Spectrum Sensing Techniques Based on Last Status Change Point Estimation for Dynamic Primary User in Additive Laplacian Noise

Abstract

A real time scenario of dynamic primary user (PU) is considered in additive Laplacian noise. Two transitions or status changes of PU in the fixed sensing time are considered. The last status change point (LSCP) is estimated with maximum likelihood estimation by using dynamic programming. We consider Cumulative Sum (CuSum) based weighted samples for detection. We consider three detection schemes such as sample mean detector, energy detection and improved absolute value cumulation detection. We derive closed form expressions of detection probability $(P_D)$ and false alarm probability $(P_F)$ for all the three schemes. We present our results with receiver operating characteristic (ROC) for the considered schemes. We also present simulation results, which are closely matching with their analytical counterparts. We compare the ROC of the considered system with the ROC of conventional techniques. In the conventional techniques, all the samples in the sensing time are used for detection without LSCP estimation and weight. It is found that the considered system outperforms the conventional schemes.

Appendix

A. Proof of $E[Z'(\mathbf{y} )|H'_o]$

From (7), we have

$$\begin{aligned} Z'(\mathbf{y} )&= \frac{1}{N-L_{scp}}\left( \sum \limits _{m = 1}^{N-L_{scp}} (N-L_{scp}-m+1) y_{N-m+1}\right) . \end{aligned}$$

(20)

Expanding the above expression, we get

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_o]&= \frac{1}{N-L_{scp}}E\Bigg [ (N-L_{scp}) w_{N}\nonumber \\&\quad +(N-L_{scp}-1)w_{N-1}+\cdots +w_{L_{scp}+1}\Bigg ]. \end{aligned}$$

(21)

As $w_N,w_{N-1},w_{N-2}\cdots$ are all i.i.ds, hence

$E\big [w_N\big ]=E[w_{N-1}\big ]=\cdots E\big [w_{L_{scp}+1}\big ]=0.$ The above expression of $E[Z'(\mathbf{y} )|H'_o]$ can be further simplified as

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_o]&=0. \end{aligned}$$

B. Proof of $var[Z'(\mathbf{y} )|H'_o]$

We have

$$\begin{aligned} var[Z'(\mathbf{y} )|H'_o]&= var\left[ \frac{1}{N-L_{scp}}\left( \sum \limits _{m = 1}^{N-L_{scp}} (N-L_{scp}-m+1)y_{N-m+1}\right) \right] \nonumber \\&= {\bigg (\frac{1}{N-L_{scp}}}\bigg ) var\Bigg [ (N-L_{scp}) w_N+(N-L_{scp}-1) w_{N-1} \nonumber \\&\quad + \cdots +w_{L_{scp}+1}\Bigg ]. \end{aligned}$$

(22)

As $w_N,w_{N-1},w_{N-2}\cdots$ are all i.i.ds, hence $var\big [w_N\big ]=var[w_{N-1}\big ]=\cdots var\big [w_{L_{scp}+1}\big ]=2b^2.$ The above expression of $var[Z'(\mathbf{y} )|H'_o]$ can be further simplified as

$$\begin{aligned} var[Z'(\mathbf{y} )|H'_o]&= \frac{2b^2}{{(N-L_{scp})}^2}\Bigg [ (N-L_{scp})^2\\&\quad +(N-L_{scp}-1)^2+ \cdots +1^2\Bigg ]\\&= \frac{2b^2}{{(N-L_{scp})}^2}\Bigg [1^2+2^2+\cdots + (N-L_{scp-2})^2\\&\quad +(N-L_{scp}-1)^2+(N-L_{scp})^2\Bigg ]\\&=\frac{2b^2}{6}\Bigg [\frac{(N-L_{scp}+1)\big \{(2(N-L_{scp})+1\big \}}{N-L_{scp}}\Bigg ]. \end{aligned}$$

As $L_{scp}=\hat{N_2}$, the final expression becomes

$$\begin{aligned} var[Z'(\mathbf{y} )|H'_o]= \frac{2b^2}{6}\Bigg [\frac{(N-\hat{N_2}+1)\big \{(2(N-\hat{N_2})+1\big \}}{N-\hat{N_2}}\Bigg ] \end{aligned}$$

