Digitalisation - Signals, Processes, and Systems - page 302

Signal distortion as a result of signal windowing
In conclusion we should like to return to the digital processing of long-lasting analog
signals - for instance an audio signal.

The digital processing of long-lasting real analog signals always
implies - as in the process of hearing - the signal processing of
equally long overlapping signal segments.

Here is a brief summary based on the descriptions in Chapter 3 (see Illustration 51 -
Illustration 54) and in Chapter 4 (see Illustration 68 - Illustration 72):

• Long-lasting signals must be analysed and processed in blocks. This block length must
always be presentable as a power of two (e.g. 1024 = 2 10 ) because the FFT is only
optimised for these block lengths.

• The block length and sampling rate - in accordance with the Sampling Principle -
should as far as possible be synchronised with each other so as not to conflict with
period length T D of the digital signal. This always applies if the signal processing in-
cludes the frequency domain. Remember that only integer multiples of the basic
frequency f D = 1/T D can be shown in the spectrum. Select the sampling rate f S if
possible as a power of 2.

• The windows must strongly overlap otherwise information is lost which was contained
in the separate time segments. The information is contained in the overall signal.

• The problem can be solved by recourse to our fundamental principles. As you know
information-bearing signals can according to the FOURIER Principle be understood
as consisting of nothing but sinusoidal signals of a certain bandwidth. If the transmis-
sion of these sinusoidal signals is guaranteed this also holds for the information they
are transporting.

• For long-lasting signals it is only possible to use a window function in which the
windowed signal begins and ends gently. Rectangular windows produce signal steps
which have nothing to do with the original curve (see Illustration 203). GAUSSian
windows are an appropriate example.

• The overlapping required can be visualised and assessed via a frequency-time
landscape. Thus it can be seen from Illustration 73 that a shorter overlapping of the
windows would not give more information on the frequency-time landscape. In
Illustration 72 there is by contrast a need for more information. Apparently the
overlapping of the windows is still too far apart.

• The overlapping required can be more precisely estimated by means of the Uncertainty
Principle:

• A window with the length Δ t necessarily produces a frequency uncer-

tainty t. The selection of the “window length” determines the

frequency resolution - independently of the bandwidth B of the signal!

Δ f ≥ 1/ Δ

• The defining of the bandwidth B also determines the highest frequency

f max which is to be recorded in terms of information. The fastest time