## 3 Performance in Selective Fading Conditions
### Test Setup  
The block diagram of the study is shown in Figure 6. The study uses proposed FM-CSK and QCPSK communication systems, multipath fading simulator, and MATLAB software  
Figure 5: Hardware implementation of the QCPSK (**a**) and FM-CSK (**b**) data transmission system. for data generation and performance estimation. The general model is similar for both chaos-based communication systems. First, the data signal \(b(t)\) is created, and the chaos signals of the transmitter's chaos oscillator are acquired. Then the next step applies chaos shift keying using the \(b(t)\) and corresponding chaos signals: phase manipulation of \(X_{1}\) in the case of QCPSK and chaos shift keying between \(X_{1}\) and \(X_{3}\) in the case of FM-CSK. The signals are then used for the transmitter signal processing and modulation following their modulation schemes in Figures 1 and 2. The carrier frequency is shifted to the intermediate frequency (IF) of 140 MHz to ensure consistency of the fading simulator input signals for both systems, which is designated by multiplication with the local oscillator (LO) signal in Figure 6. For this study, the two LOs are synchronous, so the carrier recovery is unnecessary in the receiver. The AWGN of the set signal-to-noise ratio (SNR) is applied to the signals before demodulation in the receiver. The received and processed synchronization signal \(X^{\prime}{}_{2}\) is then passed to the receiver's chaos oscillator, and the chaos signals of the oscillator are acquired. The signals are then used to detect the information signal \(b^{\prime}(t)\), following the schemes in Figures 1 and 2. Finally, the bit error rate (BER) is estimated for several SNR levels of the noise signals and different parameters of the noise channel. The multipath fading simulator is configured to establish two-ray propagation emulation, i.e., the output signal is a weighted sum of the line-of-sight \(G_{\textit{LOS}}\) and reflected \(G_{\textit{MUL}}\) rays. One can easily prove that the model of the emulated channel shown in Figure 7 allows the creation of a notch in the spectra of the transmitted signal. For the presented study, the notch frequency equals the IF of 140 MHz. In the number of tests, the depth of the notch differs; however, the system emulates a time-invariant multipath channel, which means that during the operation of a communication system, the configuration of the multipath fading simulator is kept. The performance of QCPSK and FM-CSK in selective fading conditions is studied in comparison to 4-QAM and FSK systems. The 4-QAM and FSK are the closest non-chaotic counterparts to the studied chaos-based systems, thus providing a good benchmark. Figure 8a,b show the spectra of 4-QAM and QCPSK alongside the estimated amplitude transfer function \(|\)K(\(f\))\(|\) of the multipath channel. The figure shows that 4-QAM and  
Figure 6: Test setup for the QCPSK and FM-CSK systems’ performance estimation in selective fading conditions. Figure 7: The model of the emulated multipath channel. QCPSK have the same frequency bandwidth at a -20 dB level. However, in the case of QCPSK, the carrier's amplitude is smaller than the sideband amplitudes, which is not the case in 4-QAM. This difference causes different notches in the spectra of 4-QAM and QCPSK using the same multipath channel parameters. For this reason, the channels' depths are configured differently for 4-QAM and QCPSK but result in the same notch for the two signals at 140 MHz frequency. The FSK system used for the comparison has two carriers that are offset by \(\pm 100\) kHz from 140 MHz. This configuration makes the FSK's bandwidth match the FM-CSK information-carrying signal's spectra, as seen in Figure 9a,b. The notch frequency for the FSK is set to the lower subcarrier. As in the case of 4-QAM and QCPSK, the channel depths are configured differently yet produce the same depth notch in the signals' spectra. Figure 8: The effect of a two-ray channel on 4-QAM (**a**) and QCPSK (**b**).