causality

stability

bounded-input

absolutely summable

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 .

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:

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.

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.

sponse converges toward zero for time instants diverging from zero.

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.

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