40120130405003 2

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –

6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME

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ANALYSIS OF HIGH-SNR PERFORMANCE OF MIMO MULTI-CHANNEL

BEAM FORMING IN DOUBLE-SCATTERING CHANNELS

YARRANUKALA HAREESHA M.Tech (Wireless and Mobile Communication) VAAGDEVI College of Engineering

ABSTRACT

This Work investigates the symbol error rate (SER) performance of the multiple-input

multiple-output (MIMO) multi-channel beam forming (MB) in the general double scattering channel.

We derive an asymptotic expansion on the marginal Eigen value distribution of the MIMO channel

matrix, and apply the result to get an approximate expression on the average SER at high signal-to-

noise ratio (SNR). Two parameters pertaining to the SER, i.e., the diversity gain and the array gain,

are analyzed. Our results show that it suffices for the double scattering channel to have only limited

scatterers, if the same diversity gain as the Rayleigh channel is desired; however, once the number of

scatterers is below a certain level, the array gain in the double-scattering channel will vary with the

SNR logarithmically.

Index Terms: Double-scattering, Eigen value distribution, diversity gain, Beam forming, Channel

Capacity, Diversity, MIMO, Multicast, Water filling power allocation, SNR

INTRODUCTION

Over the last decade the demand for service provision by wireless communications has risen

beyond all expectations. As a result, new improved systems emerged in order to cope with this

situation. Global system mobile, (GSM) evolved to general packet radio service (GPRS) and

enhanced data rates for GSM evolution (EDGE) and “narrowband” CDMA to wideband code

division multiple accesses (CDMA). Each new system now faces different challenges: (1) GPRS

consumes GSM user capacity as slots are used to support higher bit rates. (2) EDGE faces a similar

challenge with GPRS in addition to this it requires higher SINR to support higher coding schemes

i.e., it also has range problems. (3) WCDMA performance depends on interference and hence

coverage and capacity are interrelated.

Multiple Input Multiple Output (MIMO) multi channel beam forming (MB), also known as

MIMO singular value decomposition and MIMO spatial multiplexing is a linear transmission

scheme that applies perfect channel state information (CSI) at the transmitter and receiver to steer

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multiple data streams along the strongest eigen -directions of the MIMO channel 1. Early studies

showed that MIMO MB could achieve the MIMO channel capacity if Gaussian codes, along with

water-filling power allocation, were employed. It was also shown that, even for non- Gaussian codes,

MIMO MB still corresponds to the optimal choice of linear transmit-receive processing under

various practical criteria, such as symbol error rate (SER) and mean square error . Due to its

theoretical importance, MIMO MB has been well investigated in various Rayleigh and Rician fading

channel scenarios, evaluating the performance in terms of average SER, outage probability, and

diversity-and-multiplexing tradeoff. These prior studies, however, all made the key assumption that

the scattering environment was sufficient enough to render full-rank MIMO channel matrices. It has

been shown recently via experimental studies that, for various practical environments (such as indoor

keyhole propagation, outdoor large-distance propagation and rooftop-diffracting propagation, the

channel may in fact exhibit reduced-rank behavior due to a lack of scattering around the transmitter

and the receiver. A more general channel model that embraces this aspect of the MIMO channel had

been proposed. This model, referred to as the double-scattering model, is characterized as the matrix

product of two statistically independent complex Gaussian matrices.

Despite its generality and practical significance, there are very few analytical results on

pertaining to the double scattering model. These few results mainly focus on single stream beam

forming, space-time block codes, ergodic channel capacity and diversity-multiplexing tradeoff.

None of them studied the performance of MIMO MB. In this context, we presented in some

analytical results on the average SER of the MIMO MB system. These results, though applicable to

the whole range of SNR, are extremely complex, and thus provide very few insights. To gain more

insights into the system and the channel, in this paper we focus on the SER performance in the high-

SNR regime. Our purpose is to get an approximate expression for the average SER, which becomes

accurate at high SNR. The main difficulty in doing this is to derive the asymptotic expansion on the

eigen value distribution of the channel matrix. To solve the problem, we herein propose a new

technique, called the Expand-Remove-Omit method, which can be applied to both differentiable and

non-differentiable functions. By applying the new technique, we get the desired asymptotic

expansion, as well as the approximate SER expression. The average SER at high-SNR turns out to be

completely characterized by two parameters, the diversity gain and the array gain, where the

diversity gain determines the slope of the SER curve (on a log-to-log scale), while the array gain

determines the SNR gap between the SER curve and the benchmark curve. We prove that the

diversity gain of MIMO MB in the double-scattering channel is upper bounded by the diversity gain

in the corresponding Rayleigh channel. If the number of scatterers in the double-scattering channel is

above a certain level, the same diversity gain as the Rayleigh channel can be achieved. We also show

that the double-scattering channel is distinctly different from the Rayleigh and Rician channels in

terms of array gain. Although the array gain of MIMO MB in Rayleigh and Rician channels is well

known to be a constant independent of the SNR, the array gain in the double-scattering channel will

vary with the SNR logarithmically, if the number of scatterers is below a certain level.

