PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 467

Observe that the convolution coeffi cients can be evaluated by following the pattern

illustrated as follows:

g(0) h(0) f(0)

g(1) h(1) f(0) f(1)h(0)

g(2) h(2) f(0) h(1)f(






⋅

⋅

⋅ 1 1) h(0)f(2)

g(3) h(3) f(0) h(2)f(1) h(1)f(2) h(0)f(3)

g(k)




 ⋅

...



h(k) f(0)⋅ h(k 1)f(1) 
 h(1)f(k 1) h(0)f(k), for any valuees of k

R.5.22 Recall that if the impulse response of a given LTI discrete system is known (h(n)),

then the output sequence g(n) can then be evaluated as the convolution of any

arbitrary input f(n) with the system impulse response h(n).

The discrete convolution can be viewed as the system transformation in the time

domain of the input sequence f(n) into an output sequence g(n).

R.5.23 Some discrete convolution properties are as follows. Observe that they follow

closely the analog case. Let g(n) = f(n) ⊗ h(n), then

a. [f1(n) + f2(n)] ⊗ h(n) = f1(n) ⊗ h(n) + f2(n) ⊗ h(n) (linear)

b. f(n) ⊗ h(n) = h(n) ⊗ f(n) (com mutat ive)

c. [f1(n) ⊗ f2(n)] ⊗ h(n) = f1(n) ⊗ [f2(n) ⊗ h(n)] (associat ive)

FIGURE 5.7

Discrete sequences h(n) and f(n) of R.5.21.

h(k)

1

1

k

f(k)

k

− 10

0

1234

1234

5

1/2

1/4

1/16

1/8

FIGURE 5.8

Discrete sequences f(−k) of R.5.21.

k

− 3 − 2 − 1

1

0

f(−k)

FIGURE 5.9

Discrete sequences f(1 − k) of R.5.21.

1

− 2 − 1012 k

f(1−k)