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
INTERNATIONAL JOURNAL OF ELECTRONICS AND
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 5, September – October, 2013, pp. 20-38
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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

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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 ࢊ (≜ ( ˆ ݀1, . . . ,
ˆ ݀‫ܶ)ܮ‬ ) by weighting the received vector ࢘ with a spatial equalizing matrix ࡽ ∈ ℂܰ‫ܮ×ݎ‬
ࢊ = ࡽ‫,࢘ܪ‬ (3)
where the columns of ࡽ are the left singular vectors of ࡴ, which correspond to the ‫ܮ‬ 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
݀݇ =√ߣ݇ ݀݇ + ݊݇ , (݇ = 1, . . ., ‫,)ܮ‬ (4)
where ߣ݇ is the ݇-th largest eigenvalue of ࡴ‫,ࡴܪ‬ 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 ߩ݇ denote the power allocated to the ݇ th
sub-stream, the instantaneous output SNR of this sub-stream is given by
ߛ݇ = ߩ݇ߣ݇ , (݇ = 1, . . ., ‫)ܮ‬ (5)
Clearly, the output SNRs and the average SERs of the sub streams depend directly on the
distributions of the eigenvalues ߣ݇s. It is worth noting that the power allocating strategy considered
here is the so-called fixed power allocation [1], [3], i.e., ߩ݇ = ߶݇ߩ subject to Σ ߶݇ = 1, where ߩ
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/‫)ܮ‬ [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 ࡴ is given by [6], [13]
1
ࡴ = ___ ࡴ1ࡴ2, (6)
√ܰ‫ݏ‬
where ࡴ1 ∈ ℂܰ‫ݏܰ×ݎ‬ and ࡴ2 ∈ ℂܰ‫ݐܰ×ݏ‬ 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., ܰ‫,)ݏ‬ the
double scattering model embraces a broad family of fading channels. For instance, when ܰ‫ݏ‬ = 1, it
models the keyhole channel [21]; when ܰ‫ݏ‬ →∞, it models the standard Rayleigh fading (due to the
law of large numbers). For brevity, we hereafter use the three-tuple, (ܰ‫,)ݐܰ,ݏܰ,ݎ‬ 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, ‫-ܯ‬PAM, etc.) in the
AWGN channel can be expressed as a function of the instantaneous received SNR ߛ [23]
SER(ߛ) = ܽ࣫(√2ܾߛ), (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(ߛ), over all random realizations of ߛ. Assuming that the instantaneous SNR
ߛ is given by the product of a channel-dependent parameter ߦ and a deterministic positive quantity ߛ
[22], i.e.,
ߛ = ߦߛ. (8)
The average SER, denoted by SER (ߛ), is then given by
(9)
where ‫)⋅(ߦܨ‬ is the cumulative distribution function (CDF) of the random variable ߦ.
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 ߦ around zero could
be approximated by a single-term polynomial, i.e.,
‫)ݔ(ߦܨ‬ = ߙ‫݀ܩݔ‬ + ‫݀ܩݔ(݋‬ ) (10)
where ߙ and ‫݀ܩ‬ are two positive constants, ‫݀ܩݔ(݋‬ ) is the higher-order infinitesimal of ‫݀ܩݔ‬
as ‫ݔ‬ approaches zero. By substituting ‫)ݔ(ߦܨ‬ 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 ‫݀ܩ‬ being the diversity gain, and ‫ܽܩ‬ (a function of ߙ) 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 ‫,)ݔ(ߦܨ‬ our
focus in next section is on the asymptotic expansion of the marginal CDF of the eigenvalue ߣ݇ (݇ =
1, . . .,‫.)ܯ‬

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ASYMPTOTIC EXPANSION OF THE EIGEN VALUE DISTRIBUTION
First of all, we present the exact expression on the eigenvalue distribution of ࡴ‫ܪ‬ࡴ. Based on
