Z Transform and Discrete Convolution.

Cuthbert Nyack
Some of the most useful transforms used in electronics have the convolution property shown above. The Z transform of the convolution of 2 sampled signals is the product of the Z Transforms of the separate signals. Compared to the integral encountered in analog convolutions, discrete convolutions involve a summation and are much easier to understand and carry out.
The following example illustrates the relation between the Z transform and convolution.
We consider 2 sampled signals x1(k) and x2(k) shown above.
The Z transform of x1(k) is shown above.
The Z transform of x2(k) is shown above.
Multiplying the 2 transforms give X(z), the Z Transform of the convolved signal.
Taking the inverse Z Transform gives the following for the convolution of the 2 sampled signals.

Instead of using the Z transforms, we can convolve the 2 signals directly using the convolution summation as illustrated below.
In this sum m must range over all values for which the product is finite. If both signals have a total of n samples(6 for this case) then there must be n - 1 values for m(5 in this case).
The following equations shows the convolution sum being evaluated for values of m from 0 to 4 (5 terms). Only the non-zero contributions are included.

The above equations show that applying the convolution sum directly give the same result as multiplying the 2 Z Transforms and taking the inverse transform.

The aplet below shows the convolution x3(n) of 2 16 sample input sequences x1(n) and x2(n).

When enabled the gif image below show how the applet should appear.

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© 2000 Cuthbert A. Nyack.