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6.7 Impulse Response 131

Obviously this filter changes the signal dramatically. The output only con-
tains low-frequency components, whereas all higher frequencies are elim-
inated. The comparison of the periodograms of input and output reveals
that all frequencies above f=0.1 corresponding to a period of τ=10 are sup-
pressed.
[Pxx,F] = periodogram(x5,[],128,1);
[Pyy,F] = periodogram(y5,[],128,1);
plot(F,abs(Pxx),F,abs(Pyy))
Hence, we have now designed a frequency-selective fi lter, i.e., a fi lter that
eliminates certain frequencies whereas other periodicities are more or less
unaffected. The next chapter introduces tools to characterize a fi lter in the
time and frequency domain that help to predict the effect of a frequency-
selective filter on arbitrary signals.

6.7 Impulse Response

The impulse response is a very convenient way of describing the fi lter char-
acteristics (Fig. 6.3). A useful property of the impulse response h in LTI
systems involves the convolution of the input signal x(t) with h to obtain the
output signal y(t).

It can be shown that the impulse response h is identical to the fi lter weights
in the case of nonrecursive filters, but is different for recursive fi lters.

Alternatively, the convolution is often written in a short form:

In many examples, the convolution in the time domain is replaced by a sim-
ple multiplication of the Fourier transforms H(f) and X(f) in the frequency
domain.

The output signal y(t) in the time domain is then obtained by a reverse Fourier