# On Transform Domain Communication Systems under Spectrum Sensing Mismatch: A Deterministic Analysis
## 3 Spectrum Sensing Mismatch
### Deterministic Analysis of Performance in AWGN Channels  
In this discussion, it is assumed that most of the signal processing operations for TDCS remains the same, and the attention is focused on the mismatch of spectral bins. For the CCSK model, substituting the introduced \(\Omega^{Tx}\) for \(\Omega^{C}\) in [17] generates:  
\[(x_{n})^{(CCSK)}=\lambda\frac{1}{\sqrt{N}}\sum_{k\in\Omega^{Tx}}\Big{(}e^{jm_{k }}e^{-j2\pi kS_{C}/N}\Big{)}e^{j2\pi kn/N} \tag{11}\]  
After passing though the AWGN channel, the received signal \(\mathbf{r}=\{r_{0},r_{1},...,r_{N-1}\}\) can be expressed as:  
\[(r_{n})^{(CCSK)}=\lambda\frac{1}{\sqrt{N}}\sum_{k\in\Omega^{Tx}}\Big{(}e^{jm_{k }}e^{-j2\pi kS_{C}/N}\Big{)}e^{j2\pi kn/N}+w(n) \tag{12}\]  
where \(w(n)\) denotes the additive white Gaussian noise with \(\mathcal{E}\Big{\{}(w(n))(w(n))^{H}\Big{\}}=N_{0}\). Thus, the frequency domain term is given by:  
Figure 6: Simplified block diagram of TDCS transceivers. \[R_{1}(k)\left\{ \begin{array}{ll}\lambda e^{jm_{\mathrm{k}}}e^{-j \mathrm{\Delta}\kappa S_{C}/N}+n_{1}(k),&k\in\Omega^{Tx}\\ n_{1}(k),&\text{otherwise}\end{array} \right. \tag{13}\] \[n_{1}(k)=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N}w(n)e^{j\mathrm{ \Delta}\kappa n/N}\]  
The second term, \(n_{1}(k)\), denotes the output of the Gaussian noise after FFT operation. According to [PERSON]'s theorem and the property of independent Gaussian variables [18], it still maintains a Gaussian distribution. After multiplying with the local reference signal in an element-wise manner, we obtain:  
\[R_{2}(k)=\widehat{RCCSK}(k)+\widehat{n_{2}(k)} \tag{14}\]  
To be specific:  
\[R_{CCSK}(k) =\left\{ \begin{array}{ll}\lambda e^{-j\mathrm{\Delta}\kappa S_{C }/N},&k\in\Omega^{C}\\ 0,&\text{otherwise}\end{array} \right. \tag{15}\] \[n_{2}(k) =\left\{ \begin{array}{ll}e^{-jm_{k}}\cdot n_{1}(k),&k\in\Omega ^{Rx}\\ 0,&\text{otherwise}\end{array} \right.\]  
Notice that the difference of spectral bin index \(k\) in [16] between the signal term and noise term. Meanwhile, \(n_{2}(k)\) still maintains a Gaussian noise when it is valid. Thus, the related PCF vector is generated after an IFFT operation, \(\mathbf{y}=\mathcal{F}^{-1}(\mathbf{R}_{2})\), with:  
\[y_{\tau}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}\left(R_{2}(k)\right)e^{j \mathrm{\Delta}\kappa\tau/N}=\lambda\cdot R\left\langle\tau-S_{C}\right\rangle _{N}+n_{3}(\tau) \tag{16}\]  
where:  
\[R\left\langle\tau\right\rangle_{N} =\frac{1}{\sqrt{N}}\sum\limits_{k\in\Omega^{C}}e^{-j\mathrm{ \Delta}\kappa\left(\tau\right)/N} \begin{array}{ll}\text{IFFT}\\ \text{FFT}\end{array} \delta\left\langle\left\langle\tau\right\rangle_{N}\right\rangle \tag{17}\] \[n_{3}(\tau) =\frac{1}{\sqrt{N}}\sum\limits_{k\in\Omega^{Rx}}\left(n_{3}(k) \right)e^{j\mathrm{\Delta}\kappa\tau/N}\]  
Using the same CCSK detection as in [15], the estimated CCSK symbol is recovered. According to the property of IFFT/FFT, \(R\left\langle\tau\right\rangle_{N}\) is a perfect impulse response signal providing all the spectrum bins available. However, due to the insufficiency of valid spectrum bins in the presence of SSM, the impulse response shape of \(R\left\langle\tau\right\rangle_{N}\) is destroyed and its main peak is now down to \(\left(N_{C}/N\right)\). Thus, the SNR for CCSK-based TDCS under SSM is given as:  
\[xx \tag{18}\]  
where:  
\[\alpha=1-\frac{\left(N_{C}\right)^{2}}{N_{Tx}N_{Rx}} \tag{19}\]  
denotes the proposed mismatch factor as to measure the degree of SSM.