SYSTEM MODEL AND PROBLEM FORMULATION

System Model of MIMO MB Consider a MIMO channel with Nt transmit and Nr receive antennas. The received vector r is

given by

r = Hs + n (1)

Where H � ℂNr × Nt is the channel matrix, s � ℂNt × 1 is the transmitted signal vector, and

n � ℂNr × 1 is the complex additive white Gaussian noise (AWGN) vector with zero mean and

identity covariance matrix. In MIMO MB, under the assumption of perfect CSI at the transmitter, the



22

transmit vector s is formed by mapping L (≤ rank (H)) modulated symbols d (≜ (d1. . . dL)T ) onto

Nt transmit antennas via a linear preceding

s = Pd (2)

With P � ℂNt X L denoting the spatial preceding matrix. Here, the columns of P are the right

singular vectors of H, which correspond to the L largest singular values. Under the assumption of

perfect CSI at the receiver, the combiner of MIMO MB forms the decision statistics � (≜ ( ˆ �1, . . . ,

ˆ ��)� ) by weighting the received vector � with a spatial equalizing matrix � ∈ ℂ�×�

� = ��, (3)

where the columns of � are the left singular vectors of , which correspond to the � largest

singular values. After such preceding and equalization, the MIMO channel is decomposed into a set

of equivalent single-input single-output (SISO) channels, whose input-output relation is

�� =√�� + �� , (� = 1, . . ., �), (4)

where �� is the �-th largest eigenvalue of � , and �� is the complex AWGN with zero

mean and unit variance (i.e., 0.5 variance per complex dimension). In this paper, we term each SISO

channel a sub-stream of the MIMO MB system. Letting �� denote the power allocated to the � th

sub-stream, the instantaneous output SNR of this sub-stream is given by

�� = �� , (� = 1, . . ., �) (5)

Clearly, the output SNRs and the average SERs of the sub streams depend directly on the

distributions of the eigenvalues ��s. It is worth noting that the power allocating strategy considered

here is the so-called fixed power allocation [1], [3], i.e., �� = �� subject to Σ �� = 1, where �

is the total transmit power, and �� is a constant satisfying 0 < �� ≤ 1. The reason for adopting this

simple strategy is: in the high- SNR regime, the optimal water-filling strategy tends to the uniform

power allocation, i.e., a special case of the fixed allocation strategy (�� = 1/�) [3, App. IV]. As the

main focus of this paper is on the system performance at high SNR, the fixed power allocation serves

that purpose very well. It is also worth noting that the results in this paper can be extended to account

for problems with non-fixed power strategies (water filling, minimum error rate, etc.) by using

methods similar to [3], [20]. However, a thorough analysis along this direction is beyond the scope

of this paper.

In the (uncorrelated) double-scattering model, the channel matrix is given by [6], [13]

1

= ___ 1 2, (6)

√�

where 1 ∈ ℂ�×� and 2 ∈ ℂ�×� are mutually independent complex Gaussian

matrices, whose elements are independent and identically distributed (i.i.d.) with zero mean and unit

variance (0.5 variance per complex dimension). By controlling the number of scatterers (i.e., �), the

double scattering model embraces a broad family of fading channels. For instance, when � = 1, it

models the keyhole channel [21]; when � →∞, it models the standard Rayleigh fading (due to the

law of large numbers). For brevity, we hereafter use the three-tuple, (�,�,�), to denote the

double-scattering channel above.



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Problem Definition and Formulation This paper investigates the average SER of the sub-streams of the MIMO MB system above.

In particular, we study two important parameters pertaining to the average SER at high SNR, i.e., the

diversity gain and the array gain. To give definitions for the two gains, we reproduce below the

analysis framework proposed by Wang and Giannakis [22].

The instantaneous SER of general modulation formats (BPSK, BFSK, �-PAM, etc.) in the

AWGN channel can be expressed as a function of the instantaneous received SNR � [23]

SER(�) = ��(√2��), (7)

where �(⋅) is the Gaussian �-function, � and � are modulation-specific constants, e.g., � = 1 and

� = 1 for BPSK2. When channel fading is taken into account, the concept of average SER becomes

more useful as it reflects the influence of the fading. The average SER is obtained by averaging the

instantaneous SER, SER(�), over all random realizations of �. Assuming that the instantaneous SNR

� is given by the product of a channel-dependent parameter � and a deterministic positive quantity �

[22], i.e.,

� = ��. (8)

The average SER, denoted by SER (�), is then given by

(9)

where ��(⋅) is the cumulative distribution function (CDF) of the random variable �.

Generally speaking, obtaining closed form expression for the average SER is difficult as the integral

in (9) may yield no analytical result [1]. Although in a few cases closed-form results exist, the exact

expressions there provide very limited insights as they are prohibitively complex, e.g., see [24]. To

avoid such intractability and to gain more insights into the system, the approximate average SER,

which becomes accurate at high SNR, is studied instead. This is where Wang and Giannakis’s

analysis framework [22] came in. In their work, they assumed that the CDF of � around zero could

be approximated by a single-term polynomial, i.e.,

��(�) = �� + !(� � ) (10)

where � and � are two positive constants, !(� � ) is the higher-order infinitesimal of � �

as � approaches zero. By substituting ��(�) back into (9), they finally arrived at a conclusion that the

average SER at high SNR was characterized by two parameters, the diversity gain and the array

gain3, i.e.,

(11)

with � being the diversity gain, and � (a function of �) being the array gain. In this paper,

we apply Wang and Giannakis’s framework to analyze the average SERs of the MIMO MB sub

streams at high SNR. Since the key step in the framework is the asymptotic expansion of ��(�), our

focus in next section is on the asymptotic expansion of the marginal CDF of the eigenvalue �� (� =

1, . . .,�).



