In this section, we consider the analysis of stable LTI systems
with periodic inputs. We will
represent the input with a Fourier Series. We will consider the variation of
the system response to frequency, i.e. the system frequency response.

We saw that if we input a signal

to a stable LTI system, the steady-state response is

Next, if we input a periodic input signal (expressed with its Fourier Series) to
a stable LTI system, using superposition, we can write down the output as a second Fourier Series:

where

Note: y(t) is also a
Fourier Series with Fourier Series coefficients
C_{k}H(jkω_{0}).
The book refers to the Fourier Series coefficients of x(t) as
C_{kx} and those of y(t) as C_{ky}.
Notice that
C_{ky} = C_{kx}H(jkω_{0}).

Example Given an LTI system with impulse response
h(t) = α e^{-αt}u(t), α > 0.

Find the output of the system to an input

x(t) = sin^{2}(2t).

HINT: First find

and then express x(t) as a Fourier Series.

We will see that h(t) is a LOW PASS filter. If
h(t) is used as a filter, it passes the LOW frequencies and cuts out the HIGH frequencies.
In an image, examples of low frequencies are solid regions, while examples of high frequencies are
edges.

Example Given a low pass filter in h(t) =
e^{-t}u(t) and an input signal