Signals in the time and frequency domain - Signals, Processes, and Systems - page 40

The confusing phase spectrum
It is also possible to draw an important conclusion with regard to the phase spectrum. As
Illustration 34 shows, the same signal can have different phase spectra. The phase
spectrum depends on the time reference point t = 0.

By contrast, the amplitude spectrum is unaffected by time displacements.

For this reason the phase spectrum is more confusing and much less revealing than the
amplitude spectrum. Hence in the following chapters usually only the amplitude spectrum
will be demonstrated in the frequency domain.

Note:

• In spite of this, only the two spectral representations together provide all the informa-
tion on the progression of a signal/oscillation in the time domain. The inverse
FOURIER transformation IFT requires the amplitude and phase spectrum to calculate
the course of the signal in the time domain.

• The property of our ear (a FOURIER analyzer!) which scarcely perceives changes in
the phase spectrum of a signal is a particularly interesting phenomenon. Any
important change in the amplitude spectrum is immediately noticed. In this connec-
tion you should carry out acoustic experiments with DASY Lab .

Interference: nothing to be seen although everything is there.
The (periodic) rectangular pulses in Illustration 33 have a constant (positive or negative)
value during the pulse duration , but between pulses the value is zero. If we only

considered these periods of time T -

τ

, we might easily think that "there cannot be
anything there when the value is zero", i.e. no sinusoidal signals either.

τ

This would be fundamentally erroneous and this can be demonstrated experimentally. In
addition, the FOURIER Principle would be wrong (why?). One of the most important
principles of oscillation and wave physics is involved here:

(Sinusoidal) oscillations and waves may extinguish or strengthen
each other temporarily and locally (waves) by addition.

In wave physics this principle is called interference . Its importance for oscillation physics
and signal theory is too rarely pointed out.

Let us first off all look at Illustration 33 again. The cumulative curve of the first 16
harmonics has everywhere been - intentionally - included. We see that the sums of the
first 16 harmonics between the pulses equal zero only in a very few places (zero
crossings), otherwise they deviate a little from zero. Only the sum of an infinite number
of harmonics can result in zero. On the sinusoidal "playing field" we see that all the
sinusoidal signals of the spectrum remain unchanged during the entire period length T.