Savitzky-Golay filter design
b = sgolay(order,framelen)
b = sgolay(order,framelen,weights)
[b,g] = sgolay(...)
b = sgolay(order,framelen) designs
a Savitzky-Golay FIR smoothing filter with polynomial order
be less than
be odd. If
the designed filter produces no smoothing.
b, is a
whose rows represent the time-varying FIR filter coefficients. In
a smoothing filter implementation (for example,
sgolayfilt), the last
(each an FIR filter) are applied to the signal during the startup
transient, and the first
(framelen-1)/2 rows are
applied to the signal during the terminal transient. The center row
is applied to the signal in the steady state.
b = sgolay(order,framelen,weights) specifies
a weighting vector,
weights, with length
which contains the real, positive-valued weights to be used during
the least-squares minimization.
[b,g] = sgolay(...) returns
g of differentiation filters. Each column
g is a differentiation filter for derivatives
p is the
column index. Given a signal
x of length
you can find an estimate of the
xp, of its middle value from
xp((framelen+1)/2) = (factorial(p)) * g(:,p+1)' * x
Generate a signal that consists of a 0.2 Hz sinusoid embedded in white Gaussian noise and sampled five times a second for 200 seconds.
dt = 1/5; t = (0:dt:200-dt)'; x = 5*sin(2*pi*0.2*t) + randn(size(t));
sgolay to smooth the signal. Use 21-sample frames and 4th-order polynomials.
order = 4; framelen = 21; b = sgolay(order,framelen);
Compute the steady-state portion of the signal by convolving it with the center row of
ycenter = conv(x,b((framelen+1)/2,:),'valid');
Compute the transients. Use the last rows of
b for the startup and the first rows of
b for the terminal.
ybegin = b(end:-1:(framelen+3)/2,:) * x(framelen:-1:1); yend = b((framelen-1)/2:-1:1,:) * x(end:-1:end-(framelen-1));
Concatenate the transients and the steady-state portion to generate the complete smoothed signal. Plot the original signal and the Savitzky-Golay estimate.
y = [ybegin; ycenter; yend]; plot([x y]) legend('Noisy Sinusoid','S-G smoothed sinusoid')
Generate a signal that consists of a 0.2 Hz sinusoid embedded in white Gaussian noise and sampled four times a second for 20 seconds.
dt = 0.25; t = (0:dt:20-1)'; x = 5*sin(2*pi*0.2*t)+0.5*randn(size(t));
Estimate the first three derivatives of the sinusoid using the Savitzky-Golay method. Use 25-sample frames and 5th-order polynomials. Divide the columns by powers of
dt to scale the derivatives correctly.
[b,g] = sgolay(5,25); dx = zeros(length(x),4); for p = 0:3 dx(:,p+1) = conv(x, factorial(p)/(-dt)^p * g(:,p+1), 'same'); end
Plot the original signal, the smoothed sequence, and the derivative estimates.
plot(x,'.-') hold on plot(dx) hold off legend('x','x (smoothed)','x''','x''''', 'x''''''') title('Savitzky-Golay Derivative Estimates')
Savitzky-Golay smoothing filters (also called digital smoothing polynomial filters or least squares smoothing filters) are typically used to "smooth out" a noisy signal whose frequency span (without noise) is large. In this type of application, Savitzky-Golay smoothing filters perform much better than standard averaging FIR filters, which tend to filter out a significant portion of the signal's high frequency content along with the noise. Although Savitzky-Golay filters are more effective at preserving the pertinent high frequency components of the signal, they are less successful than standard averaging FIR filters at rejecting noise when noise levels are particularly high. The particular formulation of Savitzky-Golay filters preserves various moment orders better than other smoothing methods, which tend to preserve peak widths and heights better than Savitzky-Golay.
Savitzky-Golay filters are optimal in the sense that they minimize the least-squares error in fitting a polynomial to each frame of noisy data.
 Orfanidis, Sophocles J. Introduction to Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1996.