Digital filters - Signals, Processes, and Systems - page 319

the circuit. First a straightforward circuit should be developed which makes several
δ -pulse i.e. makes a longer pulse response from a

single pulse and therefore must function like a filter. This is shown by Illustration
210.You will be familiar with the course of the amplitude spectrum. The curve is based
on the response of an Si-function as in numerous examples in Chapter 2 (e.g. Illustration
38), but is here periodic because it is discrete in the time domain.

-pulses of the same level from one δ

The symmetry principle tells us that a pulse response which is almost like an Si function
would have to result in a rectangular-like filter curve. But how can an Si-shaped pulse
response be generated by a modified circuit compared with Illustration 210. This is in
principle shown by Illustration 211. By multiplication of the individual -pulses by

certain constants (“filter coefficients”) the pulse response is transformed as far as possible
into an Si function like form.

δ

This circuit has a very straightforward structure and is content with only three elementary
(linear) signal processes - addition, multiplication by a constant and delay .

But how do we arrive at the right coefficients? A potential but complicated possibility
would be - as shown in Illustration 199 and Illustration 200 - to sample an Si-function
with a periodic -pulse sequence and to have these values stored as a list. In this way the

correct constant for each module could be inputted but in a very laborious way.

δ

Please note in Illustration 210 and Illustration 211 that at the output of the adder (summa-
tion) the weighted -pulse at the lowest input is the first to appear at the output of the adder

and the uppermost which passes through all the delays appears last.

δ

Convolution
5 or 15 weighted

-pulses according to Illustration 211 and Illustration 212 are not suffi-
cient to produce an Si-shaped curve. 256

δ

-pulses would be better. Then the circuit would
be so complex that it would not fit on the screen. And setting 256 coefficients manually
would be slave labour.

δ

But this is not necessary as the schematic circuit according to Illustration 211 embodies
an important signalling process - convolution - which is available with DASY Lab , also
in the educational version, as a special module.

Convolution as a signalling process was already mentioned in Chapter 7 in the section
“Multiplication of two signals as a non-linear process”. See above all Illustration 146 and
the relevant text. In this case, however, it was a question of a convolution in the frequency
domain as a consequence of a multiplication in the time domain. This is a multiplication
in the frequency domain (“rectangular filter”) and - as a result of the symmetry principle
a convolution in the time domain.

The following result should be noted:

A multiplication in the time domain produces a convolution in the
frequency domain.