Spectrum sensing based on two state discrete time Markov chain in additive Laplacian noise

Abstract

In this paper, spectrum sensing for primary user (PU) is considered in additive Laplacian noise. Further, we consider dynamic behaviour of PU, where the transitions of PU in both the null and the alternate hypotheses have been modelled by two state discrete time Markov chain (DTMC). We assume PU signal to be quadrature amplitude modulated (M-QAM) with ‘M’ modulation order. The considered Markov parameters are $\tau _o$ and $\mu _o$, which represent the average number of samples present during active (ON) state and idle (OFF) state of the PU, respectively. Furthermore, the $\tau _o$ and $\mu _o$ are functions of transition probability matrix (TPM) in DTMC. At the cognitive terminal for spectrum sensing, we assume perfect information of the TPM or in other words $\tau _o$ and $\mu _o$. Then, we use prevailing detection schemes such as improved absolute value cumulation detection (i-AVCD) with the TPM. We refer to this scheme as modified i-AVCD and derive the resulting detection variable along with threshold. In the detection variable, the received samples at the cognitive terminal are raised to a positive exponent P with range of P defined as $0<P\le 2$. We also derive analytical expressions of the detection probability $(P_D)$ and false alarm probability $(P_F)$. We present our results with receiver operating characteristics (ROC) for the considered scheme. We also present simulation results and find close matching with the analytical counterparts. We discuss the effect of increasing modulation order M on the detection performance. We also discuss a special case of M-QAM at $M=2$ which refers to binary phase shift keying (BPSK) modulation scheme for PU. Further, we discuss special cases of the considered scheme at $P=1$ and $P=2$, i.e., modified AVCD and modified energy detection, respectively. Finally, we compare the performance of the considered scheme with the conventional i-AVCD detection scheme, where information of DTMC or in other words TPM, is not available at the cognitive terminal. We find that the considered scheme outperforms the conventional scheme.

Appendices

Appendix A Derivation of $E[Z(\mathbf{y} )\vert H_o]$

$$\begin{aligned} Z(\mathbf{y} )\vert H_o=\sum \limits _{n = 1}^{N_{00}} p_{00}^{(N-n)}\vert y_{n}\vert ^{P} + \sum \limits _{n = 1}^{N_{01}} p_{01}^{(N-n)}\vert y_n\vert ^P. \end{aligned}$$

(A1)

Hence,

$$\begin{aligned}&E[Z(\mathbf{y} )\vert H_o]= \underbrace{E\Bigg [\sum \limits _{n = 1}^{N_{00}}p_{00}^{(N-n)}w_n\Big \vert _{n=1,2,\cdots , N_{00}}\Bigg ]}_{\beta _1}\nonumber \\&\quad +\underbrace{E\Bigg [\sum \limits _{n = 1}^{N_{01}}p_{01}^{(N-n)}s_n\Big \vert _{n=1,2,\cdots ,N_{01}}\Bigg ]}_{\beta _2}. \end{aligned}$$

(A2)

Now, solving $\beta _1$ and $\beta _2$, we get

$$\begin{aligned} \beta _1&=E\Bigg [\sum \limits _{n = 1}^{N_{00}} p_{00}^{(N-n)} w_n\Big \vert _{n=1,2,\cdots , N_{00}}\Bigg ]\nonumber \\&=E\Bigg [p_{00}^{(N-1)}w_1+p_{00}^{(N-2)}w_2+\cdots +p_{00}^{(N-N_{00})}w_{N_{00}}\Bigg ], \end{aligned}$$

(A3)

where $w_1, w_2,\cdots ,w_{N_{00}}$ are assumed to be independent and identically distributed (i.i.d). Hence, $E[w_1]=E[w_2]=\cdots =E[w_{N_{00}}]=\psi _o$. Hence,

$$\begin{aligned} \beta _1&=\psi _o E\Bigg [p_{00}^{(N-1)}+p_{00}^{(N-2)}+\cdots +p_{00}^{(N-N_{00})}\Bigg ]\nonumber \\&=\psi _o\Bigg [p_{00}^{(N-1)}+p_{00}^{(N-2)}+\cdots +p_{00}^{(N-N_{00})}\Bigg ]\nonumber \\&=\psi _o\sum \limits _{n = 1}^{N_{00}}p_{00}^{(N-n)}. \end{aligned}$$

(A4)

Similarly,

$$\begin{aligned} \beta _2=\psi _1\sum \limits _{n = 1}^{N_{01}}p_{01}^{(N-n)}. \end{aligned}$$

(A5)

Appendix B Derivation of $var[Z(\mathbf{y} )\vert H_o]$

$$\begin{aligned} Z(\mathbf{y} )\vert H_o=\sum \limits _{n = 1}^{N_{10}} p_{10}^{(N-n)}\vert y_{n}\vert ^{P} + \sum \limits _{n = 1}^{N_{11}} p_{11}^{(N-n)}\vert y_n\vert ^P. \end{aligned}$$

(B6)

Hence,

$$\begin{aligned}&var[Z(\mathbf{y} )\vert H_o]= \underbrace{var\Bigg [\sum \limits _{n = 1}^{N_{10}}p_{10}^{(N-n)}w_n\Big \vert _{n=1,2,\cdots , N_{10}}\Bigg ]}_{\xi _o}\nonumber \\&\quad +\underbrace{var\Bigg [\sum \limits _{n = 1}^{N_{11}}p_{11}^{(N-n)}s_n\Big \vert _{n=1,2,\cdots ,N_{11}}\Bigg ]}_{\xi _1}. \end{aligned}$$

(B7)

Now, solving $\xi _o$ and $\xi _1$, we get

$$\begin{aligned} \xi _o&=var\Bigg [\sum \limits _{n = 1}^{N_{10}} p_{10}^{(N-n)}w_n\Big \vert _{n=1,2,\cdots , N_{10}}\Bigg ]\nonumber \\&=var\Bigg [p_{10}^{(N-1)}w_1+p_{10}^{(N-2)}w_2+\cdots \nonumber \\&\quad +p_{10}^{(N-N_{10})}w_{N_{10}}\Bigg ], \end{aligned}$$

(B8)

where $w_1, w_2,\cdots ,w_{N_{10}}$ are assumed to be independent and identically distributed (i.i.d). Hence, $var[w_1]=var[w_2]=\cdots =var[w_{N_{10}}]=\sigma _o^2$. Hence,

$$\begin{aligned} \xi _o&=\sigma _o^2 var\Bigg [p_{10}^{(N-1)}+p_{10}^{(N-2)}+\cdots +p_{10}^{(N-N_{10})}\Bigg ]\nonumber \\&=\sigma _o^2\Bigg [\Big \{p_{10}^{(N-1)}\Big \}^2+\Big \{p_{10}^{(N-2)}\Big \}^2+\cdots \nonumber \\&\quad +\Big \{p_{10}^{(N-N_{10})}\Big \}^2\Bigg ]\nonumber \\&=\sigma _o^2\sum \limits _{n = 1}^{N_{10}}\Big \{p_{10}^{(N-n)}\Big \}^2. \end{aligned}$$

(B9)

Similarly,

$$\begin{aligned} \xi _1=\sigma _1^2\sum \limits _{n = 1}^{N_{11}}\Big \{p_{11}^{(N-n)}\Big \}^2 \end{aligned}$$

(B10)