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ASYMPTOTIC EXPANSION OF THE EIGEN VALUE DISTRIBUTION

First of all, we present the exact expression on the eigenvalue distribution of � . Based on

the exact distribution, we then derive its asymptotic expansion. For notational convenience, we

define through the rest of this paper4: " ≜ min(�,�), � ≜ max(�,�), � ≜ min(",�), ≜

max(",�), # ≜ min(, �), � ≜ max(, �), $ ≜ min(�,�).

Lemma 1 (Exact Distribution [18]). The marginal CDF of �� is (� = 1, . . .,�)

Where =n!/m!/(n-m)!, the summation

Is over all combinations of (%1 < ⋅ ⋅ ⋅ < %�−&−1) and (%�−& < ⋅ ⋅ ⋅ < %�), with

' = (%1. . . %�) being a permutation of the integers (1. . . �), and

Where h (z, a, b, c) is given by

(13)

With ()(⋅) being the modified Bessel function of the second kind [25, Eq.(8.432.6)].

To see the complexity of the exact SER result, we substitute (12) back into (9), and get an

expression consisting of special functions, determinants, and integrals. Knowing this, we turn our

attention to the approximate SER. Our first step is to derive the asymptotic expansion on the

eigenvalue distribution, but, unfortunately, we find that the conventional deriving technique is not

applicable here. This conventional technique, termed the differential-based method5[1], [19], [20],

requires the function (to be expanded) to be differentiable around zero. However, the CDF here is not



25

even continuous around zero (as the modified Bessel function ()(⋅) is discontinuous at the origin for

) ∈ ℤ). Initial attempts to solve this problem can be found in [6], but only rank-1 double-scattering

channels were considered there. A deriving technique that applies for double scattering channels of

arbitrary configurations (�,�,�) is still missing. In this context, we propose here the Expand-

Remove-Omit method, which does not require the differentiability of the CDF, and, more

importantly, is applicable to arbitrary double-scattering channels. As detailed description of the

method is somewhat lengthy, we leave it to Appendix A, but present directly its expanding result.

Theorem 1. The marginal CDF �� (+) can be expanded as (k = 1…M)

(14) Where

And

!(+�� ) is the higher-order infinitesimal of +�� as + approaches zero, ,�,- is a constant

coefficient, and .� is a set of nonnegative integer numbers. Both .� and ,�,- are uniquely

determined by (21) in Appendix A.

Proof: See Appendix A.

From the proof of Theorem 1, we see that the variant + in /�(+) was introduced by (23) in

Appendix A. We also notice that if ⌈1/2⌉ − 1 < 0 (1 was defined in Appendix A), i.e., � + � + � +

1 − � ≤ 2max(�,�,�), the matrix elements corresponding to (23) will vanish, which means the

determinants will be independent of +. In that case, we have /�(+) = ,�,0 with ,�,0 being a certain

constant, and thus the CDF �� (+) is approximated by a single-term polynomial. In the corollary

below, we present the exact result for such a constant ,�,0.

Corollary.1 If and only if � + � + � + 1 − � ≤ 2max(�,�,�), the marginal CDF �� (+) is

approximated by a single-term polynomial (� = 1, . . .,�)

(15)

Where



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Proof: See Appendix B.

SYSTEM PERFORMANCE IN THE HIGH-SNR REGIME

In this section, we apply the asymptotic expansion to analyze the performance of MIMO MB

in the high-SNR regime. We express the average SER of the �-th strongest sub-stream as (� = 1,...,�)

(16)

Substituting the expansion (14) into the equation above, we get the following theorem on the

approximate average SER.

Theorem 2. At high SNR, the average SER of the �-th strongest sub-stream of the MIMO MB

system can be approximated as (� = 1. . . �)

(17) Where

�� is the fixed power allocation coefficient; �� and �� are the modulation-specific

parameters 6. In particular, if (and only if) � + � + � + 1 − � ≤ 2max (�,�,�), �(�) is a

constant independent of the SNR �, given by

With ,�,0 being defined in Corollary 1.

Proof: The desired result is easily obtained by substituting (20) into (16), invoking the binomial

theorem, and omitting the higher-order infinitesimal.



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Comparing Theorem 2 to Wang’s results in Section II-B, we find that, for those cases where

� +� + � +1−� > 2max(�,�,�), the term �(�) does not meet the definition of the array

gain. The array gain in [22] was defined as a constant independent of the SNR, but the term �(�)

here may vary with the SNR. However, despite this difference, �(�) agrees perfectly with the

conventional definition of the diversity gain [26], i.e., Diversity Gain ≜ − lim SNR→∞ log

SER(SNR) log SNR . Noticing that �(�) is exponentially 7 equal to a constant, we now extend

Wang’s definitions to cover the general double scattering channels. In the rest of this paper, we call

�(�) the diversity gain, and �(�) the array gain. Discussions on the two gains are given as follows.

Diversity Gain

According to Theorem 2, the diversity gain of the �-th substream is (� = 1. . . �)

(18)

Where the subtrahend, i.e., the ⌊⋅⌋ term, vanishes if and only if � + � + � − � ≤

2max(�,�,�).

First of all, we use a (2, 2, 2) double-scattering channel to verify the analytical expression on

the diversity gain. For simplicity, we assume that all MIMO MB sub-streams are active, upon which

uniform power allocation and coherent BPSK are employed. The average SERs of all the sub-

streams are plotted in Fig. 1, where each “Monte Carlo Result” curve is generated based on 108

channel realizations, and each “Analytical SER” curve is computed by substituting (12) into (16).

