Fundamentals of Digital Measurement - Part 17: "The Story of Complex Numbers (Part 2)"

To reiterate Euler's formula, which I explained last time:

          e^iθ=cosθ+isinθ                                                              ················· (1)

; This is the result. Complex exponential function e ^iθ It is represented by a vector rotating on the unit circle in the complex plane,
The direction of rotation is counterclockwise if the rotation angle θ is positive, and clockwise if it is negative.
As is clear from equation (1), the real part is the point projected onto the real axis of the complex plane, and its trajectory
The trace will be a cosine waveform. Similarly, the locus of the imaginary part will be a sine waveform. Figure 1 shows,
A rotation vector starting from a point (1, 0) on the real axis rotates counterclockwise around a cylinder of radius 1.
This is a 3D representation of the process.

Here, if we let the argument θ be ωt, then equation (1) can be written as equation (2);

       e^iωt=cosωt+isinωt                                                            ················· (2)

The left side of equation (2) represents a rotation vector rotating at a speed of ω radians per second, and the real and imaginary parts of the right side represent cosine and sinusoidal waveforms with period 2π/ω. ω is called angular velocity (angular frequency), and ω = 2πf (where f is frequency in Hz). Hereafter, angular frequency and frequency will be treated as synonymous. Conversely, to express the cosine and sinusoidal waveforms using complex exponential functions, we take the complex conjugate of equation (2):

       e^-iωt=cosωt-isinωt  =                                                         ················· (3)

From equations (2) and (3);

················· (4)

················· (5)

Equations (4) and (5) show that trigonometric functions (cosine and sine waves together) can be expressed as the sum and difference of the complex exponential function e ^iωt and its complex conjugate. For this reason, in the world of signal processing, e ^iωt is also called a complex sinusoidal signal and is frequently used in mathematical derivations.
Equation (4) means that the complex sinusoidal signal e ^iωt is a vector that rotates the unit circle counterclockwise with angular velocity ω, its complex conjugate e ^iωt is a rotation vector that rotates in the opposite direction with the same angular velocity ω, and cosωt (cosine wave) is the combination of these two vectors (always on the real axis) (Figure 2-(a)).

Here;

        e^-iωt=ei(−ω)t                                                                  ················· (6)

;Considering this, the component rotating in the clockwise direction can be seen as the component rotating with angular velocity -ω.
Done, from here on, Negative frequency components I'm thinking about this.
When considering a real cosine waveform on the frequency axis, it consists of a positive frequency component ω (magnitude 1/2) and a negative frequency component.
It can be said that it consists of component -ω (size 1/2). In fact, the cosine waveform is frequency
Numerical analysis reveals the result shown in Figure 2(b), where the projected components onto the imaginary axis cancel each other out due to complex conjugation.
Only the positive and negative components of the real part appear symmetrically (symmetrically with respect to the line) with respect to the origin. This is the result on the frequency axis.
Generally Frequency spectrum It is called that.

Similarly, considering a real sinusoidal waveform, there are positive frequency components ω (magnitude -1/2) and negative frequency components
It consists of -ω (size 1/2), and due to the difference in complex conjugates, the components on the real axis cancel each other out, and on the imaginary axis...
Synthetic components appear (Figure 3).

As an aside, when I first joined the company and started studying FFT technology, I was deeply confused about "what are negative frequencies?"
I remember not being able to understand it.

In Figures 2 and 3(b) (frequency spectrum), only the magnitude (amplitude) is shown,
As I mentioned last time, I will now discuss the other piece of information that complex functions represent: their phase.
As you know, a sine wave is identical to a cosine wave with a 90-degree delay, so from now on, cosine
I will discuss this using the wavetime waveform as the reference. The main reason for using the cosine wave as the reference is:

1. The beginning of the vector's rotation (t = 0) coincides with the beginning of the cosine wave.
2. Since these are components of the real number, they correspond well to the real world.

This is what is thought. The cosine wave waveform can be represented by equation (7):

················· (7)

················· (8)

; In fact, the entire contents of the parentheses on the left side of equation (7) are called the phase, but the ωt part is time
Since both are rotating vector components, the important information is the initial phase φ, so here
This φ is called the phase. This phase angle is the relationship between the rotation vector and the real axis at the starting point (t = 0).
This represents the angle (Figure 4).

When a cosine wave is used as the actual waveform, the result is the sum (realization) of a complex sinusoidal signal and its conjugate, as shown on the right-hand side of equation (7). However, the main purpose of frequency analysis is to determine the magnitude (amplitude) A and phase φ for each angular frequency (ω), so the form of equation (8) is also frequently used.
Specifically, if we obtain the frequency spectrum of a time waveform including the initial phase as shown in equation (7), we get Figure 4(b). The real part of the positive frequency component corresponds to cosφ, and the imaginary part corresponds to sinφ. From this, the phase φ is;

················· (9)

for example;

When φ = 0°, see Figure 2(b) (cosine waveform).
When φ = -90°, see Figure 3(b) (sine wave).
If φ = 30°, see (b) in Figure 4.

This is equivalent to a semicolon.

Similarly, for magnitude (amplitude);

(Note)
←Here, calculations are performed using a two-sided spectrum, so the results may differ from those of actual equipment.

This is the effective value of A.

As these examples show, the frequency spectrum of a real function is:

"The real part is an even function (symmetrical along a line), and the imaginary part is an odd function (symmetrical at a point)."

This can be understood from equations (4) and (5).

The Fast Fourier Transform (FFT) also makes good use of a rotation factor that rotates the unit circle.
This is a gorism. The rotation factor is WN;

··············· (10)

If we assume this, then this is obtained by rotating the unit circle on the complex plane from the point (1,0) (on the real axis) in the negative direction.
The circumference is divided into N equal parts and represented as a vector (Figure 5 shows an example where N = 8).

Specific numerical examples

Using the rotation factor WN, the previous equation (1) can be written as equation (11) below.

··············· (11)

For specific FFT techniques, please refer to reference (2), but using a unit circle as shown in Figure 5...
Understanding the vector-based approach clarifies the periodicity of complex exponential functions and helps in understanding FFT.
The degree will increase.
Of course, this includes not only FFT, but also an understanding of signal processing such as digital filters and Z-transforms, and mathematical formulas.
Even when open, the complex exponential function (complex sinusoidal signal) e ^iωt This expression has become an essential tool.

In the previous two installments, I discussed the basics of complex exponential functions, but for the benefit of my limited knowledge,
If you notice any mistakes, please let me know.

○ References

1. Newton magazine (Imaginary Numbers Explained Clearly), December 2008 issue
2. "Digital Fourier Analysis (I) - Fundamentals" by Kenichi Kido (Corona Publishing Co.)

(Excerpt from the email newsletter issued on February 19, 2009)