the exact distribution, we then derive its asymptotic expansion. For notational convenience, we
define through the rest of this paper4: ܵ ≜ min(ܰ‫,)ݏܰ,ݎ‬ ܶ ≜ max(ܰ‫,)ݏܰ,ݎ‬ ‫ܯ‬ ≜ min(ܵ,ܰ‫,)ݐ‬ ܰ ≜
max(ܵ,ܰ‫,)ݐ‬ ܲ ≜ min(ܰ, ܶ), ܳ ≜ max(ܰ, ܶ), ܴ ≜ min(ܶ,ܰ‫.)ݐ‬
Lemma 1 (Exact Distribution [18]). The marginal CDF of ߣ݇ is (݇ = 1, . . .,‫)ܯ‬
Where =n!/m!/(n-m)!, the summation
Is over all combinations of (ߚ1 < ⋅ ⋅ ⋅ < ߚ݇−݈−1) and (ߚ݇−݈ < ⋅ ⋅ ⋅ < ߚ‫,)ܯ‬ with
ࢼ = (ߚ1. . . ߚ‫)ܯ‬ being a permutation of the integers (1. . . ‫,)ܯ‬ and
Where h (z, a, b, c) is given by
(13)
With ‫)⋅(ߥܭ‬ 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

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even continuous around zero (as the modified Bessel function ‫)⋅(ߥܭ‬ is discontinuous at the origin for
ߥ ∈ 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 (ܰ‫)ݐܰ,ݏܰ,ݎ‬ 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 ‫݇ߣܨ‬ (‫)ݖ‬ can be expanded as (k = 1…M)
(14)
Where
And
‫݇݀ݖ(݋‬ ) is the higher-order infinitesimal of ‫݇݀ݖ‬ as ‫ݖ‬ approaches zero, ࣝ݇,݅ is a constant
coefficient, and ॺ݇ is a set of nonnegative integer numbers. Both ॺ݇ and ࣝ݇,݅ are uniquely
determined by (21) in Appendix A.
Proof: See Appendix A.
From the proof of Theorem 1, we see that the variant ‫ݖ‬ in ܿ݇(‫)ݖ‬ was introduced by (23) in
Appendix A. We also notice that if ⌈ߝ/2⌉ − 1 < 0 (ߝ was defined in Appendix A), i.e., ܰ‫ݎ‬ + ܰ‫ݏ‬ + ܰ‫ݐ‬ +
1 − ݇ ≤ 2max(ܰ‫,)ݐܰ,ݏܰ,ݎ‬ the matrix elements corresponding to (23) will vanish, which means the
determinants will be independent of ‫.ݖ‬ In that case, we have ܿ݇(‫)ݖ‬ = ࣝ݇,0 with ࣝ݇,0 being a certain
constant, and thus the CDF ‫݇ߣܨ‬ (‫)ݖ‬ 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 ܰ‫ݎ‬ + ܰ‫ݏ‬ + ܰ‫ݐ‬ + 1 − ݇ ≤ 2max(ܰ‫,)ݐܰ,ݏܰ,ݎ‬ the marginal CDF ‫݇ߣܨ‬ (‫)ݖ‬ is
approximated by a single-term polynomial (݇ = 1, . . .,‫)ܯ‬
(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,...,‫)ܮ‬
(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. . . ‫)ܮ‬
(17)
Where
߶݇ is the fixed power allocation coefficient; ܽ݇ and ܾ݇ are the modulation-specific
parameters 6. In particular, if (and only if) ܰ‫ݎ‬ + ܰ‫ݏ‬ + ܰ‫ݐ‬ + 1 − ݇ ≤ 2max (ܰ‫,)ݐܰ,ݏܰ,ݎ‬ ‫)݇(ܽܩ‬ is a
constant independent of the SNR ߩ, 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
ܰ‫ݎ‬ +ܰ‫ݏ‬ + ܰ‫ݐ‬ +1−݇ > 2max(ܰ‫,)ݐܰ,ݏܰ,ݎ‬ the term ‫)݇(ܽܩ‬ 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 ‫)݇(ܽܩ‬
here may vary with the SNR. However, despite this difference, ‫)݇(݀ܩ‬ agrees perfectly with the
conventional definition of the diversity gain [26], i.e., Diversity Gain ≜ − lim SNR→∞ log
SER(SNR) log SNR . Noticing that ‫)݇(ܽܩ‬ 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
‫)݇(݀ܩ‬ the diversity gain, and ‫)݇(ܽܩ‬ 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. . . ‫)ܮ‬
(18)
Where the subtrahend, i.e., the ⌊⋅⌋ term, vanishes if and only if ܰ‫ݎ‬ + ܰ‫ݏ‬ + ܰ‫ݐ‬ − ݇ ≤
2max(ܰ‫.)ݐܰ,ݏܰ,ݎ‬
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 (ܰ‫)ݐܰ,ݏܰ,ݎ‬ double-scattering channel is
smaller than or equal to (ܰ‫ݎ‬ +1−݇)(ܰ‫,)݇−1+ݐ‬ which is the diversity gain of the corresponding
Rayleigh channel (ܰ‫.)ݐܰ,∞,ݎ‬ 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).

