This chapter discusses filter transfer functions and associated analysis. The transfer function provides an algebraic representation of a linear, time-invariant (LTI) filter in the frequency domain:
The transfer function is also called the system function .
Let denote the impulse response of the filter. It turns out (as we will show) that the transfer function is equal to the z transform of the impulse response :
Since multiplying the input transform by the transfer function gives the output transform , we see that embodies the transfer characteristics of the filter--hence the name.
It remains to define ``z transform'', and to prove that the z transform of the impulse response always gives the transfer function, which we will do by proving the convolution theorem for z transforms.