Digital filters - Signals, Processes, and Systems - page 320

Important example: The sampling of an analog signal with a periodic -pulse sequence as

in Illustration 146. The spectrum of the analog signal is “convoluted” at each frequency
of the

δ

δ -pulse sequence.

For reasons of symmetry the following must hold:

A multiplication in the frequency domain (as with the filter) pro-
duces a convolution in the time domain.

Creating a good filter means from a mathematical point of view the multiplication of the
frequency spectrum by a rectangle-like function. This means however that in the time do-
main convolution must take place with an approximated (above all time-limited) Si
function because rectangles and Si functions are inseparably linked by a FOURIER trans-
formation (see Illustration 91).

While multiplication is a familiar mathematical operation this is not true of convolution.
It is therefore important to illustrate it by suitable processes of visualisation. The basis for
this is the combination of the three basic linear processes of delay, addition and multipli-
cation by a constant. The signal flow in convolution produces a block diagram with a very
simple structure which is again underlined in Illustration 212.

Note the delay between the input signal and the filtered output signal (see designation A,
B….., E). It can be seen that all rapid changes in the imput signal are “swallowed up” by
the filter or disappear via the weighted averaging.

Unlike the FFT filter the output signal is here strictly causal, i.e. something only appears
at the output after a signal was connected with the input. All in all, there are the following
advantages compared with the FFT filter:

• The filter process takes place in the time domain and as a result of the

elementary processes involved does not require a great deal of calcula-
tion. As a result real time filtering in the audio field is now perfectly
possible. The amount of calculation necessary increases “linearly” with
the block length of the Si-function or with the precision or quality of
the filter.

• It is possible to filter continuously - i.e. not in blocks. As a result all the

problems which occurred with the “overlapping windowing” (see
Chapter 4: “Frequency-time landscapes”) of signal segments do not
arise.

• The filter functions causally like an analog signal.

Note:

The type of filter described here is called FIR filter (finite impulse response) in the
literature. This type of filter generates a pulse response of a finite length (e.g. in the
case of a block length of n = 64 or n = 256). In addition so-called IIR filters (infinite
impulse response) are also used. Here the same elementary processes are used but
as a result of feedback effects fewer processes - i.e. less calculation - are required
all in all for the design of the filter. However, the phase curve is no longer linear.
IIR filters are not dealt with here.