## 3 Performance in Selective Fading Conditions
### Performance Analysis in the Selective Fading Conditions  
The results of the study for the QCPSK and matching 4-QAM systems are presented in Figures 11 and 12. The results for the FM-CSK and FSK systems are presented in Figures 13 and 14. The figures show the SNR influence on BER for several multipath channel spectral notch levels. For the comparability issues and to show the performance degradation dynamics, the SNR is normalized using the approach described for the AWGN channel. Figures 11 and 13 show the approximated curves constructed from the data points.  
Figure 11: QCPSK communication system’s performance for the two-ray channel.  
Figure 10: QCPSK and FM-CSK communication systems’ performance in the AWGN channel.  
As one can see in Figures 11-14, SNR is normalized for a 0.00 dB notch at BER = 10\({}^{-3}\) for all studied communication systems for better side-by-side comparison. QCPSK and 4-QAM data transmission system BER performance (Figures 11 and 12) for a 0.10 dB notch is 3.71 \(\times\) 10\({}^{-3}\) and 1.45 \(\times\) 10\({}^{-3}\), respectively. Increasing the notch value to 0.48 dB leads to QCPSK communication system BER performance reduction to 7.37 \(\times\) 10\({}^{-3}\), while BER performance in the 4-QAM data transmission system is 3.42 \(\times\) 10\({}^{-3}\). A further increase of the notch to 3.25 dB and 7 64 dB significantly reduces the BER performance of the 4-QAM communication system to 9.63 \(\times\) 10\({}^{-2}\) and 3.29 \(\times\) 10\({}^{-1}\), respectively. At the same notch values, QCPSK data transmission system BER performance is 1.49 \(\times\) 10\({}^{-2}\) and 2.93 \(\times\) 10\({}^{-2}\), respectively. While at lower notch values (0.10 dB and 0.48 dB) 4-QAM communication system shows better performance, at higher notch values (3.25 dB and 7.64 dB) QCPSK communication system shows significantly better performance.  
In the case of FM-CSK and FSK data transmission systems, BER performance (Figures 13 and 14) for a 3.25 dB notch is 1.52 \(\times\) 10\({}^{-3}\) and 1.79 \(\times\) 10\({}^{-3}\), respectively. For a 5.00 dB notch, both FM-CSK and FSK communication systems, BER performance is similar: 3.92 \(\times\) 10\({}^{-3}\) and 3.83 \(\times\) 10\({}^{-3}\), respectively. When increasing the notch to 7.64 dB, BER performance is 1.47 \(\times\) 10\({}^{-2}\) and 6.82 \(\times\) 10\({}^{-3}\) for FM-CSK and FSK communication systems.  
Figure 14: FSK communication system’s performance for the two-ray channel.  
Figure 12: 4-QAM communication system’s performance for the two-ray channel.  
Figure 13: FM-CSK communication system’s performance for the two-ray channel.  
For the highest investigated notch, 9.08 dB, the BER for FM-CSK and FSK is 2.55 \(\times\) 10\({}^{-2}\) and 1.18 \(\times\) 10\({}^{-2}\) correspondingly. The FM-CSK communication system has a lower BER in the case of a 3.25 dB notch compared to the FSK communication system, while at a 5.00 dB notch the systems performed similarly. However, at notch values 7.64 and 9.08 dB, FSK performs better than FM-CSK.  
Comparing communication systems based on chaotic oscillator QCPSK and FM-CSK at a 3.25 dB notch BER is 1.49 \(\times\) 10\({}^{-2}\) and 1.52 \(\times\) 10\({}^{-3}\), while at a 7.64 dB notch BER is 2.93 \(\times\) 10\({}^{-2}\) and 1.47 \(\times\) 10\({}^{-2}\). It shows that the FM-CSK communication system for deeper notches performs better than the QCPSK.