Signals and Systems - Electrical Engineering

5.7 Convolution and Filtering 327

This is also a low-pass signal likex 2 (t)in Example 5.11, but this is “smoother” than that one

because the magnitude response is more concentrated in the low frequencies. Compare the

values of the magnitude responses to verify this. Also this signal has zero phase, because its

Fourier transform is real and positive for all values of.

(b) The signaly(t)= 2 x(t)cos( 0 t)is aband-passsignal. It is not as smooth as the signals in the

above example given that the concentration of

Y()=X(− 0 )+X(+ 0 )

is around the frequency 0 , a frequency typically higher than the frequencies inx(t). The

higher this frequency, the more variation is displayed by the signal. In communications, low-

pass signals are calledbase-bandsignals. n

The bandwidth of a signalx(t)is the support—on the positive frequencies—of its Fourier transformX().

There are different definitions of the bandwidth of a signal depending on how the support of its Fourier trans-

form is measured. We will discuss some of the bandwidth measures used in filtering and in communications

in Chapter 6.

The bandwidth together with the information about the signal being low-pass or band-pass provides
a good characterization of the signal. The concept of the bandwidth of a filter that was discussed
in circuit theory is one of its possible definitions; other possible definitions will be introduced in
Chapter 6. The spectrum analyzer, a device used to measure the spectral characteristics of a signal,
will be presented in section 5.7.4 after considering filtering.

5.7 Convolution and Filtering.....................................................................

The modulation and the convolution integral properties are the most important properties of the
Fourier transform. Modulation is essential in communications, and the convolution property is basic
in the analysis and design of filters.

If the inputx(t)(periodic or aperiodic) to a stable LTI system has a Fourier transformX(), and the system has

a frequency responseH(j)=F[h(t)]whereh(t)is the impulse response of the system, the output of the LTI

system is the convolution integraly(t)=(x∗h)(t), with Fourier transform

Y()=X()H(j) (5.18)

In particular, if the input signalx(t)is periodic, the output is also periodic with Fourier transform

Y()=

∑∞

k=−∞

2 πXkH(jk 0 )δ(−k 0 ) (5.19)

whereXkare the Fourier series coefficients ofx(t)and 0 are its fundamental frequency.