Convolution obeys commutative, distributive (over addition) and
associative properties in both continuous and discrete domains.
Commutativity implies the system with input signal x(t) and impulse response h(t) and the other with input signal h(t) and impulse response x(t) both give the same output y(t).
Distributivity implies a parallel combination of LTI systems can be replaced by an equivalent LTI system which is described by the sum of the individual impulse responses in the parallel combination.
Associativity implies the unit impulse response of a cascaded LTI system is independent of the order in which the individual LTI systems are connected.
A system is
memoryless if and only if h[n] = 0for all non-zero n
.
LTI system is invertible if the the convolution of the
impulse response and its inverse results in unit impulse
For a causal discrete time LTI system,
h[n] = 0 for all n<0. (similarly for continuous time)
For a stable system ,the impulse response
must be absolutely integrable.
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