Time-domain analysis of discrete-time systems - Continuous and Discrete Time Signals and Systems - page 441

summation, Eq. (10.16), has the following values:

( k = 0) y p [0] = x p [0] h p [0] + x p [1] h p [ − 1] + x p [2] h p [ − 2] + x p [3] h p [ − 3]

= 0 5 + 1 0 + 2 0 + 3 5 = 15;

( k = 1) y p [1] = x p [0] h p [1] + x p [1] h p [0] + x p [2] h p [ − 1] + x p [3] h p [ − 2]

= 0 0 + 1 5 + 2 0 + 3 0 = 5;

( k = 2) y p [2] = x p [0] h p [2] + x p [1] h p [1] + x p [2] h p [0] + x p [3] h p [ − 1]

= 0 0 + 1 5 + 2 5 + 3 0 = 15;

( k = 3) y p [3] = x p [0] h p [3] + x p [1] h p [2] + x p [2] h p [1] + x p [3] h p [0]

= 0 0 + 1 0 + 2 5 + 3 5 = 25 .

The remaining values of y p [ k ] are easily determined by exploiting the period-
icity property of y p [ k ]. The output y p [ k ] is plotted in Fig. 10.9, step (8).

An alternative procedure for computing the periodic convolution can be
obtained by calculating the limits of Eq. (10.15) for m

= 0to m = − 1.

K 0

The resulting expression is given by
y p [ k ] = K 0 − 1 x p [ m ] h p [ k − m ]

m = 0

or

y p [ k ] = x p [0] h p [ k ] + x p [1] h p [ k − 1] + x p [2] h p [ k − 2]

++ x p [ K 0

− 1] h p [ k − ( K 0 − 1)] ,

for 0 ≤ k ≤ K 0 − 1. Expanding the above equation in terms of the time index
k yields

y p [0] = x p [0] h p [0] + x p [1] h p [ − 1] + x p [2] h p [ − 2] +
+ x p [ K 0 − 1] h p [ − ( K 0 − 1)] ,
y p [1] = x p [0] h p [1] + x p [1] h p [0] + x p [2] h p [ − 1]
++ x p [ K 0 − 1] h p [ − ( K 0 − 2)] ,
y p [2] = x p [0] h p [2] + x p [1] h p [1] + x p [2] h p [0] +
+ x p [ K 0 − 1] h p [ − ( K 0 − 3)] ,

.
y p [ K 0 − 1] = x p [0] h p [ K 0 − 1] + x p [1] h p [ K 0 − 2]
+ x p [2] h p [ K 0 − 3] ++ x p [ K 0 − 1] h p [0] .

(10.17)

Since h p [ k ] is periodic,

h p [ k ] = h p [ k + K 0 ]or h p [ − k ] = h p [ K 0 − k ] . (10.18)