ELEC 320

Prof. Rich Kozick

Prof. Rich Kozick

**Problem:**
Prove that for a continuous-time system that
is *linear* and *time-invariant*,
the zero-state response (ZSR) of the system to a sinusoidal input
is a sine wave with the same frequency as the input wave,
but a different amplitude and phase shift.
Also, find an expression for the *frequency response*
of the system in terms of the *impulse response* .

Use the following approach.

- Please explain why it is true that the ZSR of any
linear, time-invariant (LTI) system is completely described by
the impulse response of the system.
(Are there any LTI systems for which this is not true?)
If the impulse response is known, then the system
output due to any input is given by

- Now consider a particular input
that is applied to a LTI system with impulse response .
Put this into the convolution integral,
and look at the resulting .
You should be able to recognize that is a sine
wave with the same frequency , but with
a different amplitude and phase shift.
The trigonometric identities at the bottom of the page
will be helpful.
- In terms of the frequency response of the system
, recall that we expect that the system output
has the form

Use your result from item 2 to relate the frequency response of the system to the impulse response . This provides a mathematical connection between the frequency domain and time domain descriptions of a system. - You now understand the very important result
that a sine wave input to a LTI system produces a
sine wave output with the same frequency but different
amplitude and phase shift!

Here are some useful identities:

where and .