TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 171

It follows that the matrix argument I þH T H / s 2

from (3.6) is also a symmetric

Toeplitz matrix

ij ¼

H T H
s 2

r jijj =s 2 , i = j ;

Iþ

1 þr 0 =s 2 ,

i ¼ j:

We then have the following upper bound on capacity.

Theorem 1 The capacity is upper bounded in terms of the SNR as

N
2 log 2 1 þ

r 0
s 2

Capacity ( in bits per block use )

with the upper bound attained if and only if r k ¼ 0 for k = 0 .

We note that, since the channel is assumed finite impulse response (FIR), the auto-
correlation lags r k will vanish for k = 0 if, and only if, the channel impulse response
has a sole nonzero term; this gives a channel with no intersymbol interference.
As such, subject to a power constraint on r 0 ¼ P k h k , the presence of intersymbol
interference can only reduce channel capacity.
To verify the theorem, we recall that Hadamard's inequality of matrix theory ([47],
p. 477), [44] asserts that the determinant of a positive definite matrix is upper bounded
by the product of its diagonal elements, with equality if, and only if, the matrix is
diagonal. Applying this inequality to I þH T H / s 2 ,wehave

H T H
s 2

1
2 log 2 det Iþ

Capacity ¼

N

1
2 log 2 1 þ

r 0
s 2

N
2 log 2 1 þ

r 0
s 2

¼ :

Since the off-diagonal elements of I þH T H / s 2 are r jijj =s 2 : , equality holds in the

bound if, and only if, r k ¼ 0 for k = 0.
We remark that the bound gives the capacity for N uses of the channel; upon nor-
malizing by the block length N , we recover Shannon's expression 2 log 2 (1 þr 0 =s 2 )
for the per-symbol capacity of an additive white Gaussian noise channel.

B EXAMPLE 3.8
Diversity versus Signal Power. Consider an ideal channel with h 0 ¼ 1 and h k ¼ 0
for k 1, and a modified channel with g 0 ¼ g 1 ¼ 1, and g k ¼ 0 for k 2, having
thus an additional impulse response term which induces intersymbol interference.
Taking s 2

¼ 1, the capacity for the first channel, divided by N , evaluates to 0.5 bits
per channel use. In the second case, the capacity increases beyond 0.7 bits per
channel use, which is sometimes attributed to increased diversity brought from
the presence of the second impulse response term g 1 . In fact, the seeming increase
in capacity comes from a larger SNR, as g 0 þg 1 . h 0 .