2.4 Signal Processing Transformations
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41
time t/s
Fig. 2.22: Window functions for the short-time Fourier transform (STFT): The window width corres-
ponds to the time resolution ∆t of the STFT, ∆t can be varied via the parameterisation in the func-
tional description of the window functions.
time information, i.e. no information about which part of the spectrum originates from
which time section of the signal. In order to determine the spectrum of a signal at least
in sections, the short-time Fourier transform (STFT) is available. For this purpose, the
signal is multiplied by a window function w(t), which sets all signal components out
of the window function to zero. With the signal "windowed" in this way, the Fourier
transform yields only the spectrum of the section that lies in the window. Then the
window is shifted and the spectrum of the next section is calculated. The window
width is freely selectable and corresponds to the time resolution ∆t of the STFT. Com-
monly used window functions are known as Hamming, Hanning, Blackman-Harris or
Gaussian windows (see Figure 2.22).
An important property of such windows is that the signal is gently guided towards
zero at the edges of the window. Otherwise, jumps could occur at the edges, which
would result in an infinitely extended spectrum. Therefore, the square wave function
is usually ruled out as a window function. Limiting the time resolution to ∆t, accord-
ing to the uncertainty principle of communications engineering¹⁶, a limited frequency
resolution results, since the product of time and frequency resolution cannot fall be-
low a certain value. A frequently chosen definition of bandwidth and signal duration
leads to the uncertainty condition
∆t ∆f = 1 .
(2.73)
Accordingly, frequency and time accuracy cannot be high for the same windows para-
meters. If a high time resolution is required, i.e. ∆t is small, the frequency resolution
16 The uncertainty principle of communications engineering was formulated by Karl Küpfmüller
(1897 - 1977).