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It says that, as far as ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫ݐ‬ − 1, the upper-bound diversity gain is attained, which indicates,
for finite ܰ‫ݎ‬ and ܰ‫,ݐ‬ it suffices for the double scattering channel to have only ܰ‫ݎ‬ + ܰ‫ݐ‬ − 1 scatterers,
if the full diversity gain ܰ‫ݎ‬ × ܰ‫ݐ‬ 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 (ܰ‫,ݎ‬ ∞,ܰ‫)ݐ‬ Rayleigh channel, there exists ܰ‫ݎ‬ × ܰ‫ݐ‬ independent fading coefficients.
Thus, the maximum diversity gain of the channel equals ܰ‫ݐܰ×ݎ‬ [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 ܰ‫ݎ‬ ×ܰ‫.ݐ‬ 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 (ܰ‫,ݎ‬ 1,ܰ‫,)ݐ‬ where
only min(ܰ‫)ݐܰ,ݎ‬ 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., ܰ‫ݏ‬ <
ܰ‫,)1−ݐܰ+ݎ‬ 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.,
ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫ݐ‬ − 1), the correlation is mild and can be removed. In that case, the maximum diversity
gain ܰ‫ݎ‬ × ܰ‫ݐ‬ is achieved.
To see the impact of the scatterer number on the diversity gain, we fix ܰ‫ݎ‬ and ܰ‫ݐ‬ both at 2,
and increase ܰ‫ݏ‬ 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 ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫ݐ‬ − 1.
Another observation from (18) is that the diversity gain is independent of the order of
(ܰ‫.)ݐܰ,ݏܰ,ݎ‬ 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 ܰ‫ݎ‬ with ܰ‫ݐ‬ 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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(ܰ‫)ݐܰ,ݏܰ,ݎ‬ channel is indeed equivalent to that of the first sub stream in a (ܰ‫ݎ‬ + 1 − ݇,ܰ‫ݏ‬ + 1 −
݇,ܰ‫ݐ‬ + 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. . . ‫)ܮ‬
(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 ߩ. If ܰ‫ݎ‬ + ܰ‫ݏ‬ + ܰ‫ݐ‬ + 1 − ݇ ≤ 2max (ܰ‫,)ݐܰ,ݏܰ,ݎ‬ this function will be independent of ߩ;
otherwise, it varies logarithmically with ߩ. 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 ߩ −‫)݇(݀ܩ‬ ݇ .)
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 (ܰ‫)ݐܰ,ݏܰ,ݎ‬ and its
Rayleigh counterpart (ܰ‫.)ݐܰ,∞,ݎ‬ We notice from (19) that if and only if ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫,ݐ‬ 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 ܰ‫ݏ‬ is above a certain

30
level (i.e., ܰ‫ݎ‬ + ܰ‫ݐ‬ − 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 ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫ݐ‬ − 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 ܰ‫ݏ‬ ≥ ܰ‫ݎ‬ + ܰ‫,ݐ‬ the array gains became independent of the SNR.
APPENDIX A
In this appendix, we derive the asymptotic expansion on the marginal eigenvalue distribution
‫݇ߣܨ‬ (‫.)ݖ‬ 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.

31
The interim result is given as Lemma 2, which uses the same notations as Section III.
