Signals in the time and frequency domain - Signals, Processes, and Systems - page 44

Illustration 37: Signal synthesis by means of δ -pulses

Here a sine wave is "assembled" from -pulses of an appropriate magnitude following on each other. This

is exactly equivalent to the procedure in "digital signal processing" (DSP). Their signals are equivalent to
"strings of numbers" which, seen from a physical point of view, are equivalent to a rapid sequence of
measurements of an analog signal; every number gives the "weighted" value of the

δ

δ -pulse at a given point

of time t.

This strange relationship between sinusoidal and needle functions (Uncertainty Principle)
will be looked at more closely and evaluated in the next chapter.

Note:

Certain mathematical subtleties result in the -pulse being theoretically given an

amplitude tending to infinity. Physically this also makes a certain sense. An
"infinitely short" needle pulse cannot have energy unless it were "infinitely high".
This is also shown by the spectra of narrow periodic rectangular pulses and the
spectra of

δ

-pulses. The amplitudes of individual sinusoidal signals are very small
and hardly visible in the Illustrations, unless we increase the pulse amplitude (to
extend beyond the screen of the PC).

δ

For purposes of Illustration we normally choose δ -pulses of magnitude "1" in this topic.