Frequency Response

We now summarize our definition of frequency response.

Frequency Response of a Continuous Time LTI System:

A function of frequency . The frequency response tells us what the response is to the input signal , namely the response is . If the region of convergence of the transfer function contains the imaginary axis, the frequency response is simply the restriction of the transfer function to the imaginary axis. If the system maps real input signals into real output signals, it's easy to show that

In other words, the magnitude of tells us how much the system amplifies the frequency , and the angle of tells us how much the system phase shifts the frequency , roughly speaking.

Frequency Response of a Discrete Time LTI System:

A function of frequency . The frequency response tells us what the response is to the input signal , namely the response is . If the region of convergence of the transfer function contains the unit circle, the frequency response is simply the restriction of the transfer function to the unit circle. If the system maps real input signals into real output signals, it's easy to show that

In other words, the magnitude of tells us how much the system amplifies the frequency , and the angle of tells us how much the system phase shifts the frequency , roughly speaking.

Since essentially arbitrary signals can be expressed as superpositions of complex exponentials, we can use superposition and knowledge of the frequency response to determine the response to an arbitrary input signal. Thus, frequency response completely specifies an LTI system.