Lemma 2. The marginal CDF ‫݇ߣܨ‬ (‫)ݖ‬ in (12) can be expanded as (݇ = 1, . . .,‫)ܯ‬
(20)
(21)
ࢻ = (ߙ1. . . ߙ‫)݇−1+ܯ‬ is a permutation of (1, . . .,‫ܯ‬ +1− ݇), and the summation Σ ࢻ is over all
possible permutations. Letting ߬ ≜ min(‫ܯ‬ + 1 − ݇,ܳ − ܲ) and ߝ ≜ ‫ܯ‬ + 1 − ݇ − ߬, the matrix Ξܵ(ࢻ, ‫)ݖ‬
is given as follows (߰(⋅) 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 ܿ݇(‫)ݖ‬ is the sum of multiple determinants, where some matrix elements are linear
combinations of ln ‫.ݖ‬ Since the determinant is a linear combination of the product of its elements, the
term det[Ξܵ(ࢻ, ‫])ݖ‬ can be rewritten as a linear combination of (ln ‫݊)ݖ‬ for ݊ = 0, 1, 2, . . .. In this
context, the function ܿ݇(‫)ݖ‬ can be re-expressed as Σ ݅∈ॺ݇ ࣝ݇,݅(ln ‫݅)ݖ‬ (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, ܿ݇(‫)ݖ‬ and ݀݇, such that the following equality holds8

32
(27)
We notice that det[Υܵ(‫,ݖ‬ ݈, ࢼ)] is a higher-order infinitesimal of det[Υܵ(‫,ݖ‬ ݈ − 1, ࢼ)] for
arbitrary ݈ = 1, . . . , ݇ − 1. The original problem (27) can be simplified to
(28)
Since det [Υܵ(‫,ݖ‬ 0, ࢼ)] is non-differentiable (the modified Bessel function in the matrix Υܵ(‫,ݖ‬
0, ࢼ) 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 ‫݇݀ݖ‬ via an Expand-Remove-Omit process (rather than
differentiation), detailed as below.
1) For a given vector ࢼ, let (ߙ1. . . ߙ‫)݇−1+ܯ‬ be a permutation of (ߚ݇. . . ߚ‫,)ܯ‬ △ 0 = Υܵ(‫,ݖ‬ 0,
ࢼ), and ݅ = 1;
2) Expand the ߙ݅-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 ℏ݅,݆(⋅) 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 ݅ ≤ ‫ܯ‬ + 1 − ݇, go back to 2); otherwise, continue;
7) If all permutations of (ߚ݇. . . ߚ‫)ܯ‬ have been used, continue; otherwise, update (ߙ1, . . . ,
ߙ‫)݇−1+ܯ‬ with a new permutation of (ߚ݇, . . . , ߚ‫)ܯ‬ and go back to 2);
8) Sum up determinants of the remaining matrices for all possible (ߙ1, . . ., ߙ‫,)݇−1+ܯ‬ and get
the following equality: det[△0] = Σ det[△ ‫]݇−1+ܯ‬ + ࣩ,
ߙ1,...,ߙ‫݇−1+ܯ‬
Where ࣩ denotes the higher-order infinitesimal9
9) Factor out all exponential terms of det[△ 0] into , minimize over all possible ࢼ,
and finally get the desired term ݀݇. The remaining part of det[△0] after the factorization then
equals ܿ݇(‫.)ݖ‬
It is worth noting that, specific procedures of the Expand- Remove-Omit process may differ
from one another if different configurations of (ܰ‫)ݐܰ,ݏܰ,ݎ‬ are considered. However, extending the
result from one configuration to another is easy by using the symmetry of the modified Bessel
function ‫,)⋅(ߥܭ‬ i.e., ‫)⋅(ߥܭ‬ = ‫)⋅(ߥ−ܭ‬ for ߥ ∈ 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 ࢼ, we denote △0 = Υܵ(‫,ݖ‬ 0, ࢼ) and apply the series representation of the
modified Bessel function ‫)⋅(ߥܭ‬ [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 ߬ ≜ min(‫ܯ‬ + 1 − ݇, ܶ − ܰ) columns
In this step, ߬ is greater than zero (otherwise, no column is processed). Hence, ܶ is greater than ܰ,
which means ܶ > ܰ‫.ݐ‬ Knowing this, the summation of (29) can be rewritten as
, upon which we process the ߙ1-th... and ߙ߬ -th columns.
1-i): Processing the ߙߙߙߙ1-th column·
Expand: det[△ 0] = + det[ࡳ1(‫,])ݖ‬ where ࡲ1(݉, ‫)ݖ‬ and ࡳ1(‫)ݖ‬
are identical to △ 0, except that their elements in the ߙ1-th column are given by
{ࡲ1(݉, ‫݆,݅})ݖ‬ = ݂݅,݆(݉, ‫,)ݖ‬ and {ࡳ1(݉, ‫݆,݅})ݖ‬ = + ,
respectively.
