# ZINTERP

## Purpose:

Interpolates a series to a new sample rate by FFT zero insertion.

## Syntax:

ZINTERP(series, r)

 series - An input series or array. r - A real. The new sample rate of the interpolated series, r > rate(series). Defaults to 2*rate(series).

## Returns:

A series or array.

## Example:

W1: gsin(64, 1/64, 3)

W2: zinterp(W1, 4*rate(W1))

W1 contains 64 samples of a 3 Hz sine wave sampled at 64 Hz.

W2 produces a 3 Hz sine wave with an interpolated sample rate of 64 * 4 = 256 Hz. The length is 253 samples.

## Example:

W3: zinterp(W1, 100)

produces a 99 point interpolated 3 Hz sine wave with a sample rate of 100 Hz.

## Remarks:

ZINTERP effectively resamples the input series to the higher rate R using ideal sinx/sin interpolation. The interpolation is calculated by the following remarkably simple and efficient method:

 1 For series s, the FFT is calculated. 2 N zeros are inserted into the FFT starting at the Nyquist frequency, Fn = 0.5 * rate(s).   N is determined such that:     L / R = length(s) / rate(s)   where L is the length of the output series. Since:     L = length(s) + N   we have:      N = ((R * length(s)) / rate(s)) - length(s) 3 The IFFT of the inserted series is computed to produce the interpolated time domain series.

The zero insertion step is equivalent to convolving the input series with a symmetric "continuous" periodic sinx/sin window of the same length as the output series and then sampling this "continuous" waveform at the new rate. This is the precise definition of ideal sinx/sin interpolation for a periodic time series. If the input series is band limited, that is, if the series can be thought of as having been obtained by sampling a continuous time signal at rate Fs and

X(f) = 0 for f > 0.5 * Fs

where X(f) is the Fourier Transform of the continuous time signal, then the interpolation will be exact (within numerical roundoff errors).

Although the output rate R is NOT required to be an integer multiple of the input sample rate, the relation:

R / rate(s) = L / length(s)

must hold, so the actual output rate might differ from R.

Sinx/sin interpolation can be thought of as periodic sinx/x interpolation, i.e. for periodic waveforms, sinx/x interpolation is identical to sinx/sin interpolation. The sinx/sin function acts as a periodic version of the sinc (sinx/x) function.

For non-periodic waveforms, sinx/sin interpolation produces the same result as sinx/x interpolation to within a few percent.

ZINTERP also works on arrays.

See RESAMPLE to resample a series to a specific sample rate using a variety of optional methods.