Clearly, we can see that two diversity gains, 3 and 1, are attained by the two sub-streams,

respectively, which is in perfect agreement with our theoretical result (18).

From (18), we also see that the diversity gain of a (�,�,�) double-scattering channel is

smaller than or equal to (� +1−�)(�+1−�), which is the diversity gain of the corresponding

Rayleigh channel (�,∞,�). Since poor scattering has long been known to be damaging, the result

above is easy to understand. However, the question that follows is not as intuitive and deserves more

discussions. Whether or not a double-scattering channel can attain the same diversity gain as the

Rayleigh channel, if the number of scatterers is limited? To give an answer, we need to revisit (18).



28

It says that, as far as � ≥ � + � − 1, the upper-bound diversity gain is attained, which indicates,

for finite � and �, it suffices for the double scattering channel to have only � + � − 1 scatterers,

if the full diversity gain � × � is required. The explanation for this result lies in the basic idea of

diversity, i.e., diversity is achieved by collecting multiple independently faded replicas of the same

information symbol [26]. General speaking, the diversity gain is proportional to the number of

independent fading coefficients in the MIMO channel matrix. For example,

In a (�, ∞,�) Rayleigh channel, there exists � × � independent fading coefficients.

Thus, the maximum diversity gain of the channel equals �×� [3, Theo. 2]. The situation in the

double-scattering channel is quite similar, except that the maximum number of independent fading

coefficients may be smaller than � ×�. This is because, during the double scattering

Process, the faded replicas are added up at the scatterers, which may brake the independence

between the received replicas. An example on this point is the keyhole channel (�, 1,�), where

only min(�,�) independent fading paths exist. Clearly, the double-scattering process has imposed

some kind of correlation to the received replicas. When the scattering condition is poor (i.e., � <

�+�−1), the correlation imposed is so severe that only a (small) portion

of the independent replicas can be extracted. By contrast, when the scattering condition is good (i.e.,

� ≥ � + � − 1), the correlation is mild and can be removed. In that case, the maximum diversity

gain � × � is achieved.

To see the impact of the scatterer number on the diversity gain, we fix � and � both at 2,

and increase � from 2, to 3, 4, and ∞. The average SER of the strongest sub-stream is plotted in

Fig. 2. In the (2, 2, 2) case, we observe a diversity gain of 3, which is exactly the same as we

expected from (18). In the remaining cases, we notice that, once the number of scatterer is greater

than 3, adding more scatterers into the channel will not change the diversity gain. This is in line with

our earlier analysis that the diversity gain reaches its upper bound whenever � ≥ � + � − 1.

Another observation from (18) is that the diversity gain is independent of the order of

(�,�,�). In other words, letting �, �, and / be three natural numbers, the diversity gains of these

double-scattering channels, (�, �, /), (�, /, �),(/, �, �), (�, /, �), (/, �, �), and (�, �, /), are indeed

equivalent. This is an extension to the results of [3, Theo. 2] and [1, Theo. 4], where they showed

that interchanging � with � would not change the diversity gain of the Rayleigh/Rician channel.

Besides the rotational symmetry, we also observe that the diversity gain of the �-th sub-stream in a



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(�,�,�) channel is indeed equivalent to that of the first sub stream in a (� + 1 − �,� + 1 −

�,� + 1 − �) channel. It indicates

That reducing the sub-stream index by one is equivalent, in the sense of diversity gain, to

reducing the numbers of transmits antennas; receive antennas and scatterers all by one.

Array Gain

According to Theorem 2, the array gain of the �-th MIMO

MB sub-stream is (� = 1. . . �)

(19)

Where the expression simplifies to = if and only if if and only

Generally speaking, the array gain in the double-scattering channel is a function of the

average SNR �. If � + � + � + 1 − � ≤ 2max (�,�,�), this function will be independent of �;

otherwise, it varies logarithmically with �. The phenomenon of the SNR-varying array gain was first

reported by [27] when studying SISO double-scattering channels. By contrast, our result here

provides a whole picture of the array gain. To verify our more general result, we present in Fig. 3 the

average SER of the MIMO MB system in a (2, 2, 4) double-scattering channel. (The “approximate

SER” is computed based on Corollary 1, and the “Benchmark” curve is computed with � − �(�) � .)

Obviously, we see that the approximate SER results agree with actual curves very well, especially in

the high-SNR regime.

Given the array gain in (19), we revisit the double scattering channel (�,�,�) and its

Rayleigh counterpart (�,∞,�). We notice from (19) that if and only if � ≥ � + �, the diversity

gains of all the MIMO MB sub streams are independent of the SNR. We also know that each sub-

stream attains its upper-bound diversity gain whenever the number of scatterers � is above a certain



30

level (i.e., � + � − 1). By putting these together, we can draw a conclusion that, given the array

gain being independent of the SNR, increasing the number of scatters will not change

The diversity gain of the sub-stream. Although the diversity gain remains unchanged, the

increase in the scatterer number certainly brings advantages to the array gain, causing a horizontal

(Left ward) shift of the SER curve. This idea is confirmed by Fig. 4, where three double-scattering

channels (2, 5, 3), (2, 8, 3), and (2,∞, 3) are considered. (The asymptotic SER curve of the Rayleigh

faded case is computed based on [3, Eq. (34)].) In the figure, the array gain becomes larger and

larger as the scatterer number increases from 5 to 8 and infinity. Although formal proof of the

monotonicity of the array gain in the scatterer number is beyond the scope of this paper, the

interesting problem is of great importance as it may provide more insights into the double-scattering

process.

