## 2 Structure of DSSS Signals  
The process of communication signal generation in the transmitter of the DSSS UAC systems is as follows. Figure 1a shows a block diagram of a DSSS modulator. The input data stream \(d(t)\) is formed into real-value Binary Phase Shift Keying (BPSK) constellation symbols from a set \(\{-1;1\}\). Next, the symbols are combined with PN sequence \(m(t)\) in such a way that a single PN sequence is multiplied by a single BPSK symbol. The product is upsampled by a factor of \(R=f_{s}/B\), where \(f_{s}\) is the sampling frequency equal to 200 kHz, and \(B\) denotes the system bandwidth. Such a prepared digital signal \(x(t)\) modulates the phase of the carrier wave of frequency \(f_{c}\) equal to 30 kHz. The choice of carrier frequency was determined by the configuration of the laboratory model of the UAC system, which was used to perform experimental tests in the measurement pool--in particular, by the operating bandwidth of the ultrasonic transducers. Three types of PN code were applied in such a configured DSSS system: m-sequences, Kasami codes, and Gold codes, all of rank 8, so their length was equal to 255 samples. Test signals occupying different frequency bands were prepared using each type of PN sequence. The values of bandwidth \(B\) are shown in Table 1. The corresponding upsampling factor \(R\), the length of a single modulation symbol \(N_{s}\), and its duration \(T_{s}\) are also presented for each value of \(B\). Sampling frequency \(f_{s}\) is assumed to be equal to 200 kHz. The symbol duration \(T_{s}\) determines the data transmission rate, calculated as a number of DSSS symbols per second (each of the DSSS symbols represents a single bit of information). The symbols are organized in data frames, whose duration is equal to 6 s. The number of symbols \(K\) in a single data frame is also shown in Table 1.  
At the receiver side (Figure 1b), input signal \(y(t)\) is passed through the passband quadrature demodulator. Next, the complex-value digital signal is passed through the matched filter with complex-value coefficients \(m_{c}(t)=m(t)+j\hat{m}(t)\), the real part of which corresponds to the PN sequence \(m(t)\) used in the transmitter, and the imaginary part \(\hat{m}(t)\) is equal to the Hilbert transform of \(m(t)\). The output of the matched filter \(r(t)\) represents the cross-correlation function of signal \(y(t)\) and the sequence \(m_{c}(t)\). It is passed to the input of the detection algorithm. Figure 2 shows real parts of exemplary signals \(r(t)\). They are divided into segments, each of length \(N_{s}\), which correspond to modulation symbols. The algorithm used for making the decision on the information represented by a single modulation symbol is as follows. The minimum \(A_{min}\) and maximum \(A_{max}\) values are determined for the real part of each segment, and their absolute values are compared with each other. If the maximum value \(A_{max}\) is greater than the absolute value of \(A_{min}\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Bandwidth**} & **Upsampling Factor** & **Symbol Length** & **Symbol Duration** & **Symbols per Frame** & **Data Transmission Rate** \\  & \(R\) & \(N_{s}\) & \(T_{s}\) & \(K\) & **Rate** \\ \hline
1 kHz & 200 & 51,000 & 255.00 ms & 23 & 3.92 bps \\
2 kHz & 100 & 25,500 & 127.50 ms & 47 & 7.84 bps \\
4 kHz & 50 & 12,750 & 63.75 ms & 94 & 15.69 bps \\
5 kHz & 40 & 10,200 & 51.00 ms & 117 & 19.61 bps \\
8 kHz & 25 & 6375 & 31.88 ms & 188 & 31.37 bps \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DSSS signals parameters. Figure 1: Modulator (**a**) and demodulator (**b**) block schemes of the DSSS UAC system. Figure 2: Real parts of exemplary signals at the output of a filter matched to m-sequence of rank 8 (**a**) and Kasami code of rank 8 (**b**); B = 1 kHz. then the modulation symbol represents a \"1\" bit, otherwise, a \"0\" bit. The rule for making the decision on received data \(d_{r}\) can be expressed as  
\[d_{r}[n]=\left\{