# Time Modulated Arrays: From their Origin to Their Utilization in Wireless Communication Systems
## 1 Introduction
### What is a TMA? TMA, in their simplest configuration, are antenna arrays whose radiated power patterns are controlled by periodically enabling and disabling the excitations of the individual array elements, as illustrated in Figure 1. Hardware implementations of TMA simplify considerably since they can be done with switches rather than the complex weight multiplications of conventional arrays. The TMA radiated field with \(N\) isotropic elements distributed along the \(z\)-axis is given by [5]  
\[F(\theta,t)=\mathrm{e}^{\mathrm{i}2\pi/\xi_{n}}\sum_{n=0}^{N-1}g_{n}^{T_{0}}(t) I_{n}\mathrm{e}^{\mathrm{i}k_{n}\cos\theta}\in\mathbb{C}, \tag{1}\]  
where \(g_{n}^{T_{0}}(t)\) is a periodic function with a fundamental period \(T_{0}=1/f_{0}\), which governs the \(n\)-th array element, with \(n=0,\ldots,N-1\); \(I_{n}=|I_{n}|\mathrm{e}^{\mathrm{i}\varphi_{n}}\) and \(z_{n}\) represent the \(n\)-th antenna element complex static current excitation in its polar form (\(|I_{n}|\) is its modulus and \(\varphi_{n}\) is its phase) and the position on the \(z\)-axis, respectively; \(f_{c}\) is the carrier frequency of the incoming signal; \(\theta\) is the angle with respect to the array main axis; and \(k=2\pi/\lambda\) is the wave number for a wavelength \(\lambda=c/f_{c}\). Since \(g_{n}^{T_{0}}\) is a periodic signal, it can be represented by the following exponential Fourier series:  
\[g_{n}^{T_{0}}(t)=\sum_{q=-\infty}^{\infty}G_{nq}\mathrm{e}^{\mathrm{i}q2\pi/ \xi_{n}t}\in\mathbb{R}, \tag{2}\]  
where, in a conventional TMA, \(g_{n}^{T_{0}}(t)\) is a periodic rectangular pulse train with duration \(\tau_{n}\) and delay \(\delta_{n}\) (see Figure 1). In such a case, \(G_{nq}\) is given by [6]  
\[G_{nq}=\xi_{n}\mathrm{sinc}(q\pi\xi_{n})\mathrm{e}^{-\mathrm{i}q\pi(\xi_{n}+2 \theta_{n})}\in\mathbb{C}, \tag{3}\]  
Figure 1: On the left hand side, the block diagram of a linear time-modulated array (TMA) with \(N\) elements implemented with radio-frequency (RF) switches, with \(I_{n}\) being the static excitation of the \(n\)-th element of the array. On the right hand side, the periodic (\(T_{0}\)) pulse train that governs the \(n\)-th switch, \(g_{n}^{T_{0}}(t)\), which is characterized by the normalized pulse duration \(\xi_{n}\) and the normalized switch-on instant \(o_{n}\). where \(\mathrm{sinc}(x)=\sin(x)/x\) denotes the sinc function, \(\xi_{n}=\tau_{n}/T_{0}\in(0,1)\subset\mathbb{R}\) are the normalized pulse durations, and \(o_{n}=\delta_{n}/T_{0}\in[0,1-\xi_{n}]\subset\mathbb{R}\) are the normalized pulse delays. In some cases, and according to the properties of the pulses, \(g_{n}^{T_{0}}(t)\), the dependency on \(o_{n}\) can be avoided, yielding \(G_{nq}=\xi_{n}\mathrm{sinc}(q\pi\xi_{n})\mathrm{e}^{-j\pi\xi_{n}}\), but, in this work, we consider the general form. Note that for \(q=0\) we have \(G_{n0}=\xi_{n}\). Substituting Equation (2) into Equation (1) allows for rewriting the TMA radiated field as  
\[F(\theta,t) =\underbrace{\left[\sum_{n=0}^{N-1}I_{n}\xi_{n}\mathrm{e}^{jk\xi_{ n}\cos\theta}\right]\mathrm{e}^{j2\pi f_{c}t}}_{q=0}+\] \[+\left[\sum_{n=0}^{N-1}\sum_{ \begin{subarray}{c}q=-\infty\\ q\
eq 0\end{subarray} }^{\infty}I_{n}\underbrace{\xi_{n}\mathrm{sinc}(q\pi \xi_{n})\mathrm{e}^{-j\pi(\xi_{n}+2o_{n})}}_{G_{nq},\ q\
eq 0}\mathrm{e}^{j \xi_{2n}\cos\theta}\right]\mathrm{e}^{j2\pi(f_{c}+qf_{0})t}. \tag{4}\]  
Observe that the first term in Equation (4) is the array factor at the fundamental mode (\(q=0\)) frequency \(f_{c}\), which shows that, by means of the normalized pulse durations \(\xi_{n}\), we are able to reconfigure exclusively the magnitude of the excitations and thus control the radiation pattern. On the other hand, the second term in Equation (4) reveals the existence of sideband radiation (SR) at frequencies \(f_{c}+qf_{0}\) (harmonics). Note that by modifying \(\xi_{n}\) and/or \(o_{n}\) we can also control the radiation pattern at each sideband harmonic, in this case being able to simultaneously reconfigure the magnitude and the phase of the excitations.