Hioki MR8827, Instruction Manual

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FFT Definitions
Appx. 4 FFT Definitions
■ What Is FFT?
FFT is the abbreviation for Fast Fourier Transform, an efficient method to calculate the DFT (Discrete
Fourier Transform) from a time-domain waveform. The reverse process of transforming frequency data
obtained by the FFT back into its original time-domain waveform is called the IFFT (Inverse FFT). The
FFT functions perform various types of analysis using FFT and IFFT.
■ Time and Frequency Domain Considerations
All signals are input to the instrument as a function of the time domain. This function can be considered as
a combination of sine waves at various frequencies, such as in the following diagram. The characteristics
of a signal that may be difficult to analyze when viewed only as a waveform in the time domain can be
easier to understand by transforming it into a spectrum (the frequency domain).
Amplitude
Time
Time-Domain Waveform
Frequency
■ Discrete Fourier Transforms and Inverse FFT
For a discrete signal x(n), the DFT is X(k) and the number of Analysis points is N, which are related as
follows:
𝑋𝑋(𝑘𝑘) = 𝐷𝐷𝐷𝐷𝐷𝐷{𝑥𝑥(𝑥𝑥)} = ∑ 𝑋𝑋( 𝑛𝑛 )𝑊𝑊
𝑁𝑁
𝑘𝑘𝑘𝑘
𝑁𝑁𝑁𝑁
𝑘𝑘𝑛𝑛
........................................ (1)
𝑥𝑥(𝑛𝑛) = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼{𝑋𝑋(𝑘𝑘)} =
1
𝑁𝑁 ∑ 𝑋𝑋
(𝑘𝑘)𝑊𝑊
𝑁𝑁
−𝑘𝑘𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0
..................................... (2)
𝑊𝑊
𝑁𝑁
= exp (−𝑗𝑗
2𝜋𝜋
𝑁𝑁 )
....................................................................... (3)
X(k) is typically a complex number, so 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)|
: Amplitude spectrum,
𝜙𝜙(𝑘𝑘) = tan
−1
Im{𝑋𝑋(𝑘𝑘)}
Re{𝑋𝑋(𝑘𝑘)}
: Phase spectrum
Appx.