468 Practical MATLAB® Applications for Engineers
R.5.24 The convolution of a sequence f(n) with an impulse δ(n − k), where k is an integer,
results in the sequence f(n) shifted by the integer k. For example
a. δ(n) ⊗ f(n) = f(n)
b. δ(n − 1) ⊗ f(n) = f(n − 1)
c. δ(n − k) ⊗ f(n) = f(n − k)
R.5.25 A direct consequence of the convolution properties of LTI systems with an
arbitrary input sequence f(n) are summarized graphically in Figure 5.10.
R.5.26 Often, frequency domain analysis is important to explore the system behavior in
the continuous time case, and it is equally important in the analysis and synthesis
of discrete-time systems as well. Recall that in the continuous case, the FS and FT
were used to represent a time function into representing the same function by the
information contained or concentrated at the different frequencies (Chapter 4 of
this book).
DTFT is introduced next to play a similar role for the case of a discrete-time
sequence or system.
R.5.27 DTFT of f(n) denoted by F(ejW) is the Fourier representation of the discrete-time
sequence given by the following equation:
FejW f nejWn
n() ()
∑The discrete sequence f(n) can be reconstructed from the coeffi cients of F(ejW) by
using IDFT, given by
fn() Fe e dW( )jW jWn
1
2
∫FIGURE 5.10
System examples of R.5.25.
f(n)f(n)g(n) = f(n) ⊗ h2(n) ⊗ h1(n)f(n)f(n)f(n)g(n) = f(n) ⊗ h1(n) ⊗ h2(n)g(n) = h1(n) ⊗ f(n) ⊗ h2(n)g(n) = f(n) ⊗ h1 (n) + f(n) ⊗ h2(n)
g(n) = [h1(n) + h2(n)] ⊗ f(n)g(n) = f(n) ⊗ [ h1(n) + h2(n)]+h1(n)h1(n)h1(n)h2(n)h1(n) ⊗ h2(n)h1(n) + h2(n)h2(n)h2(n)