CONCLUSION

In this paper, we studied the average SER performance of MIMO MB, assuming the general

double-scattering channel. We focused on two performance parameters, i.e., the diversity gain and

the array gain, which characterized the SER of the system in the high-SNR regime. To get analytical

results on the two gains, we derived asymptotic expansions on the eigenvalue distribution of the

MIMO channel matrix, using a new method proposed. The asymptotic expansion was then applied to

get the approximate expression for the average SER. Our results showed that the diversity gain of the

double scattering channel was upper bounded by the diversity gain of the corresponding Rayleigh

channel. If and only if the condition � ≥ � + � − 1 was satisfied, the upper bound diversity gain

could be achieved. We also proved that, unlike conventional Rayleigh and Rician channels, where

the array gain was a constant number, the array gain of the double scattering channel was indeed a

function of the SNR. Only when � ≥ � + �, the array gains became independent of the SNR.

APPENDIX A

In this appendix, we derive the asymptotic expansion on the marginal eigenvalue distribution

�� (+). To that end, we present first an interim expansion result, and then rewrite itinto the desired

form (14). After that, we provide detailed proof of the interim expansion.



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The interim result is given as Lemma 2, which uses the same notations as Section III.

Lemma 2. The marginal CDF �� (+) in (12) can be expanded as (� = 1, . . .,�)

(20)

(21)

5 = (�1. . . ��+1−�) is a permutation of (1, . . .,� +1− �), and the summation Σ 5 is over all

possible permutations. Letting 6 ≜ min(� + 1 − �,� − #) and 1 ≜ � + 1 − � − 6, the matrix Ξ"(5, +)

is given as follows (7(⋅) denotes the digamma function [25, Eq. (8.362.1)])

(22)

(23)

(24)

(25)

(26)

Given Lemma 2, we now rewrite the interim result into the desired form (14). First of all, we

notice that /�(+) is the sum of multiple determinants, where some matrix elements are linear

combinations of ln +. Since the determinant is a linear combination of the product of its elements, the

term det[Ξ"(5, +)] can be rewritten as a linear combination of (ln +)� for � = 0, 1, 2, . . .. In this

context, the function /�(+) can be re-expressed as Σ -∈.� ,�,-(ln +)- (with ,�,- being a certain

constant coefficient), which yields our desired result (14).

In the remaining part of this appendix, we present the proof of Lemma 2. Our purpose here is

to get the two terms, /�(+) and ��, such that the following equality holds8



32

(27)

We notice that det[Υ"(+, &, ')] is a higher-order infinitesimal of det[Υ"(+, & − 1, ')] for

arbitrary & = 1, . . . , � − 1. The original problem (27) can be simplified to

(28)

Since det [Υ"(+, 0, ')] is non-differentiable (the modified Bessel function in the matrix Υ"(+,

0, ') is discontinuous at zero), the conventional differential-based method [1], [19], [20] is not

applicable here. To solve the problem, we develop here the Expand-Remove-Omit method, which

factors out the desired exponential term +�� via an Expand-Remove-Omit process (rather than

differentiation), detailed as below.

1) For a given vector ', let (�1. . . ��+1−�) be a permutation of (%�. . . %�), △ 0 = Υ"(+, 0,

'), and - = 1;

2) Expand the �--th column of the matrix △-−1 using the multi-linear property of the

determinant [28] (see below), and get multiple matrices with exponential terms of different

orders (let ℏ-,:(⋅) denote a generic function, and “∖” denote “except”)

3) Remove matrices with co-linear columns as their determinants are zero-valued;

4) Omit other matrices, leaving only the one with the lowest-order exponential term; denote the

remaining matrix as △-;

5) Let - = - + 1;

6) If - ≤ � + 1 − �, go back to 2); otherwise, continue;

7) If all permutations of (%�. . . %�) have been used, continue; otherwise, update (�1, . . . ,

��+1−�) with a new permutation of (%�, . . . , %�) and go back to 2);

8) Sum up determinants of the remaining matrices for all possible (�1, . . ., ��+1−�), and get

the following equality: det[△0] = Σ det[△ �+1−�] + <,

�1,...,��+1−�

Where < denotes the higher-order infinitesimal9

9) Factor out all exponential terms of det[△ 0] into , minimize over all possible ',

and finally get the desired term ��. The remaining part of det[△0] after the factorization then

equals /�(+).

It is worth noting that, specific procedures of the Expand- Remove-Omit process may differ

from one another if different configurations of (�,�,�) are considered. However, extending the

result from one configuration to another is easy by using the symmetry of the modified Bessel

function ()(⋅), i.e., ()(⋅) = (−)(⋅) for ) ∈ ℤ [25]. For this reason, we only provide here details on

the configuration of � ≥ , as the other configuration (� <) follows easily.



33

For a given vector ', we denote △0 = Υ"(+, 0, ') and apply the series representation of the

modified Bessel function ()(⋅) [25, Eq.(8.446)] to rewrite the (-, :)-th element of △0 as

(29)

Where

The desired asymptotic expansion then follows after the Expand-Remove-Omit process below.

Step 1: Processing the first 6 ≜ min(� + 1 − �, � − ) columns

In this step, 6 is greater than zero (otherwise, no column is processed). Hence, � is greater than ,

which means � > �. Knowing this, the summation of (29) can be rewritten as

, upon which we process the �1-th... and �6 -th columns.

