Improved Spectrum Sensing Schemes Using Prewhitening and Weights Under Spatially Correlated Noise

Abstract

Conventional spectrum sensing (SS) schemes in a multiantenna cognitive radio utilize matrix-inverse based prewhitening to decorrelate the spatially correlated received signals. However, the improved SS schemes can be proposed by efficiently exploiting the spatial correlation information. In this paper, two detectors—weighted cross-correlation absolute value detector (WCCAVD) and weighted energy detector (WED), are proposed by exploiting the spatial correlation of noise. It is shown that in the presence of high spatial correlation and low signal-to-noise ratio, the proposed WCCAVD and WED outperform the conventional cross-correlation absolute value detector and energy detector (ED), respectively, by more than 1 dB. For the spatial correlation below 0.5, both the proposed detectors are shown to have comparable performance with respect to the corresponding conventional detectors. The analytical expressions for the decision threshold, the false-alarm and the detection probabilities of the proposed detectors are derived. The analytical results are validated by Monte-Carlo simulations.

Appendix

1.1 Calculation of Mean and Variance of $g_{ij}$

For large values of N, $g_{ij}$ can be approximated to Gaussian distributed with mean and variance determined as

$$\begin{aligned} \mathrm{E}[g_{ij} ]&= {N \over {\lambda _i \lambda _j }}(\left| {h_i } \right| \left| {h_j } \right| \varepsilon _s + \sigma _w^2 \lambda _{i( = j)} \delta _{ij} ) \end{aligned}$$

(31)

$$\begin{aligned} {\mathrm{var}} [g_{ij} ]&= {1 \over {\lambda _i^2 \lambda _j^2 }}\sum _{n_1 = 1}^N {\sum _{n_2 = 1}^N {{\mathrm{cov}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ,z_i^* (n_2 )z_j (n_2 )e^{\mathbf{j }\phi _{ij} } ]} } \\&= {1 \over {\lambda _i^2 \lambda _j^2 }}\left( {\sum _{n_1 = 1}^N {{\mathrm{var}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ]} } \right. \\&\quad +\left. {2\sum _{n_1 < n_2 } {{\mathrm{cov}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ,z_i^* (n_2 )z_j (n_2 )e^{\mathbf{j }\phi _{ij} } ]} } \right) . \end{aligned}$$

(32)

The first term in (32) can be calculated as

$$\begin{aligned} {\mathrm{var}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ] = \mathrm{E}\left[ \left| {z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } } \right| ^2\right] - \left| {\mathrm{E}\left[ z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} }\right] } \right| ^2. \end{aligned}$$

We use an identity [22] $\text{ E }[x_1 x_2 x_3 x_4 ] = \text{ E }[x_1 x_2 ]\text{ E }[x_3 x_4 ] + \text{ E }[x_1 x_3 ]\text{ E }[x_2 x_4 ] + \text{ E }[x_1 x_4 ]\text{ E }[x_2 x_3 ] - 2\text{ E }[x_1 ]\text{ E }[x_2 ]\text{ E }[x_3 ]\text{ E }[x_4 ]$ to calculate $\mathrm{{E}}[\left| {z_i^* (n_1 )z_j (n_1 )e^{\mathbf{{j}}\phi _{ij} } } \right| ^2 ] = (\left| {h_i } \right| \left| {h_j } \right| \varepsilon _s + \sigma _w^2 \lambda _{i( = j)} \delta _{ij} )^2 + (\left| {h_i } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _i )(\left| {h_j } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _j )$. Also, it can be easily obtained $\left| {\mathrm{E}[z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ]} \right| ^2 = (\left| {h_i } \right| \left| {h_j } \right| \varepsilon _s + \sigma _w^2 \lambda _{i( = j)} \delta _{ij} )^2.$ Hence, the first term of (32) is

$$\begin{aligned} {\mathrm{var}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ] = (\left| {h_i } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _i )(\left| {h_j } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _j ). \end{aligned}$$

The second term of (32) is calculated as

$$\begin{aligned}&{\mathrm{cov}} [z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ,z_i^* (n_2 )z_j (n_2 ) e^{\mathbf{j }\phi _{ij} } ] \\&\quad = \text{ E }[z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } z_i (n_2 )z_j^* (n_2 )e^{ - \mathbf{j }\phi _{ij} } ] - \text{ E }[z_i^* (n_1 )z_j (n_1 )e^{\mathbf{j }\phi _{ij} } ]\text{ E }[z_i (n_2 )z_j^* (n_2 )e^{ - \mathbf{j }\phi _{ij} } ] \\&\quad = 0. \end{aligned}$$

Finally, (32) can be determined as

$$\begin{aligned} {\mathrm{var}} [g_{ij} ] = {{N(\left| {h_i } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _i )(\left| {h_j } \right| ^2 \varepsilon _s + \sigma _w^2 \lambda _j )} \over {\lambda _i^2 \lambda _j^2 }}. \end{aligned}$$

(33)