(23)

C. Proof of $E[Z'(\mathbf{y} )|H'_1]$

We have

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_1] = E\left[ \frac{1}{N-L_{scp}}\left( \sum \limits _{m = 1}^{N-L_{scp}} (N-L_{scp}-m+1) y_{N-m+1}\right) \right] . \end{aligned}$$

(24)

Expanding the above expression, we get

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_1]&= \frac{1}{N-L_{scp}}E\Bigg [ (N-L_{scp}) \big \{w_{N}+C\big \}\\&\quad +(N-L_{scp}-1)\big \{w_{N-1}+C\big \}+\cdots +\big \{w_{L_{scp}+1}+C\big \}\Bigg ]. \end{aligned}$$

As $w_N,w_{N-1},w_{N-2}\cdots$ are all i.i.ds, hence $E\big [w_N\big ]=E[w_{N-1}\big ]=\cdots E\big [w_{L_{scp}+1}\big ]=0.$ The above expression of $E[Z'(\mathbf{y} )|H'_1]$ can be further simplified as

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_1]&= \frac{1}{N-L_{scp}}\Bigg [ C(N-L_{scp}) \\&\quad +C(N-L_{scp}-1)+C(N-L_{scp}-2)+\cdots +C\Bigg ]\\&=\frac{C}{N-L_{scp}}\Bigg [ (N-L_{scp}) \\&\quad +(N-L_{scp}-1)+(N-L_{scp}-2)+\cdots +1\Bigg ]\\&=\frac{C}{2}\bigg \{N-L_{scp}-1\bigg \}. \end{aligned}$$

As $L_{scp}=\hat{N_2'}$, the final expression becomes

$$\begin{aligned} E[Z'(\mathbf{y} )|H'_1]= \frac{C}{2}\bigg \{N-\hat{N_2'}-1\bigg \}. \end{aligned}$$

(25)

D. Proof of $var[Z'(\mathbf{y} )|H'_1]$

We have

$$\begin{aligned} var\left[ Z'(\mathbf{y} )|H'_1\right]&= var\left[ \frac{1}{N-L_{scp}}\left( \sum \limits _{m = 1}^{N-L_{scp}} (N-L_{scp}-m+1) y_{N-m+1}\right) \right] \nonumber \\&= {\bigg (\frac{1}{N-L_{scp}}}\bigg )var\Bigg [ (N-L_{scp}) \big \{w_N+C\big \}+(N-L_{scp}-1)\big \{w_{N-1}+C\big \} \nonumber \\&\quad + \cdots +\big \{w_{L_{scp}+1}+C\big \}\Bigg ]. \end{aligned}$$

(26)

As $w_N,w_{N-1},w_{N-2}\cdots$ are all i.i.ds, hence $var\big [w_N\big ]=var[w_{N-1}\big ]=\cdots var\big [w_{L_{scp}+1}\big ]=2b^2.$ The above expression of $var[Z'(\mathbf{y} )|H'_1]$ can be further simplified as

$$\begin{aligned}&var[Z'(\mathbf{y} )|H'_1]= \frac{2b^2}{{(N-L_{scp})}^2}\Bigg [ (N-L_{scp})^2\nonumber \\&\quad +(N-L_{scp}-1)^2+ \cdots +1^2\Bigg ]\nonumber \\&= \frac{2b^2}{{(N-L_{scp})}^2}\Bigg [1^2+2^2+\cdots + (N-L_{scp-2})^2\nonumber \\&\quad +(N-L_{scp}-1)^2+(N-L_{scp})^2\Bigg ]\nonumber \\&=\frac{2b^2}{6}\Bigg [\frac{(N-L_{scp}+1)\big \{(2(N-L_{scp})+1\big \}}{N-L_{scp}}\Bigg ]\nonumber \\&=\frac{2b^2}{6}\Bigg [\frac{(N-\hat{N_2'}+1)\big \{(2(N-\hat{N_2'})+1\big \}}{N-\hat{N_2'}}\Bigg ]. \end{aligned}$$

(27)