Cooperative Spectrum Sensing: From Fundamental Limits to Practical Designs

Abstract

The first and foremost function in cognitive radio is spectrum sensing. To overcome the performance bottleneck created by fading and shadowing effects of the wireless channel, cooperation among sensing users is proposed as a promising solution. However, most designs of cooperative sensing strategies are developed in a rather ad hoc manner due to the lack of the fundamental knowledge on the cooperation gain. Hence, in this chapter, the cooperation gain in the context of cooperative spectrum sensing is rigorously quantified via diversity and error exponent analyses. The fundamental limits of the diversity and error exponent are derived as functions of key system parameters. After that, a couple of case studies are presented to illustrate how the concepts of diversity and error exponent can guide practical designs of cooperative spectrum sensing.

Appendices

Appendices

Appendix 1: Proof of Theorem 2

With $\theta _{s} = N(1 + \frac{1} {\gamma } )\log (\gamma +1)$, as γ → +∞, $e^{-\theta _{s}} = (1+\gamma )^{-N(1+\frac{1} {\gamma } )} \sim (\gamma +1)^{-N}$ and $\sum _{i=0}^{N-1}\theta _{s}^{i}/i! =\sum _{ i=0}^{N-1}N^{i}(1 + \frac{1} {\gamma } )^{i}(\log (\gamma +1))^{i}/i! \sim N^{N-1}(\log (\gamma +1))^{N-1}/(N - 1)!\;$. Thus, according to (14), $P_{f,s} \sim \frac{N^{N-1}} {(N-1)!}(\log (\gamma +1))^{N-1}(\gamma +1)^{-N}$ and

$$\displaystyle{ d_{f,s} = -\lim _{\gamma \rightarrow +\infty }\frac{\log P_{f,s}} {\log \gamma } = N. }$$

Also, with θ _s defined above, as γ → +∞, $e^{-\theta _{s}/(\gamma +1)} = e^{-\frac{N} {\gamma } \log (\gamma +1)} \rightarrow 1$ and $\sum _{i=N}^{+\infty }\theta _{s}^{i}/(i!(\gamma +1)^{i}) =\sum _{ i=N}^{+\infty }N^{i}(1+ \frac{1} {\gamma } )^{i}(\log (\gamma +1))^{i}/(i!(\gamma +1)^{i}) \sim N^{N}(\log (\gamma +1))^{N}(\gamma +1)^{-N}/N!\;$. Thus, according to (15), $P_{md,s} \sim \frac{N^{N}} {N!} (\log (\gamma +1))^{N}(\gamma +1)^{-N}$ and

$$\displaystyle{ d_{md,s} = -\lim _{\gamma \rightarrow +\infty }\frac{\log P_{md,s}} {\log \gamma } = N. }$$

Accordingly, d _e, s = min(d _f, s , d _md, s ) = N.

Appendix 2: Proof of Theorem 3

From (18) and (19),

$$\displaystyle{ P_{f,h} = P(\lambda _{h} \geq \theta _{h}\vert H_{0}) =\sum _{ i=\theta _{h}}^{N}\binom{N}{i}P_{ f,l}^{i}(1 - P_{ f,l})^{N-i}. }$$

The false alarm diversity at the local sensing decision is d _f, l , so as γ → +∞, $P_{f,l} \sim (\gamma +1)^{-d_{f,l}}$, 1 − P _f, l → 1; thus $P_{f,l}^{i}(1 - P_{f,l})^{N-i} \sim (\gamma +1)^{-id_{f,l}}$. In the summation of P _f, h , as γ → +∞, the term with lowest power order of (γ + 1)⁻¹ will dominate; thus $P_{f,h} \sim \binom{N}{\theta _{h}}(\gamma +1)^{-\theta _{h}d_{f,l}}$, and the false alarm diversity order is

$$\displaystyle{ d_{f,h} = -\lim _{\gamma \rightarrow +\infty }\frac{\log P_{f,h}} {\log \gamma } =\theta _{h}d_{f,l},\ \ \theta _{h} = 1,2,\ldots,N. }$$

From (18) and (19),

$$\displaystyle{ P_{md,h} = P(\lambda _{h} <\theta _{h}\vert H_{1}) =\sum _{ i=0}^{\theta _{h}-1}\binom{N}{i}(1 - P_{ md,l})^{i}P_{ md,l}^{N-i}. }$$

