we have the estimate Rk, and thus also R12k , we define the ”Estimated R” estimator,

by substituting the actual R in (23) for the estimated R:

AvR

= R12 (QR

12 )†Xv = R

12y + R

12 (QR

12 )†Uv. (27)

D. Comparison of AR, AR and A estimators

The different estimators are summarized in Table I. In the case of point sources, the

Known R estimator (A) is defined by Γk up to an unknown complex scaling factor. The

training symbols are used only to determine this factor. In the case of isotropic scattering

the three estimators are identical because R12k = I.


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IV. The SINRO of the Different Receivers

Next, we evaluate the output signal to interference plus noise ratio (SINRO), of receivers

using the different estimators. Using the partition, A = [D,v], where D represents the

first K − 1 columns of A and v the last column, we can define the following matrices :

T = I−D(D′D)−1D′ (28)

D = (D′D)−1D′. (29)

Here T is the projection matrix on the null space of D, and D is the left pseudo inverse

of D. It can be shown that

(A′A)−1A′ =1

v′Tv

D[v′TvI− vv′T]

v′T

(30)

and therefore the K’th detected signal can be expressed as

sK(tn) =v′Tx(tn)

v′Tv=

v′T[As(tn) + u(tn)]

v′Tv

=v′Tv

v′TvsK(tn) +

K−1∑

k=1

v′Tak

v′Tvsk(tn) +

v′Tu(tn)

v′Tv

= αsK(tn) +K−1∑

k=1

βksk(tn) + µ′u(tn). (31)

Here any hatted symbol represents an estimated value, α represents how well we matched

the received signal and βk is the component of the interferer that was not canceled. We

can now define the signal to noise ratio at the output of the beamformer (SNRO) as

SNRO =pKE[||α||2]E[||µ||2]η , (32)

where pK is the K’th signal power. The signal to interference ratio at the output (SIRO)

is

SIRO =pKE[||α||2]

∑K−1k=1 pkE[||βk||2]

, (33)

and the signal to interference and noise ratio at the output SINRO is

SINRO = [SIRO−1 + SNRO−1]−1. (34)


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A. The Perfect Channel Estimate

When A is known perfectly there is no residual interference power and β = β = 0,

α = α = 1 in (31). We can therefore write the SNRO in (32) as

SNRO = ρKv′Tv (35)

where ρK = pK/η is the input SNR of the K’th signal.

B. The Estimate when R is known (AR)

An expression for the SINRO of this estimator is derived using a perturbation analysis.

We denote <[Apq] as Apq, =[Apq] as Apq. Since βk = 0, a first order Taylor series expansion

of βk is:

βk 'M∑

p=1

K∑

q=1

∂βk

∂Apq

δApq +M∑

p=1

K∑

q=1

∂βk

∂Apq

δApq. (36)

Using lengthy but straightforward algebraic manipulations we obtain the following expres-

sions:

∂βk

∂Apq

= −δqkv′Tep

v′Tv(37)

∂βk

∂Apq

= −jδqkv′Tep

v′Tv(38)

where ep represents an M vector of zeroes, except for a one at the p-th position. If we now

define:

h(1)k =

∂βk

∂Av= −ek ⊗ (Tv)∗

v′Tv, (39)

h(2)k =

∂βk

∂Av= −jek ⊗ (Tv)∗

v′Tv, (40)

hk =

h(1)k

h(2)k

= −

ek ⊗ (Tv)∗v′Tv

jek ⊗ (Tv)∗v′Tv

, (41)

δa =

δAv

δAv

, (42)

then we can rewrite (36) as

βk ' hTk δa, (43)


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and therefore

E[||βk||2] ' hTk var(δa)h∗k. (44)

To calculate the var(δa), we use (23) to get

E[δaδa′] =η

2R

12 (R′ 12Q′QR

12 )−1R′ 12 , (45)

and this can be simplified if the training signals are orthogonal, to

E[δaδa′] =η

2N

1p1

Γ1Γ′1 0 0

0. . .

...

