TAPER Instruction

TAPER( options ) cseries start end newcseries newstart

Tapers a complex series. A taper is applied to the unpadded part of a complex series prior to taking the Fourier Transform. The taper reduces the “window leakage” by scaling the ends of the data so they merge smoothly with the zeros on either side.

Use of a taper is more important when you are smoothing (with WINDOW) using a window with a sharp cutoff such as the FLAT window. It has much less of an effect with TRIANGULAR or QUADRATIC.

Parameters

cseries	source series
start, end	range to transform. By default, range of cseries, however, you will generally need these because the taper is applied to the unpadded portion of the data.
newcseries	result series. By default, overwrite cseries
newstart	start for newcseries, by default start

Options

TYPE=[TRAPEZOIDAL]/COSINE

This gives the type of taper. See "The TYPE option" for technical details.

Note that TAPER has no option analogous to the FORM option of WINDOW. Tapering is simply the multiplication of two series, so you can implement other tapering functions fairly easily using CSET or CMULTIPLY.

FRACTION=fraction of entries affected at each end [.25]

AFFECTED=number of entries affected at each end

Use one of these options to define the number of entries at each end of the series which are affected by the taper. This is the m in the formulas below. FRACTION is the most convenient, specifying a fraction of the overall length. The default is .25 of the length.

Variables Defined

%KAPPA	the factor to divide into %EBW and %EDF (computed by WINDOW) to correct for the effects of the taper (REAL)
%SCALETAP	sum of squared taper weights (REAL). Use this rather than the number of data points when scaling the periodogram.

The TYPE option

TAPER provides two types: TRAPEZOID and COSINE. The taper multiplies the input series by the following (m in these formulas is the number of affected entries at each end):

TRAPEZOID	\(b\left( t \right) = \left\{ {\begin{array}{*{20}{c}}{{\kern 1pt} t/m} & {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \le t \le m} \\ {{\kern 1pt} 1} & {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m + 1 \le t \le N - m} \\ {{\kern 1pt} \left( {N - t + 1} \right)/m} & {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N - m + 1 \le t \le N} \\\end{array}} \right.\)
COSINE	\(b\left( t \right) = \left\{ {\begin{array}{*{20}{c}}{0.5{\kern 1pt} {\kern 1pt} \left[ {1 - \cos \left( {\pi {\kern 1pt} t/m} \right)} \right]} & {1 \le t \le m} \\ {{\kern 1pt} 1} & {m + 1 \le t \le N - m} \\ {0.5{\kern 1pt} {\kern 1pt} \left[ {1 - \cos \left( {\pi \left( {N - t + 1} \right)/m} \right)} \right]} & {N - m + 1 \le t \le N} \\ \end{array}} \right.\)

The corresponding values for \(\kappa\) (%KAPPA) are

TRAPEZOID	\({\frac{{\left[ {1 - \left( {m/N} \right)\left( {8/5} \right)} \right]}}{{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left[ {1 - \left( {m/N} \right)\left( {4/3} \right)} \right]}^2}}}}\)
COSINE	\({\frac{{\left[ {1 - 2\left( {m/N} \right)\left( {93/128} \right)} \right]}}{{{{\left[ {1 - 2\left( {m/N} \right)\left( {5/8} \right)} \right]}^2}}}}\)

Example

This uses a cosine taper affecting 20% of the data on either end.

frequency 3 768

rtoc 1956:1 2002:12

# prices

# 1

taper(type=cosine,fraction=.20) 1 1956:1 2002:12

fft 1

cmult(scale=1./(2*%pi*%scaletap)) 1 1