As above, The cosine function repeats a repeated pattern (Every cycle).
Of course, the cycle corresponds to 2π. At this point what does the Period mean?
Period means "Time required for one cycle." Therefore, a period is a unit of time(second).
Then, what is the relation between period and other parameters? Before that, Periodic means x(t+T) = x(t), T is period, which is smallest positive number, satisfing that equation.
Return to EQ.①, that equation is periodic (∵The signal has cycle and periodic)
Therefore, EQ.① has to satisfy Acos(2π*f_0*t+Φ+2π) = Acos(w_0*t+Φ)// because cycle is 2π
In terms of parameters, we can obtain another equation like that: 2π*f_0*(t+T)+Φ = w_0*t+Φ ...EQ.②
From EQ.②, after removing both sides, it is simplified: w_0*T=2π → w_0 is equal to 2πf_0
In conclusion, we can get the formula 2πf_0*T=2π, this way:
$$T=1/f_{0}$$ The relation between period and frequency in sinusoidal signals
To understand such representation, we need to know more mathematical laws such as the Taylor series, and Maclaurin's series. I will cover that in the next chapter.
$$e^jθ = cosθ+jsinθ$$ In order to express this, it is necessary to understand Taylor's series.
To be more precise, the special case of Taylor's series, "Maclaurin series".
Basic concept: "Transcendental functions or trigonometric functions can be expressed as the sum of several polynomial functions. And if such sums are infinite, they can be matched, not approximate."
Figure4. The Taylor approximations for log(1+x) (black). For x > 1, any Taylor approximation is invalid, (Sources:wikipedia)
From the above figure, it can be seen that the sum of the polynomials begins to coincide from the origin, and the further away from the origin, the more polynomials are required.
We can also apply that to trigonometric functions: Maclaurin series
(C_0, C_1, C_2 ... are all constants. It can be understood as a proportional constant for how much a curve made up of the sum of polynomials bends.)
≪Then, how should we set the constants?≫
→If you put 0 in EQ.③, you can find the constant C_0. If you want to know the constant of a polynomial, you can differentiate it until only the constant remains and then put in 0.
Figure 5. a simple example ※It is necessary to assume that the functional form of f(x) must be accurately known.
Following the example let's examine cosθ first.
Figure6. Taylor's series of Cos function;
And then, If you get the point of what I'm saying, you will get sinθ easily.
Figure7. Taylor's series of Sin function;
The following results can be obtained by summing each polynomial polynomial.
Figure8. e^j θ = cos θ + j sin function;
In conclusion, we can understand the Cartesian representation mentioned above(see Figure 3.) the real axis, and the Imaginary axis.
2-1-1. Advantages of Euler's formula
Figure9
In the case of e^jθ, it can represent It includes all areas of a circle with a center of origin and a radius of 1.
What if 2*e^jθ (=2cosθ+j2sinθ) ? It never be difficult. Just the magnitude gets doubled.
By changing only the values of magnitude and phase, all numbers, including real numbers and imaginary numbers, can be expressed.
A*e^jθ. → magnitude is A, and phase is θ.
Just think of it as a huge circle that can every area of a coordinate plane.
In addition, the Euler's formula makes calculations very easy.
For example, assume that there are three equations.
1. Z_1 = X_1 + jY_1
2. Z_2 = X_2 + jY_2
3. Z_3 = X_3 + jY_3
Z_1*Z_2*Z_3 = ??
Without Euler's formula, you have to calculate them through only the distributive law.
However, since now you can represent them like that:
1. Z_1 = A_1e^jθ_1
2.Z_2 = A_2e^jθ_2
3. Z_3 = A_3e^jθ_3
Z_1*Z_2*Z_3 = (A_1*A_2*A_3)^j(θ_1+θ_2+θ_3)
Lastly, unlike the exponential function, Euler succeeded in expressing all complex numbers by taking imaginary numbers on the exponent. the exponential function represents only a positive real number.