0 0 1pK

ΓKΓ′K

. (46)

Note that since the fading is correlated, Γk is an M × Lk matrix, where Lk is the rank of

Rk. This gets substituted into (44):

E[||βk||2] ' η

2(v′Tv)2

[eT

k ⊗ (v′T) jeTk ⊗ (v′T)

]

×<[E[δaδa′]] −=[E[δaδa′]]

=[E[δaδa′]] <[E[δaδa′]]

ek ⊗ (Tv)

−jek ⊗ (Tv)

' η

pkN(v′Tv)2v′T(ΓkΓ

′k)Tv. (47)

A similar perturbation analysis done on α, which we will not repeat here, will show that

a good approximation when the SNR is high is α ≈ 1. Next, insert the result above into

(33) to get

SIROKR ' NρK(v′Tv)2

v′T(∑K−1

k=1 ΓkΓ′k)Tv

, (48)

SINROKR ' NρK(v′Tv)2

v′T(∑K−1

k=1 ΓkΓ′k + NIM)Tv. (49)

Note that for the case of a point interferer, SIROKR = ∞, irrespective of the position of

the interferer. This is because E[||βk||2] = 0 when T projects the eigenvectors of D onto

their own null space, (TΓkΓ′k = 0). In this case the SINROKR = SNRO.

C. The case when R is Estimated (AR)

The SINRO for this estimator can be found by rewriting (47) as

E[||βk||2] ' η

Npk(v′Tv)2v′TE[ΓkΓ

′k]Tv (50)


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but note that

E[ΓkΓ′k] =

Lk∑

l=1

E[glg′l], (51)

which was shown in [9] to be

E[glg′l] =

λl

Na

Lk∑

i=1,i6=l

λi

(λl − λi)2gig

′i + glg

′l. (52)

Therefore

E[ΓkΓ′k] =

Lk∑

l=1

λl

Na

Lk∑

i=1,i 6=l

λi

(λl − λi)2gig

′i + glg

′l

= ΓkΛkΓ′k, (53)

where Λk is a diagonal matrix with elements

[Λ]ii = 1 +λi

Na

∑

j 6=i

λj

(λi − λj)2. (54)

Substituting (53) into (50) leads to

E[||βk||2] ' η

Npk(v′Tv)2v′T[ΓkΛkΓ

′k]Tv. (55)

Finally the SIRO and SINRO can be written as

SIROER ' NρK(v′Tv)2

v′T(∑K−1

k=1 ΓkΛkΓ′k)Tv, (56)

SINROER ' NρK(v′Tv)2

v′T(∑K−1

k=1 ΓkΛkΓ′k + NIM)Tv. (57)

It is clear from (57) that as Na increases, Λ → I and SINROER → SINROKR, as expected.

D. The Estimate with no a-priori Knowledge (A)

This estimator is a special case of the Estimator where R is Known with ΓkΓ′k = I in

(49). Therefore,

SINROUR = ρKv′TvN

N + K − 1, (58)

which agrees with the results in [7]. The results of this section is summarized in Table II.


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TABLE II

The SINRO for the four different estimators.

Case SINRO

1. Perfect A ρKv′Tv

2. Known R NρK(v′Tv)2

v′T(∑K−1

k=1ΓkΓ′k+NIM )Tv

3. Estimated R NρK(v′Tv)2

v′T(∑K−1

k=1ΓkΛkΓ′k+NIM )Tv

4. Unknown R ρKv′Tv NK−1+N

V. The Average SINRO

The expressions in Table II represent nonlinear combinations of multiple random vari-

ables. Evaluating the average SINRO analytically is difficult in general. Here we consider

the special cases of a single interferer, and multiple (but well separated) interferers. The

expressions for the SINRO for the cases where either A is known perfectly, or not known

at all, are identical (up to a constant). These two cases represent an upper and lower

bound and are of special interest.

A. The Average SINRO with a single interferer

We study how the SINRO changes with the position of the interferer and the angular

spread. In the case of a single interferer, D is a vector, and the average SINRO can be

written as

E[SINRO] = E[v′Tv] = E[|v|2]− E[v′DD′vD′D

]

= E[|v|2]− trace(E[DD′

D′D]E[vv′])

= trace(ΛK)− trace(RdΓ′1ΓKΛKΓ′KΓ1)

(59)

where

Rd = E[Λ

1/21 y1y

′1Λ

1/21

y′1Λ1y1

]


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=

E[rd1] 0 0

0. . . 0

0 0 E[rdL]

(60)

and where Γ′KΓ1 represents the matrix of cosines of the principle angles between the

subspaces spanned by v and D. We are interested in the mean values of the random

variables rd1 . . . rdL. The elements on the off diagonal are independent and have zero

mean. In Appendix A we show that rdi has a probability density function (pdf) of

frdi(y) =

L∑

l=1,l 6=i

αlλiλ2l

[y(λl − λi) + λi]2, (61)

αi =1

αi

L∏

l=1,l 6=i

αi

αi − αl

, (62)

where 0 < y < 1. The mean is then

E[rdi] =L∑

l=1,l 6=i

λiαlλ2l

(λl − λi)2

[λi

λl

− 1 + ln

(λl

λi

)]. (63)

