Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 165

the Parzen frequency window is plotted logarithmically, as a fraction of the peak
height of the main lobe.
The height of the n th sidelobe of the Parzen frequency window is approximately

4

3 T
8

2

, n = 1,2,3,.... (2.285)

(2 n + 1)π

The first sidelobe is only 0.2% of the height of the main lobe, and the sidelobes
fall o inversely as the fourth power of the distance along the frequency axis from

the centre point of the window. The width of the main lobe is twice that of the
Bartlett frequency window, as the width of the main lobe of the Bartlett window is
twice that of the boxcar frequency window. The suppression of frequency mixing
has been accomplished at the cost of a slight decrease in frequency resolution.
The half-power points of the Parzen frequency window are 1.82/ T apart on the
frequency axis, giving an indication of the frequency resolution limit imposed by
the use of this window to suppress frequency mixing in finite records.
Window carpentry is a highly developed art, but improvements over the Parzen
window are marginal. In the time domain the Parzen window is conveniently a
piecewise cubic. We will retain the Parzen window as our commonly used window.
The subroutine WINDOW calculates the Parzen window value W, at time T, for a
window of length AM, centred at time T0.

ff

SUBROUTINE WINDOW(T,T0,AM,W)

C
C WINDOW finds the value of the PARZEN window, W, at time T,
C for a window of length AM centred at time T0.
C

IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C Find relative time on (-1,1) time base.
X=2.D0*(T-T0)/AM
C Find unity minus absolute value of relative time.
OMX=1.D0-DABS(X)
C If absolute value of relative time is greater than 0.5,
C select alternate cubic.
IF(DABS(X).GT.0.5D0) GO TO 10
W=1.D0-6.D0*OMX*X*X
GO TO 20

10 W=2.D0*OMX*OMX*OMX

20 CONTINUE

RETURN
END

2.4.3 The e ff ects of discrete sampling
Modern data sequences are almost exclusively digital. Usually, the signals are con-
tinuous functions of time, reduced to discrete values by a digitiser.