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A More Elegant Derivation

Define the backwards difference operator $ \delta$ by

$\displaystyle \delta f(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;f(n) - f(n-1)

and the factorial polynomials (aka rising factorials or Pochhammer symbol) by

$\displaystyle x^{[N]} \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;x(x+1)(x+2)\cdots(x + N - 2) (x + N - 1)

These give a discrete-time counterpart to $ \frac{d}{dx}x^N = Nx^{N-1}$ , viz.,

$\displaystyle \delta\, x^{[N]} = N\, x^{[N-1]}

In these terms, a discrete-time Taylor series about $ n=k$ can be defined:

$\displaystyle \hat{f}(t) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sum_{n=0}^\infty [\delta^n f(k)]\frac{(t-k)^{[n]}}{n!}

$ N$ th-order Lagrange interpolation via truncated discrete-time Taylor series expansion about time $ n=k$ :

$\displaystyle \hat{f}(t) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sum_{n=0}^N [\delta^n f(k)]\frac{(t-k)^{[n]}}{n!}

Each term in the expansion can be computed recursively from the previous term:

\begin{eqnarray*}[\delta^n f(k)]\frac{(t-k)^{[n]}}{n!} &=& [\delta^{n-1} f(k)]\frac{(t-k)^{[n-1]}}{(n-1)!} \; \times\\
& & \zbox{\frac{t-k+N-1}{N}\cdot[\delta f(k)]}

This gives the same efficient computational form found previously:

\epsfig{file=eps/DepalleTassart.eps,width=\textwidth }
where $ \Delta \mathrel{\stackrel{\mathrm{\Delta}}{=}}k-t$ is the desired delay for fractional-delay filtering, and $ y_k(n)$ is the output signal for $ k$ th-order Lagrange interpolation (modular!). See also Newton's divided difference interpolation formula.

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``Bandlimited Interpolation, Fractional Delay Filtering, and Optimal FIR Filter Design'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2022-09-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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