Technical Report: About FFT Analyzers 6

The FFT of a signal that is the sum of two signals _x1 (n) and _x2 (n) is equal to the sum of the results of performing the individual FFTs.

Let F be the Fourier transform;

This holds true. This is because each spectrum can be considered a cosine waveform, so _x1 and _x2 areSetting it this way makes it easier to understand.

The relationship in the above equation can be expressed spectrally as follows:

In an FFT analyzer display, the overall spectrum appears as a complex waveform, but this can be thought of as a collection of individual spectra = individual phenomena. Furthermore, the FFT of a signal x ₁ multiplied by C (where C is a scalar quantity) is the same as the result of multiplying the FFT of x ₁ by C.

In other words;

This relationship is used as a function to calibrate the scale to m/ ^s² or dB units (as in the sound level meter) when performing FFT analysis on input signals from Accelerometer and sound level meters.

A sample value sequence x ₁ (n) is given, and its sampling value sequence is l Sample value sequence x shifted by only (L) ₂ The result of the FFT for (n) is x ₁ The power spectrum will be the same as the FFT result of (n), except that a phase change occurs.

Data 4 below shows, in the top row, a series of sample values for a 10 Hz cosine wave; in the second row, its spectrum; and in the third and fourth rows, the real and imaginary parts of the Fourier spectrum, respectively.

Data 5 is a signal with a phase shift from Data 4. The top row shows a series of sample values for a 10 Hz cosine wave, the second row shows its power spectrum, and the third and fourth rows display the real and imaginary parts of the Fourier spectrum, respectively.

Comparing data 4 and 5, the power spectrum is the same value of +0.16 dB, but the values for the real part and the imaginary part are different. This is due to the waveformWhen viewing the phase with Hz as the reference, data 5 is:

This is because there is a phase difference of +51.84 degrees. The phase spectrum in the lower part of Data 6 shows this phase relationship.

Thus, the characteristics of DFT can be mathematically demonstrated. In DFT, complex calculations (i.e., concepts) in the time domain become simple calculations in the frequency domain. In equations 3-16 and 3-17 of section 3-4 of the previous issue, x(n) and X(k) in DFT and IDFT are numerical sequences, and the only difference is whether you multiply by e^-aj or e ^aj (a = 2πkn/N). To put it simply, multiplying a sample sequence by e^-aj gives DFT, and multiplying by e ^aj gives IDFT. In fact, both DFT and IDFT utilize the FFT method.

Here's a summary of the main features of the FFT: If F[x] is the Fourier transform, then:

(If the Fourier transform X(ω) is the waveform X(t), then its inverse transform is x(-ω).)

Increasing the time-domain data size (t → at) improves frequency resolution (ω → ω/a), but the power is dispersed accordingly (1 → 1/a).

The nth derivative in the time domain is obtained in the spectrum of the current waveform.This is the result of multiplying by [a certain factor].

The n-th integral in the time domain is obtained from the spectrum of the current waveform.It will be the result of multiplying by .

The above equation is called a convolution integral, and its Fourier transform is the product of the individual Fourier transforms of _f1 and _f2. This equation represents the relationship between the input and output of a signal system and is used in measuring the frequency response function (transfer function).

This is Percival's theorem, which provides the basis for the fact that the same phenomenon can be expressed in both the time domain and the frequency domain. In other words, the left side represents the total energy of the signal f(t), and the right side represents how that total energy is distributed across each frequency component, demonstrating the law of conservation of energy. In an FFT analyzer, the overall spectrum is calculated and displayed from the frequency spectrum on the right side. It is important to note that the bandwidth is limited due to the sampling theorem.