The average SINRO is therefore a function of the squared cosines of the principle angles,

weighted by the diagonal values of ΛK and Rd. For example, if the subspaces have rank

one, there is only one principle angle, and then

E[SINRO] = 1− cos2(θ) = sin2(θ), (64)

which is a measure of the distance between the two subspaces. If each subspace is of

dimension 2, with

|g′v1gd1|2 = cos2(θ11)

|g′v1gd2|2 = cos2(θ12)

|g′v2gd1|2 = cos2(θ21) (65)

|g′v2gd2|2 = cos2(θ22)

then

E[SINRO] = trace(ΛK)− rd1λ1K cos2(θ11)− rd1λ2K cos2(θ12)

− rd2λ1K cos2(θ21)− rd2λ2K cos2(θ22). (66)


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The average SINRO in (66) can be viewed as the total received power for the signal of

interest minus the proportion of the interfering power that coexist in the subspace of the

signal of interest. In Fig. 1 we plot the normalized SINRO (E[v′Tvv′v ]) for a 10 element

uniformly spaced linear array (ULA) as the position and spread is varied. The array

beamwidth here is 10 degrees.

B. The Average SINRO for Multiple Interferers

For the case where the multiple interferers are well separated, D′D is a diagonal matrix,

and

E[SINRO] = trace(ΛK)−K−1∑

k=1

trace(RKΓ′KΓkRdkΓ′kΓK).

(67)

Here, the interpretation of the average SINRO in (67) generalizes (59) to the total received

power for the signal of interest minus the proportion of all the interfering powers that

coexist in the subspace of the signal of interest.

C. The BER Performance in a fading environment

In general it is difficult to get an expression for the error probability (Pe), since the

SINRO consists of products of multiple random variables. However, if the interferer is not

too close to the signal of interest, we can approximate the distribution of the SINRO to a

chi-square distribution, and then use the Pe expressions available in the literature. Here

we use the formulas derived by [10](in Appendix 7A), where we will use the derivation for

the Binary PSK example that uses a pilot signal estimate. These formulas are repeated

here,

γt = γtraffic + γpilot (68)

r =γtraffic

γpilot

(69)

µperfect = =

√γtraffic

γtraffic + 1(70)

µEstimate =

√rv

(r + 1)

√(1γt

+ rr+1

) (1γt

+ vr+1

) (71)


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Pe(µ) =1

2

1− µ

L−1∑

k=0

2k

k

(1− µ2

4

)k . (72)

Here, v represents the number of training symbols, r represents the ratio of the pilot power

versus the power in the traffic channel, γtraffic and γpilot are the average SNR’s for the

traffic and pilot channels respectively. The reader might be surprised to see references

to a pilot and traffic channel when we are describing a TDMA system. The reason is

that we can use these expressions (which were probably derived for a CDMA system)

with slight modification for our problem. So, rather interpret γtraffic as the average SNR

during the data burst, and γpilot as the average SNR during channel estimation. Note that

the Pe is a function of the parameter µ, so the case of the perfect estimate would use the

µperfect and the case of the KR Estimator would use µEstimate in calculating Pe. However,

these formulas were derived for the case where the fading is independent and identically

distributed (iid). Therefore, we have to redefine the variables r, v, γpilot, γperfect since the

fading is correlated here. In this paper we will simply approximate the eigenvalues as either

ones or zeros, and then transform the problem into one that approximates one where the

variables are iid. Firstly, the Pe for the perfect estimate depends only on the average SNR

for the traffic channel which we define as

γtraffic = ρKE[v′Tv], (73)

and L which is chosen to minimize the mean square error between the approximated

eigenvalues and the true eigenvalues

MSE = ||[λ1, ..λM ]− [0, ..0, 1, ..1]/L||2F . (74)

The optimum L to choose is the one that would minimize the Pe, but since the Pe in (72)

is not too accurate we use the method above. For the case of the KR estimate, we choose

L similar to the perfect estimate. Note that here we project each component that needs

to be estimated (yl) on an M dimensional orthogonal subspace before estimation, which

means that there are effectively NM independent training symbols per estimate, but at

the same time, the average SNR in the pilot channel reduces by M , thus

γtraffic = E[SINROKR] (75)


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γpilot = E[SINROKR]/M (76)

v = NM. (77)

To be accurate, these formulas have to be re-derived for the case of correlated fading. Note

that the Interference component in the SINRO is small in most cases that we describe here

since we are nulling it out, so we can simply approximate

γtraffic = ρKE[v′Tv] (78)

γpilot = ρKE[v′Tv]/M, (79)

for which we have closed form expressions. When we study the structure of µEstimate, we

see that the relationship between SINRO and Pe is not straightforward and that is why

we see a much larger performance gain by the new estimators as compared to what the

SINRO alone would suggest. The reason that we can use these expressions directly for

the KR estimator is because it projects the estimators on orthogonal subspaces, and the

variables can then be approximated as a uncorrelated and iid. However, this cannot be

done for the UR estimator and thus we do not have an expression for the BER in this

case.