1-i): Processing the ��1-th column·

Expand: det[△ 0] = + det[=1(+)], where >1(?, +) and =1(+)

are identical to △ 0, except that their elements in the �1-th column are given by

{>1(?, +)}-,: = @-,:(?, +), and {=1(?, +)}-,: = + ,

respectively.

· Remove: det[△0] = + det[=1(+)], because det[>1(?, +)] = 0 for ? = 0, . . . , " −� − 1

(existence of co-linear columns).

· Omit: det[△0] = det[>1(" − �,+)] + <. Letting △1(�1) ≜ >1(" −�,+), we get

det[△0] = det[△1]+ <.

1-ii): Processing the �2-th column



34

· Expand: det[△1] = + det[=2(+)], where >2(?, +) and =2(+) are

identical to △1(�1), except that their elements in the �2-th column are given by {>2(?, +)}-,: =

@-,:(?, +), and , respectively.

· Remove: det[△ 1] = det[>2(?, +)] ) + det[=2(+)], because det[>2(?, +)] = 0 for ? = 0, . ..

"−� (existence of co-linear columns).· Omit: det[△ 1] = det[>2(" − � + 1, +)] + <. Letting △ 2 ≜

>2("−�+1, +), we get det[△ 1] = det[△ 2]+<.

1-iii): Processing the ��3-th... ��6666 -th columns

We finally arrive at: det[△ 6−1] = det[△ 6]+<, where △ 6 is a matrix identical to △ 0 except that its

elements in the �1-th, . . .,�6 -th columns are given by {△ 6 } -,: = @-,:("−�+A−1), with - = 1, . . . ,

", : = �A, where A = 1, . . . , 6.

Step 2: Processing the remaining 1 ≜ � +1−�−6 columns the processing of the remaining 1 columns

follows the same Expand-Remove-Omit procedure as above. However, special attention should be

paid to the “Remove” procedure, which is quite different here. In Step 1, the number of zero-valued

determinants is increased by one every time a new column is expanded [compare the “Remove”

procedure of 1-i) to 1- ii) ]. However, in this step, as we will see, the number of zero-valued

determinants is increased by one only when two columns are expanded consecutively. Noticing this

difference, the remaining columns are processed as follows.

2-i) Processing the ��6666+1-th column

· Expand: det[△ 6] = (Σ�−�−1 ?=0 det[>6+1(?, +)] det[=6+1(+)], where

>6+1(?, +) and =6+1(+) are both identical to △ 6 , except that their elements in the �6+1- th column

are given by {>6+1(?, +)}-,: = @-,:(?, +), and {=6+1(?, +)}-,:

· Remove: det[△ 6] = det[=6+1(+)], because det[>6+1(?, +)] = 0 for ? = 0, . . . , � − � − 1

(existence of co-linear columns).

· Omit: Since Σ∞ B-,:(?, +) is a higher-order infinitesimal of

we see that det[△ 6] = det[△ 6+1] + <, where △ 6+1 is a matrix identical to △ 6 , except that its

elements in the �6+1-th column is given as follows: for - = 1, {△ 6+1 }-,: = B-,:(0, +); for - = 2, . . . ,

", {△ 6+1 }-,: = @-,:(� − �, +).

2-ii) Processing the ��6666+2-th column

· Expand: det[△ 6+1] = (Σ�−�−1 ?=0 det[>6+2(?, +)] ) + det[=6+2(+)], where

>6+2(?, +) and =6+2(+) are both identical to△ 6+1, except that their elements in the �6+2- th

column are given by {>6+2(?, +)}-,: = @-,:(?, +), and {=6+2(?, +)}-,: = Σ�−�+-−2 ?=�−�

@-,:(?, +) + ∞ ?=0 B-,:(?, +), respectively.

· Remove: det[△ 6+1] = det[=6+2(+)], because det[>6+2(?, +)] = 0 for ? = 0, . . . , � − � −

1 (existence of co-linear columns).

· Omit: Since Σ∞ ?=0 B-,:(?, +) is a higher-order infinitesimal of

we see that det[△ 6+1] = det[△ 6+2] + <, where △ 6+2 is a matrix identical to △ 6+1, except that its

elements in the �6+2-th column is given as follows: for - = 1, {△ 6+2 }-,: = B-,:(0, +); for - = 2, . . . ,

", {△ 6+2 }-,: = @-,:(� − �, +).



35

2-iii) Processing the ��6666+3-th column

· Expand: det[△ 6+2] = (Σ�−�−1 ?=0 det[>6+3(?, +)] ) + det[˜ =

6+3(+)] + det[=6+3(+)], where >6+3(?, +), ˜= 6+3(+), and =6+3(+) are three matrices identical to △

6+1, except that the �6+3-th column of >6+3(?, +) is given by {>6+3(?, +)}-,: = @-,:(?, +), and

that the �6+3-th column of ˜ = 6+3(+) is given by: for - = 1, {˜ = 6+3(+)}-,: =

B-,:(0, +); for - = 2, . . . , ", {˜ = 6+3(+)}-,: = @-,:(� − �, +), and that the �6+3-th column of

=6+3(+) is given by: for - = 1, {=6+3(+)}-,: = Σ∞ ?=1 B- ,:(?, +); for - = 2,

{=6+3(+)}-,: = Σ∞ ?=0 B-,:(?, + for - = 3, . . . , ", {=6+3(+)}-,: =

· Remove: det[△ 6+2] = det[=6+3(+)], because det[˜ = 6+3(+)] = 0 and det[>6+3(?, +)] = 0 for ? =

0, . . . , � − � − 1 (existence of co-linear columns).

