How Do Beams Look Like?

In the article on beamforming, we discussed the interaction of the electromagnetic waves with the antenna array without any description of what the beam shape looks like. As we explore below now, the beam shape is given by the Fourier Transform of individual antenna intensities but the reason behind this is not always explained in most of the textbooks and tutorials on this topic. Where exactly does the Fourier Transform, a conversion tool from time $t$ to frequency $\omega=2\pi F$ domain, come into the picture? And how does the frequency $\omega$ for time domain correspond to phase shift $u$ of

Proof of Poisson Sum Formula

The Poisson sum formula was discovered by the French mathematician and physicist Siméon Denis Poisson. It has several applications in digital signal processing, among which our concern is the periodic summation of modulated pulses in digital communication systems. Assume that $p(t)$ is a pulse shape (or any continuous-time function if you are not familiar with digital communications) and $P(f)$ is its Fourier Transform. The pulse is sampled at a rate of $f_s$ to produce its discrete version $p(nT_s)$ where $T_s=1/f_s$ is the duration between two samples. The Poisson summation formula relates these two quantities as \label{equation-poisson-sum-formula} \frac{1}{T_s}\sum _{k=-\infty}^{\infty} P\left(f+\frac{k}{T_s}\right) =

Windowing an OFDM Signal in Time Domain

Orthogonal Frequency Division Multiplexing (OFDM) has been introduced in a previous article as a technique suitable for high data-rate transmissions over a wireless channel. The two main advantages I mentioned were as follows: Simple one-tap equalization, and Ability to slice the spectrum and utilize each slice in an independent manner. Due to these advantages, it was adopted as the preferred modulation in WiFi and 4G-LTE systems. The interesting part is that while many new waveforms were proposed to replace it in 5G NR, OFDM was still the modulation of choice for both downlink and uplink directions with some minor changes.