Technical Report on FFT Analyzer 6

The FFT of the sum of two signals x ₁ (n) and x ₂ (n) is equal to the sum of the results of the individual FFTs.

If F is subjected to a Fourier transform;

is established. This is because individual spectra can be considered as cos waveforms, so x ₁ and x ₂ areIf you set it this way, it will become easier to understand.

When the relationship of the above equation is expressed in terms of spectrum;

In the display of the FFT analyzer, it appears as a complex spectral waveform as a whole, but it can be considered as a collection of individual spectra = individual phenomena. Also, the FFT of the signal x ₁ multiplied by C (where C is a scalar quantity) will yield the same result as the FFT of x ₁ multiplied by C.

that is

This relationship is used as a function to calibration scale to m/s ² or sound level meter dB units when FFT analysis of input signals from accelerometers or sound level meter.

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.

The data below consists of four rows: the first row contains sample values of a 10 Hz cosine wave, the second row shows its spectrum, and the third and fourth rows display the real and imaginary parts of the Fourier spectrum, respectively.

Additionally, Data 5 is a signal with a phase shift from Data 4, where the first row contains sample values of 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.

When comparing Data 4 and 5, the power spectrum is the same value of +0.16 dB, but the values of the real part and the imaginary part are different. This relates to the waveform.When observing the phase based on the reference of Hz, data 5 is;

This is due to a phase difference of +51.84 degrees. The phase relationship is displayed in the phase spectrum of the lower section of data 6.

In this way, the characteristics of DFT can be mathematically demonstrated. In DFT, complex calculations in the time domain (= the way of thinking) become simple calculations in the frequency domain. In equations 3-16 and 3-17 of the previous issue, x(n) and X(k) in DFT and IDFT are numerical sequences, differing only by whether e^-aj is multiplied or e ^aj is multiplied (a = 2πkn/N). In extreme terms, it can be said that multiplying the sample sequence by e^-aj gives DFT, while multiplying by e ^aj gives IDFT. In fact, both DFT and IDFT utilize the FFT method.

Here, summarizing the main features of FFT, we have the following: If F[x] is the Fourier transform;

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

When the time axis data is increased (t → at), the frequency resolution improves (ω → ω/a), but the power is dispersed (1 → 1/a).

The n-th derivative in the time domain relates to the spectrum of the current waveform.It becomes something that has been applied.

The n-fold integration in the time domain relates to the spectrum of the current waveform.will result from the application of.

The above equation is called convolution, and this Fourier transform is the product of the individual Fourier transforms of f ₁ and f ₂. This equation represents the relationship between the input and output of a certain signal system and is used for measuring the frequency response function (transfer function).

This is Parseval's theorem, which serves as the basis for expressing the same phenomenon in both the time domain and the frequency domain. In other words, the left side represents the total energy of the signal f(t), while the right side indicates how that total energy is distributed across each frequency component, demonstrating the law of conservation of energy. The FFT analyzer calculates and displays the frequency spectrum from the right side overall. It is important to note that it is bandwidth-limited due to the sampling theorem.