Hioki 8847, Instruction Manual

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Appendix 5 FFT Definitions
A
22
Window Function ______________________________________________
The Fourier transform of a continuous system is defined by the integral Calculus
in expression (14) for the time range from minus infinity to plus infinity.
However, because expression (14) cannot be calculated with actual measure-
ments, the Analysis is performed on a segment between finite limits. Processing
the waveform segment within these limits is called window processing. For FFT
analysis, the waveform segment within these limits is assumed to repeat period-
ically (as shown below).
When the number of points for FFT analysis is an integer multiple of the input
signal frequency, a single-line spectrum is obtained. However, if it is not an inte-
ger multiple of the frequency (when the waveform assumed with FFT includes
discontinuous points), the spectrum is scattered, and a line spectrum cannot be
obtained. This phenomena is called leakage error (as shown below).
The window function was created to suppress such leakage errors. The window
function smoothly connects each end of the time-domain waveform where it is
cut off.
(14)

∞
∞ −
−
=
dt t x f X
ft
π
ε
2
) ( ) (
Original Time-Domain Waveform
Waveform to be assumed with FFT
Time-Domain Waveform
Number of Points setting
Time-Domain Waveform
Number of Points setting
Time-domain waveform when the number of analysis points is an integer
multiple of the input frequency
Spectrum
0 0.002 0.004 0.006 0.008 0.01
-0.1
0
0.1
Time [sec]
Am
p
litu
d
e [
V
]
0 10 20 30 40
-200
-100
0
Frequency [kHz]
Ma
g
n
it
u
d
e [
d
B
]
Time-Domain Waveform
Time-domain waveform when the number of analysis points is not an
integer multiple of the input frequency
Spectrum
0 0.002 0.004 0.006 0.008 0.01
-0.1
0
0.1
Time [sec]
Am
p
litu
d
e [
V
]
0 10 20 30 40
-200
-100
0
Frequency [kHz]
Mag
n
itu
d
e [
d
B
]
Time-Domain Waveform