The impulse response function 
of a linear
time-invariant system is its output when the input is the delta
function 
and the initial state is zero. The zero-state condition excludes any output caused
by stored energy or a nonzero state before the input is applied, so the response
is due to the input alone. If the system's behavior is unchanged by a shift in time,
then the response to a general input 
is the convolution
 |
For a discrete-time linear time-invariant system, the corresponding convolution is
 |
For a causal system, 
for 
and the lower limit can be replaced by 0 when the input
also vanishes before 0. The Fourier transform
of 
is the frequency response, while its Laplace
transform is the transfer function under
the same zero-state condition.
