Mathematical modelling of Signals – Processes – Systems - Signals, Processes, and Systems - page 489

Basic characteristics of real signals and systems in the time domain
The importance of the impulse response h(t) has been dealt with in detail in Chapter 6. It
describes the reaction of a linear system simultaneously energized by any frequency of
equal amplitude. In addition, there is nothing hidden behind the filter coefficients of
digital filters, but the impulse response h(t) (see Chapter 10) . We shall shed more light
mathematically on these connections.

Firstly, in Illustration 314, step function s(t) and į- impulse į(t) are defined for the “test
signals” already dealt with in detail in Chapter 6. The step function and į- impulse as
limits raise some mathematical problems due to the singularities, but not from a real
physical point of view (see Chapter 6).

The sampling of a continuous function x(t) by į - impulses is of great practical relevance
here. If you now sample “continuously” over a very long period, after Norbert WIENER
a new perspective for the synthesis of time continuous signals results (see Illustration 37):

All continuous signals x(t) can be understood as being composed
of weighted į- impulses that lie infinitely close together
(sampling frequency f A tending to infinity).

This construction rivals the FOURIER- principle. Whilst in FOURIER signal synthesis
happens by “infinitely long” lasting sine oscillations, in the new perspective signals are
synthesized by “infinitely short”, closely packed single events .

This tremendous difference allows a new statistical perspective on continuous signals
offering a new access to information theory. This is also easily comprehensible from the
content point of view (see text at the end of Chapter 12):

At the receiver of information, there is uncertainty about the con-
dition of the next signal according the impossibility of prediction.
This leads into the mathematical methods of statistics and calcu-
lus of probability. How should this uncertainty be demonstrated
by sine functions, whose curves are predictable to the infinite?

As a conclusion of WIENER´s demonstration of continuous signals

∞

( )

³

x t () = x ( ) τδ t − τ d τ

−∞

(see Illustration 315) is one of the most important processes in the time domain (as well
as in the frequency domain).

The impulse response h(t) includes all the information of the
system in the time domain and frequency domain (transfer
function!). The overlapping of all impulse responses of the closely
p acked weighted į- impulses of x(t) has therefore to demonstrate
the response y(t) of the system to the input signal x(t).