EECS20N: Signals and Systems

Convolution

Then zero-state output can be written,

y(n) = ∑ (m = 0 to n) h(n- m) x(m)

where h is the impulse response. This form of summation is called a convolution. The output is given by a convolution of the input and the impulse response. This is written using the special shorthand notation

y = x * h.

Convolution is commutative, so

y = h * x.

To see this, change variables in the summation above, letting k = n - m, to get

y(n) = ∑ (k = 0 to n) h(k) x(n- k)