Convolution and Impulse Response

Let

h(n) = d, if n = 0

h(n) = ca^{n -1}b, if n > 0

Then the zero state output response is the convolution sum

" n ³ 0

y(n) = S _{(m = 0 to n)} h(n-m) x(m) ,

and h: Integers ® Reals is the (zero state) impulse response.

Notice that if s₀ = 0 and the input is the Kronecker delta function, or impulse:

" m Î Integers,

x(m) = 1; if m = 0,
x(m) = 0; otherwise,

then " n Î Integers,

y(n) = h(n).

So h is called the (zero state) impulse response of the system.