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Quadratic Interpolation Used as Low Pass Filter

IP.com Disclosure Number: IPCOM000081329D
Original Publication Date: 1974-May-01
Included in the Prior Art Database: 2005-Feb-27
Document File: 3 page(s) / 50K

IBM

Related People

Esteban, D: AUTHOR [+2]

Abstract

Any analog signal may be recovered from its sampled form sampled at a Nyquist rate, by a so-called Shannon interpolation between the consecutive samples. Such an operation being particularly complex to implement, usually requires an approximation of the required D/A conversion or low-pass filtering through use of less complex interpolation, e.g., zero first or second-order interpolation. An improved second-order interpolation filter is suggested here.

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Quadratic Interpolation Used as Low Pass Filter

Any analog signal may be recovered from its sampled form sampled at a Nyquist rate, by a so-called Shannon interpolation between the consecutive samples. Such an operation being particularly complex to implement, usually requires an approximation of the required D/A conversion or low-pass filtering through use of less complex interpolation, e.g., zero first or second-order interpolation. An improved second-order interpolation filter is suggested here.

Consider a signal S whose samples at time t = nT, (n+1)T, etc..., where T is the sampling period are S(nT), S((n+1)T), etc... One way to obtain an analog signal from this sampled signal consists in interpolating over the known sample values. Considering the second-order interpolation, the corresponding expression is given by the following relationships: s(t) = AX/2/ + BX + C where X = t - nT 0 < or - X < the parameters A, B and C are determined by the following relationships: A = [((n+1)T) - 2 . S(nT) + S((n-1)T)] / 2T/2/ B = [((n+1(T) - S((n- 1)T)] / 2T C = S(nT)

These parameters correspond to an interpolation between instants nT and (n+1)T, the known values of the signal being S((n-1)T), S(nT) and S((n+1)T).

In fact, it is easy to find the values of parameters for an interpolation between instants (n-1)T and nT. In this way, it is sufficient to move the origin from instant nT to (n-1)T, it leads to: s(t') = A' X'/2/ + B' X' + C' where: X' = t' - (n-1)T t' = t - T then, A' = A B' = B - 2AT C' = C - BT + AT/2/.

In fact it can easily be identified that C' is equal to S((n-1)T); the corresponding impulse responses for both cases are related in such a manner that the second one is the first one but time reversed, consequently frequency responses have the same magnitude but are opposite in phase.

Moreover it can be easily shown that all these interpolations are equivalent to a linear filtering, so both impulses can be combined in order to get a minimal- phase filtering as for the first-order interpolation.

One peculiar and interesting case is the one corresponding to combining both previous impulse responses and to select the delay between these impulses. This produces a continuous function and provides a high out-of-band rejection >40 dB, instead of close to 25 dB as given by first and second-order interpolation. The...