[SS] FT: Convolution Property

1. Covolution의  FT: Multiplication

$$x(t) \leftrightarrow X(\Omega), \quad h(t) \leftrightarrow H(\Omega) \\ h(t) * x(t) \leftrightarrow H(\Omega) X(\Omega)$$

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1-1. 증명:

$$\begin{aligned}&\int_{-\infty}^\infty [x(t) * h(t)] e^{-j\Omega t} dt
\\&= \int_{-\infty}^\infty \left[ \int_{-\infty}^\infty x(\tau) h(t-\tau) d\tau \right] e^{-j\Omega t} dt
\\&= \int_{-\infty}^\infty x(\tau) \left[ \int_{-\infty}^\infty h(t-\tau) e^{-j\Omega t} dt \right] d\tau
\\&= \int_{-\infty}^\infty x(\tau) \left[ \int_{-\infty}^\infty h(\tau') e^{-j\Omega (\tau + \tau')} d\tau' \right] d\tau
\\&= \int_{-\infty}^\infty x(\tau) e^{-j\Omega \tau} \left[ \int_{-\infty}^\infty h(\tau') e^{-j\Omega \tau'} d\tau' \right] d\tau
\\&= \int_{-\infty}^\infty x(\tau) e^{-j\Omega \tau} H(\Omega) d\tau
\\&= H(\Omega) X(\Omega)\end{aligned}$$

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2. Multiplication FT: Convolution

$$x(t) \leftrightarrow X(\Omega), \quad m(t) \leftrightarrow M(\Omega)\\ x(t)m(t) \leftrightarrow \frac{1}{2\pi} X(\Omega) * M(\Omega)$$

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2-1. 증명:

$$\begin{aligned}&\frac{1}{2\pi} \mathcal{F}^{-1}[X(\Omega) * M(\Omega)]
\\&= \frac{1}{2\pi} \int_{-\infty}^\infty \left[ \int_{-\infty}^\infty X(\omega) M(\Omega - \omega) d\omega \right] e^{j\Omega t} d\Omega
\\&= \frac{1}{2\pi} \int_{-\infty}^\infty X(\omega) \left[ \int_{-\infty}^\infty M(\Omega - \omega) e^{j\Omega t} d\Omega \right] d\omega
\\&= \frac{1}{2\pi} \int_{-\infty}^\infty X(\omega) \left[ \int_{-\infty}^\infty M(\tau) e^{j\tau t} d\tau \right] e^{j\omega t} d\omega
\\&= \frac{1}{2\pi} \int_{-\infty}^\infty X(\omega) m(t) e^{j\omega t} d\omega
\\&= m(t) \cdot x(t)\end{aligned}$$

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3.  LTI System에서 출력신호의 스펙트럼

Time-domain에서

• 입력신호와 impulse response와의 convolution으로 zero-state output이 계산되는 것이

Frequency-domain에서는

• 입력신호의 스펙트럼과 freqeuncy response와의 곱으로 zero-state output의 스펙트럼이 구해짐.

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같이보면 좋은 자료

 

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