技術レポート FFTアナライザーについて 4

In the FFT analyzer, the time waveform x(t) is treated as discrete data sampled at intervals h. If we denote a finite series of N sample values, which are sampled at the interval h, as x(n) (n = 0,.....N-1), the Fourier transform also holds for this data. The Fourier expansion states that "by multiplying the waveform e^-j2πft with the time waveform x(n) and integrating over one period, we can determine the amplitude of a certain waveform." Referring to equation 3-9b, we can apply this discrete data to the equation;

The Fourier transform formula for the k-th harmonic is as follows:

(Equation 3-16)

This is represented by the formula, and this equation 3-16 is called the Discrete Fourier Transform (DFT). Furthermore, this equation holds the same relationship as equation 3-11, similar to equation 3-7b. Similarly, the Inverse Discrete Fourier Transform (IDFT) is:

Equation 3-17

It is represented as follows. Please refer to the conceptual diagram 3-5, which illustrates equation 3-16 of the DFT.

So, how far can harmonic k be calculated?

To understand the original waveform, it is necessary to sample at a frequency of more than twice the frequency of that waveform. This is called the sampling theorem. Now, let's consider the case where the number of samples N is taken for a waveform with a period T. At this time, the sampling time h and its frequency f _s are;

(Equation 3-18)

Therefore, the frequency f _m that can be analyzed according to the sampling theorem is;

Equation 3-19

That is, up to the N/2 harmonic.

Sampling a band-limited waveform from 0 to f _m at f _s and calculating the DFT yields a spectrum from frequency -∞ to +∞. Figure 3-6 schematically illustrates the spectrum from -f _m to 4f _m.

As shown in this figure, the resulting spectrum repeats the same values as if a letter were folded back at integer multiples of f _m. Also, if a waveform band-limited from f _m to 2f _m is sampled at the same f _s and subjected to a DFT, the result is likewise as shown in Figure 3-6, and components from 0 to f _m that do not actually exist are calculated. This is called the aliasing phenomenon (foldover distortion), and f _m at this point is called the Nyquist frequency. Looking only at the 0-to-f _m portion of the resulting spectrum, it is impossible to determine whether the original waveform was from 0 to f _m or from f _m to 2f _m. Therefore, an FFT analyzer passes the signal through a low-pass filter (called an anti-aliasing filter) that limits it to f _m in advance, thereby ensuring that the signal is below f _m and satisfying the sampling theorem. Also, because frequencies up to f _m + α must be considered to reach a frequency at which this low-pass filter provides sufficient attenuation, an FFT analyzer displays only the spectrum up to 1/1.28 of f _m (Figure 3-7).

● (Red circle): Sampling point
--: Waveform played at the sampling point

If the frequency of the original signal is less than half of the sampling frequency fs (i.e., fm), it can be reproduced as shown in (a). However, if it is greater, it cannot be reproduced, resulting in frequency distortion as shown by the dotted waveform in (b). Furthermore, if the frequency of the original signal matches the sampling frequency fm, it becomes a linear waveform as indicated by the dotted line in (c).

Since there is no aliasing filter applied, the time waveform is a true rectangular wave, but the FFT results show that aliasing occurs, revealing frequency components that are not originally present.

The discussion has become quite complex and mathematical, but have you understood the content so far?

Here, I would like to organize the somewhat conceptual ideas of Fourier series, Fourier transforms, DFT, and sampling from the perspective of an FFT analyzer. I will explain this by quoting Kenichi Kido's 'Introduction to Digital Signal Processing'.

Refer to Figure 3-10. The waveform labeled 'A' is a time waveform that is band-limited below the Nyquist frequency fm and is infinitely repeated. When we derive the frequency spectrum of 'A', it becomes 'B', which is distributed around the origin at ±fm, with zero values outside this range. Next, let the spectrum X(f) of 'B' be arranged infinitely on the frequency axis with a period of 2fx; we denote this as X(f).

When fx = fm, the spectral waveform of 'C'.

When fx < fm, the spectral waveform of D.

When fx > fm, the spectral waveform of E.

It can be considered. Here, the spectral waveform of "E" is in a state where each spectrum overlaps, resulting in aliasing. Also, in the spectral waveforms of "C" and "D", in the range of -f _m to +f _m, it is identical to the spectrum X (f) of x (t). Looking at it from another perspective, consider the sampling sequence x _n obtained by sampling x (t) at 2f _x. The spectrum of x _n at this time is;

Equation 3-20

This is a formulaic representation of “C,” “D,” and “E.” In other words, sampling a continuous waveform x (t) band-limited to f _x at 2f _x, and using the data from its infinite sequence of samples {x _n} to obtain the spectrum X (f), we find that the frequency range from -f _x to +f _x is the frequency spectrum X (f) of x (t) itself (“C,” “D”).
Since taking the inverse Fourier transform of X (f) yields x (t), the sequence of sample values taken every 1/(2f _x) contains sufficient information to represent the continuous waveform x (t). If the frequency of the original waveform is higher than the sampling frequency, aliasing causes the spectra to overlap, as in “E.” This spectrum then differs from X (f), and consequently x (t) can no longer be reproduced.

This part discusses the Fourier transform. Next, if the waveform of "A" repeats infinitely with a period T like "F", the frequency spectrum obtained by Fourier expansion of that waveform will be a line spectrum arranged at intervals of 1/T on the frequency axis. Since the frequency components of this waveform are limited to below f _x, it results in a spectrum like "G", and the envelope of this line spectrum corresponds to the continuous spectrum of "B". Considering the spectrum of "G" as a spectrum "H" that repeats infinitely with a period of 2fx, similar to "C" and "D", the time waveform obtained by inverse Fourier transforming "H" will be a series of sample values that repeats infinitely with a period T, like "I". In other words, both the time waveform and the spectrum will become a series of sample values that repeat periodically and infinitely. The relationships represented by this are the DFT and IDFT in equations 3-16 and 3-17.

As with Sin and Cos waveforms, it is a periodic function that takes the same value at regular intervals, so I believe you can understand that DFT and IDFT represent 'H' and 'I' in the diagram.

Also, if the spectrum of 0 to f _x is correctly obtained, you will understand that it is sufficient as information. For reference, I will show the formula to revert from the spectrum to the continuous waveform. The sample sequence {x _n} is;

Equation 3-21

It is represented by the spectrum X (f), and the -n-th Fourier coefficient of it is x _n. Furthermore, the continuous original waveform x (t) can be obtained from the sample value series {x _n} using the following equation (see Figure 3-11).

(Equation 3-22)

If a signal x(t) that repeats infinitely with a period T is band-limited to a frequency below a certain threshold, then by performing the DFT every 1/T = 1/Nh on N discrete sampling sequences {x _n} sampled at regular intervals h according to the sampling theorem for the signal with period T, we can obtain the discrete spectrum (line spectrum) X(f), and also reproduce the discrete sampling sequence {x _n} through IDFT.

The FFT analyzer displays the time series of 0 to N-1 sampling sequences as a time waveform in the time domain, and the frequency spectrum (line spectrum) from 0 to f _m /1.28 in the frequency domain.