5.7 Convolution and Filtering 329
or
H(j)=
Y()
X()
(5.20)
The magnitude and the phase ofH(j)are the magnitude and phase frequency responses of the
system, or how the system responds to each particular frequency.
Remarks
n It is important to keep in mind the following connection between the impulse response h(t), the transfer
function H(s), and the frequency response H(j)that characterize an LTI system:
H(j)=L[h(t)]|s=j
=H(s)|s=j
=
Y(s)
X(s
|s=j
n As the Fourier transform of a real-valued function, the impulse response h(t), the function H(j)has a
magnitude|H(j)|and a phase∠H(j), which are even and odd functions of the frequency.
n The convolution property relates to the processing of an input signal by an LTI system. But it is possible, in
general, to consider the case of convolving two signals x(t)and y(t)to get z(t)=[x∗y](t), in which case
we have that Z()=X()Y()where X()and Y()are the Fourier transforms of x(t)and y(t).
5.7.1 Basics of Filtering
The most important application of LTI systems is filtering. Filtering consists in getting rid of
undesirable components of a signal. A typical example is when noiseη(t)is added to a desired
signalx(t)(i.e.,y(t)=x(t)+η(t)), and the spectral characteristics ofx(t)and the noiseη(t)are
known. The problem then is to design a filter, or an LTI system, that will get rid of the noise
as much as possible. The filter design consists in finding a transfer functionH(s)=B(s)/A(s)that
satisfies certain specifications that will allow getting rid of the noise. Such specifications are typ-
ically given in the frequency domain. This is arational approximation problem, as we look for
the coefficients of the numerator and denominator ofH(s)that makeH(j)in magnitude and
phase approximate the filter specifications. The designed filter should be implementable and sta-
ble. In this section we discuss the basics of filtering and in Chapter 6 we introduce the filter
design.
Frequency-discriminating filters keep the frequency components of a signal in a certain frequency
band and attenuate the rest. Filtering an aperiodic signalx(t)represented by its Fourier transform
X()with magnitude|X()|and phase∠X(), using a filter with frequency responseH(j), gives
an outputy(t)with a Fourier transform of
Y()=H(j)X()
Thus, the outputy(t)is composed of only those frequency components of the input that are not
filtered out by the filter. When designing the filter, we assign appropriate values to the magnitude in
