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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 `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

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.

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

`filter`

| `fir1`

| `firls`

| `sgolayfilt`

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