# 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 x_{1}(k)
and x_{2}(k) shown above.
The Z transform of
x_{1}(k) is shown above.
The Z transform of
x_{2}(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.