# Documentation

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# sgolay

Savitzky-Golay filter design

## Syntax

`b = sgolay(order,framelen)b = sgolay(order,framelen,weights)[b,g] = sgolay(...)`

## Description

`b = sgolay(order,framelen)` designs a Savitzky-Golay FIR smoothing filter with polynomial order `order` and frame length `framelen`. `order` must be less than `framelen`, and `framelen` must be odd. If `order` = `framelen-1`, the designed filter produces no smoothing.

The output, `b`, is a `framelen`-by-`framelen` matrix whose rows represent the time-varying FIR filter coefficients. In a smoothing filter implementation (for example, `sgolayfilt`), the last `(framelen-1)/2` rows (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 `framelen`, which contains the real, positive-valued weights to be used during the least-squares minimization.

`[b,g] = sgolay(...)` returns the matrix `g` of differentiation filters. Each column of `g` is a differentiation filter for derivatives of order `p-1`, where `p` is the column index. Given a signal `x` of length `framelen`, you can find an estimate of the `p`th order derivative, `xp`, of its middle value from

`xp((framelen+1)/2) = (factorial(p)) * g(:,p+1)' * x`

## Examples

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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)); ```

Use `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 `b`.

```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') ```

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### Tips

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.

## References

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