2
D. Rocchesso: Sound Processing
pole
zero
impulse response
frequency response
convolution
Here, as in most of the book, we implicitly assume that the initial conditions are
zero, otherwise eq. (2) should also contain a term in y(0). From the algebraic
equation (2) the transfer function is derived as the ratio between the output
and input transforms:
H(s) =
1
s - s
0
.
(3)
###
The coefficient s
0
, root of the denominator polynomial of (3), is called the
pole of the transfer function (or pole of the system). Any root of the numerator
would be called a zero of the system.
The inverse Laplace transform of the transfer function is an equivalent de-
scription of the system. In the case of example 1.1, it takes the form
h(t) =
e
s
0
t
t 0
0
t < 0
,
(4)
and such function is called a causal exponential.
In general, the function h(t), inverse transform of the transfer function, is
called the impulse response of the system, since it is the output obtained from
the system as a response to an ideal impulse
The two equivalent descriptions of a linear system in the time domain (im-
pulse response) and in the Laplace domain (transfer function) correspond to
two alternative ways of expressing the operations that the system performs in
order to obtain the output signal from the input signal.
The description in the Laplace domain leads to simple multiplication between
the Laplace transform of the input and the system transfer function:
Y (s) = H(s)X(s) .
(5)
This operation can be interpreted as multiplication in the frequency domain
if the complex variable s is replaced by j, being the real variable of the
Fourier domain. In other words, the frequency interpretation of (5) is obtained
by restricting the variable s from the complex plane to the imaginary axis. The
transfer function, whose domain has been restricted to j is called frequency
response. The frequency interpretation is particularly intuitive if we imagine
the input signal as a complex sinusoid e
j
0
t
, which has all its energy focused
on the frequency
0
(in other words, we have a single spectral line at
0
). The
complex value of the frequency response (magnitude and phase) at the point
j
0
corresponds to a joint magnitude scaling and phase shift of the sinusoid at
that frequency.
The description in the time domain leads to the operation of convolution,
y(t) = (h x)(t) =
+
-
h(t - )x( )d .
(6)
1
A rigorous definition of the ideal impulse, or Dirac function, is beyond the scope of this
book. The reader can think of an ideal impulse as a signal having all its energy lumped at the
time instant 0.
2
The convolution will be fully justified for discrete-time systems in section 1.4. Here, for
continuous-time systems, we give only the definition.