## Name

POLY_SMOOTH

## Purpose

Apply a least-squares (Savitzky-Golay) polynomial smoothing filter

## Explanation

Reduce noise in 1-D data (e.g. time-series, spectrum) but retain

dynamic range of variations in the data by applying a least squares

smoothing polynomial filter,

Also called the Savitzky-Golay smoothing filter, cf. Numerical

Recipes (Press et al. 1992, Sec.14.8)

The low-pass filter coefficients are computed by effectively

least-squares fitting a polynomial in moving window,

centered on each data point, so the new value will be the

zero-th coefficient of the polynomial. Approximate first derivates

of the data can be computed by using first degree coefficient of

each polynomial, and so on. The filter coefficients for a specified

polynomial degree and window width are computed independent of any

data, and stored in a common block. The filter is then convolved

with the data array to result in smoothed data with reduced noise,

but retaining higher order variations (better than SMOOTH).

This procedure became partially obsolete in IDL V5.4 with the

introduction of the SAVGOL function, which computes the smoothing

coefficients.

## Calling Sequence

spectrum = poly_smooth( data, [ width, DEGREE = , NLEFT = , NRIGHT =

DERIV_ORDER = ,COEFF = ]

## Inputs

data = 1-D array, such as a spectrum or time-series.

width = total number of data points to use in filter convolution,

(default = 5, using 2 past and 2 future data points),

must be larger than DEGREE of polynomials, and a guideline is to

make WIDTH between 1 and 2 times the FWHM of desired features.

## Optional Input Keywords

DEGREE = degree of polynomials to use in designing the filter

via least squares fits, (default DEGREE = 2)

The higher degrees will preserve sharper features.

NLEFT = # of past data points to use in filter convolution,

excluding current point, overrides width parameter,

so that width = NLEFT + NRIGHT + 1. (default = NRIGHT)

NRIGHT = # of future data points to use (default = NLEFT).

DERIV_ORDER = order of derivative desired (default = 0, no derivative).

## Optional Output Keyword

COEFFICIENTS = optional output of the filter coefficients applied,

but they are all stored in common block for reuse, anyway.

## Results

Function returns the data convolved with polynomial filter coefs.

## Example

Given a wavelength - flux spectrum (w,f), apply a 31 point quadratic

smoothing filter and plot

IDL> cgplot, w, poly_smooth(f,31)

## Common Blocks

common poly_smooth, degc, nlc, nrc, coefs, ordermax

## Procedure

As described in Numerical Recipies, 2nd edition sec.14.8,

Savitsky-Golay filter.

Matrix of normal eqs. is formed by starting with small terms

and then adding progressively larger terms (powers).

The filter coefficients of up to derivative ordermax are stored

in common, until the specifications change, then recompute coefficients.

Coefficients are stored in convolution order, zero lag in the middle.

## Modification History

Written, Frank Varosi NASA/GSFC 1993.

Converted to IDL V5.0 W. Landsman September 1997

Use /EDGE_TRUNCATE keyword to CONVOL W. Landsman March 2006