# Frequency Response and Impulse Response

Recall that if an LTI system *H*:[*DiscreteTime* →
*Reals*] → [*DiscreteTime* →
*Reals*] has impulse response *h*: *DiscreteTime* →
*Reals*, and if the input is *x*: *DiscreteTime* →
*Reals*, then the output is given by the convolution sum

*y*(*n*) = ∑_{(m
= − ∞
to ∞
)} *h*(*m*) *x*(*n*−*m*)

Suppose that the input is a complex exponential function, where for all *n*
∈ *Integers*,

*x*(*n*) = *e ^{jω
n}.*

Then

*y*(*n*) = ∑_{(m
= − ∞
to ∞
)} *h*(*m*)*e ^{jω
}*

^{(}

^{n−m}^{)}

*= e ^{jω n} *∑

_{(m = − ∞ to ∞ )}

*h*(

*m*)

*e*

^{ −jω m}

Recall further that when the input is the complex exponential with frequency ω , then the output is given by

*y*(*n*) = *H*(ω )*e
^{jω n}*

where *H*(ω ) is called the frequency response.
Comparing these two expressions for the output we see that the frequency response
is related to the impulse response by

*H*(*ω*) = ∑_{(m
= − ∞
to ∞
)} *h*(*m*)*e ^{−imω}*
.

*H*(ω ) is called the **discrete-time
Fourier transform (DTFT)** of *h*(*n*). We will study the DTFT
in more detail shortly, and will examine its relationship to the Fourier
series. For now, however, just notice that the impulse response fully defines
the frequency response, and in principle, if you know the impulse response,
you can calculate the frequency response.