Page 242 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

B. Signal Bandwidth:
The bandwidth of a signal can be defined as the range of positive frequencies in which
"most" of the energy or power lies. This definition is rather ambiguous and is subject to
various conventions (Probs. 5.57 and 5.76).
3-dB Bandwidth:
The bandwidth of a signal x(t) can also be defined on a similar basis as a filter
bandwidth such as the 3-dB bandwidth, using the magnitude spectrum (X(o)l of the signal.
Indeed, if we replace IH(o)l by IX(o)l in Figs. 5-5(a) to (c), we have frequency-domain
plots of low-pass, high-pass, and bandpass signals.

Band-Limited Signal:
A signal x(t) is called a band-limited signal if

Thus, for a band-limited signal, it is natural to define o, as the bandwidth.

Solved Problems

FOURIER SERIES

5.1. We call a set of signals {*,Jt)} orthogonal on an interval (a, b) if any two signals ql,,(t)
and qk(t) in the set satisfy the condition

where * denotes the complex conjugate and a + 0. Show that the set of complex
exponentials {ejk"o': k = 0, f 1, f 2,. . . ) is orthogonal on any interval over a period To,
where To = 2.rr/oU.
For any to we have

since eJm2" = 1. When m = 0, we have eJm"o'lm=o = 1 and