Hioki Memory HiCorder MR6000, Instruction Manual, Page: 377 / 392

371
FFT Definitions
16.2 FFT Definitions
The FFT, which stands for fast Fourier transform, is an efficient method to compute the DFT (discrete
Fourier transform) from a time-domain waveform. The reverse process of transforming frequency
data obtained through the FFT into its original time-domain waveform is called the IFFT (Inverse
FFT). The FFT function performs various analyses using FFT and IFFT.
Time and frequency domain considerations
All signals inputted to the instrument are functions of time. These functions can be considered as
a combination of sine waves at various frequencies, such as in the following diagram. Although
observing a signal only as a the time-domain waveform may be difficult to analyze, you can easily
understand its characteristics by transforming it into a frequency-domain graph.
Amplitude
Time
Time-domain waveform
Frequency
Discrete Fourier transform and inverse discrete Fourier transform
Let
x
(
n
) denote a discrete signal,
X
(
k
) denote its discrete Fourier transform (DFT), and
N
denote the
number of calculation points. The relationships among them are as follows:
𝑋𝑋(𝑘𝑘) = 𝐷𝐷𝐷𝐷𝐷𝐷{𝑥𝑥(𝑥𝑥)} = ∑
𝑥𝑥 (
𝑛𝑛
)𝑊𝑊
𝑁𝑁
𝑘𝑘𝑘𝑘
𝑁𝑁𝑁𝑁
𝑛𝑛𝑛𝑛
...........................................(1)
𝑥𝑥(𝑛𝑛) = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼{𝑋𝑋(𝑋𝑋)} =
1
𝑁𝑁 ∑ 𝑋𝑋
(𝑋𝑋)𝑊𝑊
𝑁𝑁
−𝑘𝑘𝑘𝑘
𝑁𝑁−𝑁
𝑘𝑘 =0
.....................................(2)
𝑊𝑊
𝑁𝑁
= exp (−𝑗𝑗
2𝜋𝜋
𝑁𝑁 )
....................................................................(3)
The discrete signal
X
(
k
) is typically a complex number, thus expression (1) can be transformed
again and written as follows:
𝐹𝐹(𝑘𝑘) = |𝐹𝐹(𝑘𝑘)|exp{𝑗𝑗𝜙𝜙(𝑘𝑘)} = |𝐹𝐹(𝑘𝑘)|∠𝜙𝜙(𝑘𝑘)
..................................(4)
𝜙𝜙(𝑘𝑘) = tan
−1
Im{𝑋𝑋(𝑘𝑘)}
Re{𝑋𝑋(𝑘𝑘)}
..............................................................(5)
|𝐹𝐹(𝑘𝑘)|
: Altitude spectrum,
𝜙𝜙(𝑘𝑘) = tan
−1
Im{𝑋𝑋(𝑘𝑘)}
Re{𝑋𝑋(𝑘𝑘)}
: Phase spectrum
Appendix
HIOKI MR6000A966-03