Technical Report: About FFT Analyzers 5

In this section, we will calculate the Fourier series for one period of a given time waveform.

Figure 3-9 below shows the procedure for calculating the Fourier coefficient of the second harmonic k = 2 from one-period waveform data with a sample size N = 12, and the Fourier series equation 3-20 of this waveform obtained by performing similar calculations for k = 0 to 6 ^*. (*: According to the sampling theorem, with a sample size N = 12, the number of analyzable harmonic orders f _m = 6)

Furthermore, Figure 3-10 below shows the data displayed on an FFT analyzer when this sampling data has a period T = Nh = 0.02 seconds (_ω0 = 50 Hz).

Performing these calculations manually, even for a mere 12 sample data points, would take an enormous amount of time. An FFT analyzer uses the FFT method where N is ²ⁿ (for example, 1024 = ^2¹⁰), processing these calculations in a short time, saving the data to memory, and displaying waveforms such as time-domain waveforms and spectra from that data, thereby conveying to us various characteristics of the input waveform.

The Fourier coefficients a _{= 0}, a _{= k}, and b _{= k} are:

Calculation table for the second harmonic

From the table above, the Fourier coefficients when k = 2 are:

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

This can be calculated as follows. Performing similar calculations for k = 0 to 6, we can obtain the following equation.

(Formula 3-20)

Figure 3-10

Let's actually analyze a simple waveform using an FFT analyzer and examine its relationship to the FFT and its characteristics, as explained earlier.

Data 1 below shows the results of analyzing one of the original waveforms of unknown origin at a frequency range of 10 kHz. Because the sampling time is short compared to the original waveform, the original waveform is observed in detail when displayed on the time axis, but on the frequency axis, the frequency resolution is insufficient, resulting in a connected spectrum (envelope).

The relationship between the X-axis of the frequency domain and the frequency range at this time can be determined as follows:

The original waveform is sampled at time-domain resolutions ΔT = approximately 0.039 ms, and 80 ms of data from 2048 points are displayed; this represents the fundamental frequency T.

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

This 12.5 Hz is the reciprocal (1/T) of the fundamental frequency of 80 ms. The following table summarizes the relationship between the number of samples, data length, and frequency resolution; please refer to it for further information.

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

<f: Frequency range>

Next, look at Data 2. This is the result of analyzing the same waveform as Data 1 at a frequency range of 1 kHz. Although the time-domain resolution is worse ^*, the frequency resolution has improved to 1.25 Hz, allowing for detailed reading of the odd-numbered harmonic components of the fundamental wave of 10 Hz. From the results, we can see that the original waveform is a 10 Hz square wave. (*Note: The time-domain waveform is not noticeably worse because the sampling frequency is sufficiently high compared to the signal frequency.)

Now let's look at Data 3, which is the same data waveform analyzed with one-quarter the number of sampling points = 512 points, and with the same frequency range as Data 2, 1 kHz. Compared to Data 2, the data length has been shortened from 800 ms to 200 ms, so the frequency resolution becomes 5 Hz, which is 1/200th of 1 kHz, and uncertainty begins to appear.

The spectral values at 50 Hz are shown in each of the spectral waveforms (lower) of data 1 to 3, and you can see that they are almost the same value. This suggests that, as explained in Figure 3-10 G in the previous issue, the envelope of the DFT spectrum matches the spectrum of the original waveform.