Technical Report on FFT Analyzer 3

In the previous issue, it was mentioned that 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 of "A" in Figure 1-1 of the previous issue. The voice of "A" can be decomposed into "A ₁", "A ₂", "A ₃"... Additionally, the waveform of "A ₁" is described in the previous issue as a Cos wave with a phase difference:

It can be expressed as. Similarly, "A ₂" and "A ₃" are also:

will become.

The sound of "A" x (t) is a composite wave of "A ₁", "A ₂", "A ₃".....:

Here, organizing f ₁, f ₂, f ₃, ... a ₁, a ₂, a ₃, ... b ₁, b ₂, b ₃, ... as f _n, a _n, b _n, (n = 1, 2, 3, ... ∞) gives:

(Equation 3-1)

will become.

You may notice that the expression 3-1 representing the voice of "A" is very similar to the definition of the Fourier series in expression 3-2 that follows. The components of "A", f ₁, f ₂, f ₃, ..., correspond to the fundamental frequency f ₀, where f ₁ = f ₀, f ₂ = 2f ₀, f ₃ = 3f ₀, ..., and correspond to the 1st, 2nd, 3rd, ... harmonics (frequencies that are integer multiples of f ₀). In the analysis example of "A" in Figure 1-2, T = 160 ms (f ₀ = 6.25 Hz) is taken, so in the figure, f ₁ = 106.25 Hz = 17 f ₀ (17th harmonic), f ₂ = 34 f ₀, f ₃ = 51 f ₀. Harmonics that do not correspond to f ₁, f ₂, f ₃, ... can be considered as other signal elements, such as background noise, separate from "A".

As you can see from here, from section 2-2 of the previous issue onwards, the ideas we have seen so far are essentially the Fourier series itself. Now, let's take a look at the definition of this Fourier series.

A waveform that changes periodically in the time domain can be denoted with a period T.

Fundamental frequency f ₀ = 1/T

Basic angular frequency    ω ₀ = 2 π f ₀

It can be expressed as follows in terms of a Fourier series.

Equation 3-2

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

Equation 3-3

a ₀ is the DC component, a _n and b _n are the amplitudes of the cosine and sine waves with angular frequency nω ₀, known as Fourier coefficients, and a _n and b _n together are referred to as a pair of Fourier coefficients.

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

The above formula 3-2 can be interpreted as follows.

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

It is represented, and the Fourier coefficients a _n and b _n can be obtained from equation 3-3. Furthermore, from the relationship between a _n and b _n shown in the figure 3-1, equation 3-3 can be expressed as follows.

(Equation 3-4)

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

The FFT analyzer stores the Fourier coefficients a _n and b _n as calculation results in memory, and computes the amplitude r _n and phase φ _n of frequency f _n from these a _n and b _n. The result of this computation, which represents the relationship between frequency f _n and amplitude r _n, is the spectral representation, and similarly, the representation of the relationship between frequency f _n and phase φ _n is called the phase spectrum. Additionally, it is possible to revert to the time waveform using equation 3-2 from a _n and b _n. Note that the spectrum is a quantity of magnitude only (it does not contain phase information <φ>), so it is not possible to revert to the original waveform with just the spectral data. One of the functions of the FFT analyzer is to obtain the time waveform from Fourier transform data (inverse Fourier transform), but to do this, it is necessary to store the Fourier spectrum with phase information (composed of real and imaginary parts). Furthermore, if the original waveform is stored, it also has the capability to perform FFT again from this data.

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

In the various functions processed and computed by the FFT analyzer, expressions such as 'real part' and 'imaginary part' appear when displaying Fourier spectra and transfer functions. These terms arise when representing Fourier series in the complex plane (Gaussian plane) and are a method of function representation, with each part holding significant meaning. This section will explain the concepts of 'real part' and 'imaginary part' by converting Fourier series into a complex exponential representation using Euler's formula.

It will become a bit more difficult, but please make sure to understand the diagram 3-3 provided below. Additionally, I will not explain complex numbers and exponents in detail here, so those interested should refer to specialized books.

