Technical Report on FFT Analyzer 5

In this section, we will calculate the Fourier series for one cycle of an actual time waveform.

Figure 3-9 shows the procedure for calculating the Fourier coefficients of the second harmonic with k = 2 from the one-cycle waveform data with a sample size of N = 12, as well as the results of performing similar calculations for k = 0 to 6 ^※, which represent the Fourier series expression 3-20 of this waveform. (※: According to the sampling theorem, for a sample size of N = 12, the highest harmonic order that can be analyzed is f _m = 6)

Also, the figure 3-10 below shows the data displayed by the FFT analyzer when this sampling data has a period T = Nh = 0.02 seconds (ω ₀ = 50 Hz).

When done manually like this, even with just 12 sampling data points, it can take a considerable amount of time. The FFT analyzer uses the FFT method with N taken as 2 ⁿ (for example, 1024 = 2 ¹⁰), allowing such calculations to be processed in a short time, storing the data in memory, and displaying waveforms such as time axis waveforms and spectra from that data, thereby conveying various characteristics of the input waveform to us.

The Fourier coefficients a ₀, a _k, b _k are:

Calculation table for the second harmonic.

The Fourier coefficient when k = 2 is from the above table;

a ₀ = 133/12 = 11.08, a ₂ = 2/N∑x(n)cos(2n) = 2.83

It can be calculated. By performing similar calculations for k = 0 to 6, the following expression can be obtained.

Equation 3-20

Figure 3-10

Let's analyze a simple waveform using an FFT analyzer while looking at the relationship and characteristics of FFT that were explained earlier.

The data below (1) is the result of analyzing one of the original waveforms of unknown characteristics at a frequency range of 10 kHz. Due to the short sampling time relative to the original waveform, it can be observed in detail on the time axis; however, on the frequency axis, the frequency resolution is insufficient, resulting in a spectrum that appears connected (envelope).

The relationship between the X-axis in the frequency domain and the frequency range can be determined as follows.

The original waveform is displayed with 2048 data points sampled at a time axis resolution of approximately 0.039 ms, covering a duration of 80 ms, which corresponds to the fundamental frequency T.

By performing a DFT on this time-domain waveform, 800 spectral data points will be displayed from 0 Hz to 10 kHz at intervals of 12.5 Hz.

This 12.5 Hz is the reciprocal (1/T) of the fundamental period of 80 ms. Please refer to the following table that summarizes the relationship between sample count, data length, and frequency resolution.

Number of sampling points	Data length T (seconds)	Frequency resolution 1/T
256
512
1024
2048
4096

<f: frequency range>

Next, please look at Data 2. This is the result of analyzing the same waveform as Data 1 in the frequency range of 1 kHz. The time axis resolution has worsened, but^＊conversely, the frequency resolution has improved to 1.25 Hz, allowing for detailed reading of the odd harmonics of the fundamental wave at 10 Hz. From the results, it can be seen that the original waveform is a 10 Hz square wave. (＊Note: The sampling frequency is sufficiently high compared to the signal frequency, so the time axis waveform has not noticeably worsened.)

Now, let's take a look at Data 3, which is analyzed with the same data waveform at a quarter of the sampling points = 512 points, and a frequency range similar to Data 2 at 1 kHz. Since the data length has been shortened from 800 ms to 200 ms compared to Data 2, the frequency resolution becomes 5 Hz, which is 1/200 of 1 kHz, leading to the emergence of uncertainty.

The spectral values at 50 Hz for each of the spectra waveforms from Data 1 to 3 (shown below) are indicated, and it can be inferred that they are almost the same. This suggests, as explained in Figure 3-10 of the previous issue, that the envelope of the DFT spectrum matches the spectrum of the original waveform.