Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 14

Practical Measurement of the Impulse Response

It can often be difficult to generate an analog impulse function accurately in
hardware. It is typically easier to generate the unit-step function. One can show
that the impulse response of the system h(t) is related to the unit-step response
according to (see Tutorial 8):

h ð t Þ¼ dq ð t Þ= dt : ð 1 : 6 Þ

1.2.2.2 Stability of Analog LTI Systems in the Time Domain

An analog system is BIBO stable if its impulse response is absolutely summable,
i.e., if

Z

1

j h ð t Þj \ 1: ð 1 : 7 Þ

1

Example 1 Consider the system described by the impulse response

h ð t Þ¼ e at ;

t 0

0 ; t\0 ;

where a is a positive constant. This system is causal (since h(t) = 0 for t \ 0) and
stable since R 1 1 j h ð t Þj¼ 1 = a\ 1: This system can be representative of a series
capacitor-resistor circuit.

Exercise. Find the output of the above system when the input is x(t) = cos(t).
Example 2 The system e -|t| is non-causal since h(t) = 0 for t \ 0, but it is stable
since R 1

1 j h ð t Þj¼ 2\ 1:
Although a time domain approach can be used for predicting the stability of
systems, it tends to be difficult to do so for complicated systems. The frequency
domain approach using the Laplace transform turns out to be more practical.
Frequency domain analysis is therefore considered next.

1.2.3 Frequency-Domain Representation

This section considers the representation and analysis of analog signals and sys-
tems in the frequency domain. Analysis in this domain is achieved with the help of
suitable transformations, which yield equivalent results to those that would be
obtained if time domain methods were used. The frequency domain, however, can
reveal further characteristics of signals and systems which are useful in dealing