Systems, Sampling and Quantization
11
kernel of the transform
causality
stability
bounded-input
bounded-output
BIBO
absolutely summable
Theorem 1.2 (Shift Theorem) Given two domains related by a transform
operator, the shift by in one domain corresponds, in the transform domain, to
a multiplication by the kernel of the transform raised to the power .
We recall that the kernel of the Laplace transform
7
is e
-s
and the kernel of
the Z transform is z
-1
. The shift theorem can be easily justified in the discrete
domain starting from the definition of Z transform. Let x(n) be a discrete-time
signal, and let y(n) be its version shifted by an integer number of samples.
With the variable substitution N = n - we can produce the following chain
of identities, which proves the theorem:
Y (z) =
n=-
y(n)z
-n
=
n=-
x(n - )z
-n
=
N =-
x(N )z
-N -
= z
-
X(z) .
(28)
1.4.3
Stability and Causality
The notion of causality is rather intuitive: it corresponds to the experience of
exciting a system and getting its response back only in future time instants,
i.e. in instants that follow the excitation time along the time arrow. It is easy
to realize that, for an LTI system, causality is enforced by forbidding non-zero
values to the impulse response for time instants preceding zero. Non-causal
systems, even though not realizable by sample-by-sample processing, can be
of interest for non-realtime applications or where a processing delay can be
tolerated.
The notion of stability is more delicate and can be given in different ways.
We define the so-called bounded-input bounded-output (BIBO) stability, which
requires that any input bounded in amplitude might only produce a bounded
output, even though the two bounds can be different. It can be shown that hav-
ing BIBO stability is equivalent to have an impulse response that is absolutely
summable, i.e.
-
|h(n)| < .
(29)
In particular, a necessary condition for BIBO stability is that the impulse re-
sponse converges toward zero for time instants diverging from zero.
It is easy to detect stability on the complex plane for LTI causal systems [58,
66, 65]. In the continuous-time case, the system is stable if all the poles are on
the left of the imaginary axis or, equivalently, if the strip of convergence (see
appendix A.8.1) ranges from a negative real number to infinity. In the discrete-
time case, the system is stable if all the poles are within the unit circle or,
equivalently, the ring of convergence (see appendix A.8.3) has the inner radius
of magnitude less than one and the outer radius extending to infinity.
Stability is a condition that is almost always necessary for practical real-
izability of linear filters in computing systems. It is interesting to note that
physical systems can be locally unstable but, in virtue of the principle of energy
conservation, these instabilities must be compensated in other points of the sys-
tems themselves or of the other systems they are interacting with. However, in
7
This is the kernel of the direct transform, being e
s
the kernel of the inverse transform.
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