A More Elegant Derivation

Define the backwards difference operator $\delta$ by

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

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

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

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

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

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

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

• Known as ``Newton's Backward Difference Formula''

• Truncating this expansion at $n=N$ again yields $N$ th-order Lagrange interpolation on uniformly spaced samples

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

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

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

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

This gives the same efficient computational form found previously:

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.