· Omit: det[△ 6+2] = det[△ 6+3] + <, where △ 6+3 is a matrix identical to △ 6+2, except that

its elements in the �6+3-th column is given as follows: for - = 1, {△ 6+3 }-,: = B-,:(1, +); for - = 2,

{△ 6+3 }-,: = B-,:(0, +); for -=3, ..., ", {△ 6+3 }-,: = @-,:(�−�+1, +).

2-iv) Processing the ��6666+4-th, ..., the ��+1−��-th columns Being aware of the difference in the “Remove” procedure, we process the remaining columns

to finally arrive at: det[△ �−�] = det[△ �+1−�] + <, where △ �+1−� is a matrix identical to △ 6 ,

except that its elements in the �6+1- th, . . ., ��+1−�-th columns are given by {△ �+1−� }-,: =

B-,:(A + 1 − -, +), where - = 1, . . . , A + 1, : = �6+1+2A, and �6+1+2A+min(1,⌊1/2⌋−A), with A = 0, 1,

. . . , ⌈ 1/2 ⌉ − 1; and by {△ �+1−� }-,: = @-,:(� − � + A, +), where - = A+2, . . . , ", : = �6+1+2A,

and �6+1+2A+min(1,⌊1/2⌋−A), with A = 0, 1, . . . , ⌈ 1/2 ⌉ − 1.

Step 3: Factorization and Minimization

So far, we have got the remaining matrix △ �+1−� for a given permutation (�1, . . .,

��+1−�) of the vector '. Our next step is to sum up determinants of the remaining matrices for all

possible (�1, . . . , ��+1−�), i.e., det[Σ 0] = �1,...,��+1−� det[△ �+1−�] + <. To further simplify

the expression above, we factor out all exponential terms in det[△ �+1−�], and obtain a new

equality: det[△ �+1−�] = det[˜Ξ(5, +)]+ ��, where

(30)

And ˜Ξ(5, +) is identical to △ �+1−� except that all exponential terms are removed. Noticing that

the term ˜ �� is independent of the order of �1, . . . , ��+1−�, the determinant det[△ 0] can be

rewritten as: det[△ 0] = (Σ �1,...,��+1−� det[˜Ξ(5, +)] ) + ˜ �� + <. We substitute det[△ 0] back

into the left hand side of (28), and then get the following equality10

(31)



36

To get the desired term ��, we still need to minimize ˜ �� overall possible ', i.e.,

(32)

(33)

(34)

Notice that Eq. (33) holds if and only if (�1, . . ., ��+1−�) is a permutation of (1, . . .,� + 1

− �). As a result, the optimal vector ' ∗ is: (%∗ 1 , . . . , %∗ �−1, %∗ �, . . . , %∗ �) = (�+2− �, . . . ,

�, 1, . . .,� +1− �). Knowing this, we can omit all terms except the optimal one in the summation

Of (31) and simplify the equality to

(35)

Where 5 is a permutation of (1, . . .,� + 1 − �), and the summation is over all possible

permutations. This simplified expression is indeed the result of Lemma 2.

REFERENCES

[1] S. Jin, M. R. McKay, X. Gao, and I. B. Collings, “MIMO multichannel beam forming: SER

and outage using new eigenvalue distributions of complex noncentral Wishart matrices," IEEE

Trans. Commun., vol. 56, no. 3, pp. 424-434, Mar. 2008.

[2] A. Zanella, M. Chiani, and M. Z. Win, “On the marginal distribution of the eigenvalues of

Wishart matrices," IEEE Trans. Commun., vol. 57, no. 4, pp. 1050-1060, Apr. 2009.

[3] L. G. Ordóñez, D. P. Palomar, A. Pages-Zamora, and J. R. Fonollosa, “High-SNR analytical

performance of spatial multiplexing MIMO systems with CSI," IEEE Trans. Signal Process.,

vol. 55, no. 11, pp. 5447- 5463, Nov. 2007.

[4] L. G. Ordóñez, D. P. Palomar, and J. R. Fonollosa, “Ordered eigenvalues of a general class of

Hermitian random matrices with application to the performance analysis of MIMO systems,"

IEEE Trans. Signal Process., vol. 57, no. 2, pp. 672-689, Feb. 2009.

[5] A. Maaref and S. Aïssa, “Closed-form expressions for the outage and ergodic Shannon capacity

of MIMO MRC systems," IEEE Trans. Commun., vol. 53, no. 7, pp. 1092-1095, July 2005.

[6] S. Jin, M. R. McKay, K.-K. Wong, and X. Gao, “Transmit beamforming in Rayleigh product

MIMO channels: capacity and performance analysis," IEEE Trans. Signal Process., vol. 56, no.

10, pp. 5204-5221, Oct. 2008.

[7] ˙I. E. Telatar, “Capacity of multi-antenna Gaussian channels," European Trans. Telecommun.,

vol. 10, no. 6, pp. 585-596, 1999.

[8] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint TX-RX beamforming design for

multicarrier MIMO channels: a unified framework for convex optimization," IEEE Trans.

Signal Process., vol. 51, no. 9, pp. 2381-2401, Sep. 2003.

[9] L. G. Ordóñez, A. P. Zamora, and J. R. Fonollosa, “Diversity and multiplexing tradeoff of

spatial multiplexing MIMO systems with CSI," IEEE Trans. Inf. Theory, vol. 54, no. 7, pp.

2959-2975, July 2008.