VI. Numerical Simulations

In this section, we simulate equal powered BPSK signals, with K = 2, that is one

signal of interest and one interferer. There are M = 6 antenna elements in a ULA with a

beamwidth of approximately 20 degrees. The data burst consists of Nd = 150 symbols. In

Fig.2 we present SINRO results from Monte Carlo simulations and compare them with the

analytic values in Table II. Here we simulated a signal of interest which was at broadside

and an interferer which was moved from -50 degrees to -5 degrees, both were spread over

4 degrees. The channel was estimated using N = 2 training symbols while the input

SNR was 10dB. In Fig. 3, we show BER simulation results where both signals were again

spread over 4 degrees and L = 1 in both cases. Here the channel was estimated with

N = 4 training symbols, while the covariance was estimated over Na = 20 timeslots. he

eigenvalues are

[0, 0, 0, 0, 0.0058, 0.9942].


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Note how close the Known R and Estimated R estimators are to the case where the

channel is known perfectly. In Fig. 4, the signals are both spread over 30 degrees, and

here L = 2 in both cases. The eigenvalues in this case is

[0, 0, 0.0025, 0.0591, 0.3480, 0.5903].

There is still a significant performance improvement here compared to the unknown R

estimator, but less than when the signal has a small angular spread. Note how the analytic

Pe differs from the Monte Carlo results in the high SNR cases. This is because the effect

of the approximation of the eigenvalues as either ones or zeros becomes significant. When

the signals are spread over a wide angle, e.g. over 100 degrees, the 3 estimators will be

identical, this should be clear from Table I.

VII. Conclusions

In this paper we investigated the ability of an array to separate co-channel signals in

a fading environment as a function of angular spread. We compared the performance of

several estimators which use the spatial covariance matrix of the array response (either

known a-priori or estimated from the data) to that of an estimator which uses only training

data. As expected, using the covariance information can improve the system performance.

The performance improvement increases as the number of array elements increases, the

number of interferes increases, and the number of training symbols decreases.

Appendices

I. Weighted Beta Function

We are interested in the distribution and the mean value of the random variable defined

as

Y =X1

X1 + X2 + . . . + XN

(80)

where

Xn ∼ αn

2χ2

2 (81)

with a pdf of

fXn(x) =1

αn

e−x

αn , (82)


19

for all unique αn’s. To find fY (y) we proceed in 2 steps. First we assume N = 2, and

generalize it later. Doing a transformation of random variables we define

Y =X1

X1 + X2

(83)

Yt = X1 + X2, (84)

and then x1 = yyt, and x2 = yt − yyt, with 0 < y < 1, and 0 < yt < ∞. The Jacobian is

J =

∣∣∣∣∣∣∣

yt y

−yt 1− y

∣∣∣∣∣∣∣= yt. (85)

The joint pdf is

fX1X2(x1, x2) =1

α1α2

e−(

x1α1

+x2α2

)(86)

fY Yt(y, yt) =1

α1α2

yte−(

yytα1

+yt−yyt

α2)

(87)

fY (y) =1

α1α2

∫ ∞

0yte

−yt(y

α1+ 1−y

α2)dyt

=α1α2

[y(α2 − α1) + α1]2. (88)

The expected value of Y is then

E[Y ] =∫ 1

0

α1α2y1

[y1(α2 − α1) + α1]2dy

=α1α2

(α2 − α1)2

[α1

α2

− 1 + ln(

α2

α1

)]. (89)

Now, let us consider the case of N > 2. Here we will keep the same definitions as above,

except that we will introduce

Xsum =N∑

n=2

Xn. (90)

The pdf of Xsum is the convolution of the respective pdf’s, namely

fXsum(x) = fX2(x) ∗ fX3(x) ∗ . . . ∗ fXN(x)

=N∑

n=2

αne− x

αn , (91)

where

αi =1

αi

N∏

n=2

αi

αi − αn

, ∀ i 6= n. (92)


20

The derivation from (86) to (89) is now repeated by replacing fX2(x) with fXsum(x) as

defined in (91) instead. This leads to the pdf and the Expected value

fY (y) =N∑

n=2

αnα1α2n

[y(αn − α1) + α1]2(93)

E[Y ] =N∑

n=2

α1αnα2n

(αn − α1)2

[α1

αn

− 1 + ln(

αn

α1

)]. (94)

References

[1] A. Paulraj and B. Papadias. Space-time processing for wireless communications. IEEE Signal Processing

Magazine, November 1997.

