]> Sound and music - The z-Transform

The z-Transform

Let δ(n) be a digital impulse signal, that is, δ(0 )=1 and is zero everywhere else. By scaling, shfting and summing δ(n) functions, we can construct any input signal. LTI filters preserve scaling, shifting and summing, thus the behaviour of such a filter on δ(n) determines the behaviour of the filter on any input signal.

We can easily compute the transfer function of a filter by looking at the output of the filter given δ(n).

Given a function x(n) defined for all n, define the z-transform of x by

X(z)= n= x(n)z n

Let h(n) be the output of an LTI filter applied to δ(n). Define the transfer function H(z) to be the z-transform of h(n).

Then by linearity and time-invariance, for any input signal x(n) and corresponding output signal y(n), we have Y(z)=H(z)X(z) where X(z),Y(z) are the z-transforms of x(n),y(n) respectively. Note also y(n)=h(n)*x(n), where * denotes convolution.