#### Natural Basis

The natural basis for a discrete-time signal is the set of shifted impulses:

 (12.108)

or,

 (12.109)

for all integers and . The basis set is orthonormal since . The coefficient of projection of onto is given by

 (12.110)

so that the expansion of in terms of the natural basis is simply

 (12.111)

i.e.,

This expansion was used in Book II [263] to derive the impulse-response representation of an arbitrary linear, time-invariant filter.

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