Digital Signal Processing Reference
In-Depth Information
Note: In this Version of DASY
Lab
there is a so-called FFT filter which (at present)
does not correspond to the procedure shown here. It simply makes “cutting out” in
the frequency domain easier.
Digital filtering in the time domain
Hopefully you have not lost the ability to feel amazement at the unexpected! You will now
be introduced to digital filters which
•
require little calculation,
•
avoid the route via the frequency domain (FFT - IFFT),
•
In principle and in practice have no fixed blocklength/signal length,
•
for this reason can filter signals of any length directly,
•
are completely phase-linear,
•
can be designed with the desired edge-steepness (the Uncertainty
Principle is the only physical limit) and
•
make do with the three most elementary (linear) signalling processes:
addition, multiplication by a constant
and
delay
.
Perhaps you already sense how that might be possible. The input signal in the case of a
lowpass characteristic simply has to be “deformabled” in such a way that it has a “ripple
effect” which, for instance, in the case of a lowpass filter is equivalent to the highest
(cutoff) frequency of this lowpass (see in this connection for example Illustration 49 and
Illustration 201). All the prerequisites for this have, of course, already been dealt with.
They are summarized briefly in the following:
All digital signals represent a discrete sequence of (weighted)
δ
-pulses (see in this context
Illustration 188 and Illustration 189).
• The pulse response of a virtually ideal, i.e. rectangular lowpass filter must look
roughly like the Si-function (see, for example, Illustration 48 and Illustration 49).
• The pulse response of a virtually ideal i.e. rectangular bandpass filter is always roughly
an amplitude-modulated Si-function (see Illustration 119 and Illustration 206). The
mid-frequency of this bandpass is equivalent to the carrier frequency from Chapter
8, under the heading “Amplitude modulation”.
• A sampled Si-function can be seen as the pulse response of a digital lowpass (with
periodic, virtually rectangular spectra) (see Illustration 215 and Illustration 216). The
Sampling Principle must be adhered to to avoid these spectra overlapping.
The following conclusion results:
As a digital signal consists purely of weighted
-pulses a signal-
ling process is required which generates a discrete but time-
limited
δ
-pulse sequence which is as similar as possible to the
Si function from each of these
δ
δ
-pulses.
