Transfer Function - Signal Processing

Signal Processing

Let be the input to a general linear time-invariant system, and be the output, and the bilateral Laplace transform of and be

.

Then the output is related to the input by the transfer function as

and the transfer function itself is therefore

.

In particular, if a complex harmonic signal with a sinusoidal component with amplitude, angular frequency and phase

where

is input to a linear time-invariant system, then the corresponding component in the output is:

and

Note that, in a linear time-invariant system, the input frequency has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response describes this change for every frequency in terms of gain:

and phase shift:

.

The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:

.

The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency ,

.

The transfer function can also be shown using the Fourier transform which is only a special case of the bilateral Laplace transform for the case where .