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Savitzky-Golay filtering

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

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

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

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

applies
a Savitzky-Golay FIR smoothing filter to the data in vector `x`

.
If `x`

is a matrix, `sgolayfilt`

operates
on each column. The polynomial order, `order`

, must
be less than the frame length, `framelen`

, and in
turn `framelen`

must be odd. If `order`

= `framelen-1`

, the
filter produces no smoothing.

`y = sgolayfilt(x,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. If `weights`

is not
specified, or if it is specified as empty, `[]`

,
it defaults to an identity matrix.

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

specifies
the dimension, `dim`

, along which the filter operates.
If `dim`

is not specified, `sgolayfilt`

operates
along the first nonsingleton dimension; that is, dimension 1 for column
vectors and nontrivial matrices, and dimension 2 for row vectors.

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.

Savitzky-Golay filters are optimal in the sense that they minimize the least-squares error in fitting a polynomial to frames of noisy data.

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

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