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sgolayfilt

Savitzky-Golay filtering

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Syntax

y = sgolayfilt(x,order,framelen)

y = sgolayfilt(x,order,framelen,weights)

y = sgolayfilt(x,order,framelen,weights,dim)

Description

example

y = sgolayfilt(x,order,framelen) applies a Savitzky-Golay finite impulse response (FIR) smoothing filter of polynomial order order and frame length framelen to the data in vector x. If x is a matrix, then sgolayfilt operates on each column.

example

y = sgolayfilt(x,order,framelen,weights) specifies a weighting vector to use during the least-squares minimization.

y = sgolayfilt(x,order,framelen,weights,dim) specifies the dimension along which the filter operates.

Examples

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Steady-State and Transient Savitzky-Golay Filters

Open Live Script

Generate a random signal and smooth it using sgolayfilt. Specify a polynomial order of 3 and a frame length of 11. Plot the original and smoothed signals.

order = 3;
framelen = 11;

lx = 34;
x = randn(lx,1);

sgf = sgolayfilt(x,order,framelen);

plot(x,':')
hold on
plot(sgf,'.-')
legend('signal','sgolay')

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent signal, sgolay.

The sgolayfilt function performs most of the filtering by convolving the signal with the center row of B, the output of sgolay. The result is the steady-state portion of the filtered signal. Generate and plot this portion.

m = (framelen-1)/2;

B = sgolay(order,framelen);

steady = conv(x,B(m+1,:),'same');

plot(steady)
legend('signal','sgolay','steady')

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent signal, sgolay, steady.

Samples close to the signal edges cannot be placed at the center of a symmetric window and have to be treated differently.

To determine the startup transient, matrix multiply the first (framelen-1)/2 rows of B by the first framelen samples of the signal.

ybeg = B(1:m,:)*x(1:framelen);

To determine the terminal transient, matrix multiply the final (framelen-1)/2 rows of B by the final framelen samples of the signal.

yend = B(framelen-m+1:framelen,:)*x(lx-framelen+1:lx);

Concatenate the transients and the steady-state portion to generate the complete signal.

cmplt = steady;
cmplt(1:m) = ybeg;
cmplt(lx-m+1:lx) = yend;

plot(cmplt)
legend('signal','sgolay','steady','complete')
hold off

Figure contains an axes object. The axes object contains 4 objects of type line. These objects represent signal, sgolay, steady, complete.

Adding weights to the minimization breaks the symmetry of B and requires extra steps for a proper solution.

Savitzky-Golay Filtering of Speech Signal

Open Live Script

Load a speech signal sampled at $F_{s} = 7418 Hz$ . The file contains a recording of a female voice saying the word "MATLAB®."

load mtlb
t = (0:length(mtlb)-1)/Fs;

Smooth the signal by applying a Savitzky-Golay filter of polynomial order 9 to data frames of length 21. Plot the original and filtered signals. Zoom in on a 0.02-second interval.

rd = 9;
fl = 21;

smtlb = sgolayfilt(mtlb,rd,fl);

subplot(2,1,1)
plot(t,mtlb)
axis([0.2 0.22 -3 2])
title('Original')
grid

subplot(2,1,2)
plot(t,smtlb)
axis([0.2 0.22 -3 2])
title('Filtered')
grid

Figure contains 2 axes objects. Axes object 1 with title Original contains an object of type line. Axes object 2 with title Filtered contains an object of type line.

Repeat the calculation, but now use a Kaiser window as a weighting vector. Specify a shape factor $β = 38$ . Plot the new filtered signal.

kmtlb = sgolayfilt(mtlb,rd,fl,kaiser(fl,38));

subplot(2,1,2)
hold on
plot(t,kmtlb)
axis([0.2 0.22 -3 2])
hold off

Figure contains 2 axes objects. Axes object 1 with title Original contains an object of type line. Axes object 2 with title Filtered contains 2 objects of type line.