The Uncertainty Principle - Signals, Processes, and Systems - page 76

On the one hand, nature appears to make use of this principle: in the case of the prism for
example sunlight is dissected into its spectral and sinusoidal components and even our ear
functions as a FOURIER-analyzer and can only hear sinusoidal signals. On the other hand
the structural components of all signals are accordingly sinusoidal signals which are, how-
ever, by definition infinitely long. But how can our ear perceive short time signals, if each
of the audible signals is of infinite duration!? At a concert we hear the music without inter-
ruption and not only after the signal or the concert has ended.
In 1946 the physicist Dennis GABOR (inventor of the holography) was the first to attempt
to deal with this enigma experimentally by dissecting longer signals in overlapping
GAUSSian time windows (see Illustration 52 to Illustration 54) then carrying out a signal
process known in the scientific literature as a STFFT (Short Time Fast FOURIER Trans-
formation).
Illustration 45 and Illustration 46 show in principle what happens in this case. A “short
time sine” (burst) creates in the frequency domain an “uncertain” line, i.e. a band of “very
many” frequencies or sinusoidal signals which lie close to each other. The overlapping
(interference) of these “infinitely long sinusoidal signals” results apparently in mathemat-
ically and physically correct fashion in a sum signal of finite duration for the period of
time in which they are audible and the mutual extinguishing of these sinusoidal signals
before and after this period of time. In the case of a glottal stop the duration is extremely
short which automatically leads to a broad frequency band.
Hence, if a short time window is selected, it is possible to deter-
mine relatively accurately at what point in time a relatively broad
band of neighbouring frequencies was audible in reality and did
not mutually extinguish each other by interference.
If, in contrast to this, a longer time window is chosen it is possible
to specify relatively imprecisely when a comparatively narrow
band of neighbouring frequencies was audible in real terms and
did not extinguish each other by interference.
Illustration 59 shows this using the example of an “artificial signal” which consists of five
samples of noise of equal length and four sinusoidal signals of different frequencies. Here
the so-called “cascade-representation” of DASY Lab is used in which the way the indivi-
dual signal segments change “over time” is shown apparently three-dimensionally in the
time domain.
In Illustration 60 the frequency domain of these numerous time-staggered GAUSSian
windows is also arranged three-dimensionally. This is referred to as a frequency-time
landscape. The difference in the frequency dissolution between short time and longer time
signal windows is clearly recognizable. More information is to be found in the caption
text.
This technique is used in a practical way in the next Chapter “Language as a carrier of
Information”. It is state of the art in all forms of voice recognition.
Short Time FFT (STFFT) has been used for roughly thirty years to make a window “slide”
over different frequency and time segments.