Page 34 - Introduction to Information Optics

1.4. Band-limited Analysis 19

1.4. BAND-LIMITED ANALYSIS

In practice, all communication channels (spatial and temporal as well) are
band limited. An information channel can be a low-pass, bandpass, or discrete
bandpass channel. But a strictly high-pass channel can never happen in
practice. A low-pass channel represents nonzero transfer values from zero
frequency to a definite frequency limit v m. Thus, the bandwidth of a low-pass
channel can be written as

On the other hand, if the channel possesses nonzero transfer values from a
lower frequency limit v l to a higher frequency limit v 2, then it is a bandpass
channel and the bandwidth can be written as

Av = v 2 -V! . (1.66)

It is trivial to note the analysis of a bandpass channel can be easily reduced
to an equivalent low-pass channel. Prior to our discussion a basic question
may be raised: What sort of response would we expect from a band-limited
channel? In other words, from the frequency domain standpoint, what would
happen if the signal spectrum is extended beyond the passband?
To answer this basic question, we present a very uncomplicated but
significant example. For simplicity, the transfer function of an ideal low-pass
channel, shown in Fig. 1.4, is given by

0, |v| > v
o !"!>'• m .
If the applied signal to this low-pass channel is a finite duration At signal, then
to have good output reproduction of the input signal the channel bandwidth
Av must be greater than or at least equal to I/At the signal bandwidth; that is,

Av^— , (J.68)
At
which can also be written as

ArAv 5s 1, (1.69)
where Av = 2v m is the channel bandwidth. The preceding equations show an
interesting duration-bandwidth product relationship. The significance is that if
the signal spectrum is more or less concentrated in the passband of the channel;