A Further Result about "On the Channel Capacity of Multiantenna Systems with Nakagami Fading"
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Letter to the Editor A Further Result about “On the Channel Capacity of Multiantenna Systems with Nakagami Fading” Saralees Nadarajah1 and Samuel Kotz2 1 School
of Mathematics, University of Manchester, Manchester M60 1QD, UK of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA
2 Department
Received 3 June 2006; Revised 18 December 2006; Accepted 23 December 2006 Recommended by Dimitrios Tzovaras Explicit expressions are derived for the channel capacity of multiantenna systems with the Nakagami fading channel. Copyright © 2007 S. Nadarajah and S. Kotz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
and 1 F1 and 2 F2 are the hypergeometric functions defined by
INTRODUCTION
The recent paper by Zheng and Kaiser [1] derived various expressions for the channel capacity of multiantenna systems with the Nakagami fading channel. Most of these are expressed in terms of the integral J(k, β) =
∞ 0
u k/2−1 log 1 + u exp(−u)du, β
(1)
see, for example, [1, equation (14)]. The paper provided a recurrence relation (see [1, equation (18)]) for calculating (1). Here, we show that one can derive explicit expressions for (1) in terms of well-known functions.
1 F1 (a; b; x)
=
∞ (a)k xk
k=0 2 F2 (a, b; c, d; x)
=
(b)k k!
,
∞ (a)k (b)k xk
k=0
(c)k (d)k k!
(4) ,
respectively, where ( f )k = f ( f +1) · · · ( f +k − 1) denotes the ascending factorial. If k = 2, then by [2, equation (2.6.23.5)] one can reduce (2) to J(2, β) = − exp(β)Ei(−β),
(5)
where Ei(·) denotes the exponential integral defined by 2.
EXPLICIT EXPRESSIONS FOR (1)
We calculate (1) by direct application of certain formulas in [2]. For k > 0, application of [2, equation (2.6.23.4)] yields J(k, β) =
2πβk/2 k sin(kπ/2)
k −Γ 2
1 F1
k k ;1 + ;β 2 2
log β − Ψ 2β
k 2
Ψ
1 F1
exp(t) dt. t
(6)
1 = −γ − 2 log 2, 2
√
πerfi
1 3 ; ;β = 2 2
(7)
β
2 β
,
,
where γ = 0.5772 · · · is the Euler’s constant and erfi(·) denotes the imaginary error function defined by
where Ψ(·) denotes the digamma function defined by d log Γ(x) , dx
(2)
−∞
If k = 1, then by using the facts that
k − 2 F2 1, 1; 2, 2 − ; β 2−k 2
Ψ(x) =
Ei(x) =
x
(3)
2 erfi(x) = √ π
x 0
exp t 2 dt,
(8)
2
EURASIP Journal on Advances in Signal Processing
one can reduce (2) to
J(1, β) = π 3/2 erfi −
√
β
3 π log β + γ + 2 log 2 − 2β2 F2 1, 1; 2, ; β 2
. (9)
If k = 3, then by using the facts that Ψ 1 F1
3 = 2 − γ − 2 log 2, 2 √
3 exp(β) 3 5 − ; ;β = 2 2 2β
3 πerfi
4β3/2
(10)
β
,
one can reduce (2) to
J(3, β) = −πβ1/2 exp(β) +
π 3/2 erfi
β
2 π 1 − log β − 2 + γ + 2 log 2+2β2 F2 1, 1; 2, ; β . 2 2 (11) √
3.
DISCUSSION
We expect that the expression given by (2) and its particular cases could be useful with respect to channel capacity modeling
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