Digital filters - Signals, Processes, and Systems - page 321

1

Ungefiltertes Signal

A

Source signal

E

D

C

B

τ τ τ τ τ

∗ C ∗ C ∗ C ∗ C ∗ C

Adder

1

+

C Gefiltertes Signal

E

Filtered signal

A B

D

Illustration 212: Illustrating the process of convolution in the time domain

The block diagram shows the development of the convolution process but it is difficult to internalise this
pictorially. Imagine the (discrete) approximated Si-function as a template which moves from the very left to
the very right by the top signal, stopping briefly at each step from measurement to measurement. At each
position of the template the convolution operation represented by the block circuit is carried out.

At first only the first two values of the signal and “template” overlap, designated in the representation by
means of “1”. Then in the next step 2. 3 … up to a maximum of 64 values (length of the template is here
n = 64). At every step a kind of “weighted average value calculation“ is carried out, i.e. in the frequency
domain a lowpass filtering process. A maximum of 64 different values are adduced to calculate the
average value, i.e. 64 different measurements are simultaneously within the block circuit diagram.

Note the Si-shaped beginning of the filtered signal. It is a result of the pulse-like beginning of the unfiltered
signal. The letters designate the comparable segments of both signals.

Case study: Design and application of digital filters
The right instruments seem to be available in the convolution module to deploy effective
digital filters. As filtering corresponds to multiplication in the frequency domain, the
equivalent operation in the time domain represents convolution. Both processes are com-
pletely equivalent as far as their effect is concerned if the convolution function is the IFFT
of the filter function.