Signals and Systems - Electrical Engineering

488 C H A P T E R 8: Discrete-Time Signals and Systems

the output of the system is given by either of the following two equivalent forms of theconvolution sum:

y[n]=

∑∞

k=−∞

x[k]h[n−k]

=

∑∞

m=−∞

x[n−m]h[m] (8.34)

The impulse responseh[n] of a discrete-time system is due exclusively to an inputδ[n]; as such,

the initial conditions are set to zero. In some cases there are no initial conditions, as in the case of

nonrecursive systems.

Now, ifh[n] is the response due toδ[n], by time invariance the response toδ[n−k] ish[n−k]. By

superposition, the response due tox[n] with the generic representation

x[n]=

∑

k

x[k]δ[n−k]

is the sum of responses due tox[k]δ[n−k], which isx[k]h[n−k] (x[k] is not a function ofn), or

y[n]=

∑

k

x[k]h[n−k]

i.e., the convolution sum of the inputx[n] with the impulse responseh[n] of the system. The second

expression of the convolution sum in Equation (8.34) is obtained by a change of variablem=n−k.

Remarks

n The output of nonrecursive or FIR systems is the convolution sum of the input and the impulse response of

the system. The input–output expression of an FIR system is

y[n]=

N∑− 1

k= 0

bkx[n−k] (8.35)

and its impulse response is found by letting x[n]=δ[n], which gives

h[n]=

N∑− 1

k= 0

bkδ[n−k]=b 0 δ[n]+b 1 δ[n−1]+···+bN− 1 δ[n−(N− 1 )]

so that h[n]=bn for n=0,...,N− 1 , and zero otherwise. Replacing the bk coefficients in

Equaiton (8.35) by h[k]we find that the output can be written as

y[n]=

N∑− 1

k= 0

h[k]x[n−k]