Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 222

q ( t )
0

x ( t )
0

(a)

q ( t )
1

x ( t )
1

noise spectrum, band 0

N / 2

signal spectrum, band 0

0

p / 2 B
00

(b)

f (Hz) f (Hz)

− B/ 20 B/ 2 − B/ 20 B/ 2

signal spectrum, band 1

p / 2 B
01

noise spectrum, band 1

N / 2 N / 2

1 1

(c)

f (Hz) f (Hz)

− B − B/ 2 0 B/ 2 B − B − B/ 2 0 B/ 2

B

Figure 6.4 . (a) Splitting the channel into subbands, (b) noise and signal spectra in
subband 0, and (c) noise and signal spectra in subband 1.

bits per second. We can rewrite the capacity of the unsplit channel (6.20) in the
form

1+ .

p 0

C unsplit = B log 2 (6 . 24)

(0 . 5 N 0 +0 . 5 N 1 ) B

Now, it turns out that the function

1+ A
x

ψ ( x )=log
2

(6 . 25)

is strictly convex in x for A, x > 0 (because ∂ 2 ψ/∂x 2 > 0; see Sec. 21.2 in Chap.
21). This implies that

1+ 1+ A

x 0

+(1 1+ A

x 1

(6 . 26)

A
αx 0 +(1 − α ) x 1

log 2 ≤ α log 2 − α )log 2

for any α in 0 ≤ α ≤ 1 , with strict inequality unless x 0 = x 1 , or α = 0 or 1 .

Using this with α =0 . 5 we see immediately that

C unsplit <C split (6 . 27)