A Rectangular Signal

A rectangular signal, also known as a rectangular window, is the signal that is most frequently encountered in DSP. This is because the fundamentals of DSP are built on the basis of continuous-time and infinite-duration signals. However, all practical signals are limited in time (say, $T$ seconds) that can be taken as a product between the same infinite-duration signal and a rectangular window of length $T$. This process is known as windowing.

Definition

In discrete time, an odd-length rectangular sequence $L$ is given by:

$$\begin{equation*} x[n] = \left\{ \begin{array}{l} 1, \quad -\frac{L-1}{2} \le n \le \frac{L-1}{2} \\ 0, \quad \textmd{otherwise} \\ \end{array} \right. \end{equation*}$$

This is drawn in the figure below for $L=7$ and $N=16$:

Spectrum

The spectrum of such a signal is not really intuitive. Let’s derive and plot this spectrum with the help of the DFT.

Since the DFT is periodic over $N$ samples, the summation can either be from $0$ to $N-1$ or $-N/2$ to $N/2$, even for $N$. We have

$$X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}$$ ...