2.24 Compute the inverse Fourier transform of the signal $X(f) = \frac{1}{2}\delta(f+2) + \frac{1}{2}\delta(f-2) + \frac{e^{j4\pi f}}{2+2\pi f}$
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Step 1: The inverse Fourier transform is given by: $$x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} df$$ We can compute the inverse Fourier transform term by term. Show more…
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(a) Find the exponential Fourier transform of $f(x)=e^{-|x|}$ and write the inverse transform. You should find $$ \int_{0}^{\infty} \frac{\cos x x}{x^{2}+1} d x=\frac{\pi}{2} e^{-|x|} $$ (b) Also obtain the result in (a) by using the Fourier cosine transform equations (4.15). (c) Find the Fourier cosine transform of $f(x)=1 /\left(1+x^{2}\right)$. Hint : Write your result in (b) with $x$ and $\alpha$ interchanged.
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(a) Find the exponential Fourier transform of $f(x)=e^{-|x|}$ and write the inverse transform. You should find $$\int_{0}^{\infty} \frac{\cos \alpha x}{\alpha^{2}+1} d \alpha=\frac{\pi}{2} e^{-|x|}$$ (b) Obtain the result in (a) by using the Fourier cosine transform equations (12.15) (c) Find the Fourier cosine transform of $f(x)=1 /\left(1+x^{2}\right)$. Hint: Write your result in (b) with $x$ and $\alpha$ interchanged.
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