Signals and Systems

7.4 DISCRETE CONVOLUTION

If the input x(n) is given to discrete time system having an impulse {h(n)}, the output is given by convolving the {h(n)} with x(n).

 

y(n) = h(n)* x(n) = x(n)* h(n)       (7.8)

 

where the symbol ‘*’ represents the convolution of x(n) with h(n) and this process is also called convolution sum. It is defined by

 

 

The limits will reduce to 0 to n for one-sided x(n) and h(n).

Proof:

Fig. 7.7 shows an input sequence {x(n)}.

 

 

Fig. 7.7 Input sequence {x(n)}

 

Let us express x(n) as a sum of delta sequences. We can write ...