[2] L. Godara. Applications of antenna arrays to mobile communications, part 1: Performance improvement,

feasibility, and system considerations. Proceedings of the IEEE, July 1997.

[3] J. Winters, J. Salz, and R. Gitlin. The impact of antenna diversity on the capacity of wireless communication

systems. IEEE Transactions on Communications, February 1994.

[4] G. Klang and B. Ottersten. A structured approach to channel estimation and interference rejection in multi-

channel systems. In Proceedings of the RVK, June 1999.

[5] R. Raich, J. Goldberg, and H. Messer. Localization of a distributed source which is partially coherent

- modeling and cramer rao bounds. International Conference on Acoustics, Speech and Signal Processing

(ICASSP), Phoenix, March 1999.

[6] V. Buchoux, O. Cappe’, E. Moulines, and A. Gorokhov. On the performance of semiblind subspace-based

channel estimation. IEEE Transactions on Signal Processing, June 2000.

[7] A. J. Weiss and B. Friedlander. Fading effects on antenna arrays in cellular communications. IEEE Transac-

tions on Signal Processing, May 1997.

[8] C. van Rensburg and B. Friedlander. Performance of antenna arrays in an urban multi-path environment.

Conference Record of GLOBECOM2000, San Francisco, CA, USA, November 2000.

[9] D. R. Brillinger. Time Series Data Analysis and Theory. San Francisco, CA: Holden-Day, 1981.

[10] J. G. Proakis. Digital Communications. McGraw-Hill, 1983.


21

−60 −40 −20 0 20 40 60−0.2

0

0.2

0.4

0.6

0.8

1

Position of Interferer (degrees)

Nor

mal

ized

SIN

RO

= vH

Tv/

vHv

Normalized SINRO as the position of the interferer is moved.

Fig. 1. The normalized SINRO as the position of the interferer is moved from -50 to +50 degrees off

broadside for an array with a 10 degree beamwidth. The signal of interest is at 0 degrees. The

narrowest curve represents a 2 degree spread of the signal and interferer, each wider curve represents

an extra 4 degree spread until the widest curve represents a 40 degree spread.

−50 −45 −40 −35 −30 −25 −20 −15 −10 −58

9

10

11

12

13

14

15

16

17

18

Position of Interferer (degrees)

SIN

RO

(dB

)

The SINROs for a mobile at broadside with an interferer at different positions

Perfect A: AnalysisPerfect A: MCKnown R: AnalysisKnown R: MCUnknown R: AnalysisUnknown R: MC

Fig. 2. SINRO Curves for 3 estimators using a 6 element ULA with a 20 degree beamwidth. The signal

of interest is at broadside and the interferer position is varied from -50 degrees to -5 degrees, both

signals are spread over 4 degrees. In the plots the lines represent the analytic curves generated from

Table II. The symbols represent the results of Monte Carlo experiments. The input SNR is 10dB. In

this plot the Analytic curves for the Perfect A and the Known R estimates are indistinguishable


22

0 1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

Input SNR

BE

R

BER curves when the signals are spread by 3 degrees

Perfect AUnknown RKnown REstimated RPe Perfect APe Known R

Fig. 3. BER Curves for the 4 estimators. There is one interferer, both signals are spread over 4 degrees,

and the interferer is 1 beamwidth away, so that there is no overlap between the 2 signals. In the plots

the lines represent the analytic curves generated from (72) (except for the curve for the Unknown R

case). The symbols represent the results of Monte Carlo experiments.

0 1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

Input SNR

BE

R

BER curves when the signals are spread by 31 degrees

Perfect AUnknown RKnown REstimated RPe Perfect APe Known R

Fig. 4. BER Curves for the 4 estimators. There is one interferer, both signals are spread over 30 degrees,

and the interferer is 1 beamwidth away, so that there is an overlap of 10 degrees between the 2 signals.

In the plots the lines represent the analytic curves generated from (72) (except for the curve for the

Unknown R case). The symbols represent the results of Monte Carlo experiments


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The Performance of a Null-Steering Beamformer in Correlated Rayleigh Fading

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