## Example of Computation of Fourier transform of a CT SIGNAL

Suppose we have a signal:

$e^{-2(t-1)}u(t-1)\,$

The formula of Fourier Transform is:

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

Substituting:

$X(w) = \int_{-\infty}^{ \infty} e^{-2(t-1)}u(t-1)e^{-jwt}dt\,$

From the step function, the range becomes 1 to $\infty$, so the equation becomes:

$X(w) = \int_{1}^{ \infty} e^{-2(t-1)}e^{-jwt}dt\,$
$X(w) = \int_{1}^{ \infty} e^{2-(2+jw)t}dt\,$

Integrating yields:

$X(w) = {\left. -\frac{e^{2-(2+jw)t}}{2+jw} \right]_{1}^{\infty}}\,$
$X(w) = 0 - -\frac{e^{2-(2+jw)}}{2+jw} \,$
$X(w) = \frac{e^{2-2-jw}}{2+jw} \,$
$X(w) = \frac{e^{-jw}}{2+jw} \,$

## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett