# Time Modulated Arrays: From their Origin to Their Utilization in Wireless Communication Systems
## 2 State of the Art
### TMA Design under a Signal Processing Perspective
#### 2.3.2 Towards Using TMAs in Wireless Communication Systems
The block diagram of the corresponding digital communication system under study is shown in Figure 3. Such a study characterized, for the first time and from a theoretical point of view, the bit error rate (BER) (Chapter 6, p. 171, [73]) of a linearly modulated digital communication system with a TMA at the receiver for the simplest case of an AWGN channel model (Chapter 1, p. 33, [76]). However, the major potentialities of TMA for wireless communications are unavoidably related to the exploitation of their harmonic patterns. In this sense, most works available in the literature address the harmonic beamforming capability exhibited by TMA focusing exclusively on a radiation point of view. Accordingly, different scenarios of harmonic beamforming have been properly investigated, e.g., the conversion of spatial diversity into frequency diversity [6; 42; 43; 44; 45], the mitigation of interfering signals [6; 42], the orthogonality between patterns [66; 69], and the advantages of time redundancy [43]. However, all of those works have neglected a crucial aspect in the system performance when communications signals are involved: the TMA efficiency. Therefore, a more ambitious extension of the analysis in [75] had to be undertaken, by involving the TMA in a multipath fading communication where the antenna efficiency played a key role. Hence, the true challenge at this point is the synthesis of an efficient harmonic beamforming where it is not enough to satisfy certain conditions in terms of locations of maxima and nulls, but to safeguard the TMA efficiency and, consequently, the average SNR at the receiver (recall that the gain per beam path is proportional to the TMA efficiency [27]). Figure 3: Receiver block diagram based on a linear digital modulation scheme incorporating a TMA. The notation is the same as in Figure 2 except \(p_{\mathrm{RX}}(t)\), which denotes the receive filter. Figure 2: Transmitter block diagram considering a linearly modulated digital signal radiated through a time-modulated array (TMA) exploiting its fundamental pattern: \(p_{\mathrm{TX}}(t)\) is the transmit filter, \(f_{c}\) is the carrier frequency, \(s(t)\) is the TMA input signal, and \(g_{0}^{T_{0}}(t),\ldots,g_{N-1}^{T_{0}}(t)\) are the periodic (\(T_{0}\)) functions governing the \(N\) array elements. The pioneering work involving the efficiency in the TMA beamforming design is [77], which is devoted to the performance analysis of TMA for the angle diversity reception of digital communication signals. Figure 4 shows the system block diagram whose performance is analyzed in [77]: a narrowband wireless communication system using a conventional \(M\)-ary linear digital modulation assuming an omnidirectional antenna at the transmitter, and a TMA, together with a maximum ratio combining (MRC) at the receiver, to exploit the channel diversity through the beamforming with the first \(L-1\) positive TMA harmonic patterns. According to Figure 4, once the transmitted signal \(s(t)\) arrives through a multipath channel at the receiver, we obtain \(L\) different replicas \(y_{0}(t),\ldots,y_{L-1}(t)\) constituting the TMA input. Each of the \(L\) replicas at the TMA output appears at a different (harmonic) frequency \(f_{l}=f_{c}+l\cdot f_{0}\), with \(l=0,\ldots,L-1\). Assuming that \(f_{0}\) has been correctly selected according to the bandwidth \(B_{s}\) of \(s(t)\), the TMA output signal occupies a total bandwidth \(B_{\text{TMA}}\geq\sum_{l=0}^{L-1}B_{s}=L\cdot B_{s}\). After the RF down-conversion at a frequency \(f_{c}-f_{l}\) (being \(f_{l}\) the lowest intermediate frequency), the obtained signal is sampled at a frequency \(f_{s}^{\text{TMA}}>2B_{\text{TMA}}\) (fulfilling the Nyquist criterion) and each of the resulting \(L\) digital signals is I/Q decomposed (including the corresponding receive filter) at a different intermediate frequency \(f_{l_{IF}}=f_{l}+l\cdot f_{0}\), with \(l=0,\ldots,L-1\). Finally, the obtained \(L\) complex-valued signals are combined in an MRC and the resulting signal is demodulated to obtain the received bits.