Technical Report: About FFT Analyzers 3

In the previous issue, we discussed how waveforms that change periodically in the time domain can be represented using Fourier series. In this section, we will actually represent such periodic waveforms using Fourier series and Fourier transforms.

First, let's consider the sound "A" in Figure 1-1 from the previous issue. The sound "A" could be broken down into "_A1", "_A2", "_A3", etc. Also, the waveform of "_A1" is a cosine wave with a phase difference, as explained in the previous issue:

It can be expressed as follows: Similarly, "_A2" and "_A3" can also be expressed as follows:

It will be.

The sound "a" x (t) is a composite wave of "_a1", "_a2", "_a3", etc.

Now, rearranging _f1, _f2, _f3, ... _a1, _a2, _a3, ... _b1, _b2, _b3, ... as _fn, _an, _bn, (n = 1, 2, 3, ...∞) respectively:

(Formula 3-1)

It will be.

If you compare equation 3-1, which represents the sound "A", with the definition of the Fourier series shown below, equation 3-2, you will notice that they are very similar. The components _f1, _f2, _f3, ... of "A" are, if the fundamental frequency is _f0, _f1 = _f0, _f2 = _2f0, _f3 = _3f0, ... and correspond to the 1st, 2nd, 3rd, ... harmonics (frequencies that are integer multiples of _f0). In the analysis example of "A" in Figure 1-2 of the previous issue, we took T = 160 ms (_f0 = 6.25 Hz), so in the figure, _f1 = 106.25 Hz = 17 f0 (17th harmonic), _f2 = 34 _f0, and _f3 = 51 _f0. Harmonics that do not fall under _f1, _f2, _f3, etc., can be considered as background noise or other signal elements separate from "A".

As you can see from this, the concept we have been looking at since section 2-2 of the previous issue is the Fourier series itself. Now let's look at the definition of this Fourier series.

A waveform that changes periodically in the time domain, if we let the period of the waveform be T,

Fundamental frequency f ₀ = 1/T

Fundamental angular frequency ω ₀ = 2 π f ₀

Therefore, it can be expressed as a Fourier series as follows:

(Formula 3-2)

Here, using the multiplication of Cos and Sin from section 2-6 of the previous issue, and the area it covers, we can find each component of equation 3-2.

(Formula 3-3)

​_a₀ is the DC component, and _a₁n and _b₁n are the amplitudes of the cosine wave and sine wave with angular frequency _nω₀, respectively. These are called Fourier coefficients, and _a₁n and _b₁n together are called a Fourier coefficient pair.

Let's summarize what we've covered so far.

Equation 3-2 above can be interpreted as follows:

nω ₀ t (n = 1, 2, 3, ...∞) are harmonics of the fundamental angular frequency ω ₀ at 1x, 2x, 3x, ..., and the waveform of the nω ₀ t component is

It is expressed as follows, and the Fourier coefficients _{a_n} and _{b_n} at that time can be found using equation 3-3. Also, from the relationship between _{a_n} and _{b_n} shown in Figure 3-1 below, equation 3-3 can be expressed as follows.

(Formula 3-4)

Here, r _n and φ _n are the amplitude and phase of the nth harmonic, respectively.

An FFT analyzer stores the Fourier coefficients _a₁n and _b₁n as calculation results in memory, and calculates the amplitude _r₁n and phase _φ₁n of frequency _f₁n from _a₁n and _b₁n. The result of this calculation is the spectral display, which shows the relationship between frequency _f₁n and amplitude _r₁n, and similarly, the phase spectrum shows the relationship between frequency _f₁n and phase _φ₁n. Furthermore, the time waveform can be reconstructed from _a₁n and _b₁n using equation 3-2. Note that the spectrum is only a quantity of magnitude (it does not have phase information <φ>), so the original waveform cannot be reconstructed from spectral data alone. One of the functions of an FFT analyzer is to obtain the time waveform from Fourier transform data (inverse Fourier transform), but in order to do this, it is necessary to store the Fourier spectrum (consisting of a real part and an imaginary part) which has phase information. It also has the function to perform another FFT from the original waveform if it is stored.

*The real and imaginary parts will be explained in the next section.

When displaying various functions calculated and processed by an FFT analyzer, such as Fourier spectra and transfer functions, the terms "real part" and "imaginary part" appear. These terms appear when representing a Fourier series in the complex plane (Gaussian plane) and are a method of function representation, with each part having an important meaning. In this section, we will convert a Fourier series into a complex exponential function representation using Euler's formula and explain the concepts of the "real part" and "imaginary part."

This will get a little more difficult, but please try to understand Figure 3-3 shown below. Note that complex numbers and exponents will not be explained in detail here, so those interested should refer to specialized books.

Euler's formula is:

(Formula 3-5)

In Euler's formula, n, e, and j are respectively:

n: Pi (3.141592)...

e: The base of the natural logarithm, e = 2.71828.... e ^t is the ^same when differentiated or integrated: d/dt (e ^t) = e ^t

j: Values such that ^j² = -1

As you know, there are real numbers and imaginary numbers. Numbers without the letter 'j' are called real numbers, and numbers with 'j' are called imaginary numbers.
On the complex plane (Gaussian plane), the complex number Z = 1 + 1j is represented as shown in Figure 3-2(a). Also, X = ^ejnωt represents a circle with radius 1 on the complex plane, as shown in Figure 3-2(b), and rotates counterclockwise (positive direction) with a velocity (angular velocity) of nω per second. This can be said to represent both the cosine and sine equations that describe the trajectory of the ball in equations 2-7 and 2-8 of the previous issue in a single equation. Similarly, Y = ^ejnωt rotates clockwise (negative direction). X and Y are symmetric with respect to the real axis (Re axis), and Y is called the complex conjugate of X, and X is called the complex conjugate of Y. Complex conjugates are represented with an asterisk, such as Z ^*.

