Fourier Transform of Double-Sided Exponential Function We have a double-sided decaying exponential time-domain function where f(t) = e^{-a|t|} = e^{at} for t < 0 e^{-at} for t > 0 where a is a positive constant and the vertical bars represent the absolute value of t. (a) Find the Fourier transform F(?) of function f(t). (b) Find the magnitude and phase of the expression you found in part (a) above.
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Step 1:** The Fourier transform of the given function \(f(t)\) can be calculated using the formula: \[F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt\] ** 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 $$\int_{0}^{\infty} \frac{\cos \alpha x}{\alpha^{2}+1} d \alpha=\frac{\pi}{2} e^{-|x|}$$ (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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(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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Derive the Fourier transform of the double-sided exponential: f(t) = e^(-at) (a > 0). b. Derive the Fourier transform of the signal: f(t) = x(3t + 7).
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