· Remove: det[△0] = + det[ࡳ1(‫,])ݖ‬ because det[ࡲ1(݉, ‫])ݖ‬ = 0 for ݉ = 0, . . . , ܵ −‫ܯ‬ − 1
(existence of co-linear columns).
· Omit: det[△0] = det[ࡲ1(ܵ − ‫])ݖ,ܯ‬ + ࣩ. Letting △1(ߙ1) ≜ ࡲ1(ܵ −‫,)ݖ,ܯ‬ we get
det[△0] = det[△1]+ ࣩ.
1-ii): Processing the ߙ2-th column

34
· Expand: det[△1] = + det[ࡳ2(‫,])ݖ‬ where ࡲ2(݉, ‫)ݖ‬ and ࡳ2(‫)ݖ‬ are
identical to △1(ߙ1), except that their elements in the ߙ2-th column are given by {ࡲ2(݉, ‫݆,݅})ݖ‬ =
݂݅,݆(݉, ‫,)ݖ‬ and , respectively.
· Remove: det[△ 1] = det[ࡲ2(݉, ‫])ݖ‬ ) + det[ࡳ2(‫,])ݖ‬ because det[ࡲ2(݉, ‫])ݖ‬ = 0 for ݉ = 0, . ..
ܵ−‫ܯ‬ (existence of co-linear columns).· Omit: det[△ 1] = det[ࡲ2(ܵ − ‫ܯ‬ + 1, ‫])ݖ‬ + ࣩ. Letting △ 2 ≜
ࡲ2(ܵ−‫,1+ܯ‬ ‫,)ݖ‬ we get det[△ 1] = det[△ 2]+ࣩ.
1-iii): Processing the ߙߙߙߙ3-th... ߙߙߙߙ߬߬߬߬ -th columns
We finally arrive at: det[△ ߬−1] = det[△ ߬]+ࣩ, where △ ߬ is a matrix identical to △ 0 except that its
elements in the ߙ1-th, . . .,ߙ߬ -th columns are given by {△ ߬ } ݅,݆ = ݂݅,݆(ܵ−‫,)1−ݍ+ܯ‬ with ݅ = 1, . . . ,
ܵ, ݆ = ߙ‫,ݍ‬ where ‫ݍ‬ = 1, . . . , ߬.
Step 2: Processing the remaining ߝ ≜ ‫ܯ‬ +1−݇−߬ columns the processing of the remaining ߝ 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 ߙߙߙߙ߬߬߬߬+1-th column
· Expand: det[△ ߬] = (Σܶ−ܰ‫1−ݐ‬ ݉=0 det[ࡲ߬+1(݉, ‫])ݖ‬ det[ࡳ߬+1(‫,])ݖ‬ where
ࡲ߬+1(݉, ‫)ݖ‬ and ࡳ߬+1(‫)ݖ‬ are both identical to △ ߬ , except that their elements in the ߙ߬+1- th column
are given by {ࡲ߬+1(݉, ‫݆,݅})ݖ‬ = ݂݅,݆(݉, ‫,)ݖ‬ and {ࡳ߬+1(݉, ‫݆,݅})ݖ‬
·
Remove: det[△ ߬] = det[ࡳ߬+1(‫,])ݖ‬ because det[ࡲ߬+1(݉, ‫])ݖ‬ = 0 for ݉ = 0, . . . , ܶ − ܰ‫ݐ‬ − 1
(existence of co-linear columns).
· Omit: Since Σ∞ ݃݅,݆(݉, ‫)ݖ‬ is a higher-order infinitesimal of
we see that det[△ ߬] = det[△ ߬+1] + ࣩ, where △ ߬+1 is a matrix identical to △ ߬ , except that its
elements in the ߙ߬+1-th column is given as follows: for ݅ = 1, {△ ߬+1 }݅,݆ = ݃݅,݆(0, ‫;)ݖ‬ for ݅ = 2, . . . ,
ܵ, {△ ߬+1 }݅,݆ = ݂݅,݆(ܶ − ܰ‫,ݐ‬ ‫.)ݖ‬
2-ii) Processing the ߙߙߙߙ߬߬߬߬+2-th column
· Expand: det[△ ߬+1] = (Σܶ−ܰ‫1−ݐ‬ ݉=0 det[ࡲ߬+2(݉, ‫])ݖ‬ ) + det[ࡳ߬+2(‫,])ݖ‬ where
ࡲ߬+2(݉, ‫)ݖ‬ and ࡳ߬+2(‫)ݖ‬ are both identical to△ ߬+1, except that their elements in the ߙ߬+2- th
column are given by {ࡲ߬+2(݉, ‫݆,݅})ݖ‬ = ݂݅,݆(݉, ‫,)ݖ‬ and {ࡳ߬+2(݉, ‫݆,݅})ݖ‬ = Σܶ−ܰ‫2−݅+ݐ‬ ݉=ܶ−ܰ‫ݐ‬
݂݅,݆(݉, ‫)ݖ‬ + ∞ ݉=0 ݃݅,݆(݉, ‫,)ݖ‬ respectively.
