[SS] Fourier Transform : Frequency Shifting

어떤 성질인가?

$$x(t)e^{j\Omega_0}t \leftrightarrow X(\Omega-\Omega_0)$$

• $x(t)$ : time domain function
• $X( \Omega)$ : Fourier representation, Foruier Transform
• $\Omega_0$ : Frequency Shift (constant)

 

Frequency Domain에서 $\Omega_0$ 만큼 shifting 시킬 경우,

Time Domain에서는 $e^{j\Omega_0}t$가 곱해지게 된다.

 

---

증명

$$\begin{aligned} \frac{1}{2\pi} \displaystyle \int^\infty_{-\infty} X(\Omega - \Omega_0)e^{j\Omega}t d\Omega &= \frac{1}{2\pi}\displaystyle \int^\infty_{-\infty}X(\omega)e^{j(\omega+\Omega_0 )t}d\omega \quad \leftarrow \omega=\Omega-\Omega_0 \\ &= e^{j\Omega_0t} \left\{ \frac{1}{2\pi} \displaystyle \int^\infty_{-\infty} X(\omega) e^{j\omega t} d\omega \right\} \\ &=e^{j\Omega_0 t} \left\{ x(t) \right\} \\ &= x(t) e^{j\Omega_0 t} \end{aligned}$$

---

관련자료

2023.10.13 - [.../Signals and Systems] - [SS] A Short Table : Fourier Transform

[SS] A Short Table : Fourier Transform

2022.09.28 - [.../Signals and Systems] - [SS] Fourier Transform of Complex Exponential Function

[SS] Fourier Transform of Complex Exponential Function