Hioki Memory HiCorder MR6000, Instruction Manual, Page: 382 / 392

376
FFT Definitions
Window function
The Fourier transform of a continuous system is defined to be the integral from negative infinity to
positive infinity with respect to time as expression (15) provides.
𝑋𝑋(𝑓𝑓) = ∫ 𝑥𝑥(𝑡𝑡)
∞
−∞
𝜀𝜀
−2𝜋𝜋𝑓𝑓𝑓𝑓
𝑑𝑑𝑡𝑡
...........................................................(15)
However, since expression (15) cannot apply to actual measurement, a certain finite interval is used
for calculation. Extracting a segment of a waveform is called window processing. The FFT function
performs a calculation assuming that this finite interval of a waveform segment repeats periodically
(as shown below).
Time-domain waveform
Number of specified points
Time-domain waveform
Number of specified points
Time-domain waveform FFT-assumed waveform
When the number of FFT calculation points is equal to an integer multiple of the input signal
frequency, a single line spectrum is obtained. However, if the number is unequal to an integer
multiple of the frequency (with the result that the FFT-assumed waveform has discontinuous
points), spectral lines spread, resulting in failure of production of a line spectrum. This phenomenon
is called the leakage error (as shown below).
Time-domain waveform when the number of analysis points is equal to
an integer multiple of the input frequency
0.1
0
−0.1
0.002 0.004 0.006 0.008 0.01 0
10 20 30 40 0
0
−200
−100
Time-domain waveform Spectrum
Time (s)
Amplitude (V)
Frequency (kHz)
Magnitude (dB)
Time-domain waveform when the number of analysis points is unequal to
an integer multiple of the input frequency
0.1
0
−0.1
0.002 0.004 0.006 0.008 0.01 0
10 20 30 40 0
0
−200
−100
Time-domain waveform Spectrum
Time (s)
Amplitude (V)
Frequency (kHz)
Magnitude (dB)
The window function has been created to suppress such leakage errors. The window function
smoothly connects each cropped end of a time-domain waveform.
HIOKI MR6000A966-03