· Remove: det[△ ߬+1] = det[ࡳ߬+2(‫,])ݖ‬ because det[ࡲ߬+2(݉, ‫])ݖ‬ = 0 for ݉ = 0, . . . , ܶ − ܰ‫ݐ‬ −
1 (existence of co-linear columns).
· Omit: Since Σ∞ ݉=0 ݃݅,݆(݉, ‫)ݖ‬ is a higher-order infinitesimal of
we see that det[△ ߬+1] = det[△ ߬+2] + ࣩ, where △ ߬+2 is a matrix identical to △ ߬+1, except that its
elements in the ߙ߬+2-th column is given as follows: for ݅ = 1, {△ ߬+2 }݅,݆ = ݃݅,݆(0, ‫;)ݖ‬ for ݅ = 2, . . . ,
ܵ, {△ ߬+2 }݅,݆ = ݂݅,݆(ܶ − ܰ‫,ݐ‬ ‫.)ݖ‬

35
2-iii) Processing the ߙߙߙߙ߬߬߬߬+3-th column
· Expand: det[△ ߬+2] = (Σܶ−ܰ‫1−ݐ‬ ݉=0 det[ࡲ߬+3(݉, ‫])ݖ‬ ) + det[˜ ࡳ
߬+3(‫])ݖ‬ + det[ࡳ߬+3(‫,])ݖ‬ where ࡲ߬+3(݉, ‫,)ݖ‬ ˜ࡳ ߬+3(‫,)ݖ‬ and ࡳ߬+3(‫)ݖ‬ are three matrices identical to △
߬+1, except that the ߙ߬+3-th column of ࡲ߬+3(݉, ‫)ݖ‬ is given by {ࡲ߬+3(݉, ‫݆,݅})ݖ‬ = ݂݅,݆(݉, ‫,)ݖ‬ and
that the ߙ߬+3-th column of ˜ ࡳ ߬+3(‫)ݖ‬ is given by: for ݅ = 1, {˜ ࡳ ߬+3(‫݆,݅})ݖ‬ =
݃݅,݆(0, ‫;)ݖ‬ for ݅ = 2, . . . , ܵ, {˜ ࡳ ߬+3(‫݆,݅})ݖ‬ = ݂݅,݆(ܶ − ܰ‫,ݐ‬ ‫,)ݖ‬ and that the ߙ߬+3-th column of
ࡳ߬+3(‫)ݖ‬ is given by: for ݅ = 1, {ࡳ߬+3(‫݆,݅})ݖ‬ = Σ∞ ݉=1 ݃݅ ,݆(݉, ‫;)ݖ‬ for ݅ = 2,
{ࡳ߬+3(‫݆,݅})ݖ‬ = Σ∞ ݉=0 ݃݅,݆(݉, ‫ݖ‬ for ݅ = 3, . . . , ܵ, {ࡳ߬+3(‫݆,݅})ݖ‬ =
· Remove: det[△ ߬+2] = det[ࡳ߬+3(‫,])ݖ‬ because det[˜ ࡳ ߬+3(‫])ݖ‬ = 0 and det[ࡲ߬+3(݉, ‫])ݖ‬ = 0 for ݉ =
0, . . . , ܶ − ܰ‫ݐ‬ − 1 (existence of co-linear columns).
