[SS] Ch03 Quiz (정답포함)

다음의 sin, cos 및 이들의 선형조합들에 대한 Fourier Transform은 다음과 같음.

1. $\sin \omega_0 t$

$$\begin{aligned} \mathcal{FT}[\sin \omega_0 t] & = \int_{-\infty}^{\infty} \sin \omega_0 t e^{-j\Omega t} dt \\ &= \int_{-\infty}^{\infty} \frac{e^{j\omega_0 t}-e^{-j\omega_0 t}}{2j} e^{-j\Omega t} dt \\ &= \frac{1}{2j} \left\{\int_{-\infty}^{\infty} e^{j\omega_0 t} e^{-j\Omega t} dt - \int_{-\infty}^{\infty} e^{-j\omega_0 t} e^{-j\Omega t} dt \right\}\\ &= \frac{-j}{2}\left\{ 2\pi \delta(\Omega -\omega_0)- 2\pi \delta(\Omega+\omega_0)\right\}\\ &= -j\pi \delta(\Omega -\omega_0) +j\pi \delta(\Omega+\omega_0) \end{aligned}$$

2. $\cos \omega_1 t$

$$\begin{aligned} \mathcal{FT}[\cos \omega_1 t] & = \int_{-\infty}^{\infty} \cos \omega_1 t e^{-j\Omega t} dt \\ &= \int_{-\infty}^{\infty} \frac{e^{j\omega_1 t}+e^{-j\omega_1 t}}{2} e^{-j\Omega t} dt \\ &= \frac{1}{2} \left\{\int_{-\infty}^{\infty} e^{j\omega_1 t} e^{-j\Omega t} dt + \int_{-\infty}^{\infty} e^{-j\omega_1 t} e^{-j\Omega t} dt \right\}\\ &= \frac{1}{2}\left\{ 2\pi \delta(\Omega -\omega_1)+ 2\pi \delta(\Omega+\omega_1)\right\}\\ &= \pi \delta(\Omega -\omega_1) +\pi \delta(\Omega+\omega_1) \end{aligned}$$

3. $1+2\sin \omega_0 t +\cos \omega_1 t$

$$\begin{aligned} & \mathcal{FT}\left[ 1+2\sin \omega_0 t +\cos \omega_1 t \right] \\ &=2\pi \delta(\Omega) +2 \left\{ -j\pi \delta(\Omega -\omega_0) +j\pi \delta(\Omega+\omega_0) \right\} +\pi \delta(\Omega -\omega_1) +\pi \delta(\Omega+\omega_1) \\ \end{aligned}$$

5. $10+3\sin \omega_0 t +\cos \omega_1 t$

$$\begin{align*} &\mathcal{FT} \left[ 10+3\sin \omega_0 t +\cos \omega_1 t \right] \\ &=20\pi\delta(\Omega) +3 \left\{ -j\pi \delta(\Omega -\omega_0) +j\pi \delta(\Omega+\omega_0) \right\} +\pi \delta(\Omega -\omega_1) +\pi \delta(\Omega+\omega_1) \end{align*}$$

 

참고