Chapitre 3

Let

\(x(n) =\begin{cases}1 & \lvert n \rvert \leq \frac{N}{2}\\\ 0 & \quad \text{else }\end{cases}\)

In this case, \(X\left(f\right)\) is the sum of N +1 terms of a geometric sequence with a common ratio of \(e^{-j\pi Nf}\) and a first of \(e^{j\pi Nf}\)

\(X\left(f\right)=e^{j\pi Nf}\frac{1-e^{-j2\pi(N+1)f}}{1-e^{-j2\pi f}}=\frac{e^{-j\pi f}\left(e^{j\pi(N+1)f}-e^{-j\pi(N+1)f}\right)}{e^{-j\pi f}\left(e^{j\pi f}-e^{-j\pi f}\right)}\ =\frac{\sin(\pi f(N+1))}{\sin(\pi f)}\) (5)