· Omit: det[△ ߬+2] = det[△ ߬+3] + ࣩ, where △ ߬+3 is a matrix identical to △ ߬+2, except that
its elements in the ߙ߬+3-th column is given as follows: for ݅ = 1, {△ ߬+3 }݅,݆ = ݃݅,݆(1, ‫;)ݖ‬ for ݅ = 2,
{△ ߬+3 }݅,݆ = ݃݅,݆(0, ‫;)ݖ‬ for ݅=3, ..., ܵ, {△ ߬+3 }݅,݆ = ݂݅,݆(ܶ−ܰ‫,1+ݐ‬ ‫.)ݖ‬
2-iv) Processing the ߙߙߙߙ߬߬߬߬+4-th, ..., the ߙߙߙߙ‫ܯ‬‫ܯ‬‫ܯ‬‫ܯ‬+1−݇݇݇݇-th columns
Being aware of the difference in the “Remove” procedure, we process the remaining columns
to finally arrive at: det[△ ‫]݇−ܯ‬ = det[△ ‫]݇−1+ܯ‬ + ࣩ, where △ ‫݇−1+ܯ‬ is a matrix identical to △ ߬ ,
except that its elements in the ߙ߬+1- th, . . ., ߙ‫-݇−1+ܯ‬th columns are given by {△ ‫݇−1+ܯ‬ }݅,݆ =
݃݅,݆(‫ݍ‬ + 1 − ݅, ‫,)ݖ‬ where ݅ = 1, . . . , ‫ݍ‬ + 1, ݆ = ߙ߬+1+2‫,ݍ‬ and ߙ߬+1+2‫+ݍ‬min(1,⌊ߝ/2⌋−‫,)ݍ‬ with ‫ݍ‬ = 0, 1,
. . . , ⌈ ߝ/2 ⌉ − 1; and by {△ ‫݇−1+ܯ‬ }݅,݆ = ݂݅,݆(ܶ − ܰ‫ݐ‬ + ‫,ݍ‬ ‫,)ݖ‬ where ݅ = ‫,2+ݍ‬ . . . , ܵ, ݆ = ߙ߬+1+2‫,ݍ‬
and ߙ߬+1+2‫+ݍ‬min(1,⌊ߝ/2⌋−‫,)ݍ‬ with ‫ݍ‬ = 0, 1, . . . , ⌈ ߝ/2 ⌉ − 1.
Step 3: Factorization and Minimization
So far, we have got the remaining matrix △ ‫݇−1+ܯ‬ for a given permutation (ߙ1, . . .,
ߙ‫)݇−1+ܯ‬ of the vector ࢼ. Our next step is to sum up determinants of the remaining matrices for all
possible (ߙ1, . . . , ߙ‫,)݇−1+ܯ‬ i.e., det[Σ 0] = ߙ1,...,ߙ‫݇−1+ܯ‬ det[△ ‫]݇−1+ܯ‬ + ࣩ. To further simplify
the expression above, we factor out all exponential terms in det[△ ‫,]݇−1+ܯ‬ and obtain a new
equality: det[△ ‫]݇−1+ܯ‬ = det[˜Ξ(ࢻ, ‫ݖ])ݖ‬ ݀݇, where
(30)
And ˜Ξ(ࢻ, ‫)ݖ‬ is identical to △ ‫݇−1+ܯ‬ except that all exponential terms are removed. Noticing that
the term ˜ ݀݇ is independent of the order of ߙ1, . . . , ߙ‫,݇−1+ܯ‬ the determinant det[△ 0] can be
rewritten as: det[△ 0] = (Σ ߙ1,...,ߙ‫݇−1+ܯ‬ det[˜Ξ(ࢻ, ‫])ݖ‬ ) ‫ݖ‬ ˜ ݀݇ + ࣩ. 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 ࢼ, i.e.,
(32)
(33)
(34)
Notice that Eq. (33) holds if and only if (ߙ1, . . ., ߙ‫)݇−1+ܯ‬ is a permutation of (1, . . .,‫ܯ‬ + 1
− ݇). As a result, the optimal vector ࢼ ∗ is: (ߚ∗ 1 , . . . , ߚ∗ ݇−1, ߚ∗ ݇, . . . , ߚ∗ ‫)ܯ‬ = (‫−2+ܯ‬ ݇, . . . ,
‫,ܯ‬ 1, . . .,‫ܯ‬ +1− ݇). Knowing this, we can omit all terms except the optimal one in the summation
Of (31) and simplify the equality to
(35)
Where ࢻ is a permutation of (1, . . .,‫ܯ‬ + 1 − ݇), and the summation is over all possible
permutations. This simplified expression is indeed the result of Lemma 2.
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