Chaptre 1

The inverse transform is derived through :

\(X(\omega)=TF^-1{X(\omega)}\)

\(x\left(t\right)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}{X\left(\omega\right)e^{j\omega t}d\omega}\)

Application

1. Calculate the Fourier transform of x(t) = rect_T(t);

2. Illustrate the amplitude and phase spectrum of x(t).

Correction

1. We apply the definition directly and obtain :

\(X\left(\omega\right)=TF\left\{x(t)\right\}=\int_{-\frac{T}{2}}^{+\frac{T}{2}}{1.e^{-j\omega t}dt}=Tsinc\left(\frac{\omega T}{2}\right)\)

2. representation of X(f) :