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Find inverse Fourier transform $X(j \Omega)=\delta(\Omega)$

written 23 months ago by prajapatijaimin • 3.2k

•   modified 23 months ago

discrete time signal processing

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written 23 months ago by prajapatijaimin • 3.2k

Solution:

$ F^{-1}[X(j \Omega)]=F^{-1}[\delta(\Omega)]\\ $

$ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \Omega) e^{j \Omega t} d \Omega\\ $

$ =\frac{1}{2 \pi} \int_{-\infty}^{\infty} \delta(\Omega) e^{j \Omega t} d \Omega=\frac{1}{2 \pi}[1]\\ $

$ F^{-1}[\delta(\Omega)]=\frac{1}{2 \pi}\\ $

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