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Savitzky-Golay filtering



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.


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.


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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);

hold on

Figure contains an axes. The axes 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');


Figure contains an axes. The axes 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;

hold off

Figure contains an axes. The axes 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.

Load a speech signal sampled at Fs=7418Hz. 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);

axis([0.2 0.22 -3 2])

axis([0.2 0.22 -3 2])

Figure contains 2 axes. Axes 1 with title Original contains an object of type line. Axes 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));

hold on
axis([0.2 0.22 -3 2])
hold off

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

Input Arguments

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Input signal, specified as a vector or matrix.

Data Types: single | double

Polynomial order, specified as a positive integer. order must be smaller than framelen. If order = framelen – 1, the filter produces no smoothing.

Data Types: single | double

Frame length, specified as a positive odd integer.

Data Types: single | double

Weighting array, specified as a real positive vector or matrix of length framelen.

Data Types: single | double

Dimension to filter along, specified as a positive integer scalar. By default, sgolayfilt operates along the first dimension of x whose size is greater than 1.

Data Types: single | double

Output Arguments

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Filtered signal, returned as a vector or matrix.


Savitzky-Golay smoothing filters are typically used to "smooth out" a noisy signal whose frequency span (without noise) is large. They are also called digital smoothing polynomial filters or least-squares smoothing filters. Savitzky-Golay filters perform better in some applications than standard averaging FIR filters, which tend to filter high-frequency content along with the noise. Savitzky-Golay filters are more effective at preserving high frequency signal components but less successful at rejecting noise.

Savitzky-Golay filters are optimal in the sense that they minimize the least-squares error in fitting a polynomial to frames of noisy data. See sgolay for more information about the Savitzky-Golay algorithm.


[1] Orfanidis, Sophocles J. Introduction to Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1996.

[2] Schafer, Ronald. “What Is a Savitzky-Golay Filter? [Lecture Notes].” IEEE Signal Processing Magazine 28, no. 4 (July 2011): 111–17.

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