Green Wireless Communications: A Time-Reversal Paradigm

WANG et al.: GREEN WIRELESS COMMUNICATIONS: A TIME-REVERSAL PARADIGM 1701
However, the Z ’s are not identically distributed, so we need
In order for the TR system and the direct transmission to
q = nL−nLR
nL
= L−LR
L . beyond the scope of the paper.
l
have the same performance, i.e., SNR TR = SNR DT ,wemust [ calibrate the results obtained in (21). An upper bound of
∑LR−1
]
have
E
P Sig = PSig DT
l=0
|h (l) | 2 can be obtained by substituting the largest
. (16) quantile in (21), i.e.,
[
Then, we can express the ratio of the transmission power of
LR−1
]
∑
the two schemes as
E Z (l) ≤ L R E [ ]
Z (0) |Z (0) ≥ z (0),q , (22)
r P =
P
∑ LR−1
P DT = l=0
|h (l) | 2
l=0
∑ L−1
, (17) and an approximation can be expressed as
l=0
|h[l]|2 [ LR−1
]
∑
L∑
R−1
and the ratio of the expected transmission power needed for E Z (l) ≈ E [ ]
Z (l) |Z (l) ≥ z (l),q , (23)
TR and direct transmission can be expressed as
l=0
l=0
[ ∑LR−1
]
τ P = E[P ] E
E[P DT ] = l=0
|h (l) | 2 where z (0),q ≥ ··· ≥ z (LR−1),q ≥ ··· ≥ z (L−1),q , and
∑ L−1
. (18) Z (0) , ··· ,Z (LR−1) are corresponding random variables.
l=0 E[|h[l]|2 ]
As defined earlier in Section II, h[l] is a CSCG random
In order to derive the numerator of (18), one needs to variable with E[|h[l]| 2 ]=e − lTs
σ T . Denote σ 2 △
l = e − lTs
σ T ,then
analyze the order statistics of the |h[l]| 2 ’s. However, since the |h[l]| 2
|h[l]| 2 σl ’s are not identically distributed and it is also unknown
(k), with k =2. In the special case of k =2,a
which L R out of all the |h[l]| 2 χ 2 (k) distribution is equivalent to an exponential distribution
’s are the L R largest channel
Exp(λ) with λ = 1 2
. After some mathematical derivation, we
taps, it is very difficult to obtain the closed-form expression of
can get the distribution function of Z l as
the numerator in (18). Therefore, we will first assume that the
|h[l]| 2 ’s are identically and independently distributed (i.i.d.),
F
and derive the numerator of (18). Then we will calibrate the
Zl (z) =
{1 − e − z
σ
l 2 ,z ≥ 0,
(24)
0, z < 0.
results for non-identically distributed |h[l]| 2 ’s.
Before we start our analysis, let us first introduce the Therefore, Z l is also exponentially distributed, with mean
concept of quantile [15] in order statistics. Denote F (z) as E[Z l ]=σl
2 = e − lTs
σ T . Solving the inverse function of F Zl (z)
the distribution function for a continuous random variable. and substituting q = L−LR
L
yields the q-quantile of Z l
Definition 1: Suppose that F (z) is continuous and strictly
increasing when 0