Performance of Low-Order Polynomial Interpolators in the Presence of Oversampled Input

this paper compares the signal-to-noise ratio performance of several polynomial interpolator algorithms. Signal-to-noise analysis is extended to the improvement thereof as the input signal is ideally oversampled. The algorithms considered are drop-sample, linear, third-order Lagrange, Hermite, second-order osculating, and Bspline. The analysis reveals an interesting performance trade-off between signal-to-noise ratio and passband flatness. Motivation Interpolation is the method by which a function of known value at a sample of points is approximated at an unknown value of the independent variable, which, in audio signal processing, is often time. The points chosen for interpolation are the ones in closest proximity to the target time value. Interpolation employs the time values of the known points relative to the unknown point to calculate weights. The interpolated result is the dot product of the chosen points and the calculated weights. Polynomial interpolation uses polynomials of the time variable to determine the weights. Its principal advantage over table-lookup interpolation is that the time variable resolution of the interpolation function is arbitrary, whereas the resolution of table-lookup is restricted to the size of the table. Two widely employed interpolation methods, drop-sample and linear, fall into the zeroth and first order classes, respectively, of polynomial interpolators. For the application of audio signal resampling, the source to interpolate employs a constant sampling interval. The spectrum of this sampled signal has a periodicity of the sampling frequency. An interpolator acts as a (theoretically) continuous filter to convert the sampled source to a continuous signal, from which samples can be taken at the desired sampling interval. Therefore, an interpolator must be judged on how good a lowpass filter the continuous representation of the system becomes. Ideally, all frequencies beyond the Nyquist frequency, or aliasing frequencies, should be completely attenuated. However, no convolution with a function of finite length can attain this ideal frequency response. A good performance metric integrates the frequency response power over the passband, from negative to positive Nyquist, and the power over all other frequencies to attain a signal-to-noise ratio.