# On Transform Domain Communication Systems under Spectrum Sensing Mismatch: A Deterministic Analysis
## 3 Spectrum Sensing Mismatch
### Semi-Deterministic Analysis of Performance in Multipath Fading Channels  
For multipath Rayleigh fading channels, a well-established discrete-time finite channel response is adopted as \(h_{l}\) with \(l\in\{0,1,...,L-1\}\) (\(L\) is the number of total channel paths). Thus, the channel response in the frequency domain is given as:  
\[H_{k}=\sum_{l=0}^{L-1}h_{l}\cdot e^{-j2\pi kl/N},\quad k=0,1,...,N-1 \tag{25}\]  
In this paper, a linear one-tap frequency-domain zero-forcing (ZF) equalizer, \(\mathbf{G}=\{G_{0},G_{1},...,G_{N-1}\}\), is employed providing perfect channel estimation:  
\[G_{k}=\frac{1}{H_{k}},\quad k=0,1,...,N-1 \tag{26}\]  
For the CCSK-based TDCS, after performing normal OFDM demodulation and equalization, the received signal in the frequency domain is given as:  
\[R_{1}(k)\Bigg{\{}\begin{array}{ll}\lambda e^{im_{k}}e^{-j2\pi kS_{C}/N}+n_{ 1}(k),&k\in\Omega^{Tx}\\ n_{1}(k),&\text{otherwise}\\ n_{1}(k)=\frac{1}{H_{k}\cdot\sqrt{N}}\sum\limits_{k=0}^{N}w(n)e^{j2\pi kn/N} \end{array} \tag{27}\]  
Compared to Equation (26), the signal term remains the same. However, the noise term \(n_{1}(k)\) is divided by the frequency channel response of \(k\)th subcarrier, resulting in the change of noise variance:\[n_{3}(\tau)=\frac{1}{\sqrt{N}}\sum\limits_{k\in\Omega^{Rx}}\Bigg{(}\frac{1}{H_{k} \cdot\sqrt{N}}\sum\limits_{k=0}^{N}w(n)e^{j2\pi kn/N}e^{-jm_{k}}\Bigg{)}e^{j2\pi k \tau/N} \tag{28}\]  
Thus, the SNR for the CCSK-based TDCS under SSM in the multipath Rayleigh fading channel is written as:  
\[(SNR)^{(CCSK)}=\frac{\left(N_{\text{C}}\right)^{2}}{N_{Tx}\Bigg{(}\sum\limits _{k\in\Omega^{Rx}}\left(1/(H_{k})^{2}\right)\Bigg{)}}\cdot\Bigg{(}2\frac{ \varepsilon_{S}}{N_{0}}\Bigg{)}=(1-\alpha)\cdot\underbrace{\Bigg{(}2\frac{ \varepsilon_{S}}{N_{0}}\Bigg{)}}_{traditional} \tag{29}\]  
Following the concept of the mismatch factor \(\alpha\) here, this value now becomes:  
\[\alpha=1-\frac{\left(N_{\text{C}}\right)^{2}}{N_{Tx}\Bigg{(}\sum\limits_{k\in \Omega^{Rx}}\left(1/(H_{k})^{2}\right)\Bigg{)}} \tag{30}\]  
The BER performance can be easily derived using the same expression in [17]. For the QPSK-based TDCS, the performance analysis can be derived by following the same idea, and the final SNR under SSM is given as:  
\[(SNR)^{(QPSK)}=\frac{\left(N_{\text{C}}\right)^{2}}{N_{Tx}\Bigg{(}\sum\limits _{k\in\Omega^{Rx}}\left(1/(H_{k})^{2}\right)\Bigg{)}}\cdot\Bigg{(}\frac{ \varepsilon_{S}}{N_{0}}\Bigg{)}=(1-\alpha)\cdot\underbrace{\Bigg{(}\frac{ \varepsilon_{S}}{N_{0}}\Bigg{)}}_{traditional} \tag{31}\]  
Likely, it is found that \(\alpha\) remains the same value as in Equation (30), and the BER performance expression still satisfies Equation (31).  
From our analysis above, a mismatch factor with an explicit mathematical expression has been introduced to precisely measure the degree of spectrum sensing mismatch. It should be noticed that in the AWGN channel, the value of \(\alpha\) only depends on the spectrum sensing result and the analysis is deterministic; whereas in the multipath Rayleigh fading channel, it is referred to as semi-deterministic analysis since perfect estimation of the channel response is required.