Natural Basis

The natural basis for a discrete-time signal $x(n)$ is the set of shifted impulses:

$$\varphi_k \isdefs [\ldots, 0,\underbrace{1}_{k^{\hbox{\tiny th}}},0,\ldots],$$ (12.108)

or,

$$\varphi_k (n) \isdefs \delta(n-k)$$ (12.109)

for all integers $k$ and $n$ . The basis set is orthonormal since $\left<\varphi_k ,\varphi_l \right>=\delta(k-l)$ . The coefficient of projection of $x$ onto $\varphi_k$ is given by

$$\left<\varphi_k ,x\right> \eqsp x(k)$$ (12.110)

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

$$x \eqsp \sum_{k=-\infty}^{\infty}x(k) \varphi_k ,$$ (12.111)

i.e.,

$$x(n) = \sum_{k=-\infty}^{\infty}x(k) \varphi_k (n)$$

$$= \sum_{k=-\infty}^{\infty}x(k) \delta(n-k) \isdefs (x \ast \delta)(n).$$

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