ELEC 320

Prof. Rich Kozick

Fall, 1998

Prof. Rich Kozick

Fall, 1998

**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* *h*(*t*).

Use the following approach.

- 1.
- Please explain why it is true that the ZSR of any
linear, time-invariant (LTI) system is completely described by
the impulse response
*h*(*t*) of the system. (Are there any LTI systems for which this is not true?) If the impulse response*h*(*t*) is known, then the system output*y*(*t*) due to any input*x*(*t*) is given by - 2.
- Now consider a particular input
that is applied to a LTI system with impulse response
*h*(*t*). Put this*x*(*t*) into the convolution integral, and look at the resulting*y*(*t*). You should be able to recognize that*y*(*t*) 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. - 3.
- 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
*h*(*t*). This provides a mathematical connection between the frequency domain and time domain descriptions of a system. - 4.
- 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: