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) |

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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