Substituting equation 3-5 above into the Fourier series equations 3-2 and 3-3, and rearranging, we get:

(Formula 3-6)

This is the result. Now, focusing on (a _n- jb _n) and (a _n + jb _n), X ₀, X _n, and X _n^* are as follows:

If we set it as follows, then equation 3-6 becomes:

(Formula 3-7)

(Formula 3-7a)

(Formula 3-7b)

(Formula 3-7c)

It can be rewritten as follows.

Considering the case where n = 0 in equation 3-7b above, since e ⁰ = 1, the right-hand side of equation 3-7b is the same as the right-hand side of equation 3-7a. Therefore, by setting n from n = 0 to ∞, we can rewrite equation 3-7 as follows.

(Formula 3-8)

Furthermore, if we set n in equation 3-7c to n = -1 to -∞, it matches the right-hand side of equation 3-7b (n = 1 to ∞), so X _n^* can be treated as X _n when n = -1 to -∞. Therefore, summarizing the above, equations 3-7, 7a, 7b, and 7c are:

(Formula 3-9a)

(Formula 3-9b)

This allows us to express it in a very clear and concise way.

Equation 3-9a is called the Fourier series in complex exponential notation, and equation 3-9b is called the Fourier expansion in complex exponential notation.

Since e ^jnω0t and e^-jnω0t in Euler's formula are made up of cos and sine, they inherit the periodicity of cos and sine that we have seen so far, and also possess the properties of exponents. If we replace e ^jnω0t and e^-jnω0t with cos and sine respectively, for example, if we replace equation 3-9b with cos, it corresponds to the equation for a _n in equation 3-3, and also;

The relationship between the Fourier series and its complex representation becomes easier to understand if we consider the cosine and sine terms of the Fourier coefficients as the real and imaginary parts of the complex number representation (compare Figures 3-1 and 3-3).

Now, the k-th harmonic at this time is expressed as follows:

(Formula 3-10)

X _k^* is the complex conjugate of X _k, and when represented on a complex coordinate system with the real part (Re) on the horizontal axis and the imaginary part (Im) on the vertical axis, it is symmetric with respect to the Re axis, as shown in Figure 3-3.

Figure 3-3 can be considered in the same way as Figure 2-9 in Section 2-6 of the previous issue, so the following equation 3-11 holds true.

(Formula 3-11)

In this case, X _k is called the Fourier spectrum, 1/2a _k is the real part (Re) of the Fourier spectrum, 1/2b _k is the imaginary part (Im), and |X _k | ² is called the power spectrum. Note that an FFT analyzer displays a _k and b _k for the Fourier spectrum, and 4|X _k | ² for the power spectrum. This is because it is convenient to represent them using r _n, a _n, and b _n in Figure 3-1.

Equation 3-9a is a Fourier series, and therefore represents a time waveform where "a certain waveform continues periodically." Substituting equation 3-9b into this equation yields equation 3-12 (note that, for ease of calculation, the waveform range from 0 to T is replaced here with -2/T to 2/T).

(Formula 3-12)

Here, let's consider extending the period T from -∞ to +∞ so that we can handle waveforms without periodicity. If we set df to be the extremely small frequency at the limit of T in equation 3-12 as we increase 1/T:

(Formula 3-13)

If we let the expression inside the braces {} be X (f) in the above equation:

(Formula 3-14)

(Formula 3-15)

Equation 3-14 represents the Fourier transform, and equation 3-15 represents the inverse Fourier transform. In the Fourier transform equation 3-14, 1/T disappears, but if we consider it as having moved to the inverse Fourier transform side of equation 3-15, we can see that this pair of Fourier transforms and inverse transforms corresponds to the pair of Fourier expansions and series in equation 3-9. Equations 3-14 and 3-15 are similar, so be careful not to confuse them. X (f) represents the frequency domain, and x (t) represents the time domain.
In the Fourier series, the spectrum is represented as discrete harmonics, but in the Fourier transform, the period is extended to infinity, so the fundamental frequency _f0 (= 1/T) becomes a very small value, resulting in a spectrum of continuous frequencies.

Now let's consider the Fourier transform of x(t) = 1. We take a portion of the infinitely continuing value of 1, cut off a certain time period T, and treat the rest as zero (0), then perform the Fourier transform. Figure 3-4 shows the cases for T = 1 and T = 5. The expected result is Figure 3-4a, but it is quite different from Figure 3-4b. However, taking a larger T results in a narrower waveform width and larger amplitude, bringing it closer to Figure 3-4a. This difference arises because we arbitrarily treated the invisible parts as 0, and naturally, taking a larger T clearly represents the characteristics of the entire waveform. As you can see from this, the larger the time period T used to cut off the waveform (called the time resolution), the higher the frequency resolution (= 1/T), allowing us to know the detailed spectrum. Conversely, if the time period used to cut off the waveform is small (high time resolution), the frequency resolution becomes coarser, and we can only understand the approximate spectrum. Thus, the Fourier transform has a property called uncertainty between time resolution and frequency resolution. In an FFT analyzer, this time length T is the data length (number of samples) and can be changed according to the purpose.