basic formula

all of formula is from wikipedia

Euler's formula

sinc function

Fourier Transform

Fourier Series

period signal to fourier series
Expand fourier series is equal finding fourier coefficient

T: period of signal [s]
f: reciprocal of T, [Hz]
a_0: DC parameter
a_n, b_n: fourier coefficient

And we gonna be familiar with following formula.
Check the sign of g_p(t) and c_n.

c_n can be complex even if g_p(t) is real function.

g_p(t) has finite power and DC(c_0), harmonic component(f, 2f, ...)

푸리어 급수를 선형대수로 이해해보자

https://darkpgmr.tistory.com/171

Fourier Transform

fourier transform is derived from coefficient of series, which means it is amplitude of periodic part. so, it can have relevance of frequency of periodic.

Signal

• Power signal
  - Periodic, Random
• Energy signal

Energy signal is satisfied with

and then, it can be transformed

Derivation

참고할 것
https://supermemi.tistory.com/entry/푸리에-변환-Fourier-Transform-mathcalF-displaystyleu

Property

• Linear
• Time Scaling
• Duality
• Time shift
• Frequency shift
• differenciation
• integration
• conjugate

Spectrum, or density

Example

Example_1 using integration property, triangle time function

Example_2 rectangular function

Convolution

Convolution theorem

참고

푸리에 급수
임의의 신호는 특정 계수 수열과 기저 신호의 곱으로 표현할 수 있다. 푸리어 급수로 주기함수 코사인과 사인의 합으로 임의의 주기 함수를 표현할 수 있다.
https://angeloyeo.github.io/2019/06/23/Fourier_Series.html