A3 Recursive Filtering

The following routine performs the in-place determination of a 1D sequence of interpolation coefficients {cn} from a sequence of data samples {fk}. The returned coefficients {cn} satisfy ft = E cnÇ(k - n) Vk eZ, neZ

where the synthesis function ^ is represented by its poles. The values of these poles for B-splines of degree n e {2, 3,4, 5} and for cubic o-Moms are available in Table 3. (The B-spline poles of any degree n> 1 can be shown to be real and lead to a stable implementation.) This routine implicitly assumes that the finite sequence of physical samples, known on the support X = [0, N — 1], is extended to infinity by the following set of boundary conventions:

This scheme is called mirror (or symmetric) boundaries. The first relation expresses that the extended signal is symmetric with respect to the origin, and the second mirrors the extended signal around x = N — 1. Together, these relations associate to any abscissa y e R\X some specific value f (y) =f (x) with xeX. The resulting extended signal is (2N — 1)-periodic, and has no additional discontinuities with respect to those that might already exist in the finite-support version of / This is generally to be preferred to both zero-padding and simple periodization of the signal, because these latter introduce discontinuities at the signal extremities.

The following routine is a digital filter. The z-transform of its function transfer is given by z(1 - z,)(1 - z-1) 5 (Z)~ Ü (z - z,)(z - z-,) •

As is apparent from this expression, the poles of this filter are power-conjugate, which implies that every second pole is outside the unit circle. To regain stability, this meromorphic filter is realized as a cascade of causal filters that are implemented as a forward recursion, and anticausal filters that are implemented as a backward recursion. More details are available in [19,20].

TABLE 3 Value of the B-spline poles required for the recursive filtering routine

Zl 22

ß4 a/664 — V4S8976 + VSOi — 19 V¡64 + V4S8976 — VSOi — 19

ß5 2 (V27O — V7O98O + v/ÎO5 — 1s) i ^27O + V7O98O — v/ÎO5 — 1s)

TABLE 4 Some synthesis functions



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