Euler's formula is:

Equation 3-5

In this 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 remains "e ^t" when differentiated or integrated d/dt (e ^t) = e ^t

j :    j ² =−1 is the value

As you know, there are real numbers and imaginary numbers, with numbers without j being real numbers and those with j being 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 = e ^jnωt represents a circle with a radius of 1 on the complex plane as shown in Figure 3-2 (b), rotating counterclockwise (in the positive direction) at a speed of nω per second (angular velocity) over time. This can be said to express both the Cos and Sin equations that represent the trajectory of the ball in the previous issue's equations 2-7 and 8 in a single equation. Similarly, Y = e^-jnωt rotates in the clockwise (negative) direction. X and Y have a symmetric relationship with respect to the real axis (Re axis), and Y is the conjugate complex number of X, while X is the conjugate complex number of Y. Conjugate complex numbers are denoted with a "*" as in Z ^*.

Substituting formula 3-5 into Fourier series formulas 3-2 and 3-3, and simplifying gives:

Equation 3-6

This will be. Here, focusing on (a _n- jb _n) and (a _n +jb _n), we have X ₀, X _n, and X _n^* respectively:

If we place it, equation 3-6 is:

Equation 3-7

(Equation 3-7a)

(Equation 3-7b)

Equation 3-7c

can be rewritten as.

Considering the case where n = 0 in equation 3-7b, since e ⁰ = 1, the right side of equation 3-7b matches the right side of equation 3-7a. Therefore, we can rewrite equation 3-7 as follows for n from n = 0 to ∞.

Equation 3-8

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

Equation 3-9a

(Equation 3-9b)

can be expressed in a very neat form.

We call the expression 3-9a the Fourier series in complex exponential form, and the expression 3-9b the Fourier expansion in complex exponential form.

Euler's formula e ^jnω0t and e^-jnω0t consist of Cos and Sin, thus inheriting the periodicity of Cos and Sin that we have seen so far, while also possessing the properties of exponents. Replacing e ^jnω0t and e^-jnω0t with Cos and Sin respectively, for example, replacing equation 3-9b with Cos corresponds to the formula for a _n in equation 3-3, and also;

From the relationship, if we consider the cosine and sine terms of the Fourier coefficients as the real and imaginary parts of a complex representation, it becomes easier to understand the relationship between Fourier series and their complex representation (please compare Figures 3-1 and 3-3).

Now, the k-th harmonic at this time is represented as follows.

Equation 3-10

X _k^* is the conjugate complex number of X _k, and when represented in complex coordinates with the horizontal axis as the real part (Re) and the vertical axis as the imaginary part (Im), 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 the previous issue, so the following equation 3-11 holds.

(Equation 3-11)

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

Equation 3-9a is a Fourier series, representing a time waveform that continues periodically. By substituting Equation 3-9b into this equation, we obtain Equation 3-12 (for ease of calculation, the range of the waveform from 0 to T has been replaced with -2/T to 2/T here).

(Equation 3-12)

Here, we consider extending the period T from -∞ to +∞ so that we can handle waveforms without periodicity. If we denote the limit of 1/T as T increases, resulting in a very small frequency df, as shown in equation 3-12:

(Equation 3-13)

If we denote the contents within the brackets as X(f) in the above equation:

Equation 3-14

Equation 3-15

Equation 3-14 represents the Fourier transform, while 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 transform and inverse transform corresponds to the pair of Fourier series and expansion in the earlier equation 3-9. Please be careful not to confuse equations 3-14 and 3-15 as they are similar. X (f) represents the frequency domain, while x (t) represents the time domain.
In addition, in the Fourier series, the spectrum was represented as discrete harmonics, but in the Fourier transform, since the period is extended to infinity, the fundamental frequency f ₀ (= 1/T) becomes a very small value, resulting in a continuous frequency spectrum.

Here, let's consider the Fourier transform of x(t) = 1. We take a segment of time T from the infinite value of 1, while considering the rest as zero (0), and perform the Fourier transform. Figure 3-4 shows the cases for T = 1 and T = 5. The expectation is shown in Figure 3-4a, but it is quite different when compared to Figure 3-4b. However, as T increases, the width of the waveform becomes narrower, and the amplitude increases, approaching Figure 3-4a. This discrepancy arises because we arbitrarily assumed the unseen parts to be 0, and naturally, a larger T better represents the overall characteristics of the waveform. As you can see from this, the Fourier transform has a property known as uncertainty between time resolution (the larger the time T, the higher the frequency resolution = 1/T) and frequency resolution. In an FFT analyzer, this time length T can be modified according to the purpose as the data length (number of samples).