Documentation

# intfilt

Interpolation FIR filter design

## Syntax

```b = intfilt(l,p,alpha) b = intfilt(l,n,'Lagrange') ```

## Description

`b = intfilt(l,p,alpha)` designs a linear phase FIR filter that performs ideal bandlimited interpolation using the nearest `2`*`p` nonzero samples, when used on a sequence interleaved with `l-1` consecutive zeros every `l` samples. It assumes an original bandlimitedness of alpha times the Nyquist frequency. The returned filter is identical to that used by `interp`. `b` is length 2*`l`*`p-1`.

alpha is inversely proportional to the transition bandwidth of the filter and it also affects the bandwidth of the don't-care regions in the stopband. Specifying alpha allows you to specify how much of the Nyquist interval your input signal occupies. This is beneficial, particularly for signals to be interpolated, because it allows you to increase the transition bandwidth without affecting the interpolation and results in better stopband attenuation for a given `l` and `p`. If you set alpha to 1, your signal is assumed to occupy the entire Nyquist interval. Setting alpha to less than one allows for don't-care regions in the stopband. For example, if your input occupies half the Nyquist interval, you could set alpha to 0.5.

`b = intfilt(l,n,'Lagrange')` designs an FIR filter that performs `n`th-order Lagrange polynomial interpolation on a sequence interleaved with `l-1` consecutive zeros every `l` samples. `b` has length `(n+1)`*`l` for `n` even, and length `(n+1)`*`l-1` for `n` odd. If both `n` and `l` are even, the filter designed is not linear phase.

Both types of filters are basically lowpass and have a gain of `l` in the passband.

## Examples

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Design a digital interpolation filter to upsample a signal by seven, using the bandlimited method. Specify a "bandlimitedness" factor of 0.5 and use $2×2$ samples in the interpolation.

```upfac = 7; alpha = 0.5; h1 = intfilt(upfac,2,alpha);```

The filter works best when the original signal is bandlimited to `alpha` times the Nyquist frequency. Create a bandlimited noise signal by generating 200 Gaussian random numbers and filtering the sequence with a 40th-order FIR lowpass filter. Reset the random number generator for reproducible results.

```lowp = fir1(40,alpha); rng('default') x = filter(lowp,1,randn(200,1));```

Increase the sample rate of the signal by inserting zeros between each pair of samples of `x`.

`xr = upsample(x,upfac);`

Use the `filter` function to produce an interpolated signal.

`y = filter(h1,1,xr);`

Compensate for the delay introduced by the filter. Plot the original and interpolated signals.

```delay = mean(grpdelay(h1)); y(1:delay) = []; stem(1:upfac:upfac*length(x),x) hold on plot(y) xlim([400 700])``` `intfilt` also performs Lagrange polynomial interpolation.

• First-order polynomial interpolation is just linear interpolation, which is accomplished with a triangular filter.

• Zeroth-order interpolation is accomplished with a moving average filter and resembles the output of a sample-and-hold display.

Interpolate the original signal and overlay the result.

```h2 = intfilt(upfac,1,'Lagrange'); y2 = filter(h2,1,xr); y2(1:floor(mean(grpdelay(h2)))) = []; plot(y2) hold off``` ## Algorithms

The bandlimited method uses `firls` to design an interpolation FIR filter. The polynomial method uses Lagrange's polynomial interpolation formula on equally spaced samples to construct the appropriate filter.