Fourier Series Coefficient : Pulse Train

period $T$, pulse width $\tau$, amplitude $A>0$ 인 periodic rectangular pulse의 Fourier Series Coefficient를 구함.

$$
x(t)= \begin{cases} A, & -\dfrac{\tau}{2}\le t\le\dfrac{\tau}{2}\\ 0, & \text{otherwise} \end{cases} \\ \qquad \\ x(t+T)=x(t)
$$

2023.07.05 - [.../Signals and Systems] - [SS] Pulse Signal : Rect function

[SS] Pulse Signal : Rect function

 

Complex Fourier Series Coefficient는 다음과 같이 정의됨:

$$
X_k = \frac{1}{T}
\int_{-T/2}^{T/2}
x(t)e^{-j\frac{2\pi k}{T}t} dt
\qquad, k\in\mathbb{Z}
$$

2025.04.26 - [.../Signals and Systems] - [SS] Fourier Series (푸리에 급수)

[SS] Fourier Series (푸리에 급수)

참고로, 정수 집합을 나타내는 $\mathbb{Z}$는 독일어 Zahlen(수들)의 첫 글자에서 온 기호임. 단수형 Zahl은 ‘수’를 뜻함.

2022.08.31 - [.../Math] - [Math] 수(Number)의 종류

[Math] 수(Number)의 종류

 

$x(t)\ne 0$인 interval만 적분하면, $k\ne 0$에서 다음과 같음:

$$
\begin{aligned} X_k &=
\frac{A}{T}
\int_{-\tau/2}^{\tau/2}
e^{-j\frac{2\pi k}{T}t} dt \\ &= \frac{A}{T}
\left[ \frac{e^{-j\frac{2\pi k}{T}t}}
{-j\frac{2\pi k}{T}}
\right]_{-\tau/2}^{\tau/2} \\ &=
\frac{A}{\pi k}
\sin\left(\frac{k\pi\tau}{T}\right) \\ &=
A\frac{\tau}{T}
\frac{\sin\left(\frac{k\pi\tau}{T}\right)}
{\frac{k\pi\tau}{T}} \\ &=
A\frac{\tau}{T}
\mathrm{Sa}\left(\frac{k\pi\tau}{T}\right) \\ &=
A\frac{\tau}{T}
\mathrm{sinc}\left(\frac{k\tau}{T}\right)
\end{aligned}
$$

 

여기서 $\mathrm{Sa}$는 sampling function이며, normalized $\mathrm{sinc}$와 다음 관계를 가짐:

$$
\mathrm{Sa}(u)=\frac{\sin u}{u}, \qquad \mathrm{sinc}(v)=\frac{\sin(\pi v)}{\pi v}
$$

$$
\mathrm{Sa}\left(\frac{k\pi\tau}{T}\right) = \mathrm{sinc}\left(\frac{k\tau}{T}\right)
$$

$\mathrm{Sa}(0)=\mathrm{sinc}(0)=1$은 극한값으로 정의함.

$k=0$일 때 coefficient는 다음과 같음.

$$
X_0 = \frac{1}{T} \int_{-\tau/2}^{\tau/2}A dt = A\frac{\tau}{T}
$$

2022.12.09 - [.../Signals and Systems] - [SS] sinc function and sampling function

[SS] sinc function and sampling function

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Magnitude Spectrum

$$
|X_k| = A\frac{\tau}{T} \left|
\mathrm{sinc}\left(\frac{k\tau}{T}\right)
\right|,
\qquad k\in\mathbb{Z}
$$

2023.10.03 - [.../Signals and Systems] - [SS] Spectrum 이란?

[SS] Spectrum 이란?

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Phase Spectrum

$x(t)$가 real and even이므로 $X_k$는 real and even임.

Real-valued signal 의 Fourier Transform은 Hermitian symmetry를 가짐.

$$X(-f)=X^*(f)$$

따라서 magnitude spectrum은 even symmetry, phase spectrum은 $2\pi$ modulo 에서 odd symmetry를 가짐.

$$|X(-f)|=|X(f)|, \qquad \angle X(-f)=-\angle X(f)\pmod{2\pi}$$

 

signal 이 complex-valued이면 일반적으로 Hermitian symmetry가 성립하지 않음.

 

https://bme808.blogspot.com/2022/11/math-hermitian-symmetry.html

Math : Hermitian Symmetry

 

Phase 는 odd symmetry 를 나타내도록 negative coefficient 에 $k>0$에서 $+\pi$, $k<0$ 에서 $-\pi$를 할당함.

$X_k=0$ 이면 phase는 undefined임.

$$
\angle X_k = \begin{cases}
0,
&
\mathrm{sinc}\left(\dfrac{k\tau}{T}\right)>0\\
\pi,
&
\mathrm{sinc}\left(\dfrac{k\tau}{T}\right)<0,\quad k>0\\
-\pi,
&
\mathrm{sinc}\left(\dfrac{k\tau}{T}\right)<0,\quad k<0\\
\text{undefined},
&
\mathrm{sinc}\left(\dfrac{k\tau}{T}\right)=0
\end{cases}
$$

$$
\angle X_0 = \angle\left(A\frac{\tau}{T}\right) = 0
$$

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Pulse Train and Spectrum

위 pulse train 은 schematic임.

아래 magnitude 와 phase plot 은 $\mathrm{sinc}$ envelope 를 나타내며,
Fourier Series spectrum 은 $u_k=k\tau/T$ 에서 이 envelope 를 sample 한 값임.

 

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같이 보면 좋은 자료

2022.09.28 - [.../Signals and Systems] - [SS] Fourier Transform of Pulse Signal

[SS] Fourier Transform of Pulse Signal

2025.04.26 - [.../Signals and Systems] - [SS] Fourier Series (푸리에 급수)

[SS] Fourier Series (푸리에 급수)

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