## 5 System Outline
### Receive Hardware Description  
The receive hardware is a Universal Software Radio Peripheral (USRP) X310 (National Instruments, Austin, TX, USA) equipped with TwinRX daughterboards. The TwinRX  
Figure 5: Block diagram (**left**) and photograph (**right**) of the BPSK FDA transmit–receive system. The blue arrow on the block diagram indicates the direction of wave propagation. Figure 6: Measured antenna radiation pattern (**left**) and measured \(S_{11}\) parameter (**right**). supports 80 MHz of instantaneous bandwidth, and our host PC can stream captured baseband data at a maximum rate of 20 Ms/s. Receiver Synchronization: Carrier Frequency Offset (CFO) Removal, Phase Rotation and Bitstream Generation  
We assume the channel to be simply additive white Gaussian noise (AWGN), as our experiments are performed in an open field at high SNR. The AWGN channel model is  
\[s[n]=x[n]+w[n]. \tag{16}\]  
where \(s[\cdot]\) is the received samples, \(x[\cdot]\) is the transmitted signal, and \(w[\cdot]\) is AWGN. We assume the transmitting FDA will send short bursts of information in an SDC setting. Thus, we take in only \(N_{rx}=5000\) IQ samples at the receiver at a rate of \(f_{s}=10\) MHz, leading to a frequency resolution \(f_{res}=1/t_{res}=f_{s}/N_{rx}=2\) kHz. We show the raw received signal in Figure 7. To acquire the carrier frequencies, we use a split-gate filter in the frequency domain. This filter is based on the fast-time chip length \(T_{c}=1\)\(\upmu\)s which yields a mainlobe bandwidth of \(B_{ML}=2\frac{1}{T_{c}}=2\) MHz. The filter equation is given as  
\[H[n]=\left\{ \begin{array}{cc}\frac{1}{H_{len}},&0\leq n\leq\frac{t_{res}}{T _{c}}\\ -\frac{1}{H_{len}},&-\frac{t_{res}}{T_{c}}\leq n<0\end{array} \right., \tag{17}\]  
where \(H_{len}=2\frac{t_{res}}{T_{c}}\) is the length of the filter \(H[\cdot]\). The split-gate filter \(H[n]\) is correlated with the magnitude of the received signal \(|s[\cdot]|\):  
\[r[k]=\sum_{n}|s[n]|\cdot H[n+k]. \tag{18}\]  
The correlation \(r[k]\) is shown in Figure 7, where its minimum value provides the approximate center of the received spectrum and location of the carrier. In an orthogonal frequency division multiplexing (OFDM) communications signal, a portion of the signal at the end of the data transmission is placed at the beginning of the signal. This is called the cylic prefix (CP) and allows for a reference signal to compare against when trying to remove CFO. The parallel between OFDM  
Figure 7: Captured baseband data before receiver synchronization (**left**) and associated spectrum (**right**) with a split gate used to find the approximate carrier location. visually in Figure 8. In our BPSK FDA signal, the regions \(b_{1}\leq n\leq b_{2}\) and \(b_{3}\leq n\leq b_{4}\) (shown in Figure 9) provide the equivalent of a CP for the system. After finding the approximate carrier location, we use the phase information from the auto-correlation of region \(b_{1}\leq n\leq b_{2}\) and \(b_{3}\leq n\leq b_{4}\) to identify and remove CFO present in Figure 7. The discrete-time baseband signal will have magnitude, phase, and CFO given by [49]  
\[s[n]=|s[n]|e^{i\angle s[n]}\cdot e^{i2\pi n\Delta\
u/K}, \tag{19}\]  
where \(K\) is the number of samples in one temporal period \(T_{p}\), and \(\Delta\
u\) is the CFO. The auto-correlation of \(s[n]\) and \(s[n+K]\) is  
\[ \begin{split} s[n]\cdot s^{*}[n+K]&=|s[n]||s[n+K] |\,e^{i\left(\angle s[n]-\angle s[n+K]\right)}\cdot e^{i2\pi\Delta\
u(n-(n+K))/K }\\ &=|s[n]|^{2}\cdot e^{i2\pi\Delta\
u}\end{split}  \tag{20}\]  
Since \(|s[n]|=|s[n+K]|\) and \(\angle s[n]=\angle s[n+K]\), the phase of the auto-correlation will allow us to remove CFO that exists in the received signal (see Figure 9 compared with Figure 10). We remove the phase offset with inspiration from a Costas loop. In our setup, we desire to keep the frequency offsets \(n\Delta f\) present in the baseband signal, and it is expected that the remaining carrier frequencies will cause a rotation of the constellation (see Figure 11).