Technical Report: About FFT Analyzers 4

In an FFT analyzer, the time waveform x(t) is treated as discrete data with a sampling interval h. If x(n) (n = 0, .....N-1) is a finite sequence of N sample values sampled at a sampling interval h, then the Fourier transform also applies to this data. The Fourier expansion is based on the principle that "multiplying the time waveform x(n) by a waveform e^-j2πft and integrating over one period allows us to find the amplitude of a given waveform." Referring to equation 3-9b, we can apply this discrete data to the formula:

Therefore, the Fourier transform equation for the k-th harmonic is:

(Formula 3-16)

This is expressed as follows, and this equation 3-16 is called the Discrete Fourier Transform (DFT). Furthermore, just like equation 3-7b, the relationship in equation 3-11 holds. Similarly, the Inverse Discrete Fourier Transform (IDFT) is:

(Formula 3-17)

It is expressed as follows. For reference, Figure 3-5 below conceptually illustrates equation 3-16 of the DFT.

So, to what extent can we calculate the harmonic k?

To understand the original waveform, it is necessary to sample at a frequency at least twice the frequency of the original waveform. This is called the sampling theorem. Now, let's consider the case where the waveform has period T and the number of samples is N. In this case, the sampling time h and its frequency f _s are:

(Formula 3-18)

Therefore, the frequency f _m that can be analyzed by the sampling theorem is:

(Formula 3-19)

In other words, it goes up to the N/2nd harmonic.

When a waveform band-limited from 0 to f _m is sampled at f _s and a DFT is calculated, a spectrum from frequency -∞ to +∞ is obtained. Figure 3-6 schematically illustrates the spectrum from frequency -f _m to 4f _m.

As can be seen in this figure, the resulting spectrum repeats the same values exactly at integer multiples of _fm, as if a letter had been folded back. Similarly, if a waveform band-limited to _fm to _2fm is sampled at the same _fs and subjected to a DFT, the result will be as shown in Figure 3-6, and components from 0 to _fm that do not actually exist will be calculated. This is called aliasing (aliasing distortion), and the _fm at this point is called the Nyquist frequency. Looking only at the 0 to _fm portion of the obtained spectrum, it is impossible to determine whether the original waveform is 0 to _fm or _fm to _2fm. Therefore, FFT analyzers pre-pass the signal through a low-pass filter (called an aliasing filter) that limits it to _fm, ensuring that the signal is below _fm and satisfying the sampling theorem. Furthermore, since it is necessary to consider frequencies up to _fm + α in order to obtain a frequency attenuated sufficiently by this low-pass filter, FFT analyzers only display the spectrum up to 1/1.28 of _fm (Figure 3-7).

Furthermore, Figure 3-8 shows the sampling theorem considered from the perspective of the time waveform, and Figures 3-9a and 3-9b show the waveforms displayed on the FFT analyzer when the aliasing filter is turned ON/OFF.

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

If the frequency of the original signal is less than half the sampling frequency fs (= fm), the original signal can be reproduced as in (a). However, if it is greater, it cannot be reproduced, resulting in frequency distortion as shown by the dotted waveform in (b). Also, when the frequency of the original signal matches the sampling frequency fm, a straight waveform is produced as shown by the dotted line in (c).

Due to the effect of the aliasing filter, aliasing does not occur. However, the square wave time waveform is affected by this aliasing filter because high-frequency components are cut off, so it does not become a perfect square wave.

Because no aliasing filter is applied, the time waveform is the original square wave, but the FFT result shows aliasing, and frequency components that are not originally present appear.

The discussion has become quite complex and mathematical, but I hope you have understood everything so far.

Here, I'd like to introduce some conceptual ideas and organize the concepts of Fourier series/transforms, DFT, and sampling from the perspective of an FFT analyzer. I will explain these concepts by quoting from Kenichi Kido's "Introduction to Digital Signal Processing."

Please see Figure 3-10. Waveform "A" is an infinitely repeating time waveform band-limited to below the Nyquist frequency fm. The frequency spectrum of "A" is obtained as "B," which is distributed around the origin within ±fm, with zero outside this range. Next, let's assume that the spectrum X(f) of "B" is arranged infinitely on the frequency axis with a period of 2fx, and let this be X(f):

When fx = fm, the spectral waveform is "C".

When fx < fm, the spectral waveform is "D".

When fx > fm, the spectral waveform is "E".

This is a possible explanation. Here, the spectral waveform of "E" is a state where each spectrum overlaps and aliasing occurs. Also, in the spectral waveforms of "C" and "D", the spectrum X(f) of x(t) is identical in the range of -f _m to +f _m. Looking at it from another perspective, consider the sampling sequence _xn obtained by sampling x(t) with 2f _x. The spectrum of _xn in this case is:

(Formula 3-20)

This is an equation representing "C," "D," and "E." That is, a continuous waveform x(t) band-limited by _fx is sampled with _2fx, and from the data of that infinite sequence of samples {x _n}, the spectrum X(f) is obtained, and when we look at its frequency range from_-fx to + _fx, it is the frequency spectrum X(f) of x(t) itself ("C" and "D").
The inverse Fourier transform of X(f) yields x(t), so a series of sample values every 1/(2f _x) contains enough information to represent the continuous waveform x(t). If the frequency of the original waveform is higher than the sampling frequency, aliasing will result in an overlapping spectrum like "E," and this spectrum will differ from X(f), making it impossible to reproduce x(t).

Up to this point, we have discussed the Fourier transform. Next, if we assume that waveform "A" is repeated infinitely with period T as in "F", then the frequency spectrum obtained by Fourier expanding this waveform will be a line spectrum with intervals of 1/T on the frequency axis. Since the frequency components of this waveform are restricted to f _x or less, it will be a spectrum like "G", and the envelope of this line spectrum will be the same as the continuous spectrum of "B". Consider a spectrum "H" in which the spectrum of "G" is repeated infinitely with period 2fx, similar to "C" and "D", and the time waveform obtained by inverse Fourier transform of "H" will be a sample value series that repeats infinitely with period T, as in "I". In other words, both the time waveform and the spectrum are sample value series that repeat infinitely with periodicity. This relationship is expressed in equations 3-16 and 3-17, DFT and IDFT. Equations 3-16 and 3-17:

Since these are periodic functions that repeatedly take the same value at a certain period, just like sine and cosine waveforms, you can see that DFT and IDFT represent "H" and "I" in the diagram.

Furthermore, you will understand that if the spectrum from 0 to f _x is correctly obtained, it is sufficient information. For reference, the formula for converting the spectrum back to a continuous waveform is shown below. The sample sequence {x _n} is:

(Formula 3-21)

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

(Formula 3-22)

Finally, let's summarize DFT and conclude this issue's explanation of Fourier series and Fourier transform.

If a signal x(t) that repeats infinitely with period T is band-limited to a certain frequency, then a discrete spectrum (line spectrum) X(_f) can be obtained from N discrete sample sequences {xn} obtained by sampling the signal with period T at regular intervals h according to the sampling theorem. By performing DFT every 1/T = 1/Nh, the discrete sample sequences {_xn} can be reconstructed using IDFT.

An FFT analyzer displays a time waveform in the time domain, consisting of 0 to N-1 sample sequences, and a frequency spectrum (line spectrum) in the frequency domain, ranging from 0 to f _m /1.28.