# Chapter 5

The Fourier Transform:
$$X(\omega) = F[x(t)] = \int\limits_{-\infty}^\infty x(t) e^{-j\omega t} dt$$

The Inverse Fourier Transform:
$$x(t) = F^{-1}[X(\omega)] = {1 \over 2\pi} \int\limits_{-\infty}^{\infty} X(\omega)e^{j\omega t} d\omega$$

The Fourier transform exists if:
1. $x(t)$ is abolutely integrable
2. $x(t)$ has only a finite number of discontinuities and a finite number of minima and maxima in any finite interval.

If $x(t)$ is even, $X(\omega)$ is real.

If $x(t)$ is odd, $X(\omega)$ is imaginary.

Otherwise, $X(t)$ has a real and imaginary part.

### Example
Consider $x(t) = \delta(t)$:
$$X(\omega) = \int\limits_{-\infty}^\infty \delta(t)e^{-j\omega t} dt = 1$$

### Example
Consider $x(t) = \delta(t - \alpha)$:
$$X(\omega) = \int\limits_{-\infty}^\infty \delta(t - \alpha)e^{-j\omega t}dt = e^{-j\omega \alpha}$$

### Example
Consider $x(t) = u(t+T) - u(t-T)$:

This is a pulse whose value is 1 on the interval $[-T,T]$.
$$X(t) = \int\limits_{-T}^T e^{-j\omega t}dt = -{1 \over j \omega} \left[e^{-j\omega T} - e^{-j\omega (-T)}\right] = {2 \over \omega}\left[{e^{j\omega T} - e^{-j\omega T} \over 2j}\right] = 2T\operatorname{sinc}(\omega T)$$

### Example
Consider $x(t) = e^{-|t|}$
$$X(\omega) = \int\limits_

## Method 2 - Laplace Transfor
$$x(t) \to X(s) \to X(\omega)$$
$$s = j\omega$$

Using a Laplace transform requires that $x(t)$ is a causal signal and the region of convergence of $X(s)$ includes the imaginary axis.

### Example - Finite Support Signals
The region of convergence is the entire s-plane (must check $s=0$). It definitely includes $s=j\omeaga$.

$$X(s) = {0.5 \over s^2} - {0.5e^{-2s} \over s^2} - {e^{-2s} \over s}$$
$$X(j\omega) = {0.5 \over j\omega^2} - {0.5e^{-2j\omega} \over j\omega^2} - {e^{-2j\omega} \over j\omega}$$