"Waveforms and FFT-7" 7. FFT Analyzer

7. FFT Analyzer

Last time, we encountered Fourier series and Fourier transform formulas, which made things a bit complicated. However, since an FFT analyzer calculates Fourier coefficients, it should be easier to understand if you think of Fourier series and Fourier coefficients. The Fourier coefficient Cn is called the "spectrum," and calculating Cn is referred to as "calculating the spectrum" or "decomposing into a spectrum." In an FFT analyzer, we consider the power spectrum obtained as the basis for our analysis. The power spectrum is Cn squared, and Cn is read as the "linear representation of the power spectrum" or "Fourier spectrum." Let's consider the data screen of the DS-0221 FFT analysis software from this perspective.
We will continue our explanation using the same sample values for the original waveform f(t) created with the following formula as in the previous lesson.

An FFT analyzer calculates the Fourier coefficients as integer multiples of f0 (= 1/T).

As some of you may have noticed last time, with a sampling frequency of 1000Hz and 2048 data points, T = 2.048s and f0 = 1/T = 0.488Hz, so 10Hz is not a harmonic of f0. However, we calculated the Fourier coefficients at 10Hz. Figure 7-1 is the power spectrum screen obtained by FFT with DS-0221. Looking at this data, there is no point at 10Hz, and it is displayed as two harmonics of f0, the 20th order at 9.766Hz and the 21st order at 10.254Hz. 20Hz is almost identical to the 41st order at 20.020Hz, so it appears to be displayed as 20.020Hz.

In an FFT analyzer, please understand that the X-axis scale represents the harmonics of f0.

At a sampling frequency of 1000Hz and 2048 sample points, the 10Hz harmonic of f0 is absent.

7-1 Time Waveform

Next, let's look at Figure 7-2. When starting an analysis with the DS-0221FFT analysis software, you first set the frequency range. The sampling frequency is set to 2.56 times the frequency range. If the frequency range is 100Hz, the sampling frequency is 256Hz, and the sampling interval [time resolution] is 1/256 (s).
Converting the sample score of 2048 points to time T gives us 2048 × (1/256) = 8 (s).

Sample frequency = frequency range × 2.56 = 100Hz × 2.56 = 256 Hz
T = 2048 points ÷ 256Hz = 8 s

The time waveform displays exactly these 2048 sample points.

FFT (Fast Fourier Transform) is an algorithm for rapidly calculating the discrete Fourier transform.
Therefore, the number of sample points needs to be 2 to the power of n.
The scores are 2048 points and 1024 points, etc. And the sample score (2048 points in this case)
We assume that this waveform repeats, with time being one period T. FFT is complex.
We are calculating the real part (Real) and the imaginary part (Img) of a function, and here we will explain this by comparing it to a Fourier series.
Therefore, we will continue to use the terms "Fourier series, Fourier coefficients" and "Real, Img" in the future.
Let's proceed. Note that the coefficient Bn of the sine term in the Fourier series is -Bn in Img.
please.

7-2 X-axis of Fourier spectrum and power spectrum

When you move the cursor on the Fourier spectrum and power spectrum screens, the frequency on the X axis is
These are discrete values for each f0 [frequency resolution], and they are n times the value of f0 (harmonics).
For example, consider the case of "frequency range 100Hz, number of sample points 2048".

10Hz corresponds to 80 times f0, and 20Hz corresponds to 160 times f0, indicating that 10Hz and 20Hz are on the ω0, 2ω0, 3ω0, ... of the Fourier series.
Figure 7-2 shows an example where the sample sequence in equation (20) was created with a sampling frequency of 256 Hz instead of 1000 Hz, and analyzed using DS-0221.
The analysis covers frequencies from 0Hz to 100Hz, but for easier viewing, the X-axis scale has been enlarged to 0Hz to 50Hz.

Fourier series

····································· （21）

The number of Fourier coefficients N that can be obtained by FFT calculation is half the number of sample points; in the case of 2048 sample points, it is up to 1024. This is related to the sampling theorem, which states that sampling must be done at more than twice the highest frequency component contained in the waveform. Specifically, the input signal is processed through a low-pass filter (anti-aliasing filter) before sampling. The cutoff frequency of this low-pass filter is set to L f0. (Here, the value of L is 800 < L < 1024) To avoid aliasing, where components above 1024 f0 appear as components folded around 1024 f0, we display n = 0 to 800 points, and do not display n = 801 to 1024. When n = 800, it is exactly 100 Hz, which matches the frequency range. The X axis displays the spectrum for each f0, so the frequency resolution is f0. The frequency resolution can be calculated from the FFT frequency range and the number of sample points using the following formula, and with 2048 sample points, it becomes 1/800 of the frequency range.

Frequency resolution f0 = Frequency range ÷ (Number of sample points ÷ 2.56) Hz ········· (22)

7-3 Y-axis values of Fourier spectra and power spectra

Reading the Y-axis values for frequencies of 10 Hz and 20 Hz in the Fourier spectrum of Figure 7-3, we can determine the phase θ and oscillation.
The width C can be calculated. As a repetition of the previous example, the calculation example for 10Hz is shown below.

Frequency range 100Hz, sample frequency 100 × 2.56 = 256Hz

Power spectrum (linear): n C
Fourier spectrum Real: n A
Fourier spectrum Img: Displaying n − B

<Key Points>
The FFT is calculated using a period T for the time interval (the time window for acquiring the sample points).
The frequency resolution f0 is
f0 = 1/T = Frequency range ÷ (Number of sample points ÷ 2.56)

(Excerpt from the email newsletter issued on June 21, 2007)