Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform



$ x(t) = e^{-|t-1|} \, $

$ X(w) = \int_{-\infty}^{\infty}e^{-|t-1|}e^{-jwt}dt $

$ X(w) = \int_{-\infty}^{1}e^{(t-1)}e^{-jwt}dt+\int_{1}^{\infty}e^{-(t-1)}e^{-jwt}dt $

$ X(w) = \int_{-\infty}^{1}e^{-1}e^{(1-jw)t}dt+\int_{1}^{\infty}e^{1}e^{-(1+jw)t}dt $

$ X(w) = {\left.\frac{e^{-1}e^{(1-jw)t}}{1-jw}\right]^1_{-\infty} }+{\left.\frac{-e^{1}e^{-(1+jw)t}}{1+jw}\right]^{\infty}_1 } $$ = e^{-1}\frac{e^{(1-jw)}}{1-jw}+e^{1}\frac{e^{-(1+jw)}}{1+jw} $

$ X(w) = \frac{e^{-jw}}{1-jw}+\frac{e^{-jw}}{1+jw} $

$ X(w) = \frac{2e^{-jw}}{1+w^2} $


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Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva