FREQUENCY D O M A I N METHOD FOR
WINDOWING I N FOURIER ANALYSIS
T. L. J. Ferris a n d A. J. G r a n t
Indexing term: Fourier transforms
arithmetic redundancy in the exponential terms, N real multiplications, 3N real additions and N real divisions by 2.
If the application requires comparison of the effect of two
windows, the convolution process is faster than performing
time domain windowing and an FFT for each additional
window. The convolution method requires 2N real multiplications. The F F T method requires N 2 . N . log, N real multiplications, resulting in a factor of improvement k:
+
DFT windows are applied by frequency domain convolution.
Suitable windows, transformed to sums of delta functions,
result in convolutions requiring the addition of few terms.
These convolutions, applied to the DFT output, provide
more accurate results and less computation if multiple
windows need to be applied to the same data than does time
domain windowing.
Introduction: The discrete Fourier transform in its various
algorithmic forms is frequently implemented in conjunction
with time domain window functions to reduce the problem of
spectral leakage. It is well known that time domain multiplication corresponds to frequency domain convolution [l]. A
sliding window Fourier transform algorithm has been
developed* to assist in identifying spectral changes in machine
tool monitoring and clinical EEG measurement. This Letter
outlines a simple method of effecting window functions on
completed DFT/FFT outputs, allowing convenient comparison of the effects of multiple window functions.
Let d(t) be the time domain function, after sampling and
rectangular windowing, submitted to the DFT. Let w ( t ) be the
window function to be multiplied by d(t). Let o(t) be the actual
function of which the transform is found; then
k=
N
+2.N
. log, N
2 1
2.N
V N t 2 (7)
Extra windows: A list of additional windows usable by this
technique follows. If a sliding window is not used the exponential terms are replaced by 1.
(i) Cosine tofourth power:
(ii) Hamming window:
o(t) = w(t) . d ( t )
(1)
Let O(o),W(w),D(w) be the Fourier transforms of o(t), w(t),
and d(t),respectively; then
Nt) = a + ( 1 - a ) . c o s
(?)
where '8 is the circular convolution operator. If we let w ( t ) be
the raised cosine window
w(t) = 1
+ cos
(F)
(iii) Blackman windows:
(3)
where 7 is the nonzero duration of the rectangular window
contributing to d(t), then
(
W ( w ) = 2 . x . 6 ( w ) + n . Sw - -
( ):
+n.6 w+-
2):
Nt) =
cos
m=o
1
(F)
"/2
O(o)=-.C(-I)".(I"
2
b
m=o
(4)
and
( :") + ' ( ):
O(w)=D(w)+t.D
U--
T.
D U+-
This convolution is the sum of the D F T and two frequency
shifts of the DFT.
Where a sliding window is implemented, and D ( o ) has
phase referred to t = 0, then when N is the length of the
transform, and n is the number of the first data point in the
window referred to n = 0 at t = 0, eqn. 5 becomes
(
:RI
O(O)= D(o) - f . D w - -
Table 1 VALUES OF a,,,AND b FOR
BLACKMAN WINDOWS [I]
a,
a,
a,
3 term
-67dB
3 term
-61dB
4 term
-92dB
4 term
-74dB
0.42323
0.49755
0.07922
04959
0.49364
0.05677
0.35875
0.48829
0.14128
0.01168
0.36
0.40217
0.49703
0.09392
0.00183
0.40
-
i3 0.42
-
0.45
Table 1 contains values of a,,, and b for use in eqn. 13.
1st June 1992
T. L. J. Ferns and A. J. Grant (Sensor Science and Engineering Group,
SigniJicance: For real input data, resulting in D(w) = D*( -o),
the convolution represented by eqn. 6 requires, using the
* FERRIS, T.
L. J., and GRANT, A. I.: 'Sliding window Fourier transform
algorithm', submitted to IEEE Trans. Instrumentailon and Measure-
ments
1440
School of Electronic Engineering, University of South Australia, PO
Box I , Ingle Farm 5098, South Australia, Australia)
Reference
1
HARRIS, F. I.: 'On the use of windows for harmonic analysis with
discrete Fourier transform', Proc. IEEE., 1978,66, pp. 51-83
ELECTRONICS LE77ERS
16th July 1992
Vol. 28 No. 15