336 Practical MATLAB® Applications for Engineers
The cross correlation of two signals f 1 (t) with f 2 (t) provides an indication of
the degree of similarity between them. If f 1 (t) = f 2 (t), then the cross correlation
becomes an autocorrelation. Note the similarity between the correlation and
convolution process.
m. Time reversalft Fw()↔ ( )
R.4.49 Recall that the integral equation
fftd 12 () ( ⋅ )
∞∞
∫
is called the convolution integral of f 1 (t) with f 2 (t), and is indicated using the
notation
ft^12 ()⊗ft()
Recall also that the convolution integral is a powerful way of defi ning the input–
output relations of a time invariant linear system (see Chapter 1).
R.4.50 From the transform equation pair, and their properties, the following can be
observed: if f(t) is a real and an even function of t, then F(w) is also a real and an
even function of w.
R.4.51 If f(t) is a real and an odd function of t, then F(w) is an imaginary function of w.
R.4.52 If f(t) is a periodic function in the time domain, then F(w) is a discrete function in
the frequency domain.
R.4.53 If f(t) is a periodic function in the time domain, then F(w) is a nonperiodic function
in the frequency domain.
R.4.54 If f(t) is a continuous and nonperiodic function in the time domain, then F(w) is a
nonperiodic and continuous function in the frequency domain.
R.4.55 Let the input to a given system be x(t), and its impulse system response be h(t),
as indicated in Figure 4.3. Recall then that the output y(t) of the system is
given by
yt()xt()⊗ht() x h( ) ( td)
∞∞
∫^
R.4.56 Let x(t) be the input to a given system (as indicated in Figure 4.3), and let
x(t) ↔ X(w) and h(t) ↔ H(w)
Then the general block box system diagram given in the time domain indicated
in Figure 4.3 can be transformed to the equivalent block box system diagram in the
frequency domain indicated in Figure 4.4.
Recall that the convolution integral in the time domain is transformed into a
product of its spectrums in the frequency domain (R.4.48, property k).