From the same argument, the missed detection diversity order is

$$\displaystyle{ d_{md,h} = - \lim _{\gamma \rightarrow +\infty } \frac{\log P_{md,h}} {\log \gamma } = (N -\theta _{h} + 1)d_{md,l},\ \theta _{h} = 1,2,\ldots,N. }$$

Appendix 3: Proof of Corollary 3

From Corollary 1, we know that the local threshold can be chosen as θ _l = d ₀ θ ^o (d ₀ > 0). In this case, d _f, l = d ₀ and d _md, l = 1. By Theorem 3, if the hard decision threshold is θ _h (θ _h ∈ {1, 2, …, N}), the average error diversity is d _e, h = min(θ _h d ₀, N −θ _h + 1). To maximize d _e, h , we need to maximize both θ _h d ₀ and (N −θ _h + 1) simultaneously. By maximizing the latter, we obtain θ _h = 1. Then d _e, h = min(d ₀, N). Thus, as long as d ₀ ≥ N, we obtain the maximum diversity d _e, h = N. However, as stated in Corollary 1, higher d ₀ will cause higher SNR loss for the missed detection performance. Thus, we choose d ₀ = N to minimize the SNR loss for the missed detection performance while achieving the maximum average error diversity.

Appendix 4: Proof of Theorem 4

If $\boldsymbol{\mathcal{R}}_{1} =\{ (n_{0},n_{1}): n_{1} \geq \theta _{t}\}$, then:

$$\displaystyle{ \begin{array}{rl} P_{f,t}& = \sum _{ n_{1}=\theta _{t}}^{N}\sum _{n_{0}=0}^{N-n_{1}} \frac{N!} {n_{0}!(N-n_{0}-n_{1})!n_{1}!}\alpha _{1}^{n_{0}}\alpha _{ 2}^{N-n_{0}-n_{1}}\alpha _{ 3}^{n_{1}} \\ & = \sum _{ n_{1}=\theta _{t}}^{N} \frac{N} {n_{1}!(N-n_{1})!}(1 -\alpha _{3})^{N-n_{1}}\alpha _{ 3}^{n_{1}}, \end{array} }$$

and

$$\displaystyle{ \begin{array}{rl} P_{md,t}& = \sum _{ n_{1}=0}^{\theta _{t}-1}\sum _{n_{ 0}=0}^{N-n_{1}} \frac{N!} {n_{0}!(N - n_{0} - n_{1})!n_{1}!}\beta _{1}^{n_{0}}\beta _{ 2}^{N - n_{0} - n_{1}}\beta _{ 3}^{n_{1}} \\ & = \sum _{ n_{1}=0}^{\theta _{t}-1} \frac{N} {n_{1}!(N-n_{1})!}(1 -\beta _{3})^{N-n_{1}}\beta _{ 3}^{n_{1}}. \end{array} }$$

This is equivalent to BT with θ _l, B = θ _l, 2 and θ _f, B = θ _t .

If $\boldsymbol{\mathcal{R}}_{0} =\{ (n_{0},n_{1}): n_{0} \geq N + 1 -\theta _{t}\}$, then:

$$\displaystyle{ \begin{array}{rl} P_{f,t}& = \sum _{ n_{0}=0}^{N - \theta _{t} + 1}\sum _{n_{ 1}=0}^{N - n_{0}} \frac{N!} {n_{0}!(N - n_{0} - n_{1})!n_{1}!}\alpha _{1}^{n_{0}}\alpha _{ 2}^{N-n_{0}-n_{1}}\alpha _{ 3}^{n_{1}} \\ & = \sum _{ n_{0}=0}^{N-\theta _{t}+1} \frac{N} {n_{0}!(N-n_{0})!}\alpha _{1}^{n_{0}}(1 -\alpha _{ 1})^{N-n_{0}}, \end{array} }$$

and

$$\displaystyle{ \begin{array}{rl} P_{md,t}& = \sum _{ n_{0} = N - \theta _{t} + 1}^{N}\sum _{n_{1} = 0}^{N-n_{1}} \frac{N!} {n_{0}!(N - n_{0} - n_{1})!n_{1}!}\beta _{1}^{n_{0}}\beta _{ 2}^{N - n_{0} - n_{1}}\beta _{ 3}^{n_{1}} \\ & = \sum _{ n_{0} = N - \theta _{t} + 1}^{N} \frac{N} {n_{0}!(N-n_{0})!}\beta _{1}^{n_{0}}(1 -\beta _{ 1})^{N-n_{0}}. \end{array} }$$

This is equivalent to BT with θ _l, B = θ _l, 1 and θ _f, B = θ _t .