37

[10] P. Almers, F. Tufvesson, and A. F. Molisch, “Keyhole effect in MIMO wireless channels:

measurements and theory," IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3596-3604,

Dec. 2006.

[11] J. B. Andersen and I. Z. Kovács, “Power distributions revisited," in Proc. COST273 3rd

Manage. Committee Meeting, 2002, TD (02)004.

[12] V. Erceg, S. J. Fortune, J. Ling, A. Rustako, and R. A. Valenzuela, “Comparisons of a

computer-based propagation prediction tool with experimental data collected in urban

microcellular environments," IEEE J. Sel. Areas Commun., vol. 15, no. 4, pp. 677-684, May

1997.

[13] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels:

models and performance prediction," IEEE Trans. Commun., vol. 50, no. 12, pp. 1926-1934,

Dec. 2002.

[14] H. Shin and M. Z. Win, “MIMO diversity in the presence of doublescattering," IEEE Trans.

Inf. Theory, vol. 54, no. 7, pp. 2976-2996, July 2008.

[15] H. Shin and J. H. Lee, “Capacity of multiple-antenna fading channels: spatial fading

correlation, double-scattering, and keyhole," IEEE Trans. Inf. Theory, vol. 49, no. 10, pp.

2636-2647, Oct. 2003.

[16] A. Maaref and S. Aïssa, “Impact of spatial fading correlation and keyhole on the capacity of

MIMO systems with transmitter and receiver CSI," IEEE Trans. Wireless Commun., vol. 7, no.

8, pp. 3218-3229, Aug. 2008.

[17] S. Yang and J. C. Belfiore, “Diversity-multiplexing tradeoff of doublescattering MIMO

channels," submitted to IEEE Trans. Inf. Theory, Mar. 2006. Available:

http://arxiv.org/abs/cs/0603124.

[18] H. Zhang, S. Jin, X. Zhang, and D. Yang, “On marginal distributions of the ordered

eigenvalues of certain random matrices," EURASIP J. Advances Signal Process., 2010, article

ID 957243, 12 pages.

[19] S. Jin, M. R. McKay, X. Gao, and I. B. Collings, “Asymptotic SER and outage probability of

MIMO MRC in correlated fading," IEEE Signal Process. Lett., vol. 14, no. 1, pp. 9-12, Jan.

2007.

[20] L. G. Ordóñez, D. P. Palomar, A. Pagès-Zamora, and J. R. Fonollosa, “Minimum BER linear

MIMO transceivers with adaptive number of substreams," IEEE Trans. Signal Process., vol. 57,

no. 6, pp. 2336-2353, June 2009.

[21] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela, “Keyholes, correlations, and

capacities of multielement transmit and receive antennas," IEEE Trans. Wireless Commun.,

vol. 1, no. 2, pp. 361-368, Apr. 2002.

[22] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying

performance in fading channels," IEEE Trans. Commun., vol. 51, no. 8, pp. 1389-1398, Aug.

2003.

[23] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill, 2001.

[24] M. R. McKay, A. J. Grant, and I. B. Collings, “Performance analysis of MIMO-MRC in

double-correlated Rayleigh environments," IEEE Trans. Commun., vol. 55, no. 3, pp. 497-507,

Mar. 2007.

[25] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th edition.

Academic Press, 1996.

[26] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradoff in multiple-

antenna channels," IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073-1096, May 2003.

[27] J. Salo, H. M. El-Sallabi, and P. Vainikainen, “Impact of double- Rayleigh fading on system

performance," IEEE 2006 1st Int. Symp. Wireless Pervasive Comput., Jan. 2006.

[28] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. SIAM, 2000.



38

[29] A. Maaref and S. Aïssa, “Eigenvalue distributions of Wishart-type random matrices with

application to the performance analysis of MIMO MRC systems," IEEE Trans. Wireless

Commun., vol. 6, no. 7, pp. 2678- 2689, July 2007.

[30] ——, “Joint and marginal eigenvalue distributions of (non)central complex Wishart matrices

and PDF-based approach for characterizing the capacity statistics of MIMO Ricean and

Rayleigh fading channels," IEEE Trans. Wireless Commun., vol. 6, no. 10, pp. 3607-3619, Oct.

2007.

[31] David Tse and Pramod Viswanath, Fundamentals of wireless Communication, 1st edition

Cambridge University Press 2005.

[32] Sharon. P. S, M.Vanithalakshmi, Arun.S and G. Dharini, “Spectrum Management and Power

Control in MIMO Cognitive Radio Network and Reduction of Power Optimization Problem

Using Water-Filling Method”, International journal of Electronics and Communication

Engineering & Technology (IJECET), Volume 3, Issue 1, 2012, pp. 160 - 170, ISSN Print:

0976- 6464, ISSN Online: 0976 –6472.

[33] Bharti Rani and Mrs Garima Saini, “Cooperative Partial Transmit Sequence for PAPR

Reduction in Space Frequency Block Code MIMO-OFDM Signal”, International journal of

Electronics and Communication Engineering & Technology (IJECET), Volume 3, Issue 2,

2012, pp. 321 - 327, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.

[34] Er. Ravi Garg and Er. Abhijeet Kumar “Compression Of Snr And Mse For Various Noises

Using Bayesian Framework” International journal of Electronics and Communication

Engineering &Technology (IJECET), Volume 3, Issue 1, 2012, pp. 76 - 82, ISSN Print: 0976-

6464, ISSN Online: 0976 –6472, Published by IAEME.

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