B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

13 .7 Frequency-Selective Channel Capacity 777
This channel has a constant gain of H across the bandwidth. Based on Eq. (13.63) this bandlimited
(low-pass) AWGN channel with bandwidth B has capacity
bit/s
(13.81)
in which S and N are the signal power and the noise power, respectively. Furthermore, in
Chapter 4 and Chapter 9, we have demonstrated the equivalence of baseband and passband
channels through modulation. Therefore, given the san1e noise spectrum and bandwidth,
AWGN low-pass, band-limited channels and AWGN bandpass channels possess identical
channel capacity. We are now ready to describe the capacity of frequency-selective channels.
Consider a bandpass channel of infinitesimal bandwidth l'lf centered at a frequency f;.
Within this small band, the channel gain is H (f;), the signal power spectral density (PSD) is
Sx (j;), and the Gaussian noise PSD is Sn (j;). Since this small bandwidth is basically a band
limited AWGN channel, according to Eq. (13.63), its capacity is
bit/s (13.82)
This means that we can divide a frequency-selective channel H (f) into small disjoint
A WGN bandpass channels of bandwidth fl/ . Thus, the sum channel capacity is simply
approximated by
In fact, the practical OFDM (or DMT) system discussed in Chapter 12 is precisely such a
system, which consists of a bank of parallel flat channels with different gains. This capacity
is an approximation because the channel response, the signal PSD, or the noise PSD, may
not be constant over a nonzero flf . By taking flf ➔ 0, we can determine the total channel
capacity as
Maximum Capacity Power Loading
C = J oo log [ 1 + I H (f) l 2 Sx if) ]
df
-oo
Sn if)
(13.83)
In Eq. (13.83), we have established that the capacity of a frequency-selective channel with
response H (f) under colored Gaussian noise of power spectral density (PSD) Sn (f) depends
on the input PSD Sx (J). For the transmitter to utilize the full channel capacity, we now need
to find the optimum input power spectral density (PSD) Sx (j) that can further maximize the
integral capacity
J oo log
[1 +
IH (j) 1 2 Sx (f) ]
df
-oo
Sn if)
To do so, we have noted that it would not be fair to consider arbitrary input PSD Sx (f) because
different power spectral densities may lead to different values of total input power. Given two