Interpolation FIR filter design
b = intfilt(l,p,alpha)
b = intfilt(l,n,'Lagrange')
b = intfilt(l,p,alpha) designs
a linear phase FIR filter that performs ideal bandlimited interpolation
using the nearest
samples, when used on a sequence interleaved with
l samples. It assumes an original bandlimitedness
of alpha times the Nyquist frequency. The returned filter is identical
to that used by
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
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
nth-order Lagrange polynomial interpolation on a sequence interleaved
l-1 consecutive zeros every
b has length
n even, and length
n odd. If both
l are even, the
filter designed is not linear phase.
Both types of filters are basically lowpass and have a gain
l in the passband.
Design a digital interpolation filter to upsample a signal by seven, using the bandlimited method. Specify a "bandlimitedness" factor of 0.5 and use 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
xr = upsample(x,upfac);
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
The bandlimited method uses
design an interpolation FIR filter. The polynomial method uses Lagrange's
polynomial interpolation formula on equally spaced samples to construct
the appropriate filter.
Usage notes and limitations:
All inputs must be constant. Expressions or variables are allowed if their values do not change.