10
D. Rocchesso: Sound Processing
impulse response
convolution
transfer function
shift operation
As we have already mentioned for continuous-time systems, there are two
important system descriptions: the impulse response and the transfer function.
LTI discrete-time systems are completely described by either one of these two
representations.
1.4.1
The Impulse Response
Any input sequence can be expressed as a weighted sum of discrete impulses
properly shifted in time. A discrete impulse is defined as
(n) =
1 n = 0
0 n = 0
.
(25)
If the impulse (25) gives as output a sequence (called, indeed, the impulse re-
sponse) h(n) defined in the discrete domain, then a linear combination of shifted
impulses will produce a linear combination of shifted impulse responses. There-
fore, it is easy to be convinced that the output can be expressed by the following
general convolution
6
:
y(n) = (h x)(n) =
m
x(m)h(n - m) =
m
h(m)x(n - m) ,
(26)
which is the discrete-time version of (6).
The Z transform H(z) of the impulse response is called transfer function of
the LTI discrete-time system. By analogy to what we showed in sec. 1.1, the
input-output relationship for LTI systems can be described in the transform
domain by
Y (z) = H(z)X(z) ,
(27)
where the input and output signals X(z) and Y (z) have been capitalized to
indicate that these are the Z transforms of the signals themselves.
The following general rule can be given:
A linear and time-invariant system working in continuous or discrete time
can be represented by an operation of convolution in the time domain or,
equivalently, by a complex multiplication in the (respectively Laplace or
Z) transform domain. The results of the two operations are related by a
(Laplace or Z) transform.
Since the transforms can be inverted the converse statement is also true:
The convolution between two signals in the transform domain is the trans-
form of a multiplication in the time domain between the antitransforms
of the signals.
1.4.2
The Shift Theorem
We have seen how two domains related by a transform operation such as the
Z transform are characterized by the fact that the convolution in one domain
corresponds to the multiplication in the other domain. We are now interested
to know what happens in one domain if in the other domain we perform a shift
operation. This is stated in the
6
The reader is invited to construct an example with an impulse response that is different
from zero only in a few points.
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