Time Convolution Theorem

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Convolution

The convolution of two signals 𝑥(𝑡) and ℎ(𝑡) is defined as,

$$\mathrm{y(t) \:=\: x(t)\:\ast\: h(t)\:=\:\int_{-\infty }^{\infty}\:x(\tau)h(t\:-\:\tau)\:d\tau}$$

This integral is also called the convolution integral.

Time Convolution Theorem

Statement – The time convolution theorem states that the convolution in time domain is equivalent to the multiplication of their spectrum in frequency domain. Therefore, if the Fourier transform of two time signals is given as,

$$\mathrm{x_{1}(t)\overset{FT}{\leftrightarrow}X_{1}(\omega)}$$

And

$$\mathrm{x_{2}(t)\overset{FT}{\leftrightarrow}X_{2}(\omega)}$$

Then, according to the time convolution theorem,

$$\mathrm{x_{1}(t)\:\ast\: x_{2}(t)\overset{FT}{\leftrightarrow}X_{1}(\omega)\:\cdot\: X_{2}(\omega)}$$

Proof

From the definition of Fourier transform, we have,

$$\mathrm{F[x(t)]\:=\:\int_{-\infty }^{\infty }\:x(t)e^{-j\omega t}dt}$$

Therefore,

$$\mathrm{F[x_{1}(t)\:\ast\: x_{2}(t)]\:=\:\int_{-\infty }^{\infty }[x_{1}(t)\:\ast\: x_{2}(t)]\:e^{-j\omega t}\:dt}$$

Also, by the definition of convolution, we have,

$$\mathrm{x_{1}(t)\:\ast\: x_{2}(t) \:=\:\int_{-\infty }^{\infty }x_{1}(\tau)x_{2}(t\:-\:\tau)d\tau }$$

$$\mathrm{\therefore\: F[x_{1}(t)\:\ast\: x_{2}(t)]\:=\:\int_{-\infty }^{\infty }[\int_{-\infty }^{\infty }x_{1}(\tau) x_{2}(t\:-\:\tau)d\tau]e^{-j\omega t}dt}$$

By rearranging the order of integrations, we get,

$$\mathrm{F[x_{1}(t)\:\ast\:x_{2}(t)]\:=\:\int_{-\infty}^{\infty}\:x_{1}(\tau)[\int_{-\infty}^{\infty}\:x_{2}(t\:-\:\tau) e^{-j\omega t}dt]d\tau}$$

On substituting (t − τ) = u, in the second integration, we get, t = u + τ and dt = du

$$\mathrm{\therefore\: F[x_{1}(t)\:\ast\: x_{2}(t)]\:=\:\int_{-\infty}^{\infty}x_{1}(\tau)[\int_{-\infty}^{\infty}x_{2}(u) e^{-j\omega(u\:+\:\tau)}du]d\tau}$$

$$\mathrm{\Rightarrow\: F\left [ x_{1}\left ( t \right )\:\ast\: x_{2}\left ( t \right ) \right ]\:=\:\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )\left [ \int_{-\infty }^{\infty }x_{2}\left ( u \right )e^{-j\omega u}du \right ]e^{-j\omega \tau }d\tau}$$

$$\mathrm{\Rightarrow\: F\left [ x_{1}\left ( t \right )\:\ast\: x_{2}\left ( t \right ) \right ]\:=\:\int_{-\infty }^{\infty }x_{1}\left ( \tau \right ) X_{2}\left ( \omega \right )e^{-j\omega \tau }d\tau \:=\:X_{2}\left ( \omega \right )\int_{-\infty}^{\infty}x_{1}\left ( \tau \right )e^{-j\omega \tau }d\tau }$$

$$\mathrm{\Rightarrow\: F\left [ x_{1}\left ( t \right )\:\ast\: x_{2}\left ( t \right ) \right ]\:=\:\int_{-\infty }^{\infty }x_{1}\left ( \tau \right ) X_{2}\left ( \omega \right )e^{-j\omega \tau }d\tau \:=\:X_{2}\left ( \omega \right )\int_{-\infty}^{\infty}x_{1}\left ( \tau \right )e^{-j\omega \tau }d\tau }$$

$$\mathrm{\Rightarrow\: F\left [ x_{1}\left ( t \right )\:\ast\: x_{2}\left ( t \right ) \right ]\:=\:X_{2}\left ( \omega \right )X_{1}\left ( \omega \right )}$$

Therefore, it proves that,

$$\mathrm{x_{1}\left ( t \right )\:\ast\: x_{2}\left ( t \right )\overset{FT}{\leftrightarrow}X_{1}\left ( \omega \right )\:\cdot\: X_{2}\left ( \omega \right )}$$

The above expression is known as Time Convolution Theorem.