Consider the PU transmission in

*K* frequency slots with equal probability

*λ* and equal power

*S*_{
P
}. The SU keeps sensing every frequency slot and chooses the minimum sampled power to compare with the optimal threshold. If the sampled power is smaller than the predefined threshold, then the SU will perceive the corresponding frequency slot which is idle and initiate its communication. However, in this cognitive process, the SU will not be able to perfectly detect the idle/occupied frequency slot. Since

*miss detection* will bring interference to the PU and

*false alarm* will waste SU’s precious transmission opportunities, they will both have the impact on the network capacity. For a threshold

*γ* used in the selected frequency slot, the

*conditional* probabilities of miss detection

*P*_{m}(

*γ*) and false alarm

*P*_{f}(

*γ*) can be derived as

${P}_{m}\left(\gamma \right)={\displaystyle {\int}_{0}^{\gamma}\frac{{e}^{-\frac{\xi}{{\sigma}_{0}^{2}+{\sigma}_{X}^{2}}}}{{\sigma}_{0}^{2}+{\sigma}_{X}^{2}}d\xi =1-{e}^{\frac{-\gamma}{{\sigma}_{0}^{2}+{\sigma}_{X}^{2}}}\text{,}}$

(7)

and

${P}_{f}\left(\gamma \right)={\displaystyle {\int}_{\gamma}^{\infty}\frac{{e}^{-\frac{\xi}{{\sigma}_{0}^{2}}}}{{\sigma}_{0}^{2}}d\xi}={e}^{-\frac{\gamma}{{\sigma}_{0}^{2}}}\text{.}$

(8)

For the sensing structure and spectrum accessing strategy shown in Figure

2, and independently and exponentially distributed |

*Y*_{
k
}|

^{2},

*k* = 1, 2, …,

*K*, the probability that the minimum power sample is in the frequency slot

*k** can be derived as

$P\left({\left|{Y}_{{k}_{*}}\right|}^{2}\le \forall {\left|{Y}_{k}\right|}^{2}\right)={\displaystyle {\int}_{0}^{\infty}{r}_{k*}{e}^{-{\displaystyle {\sum}_{k=1}^{K}{r}_{k}y}}dy=\frac{{r}_{k*}}{{\displaystyle {\sum}_{j=1}^{K}{r}_{j}}}}$. In a given time, the PU may occupy

*q* frequency slots, where 0 ≤

*q* ≤

*K* with the probability of

*P*_{
q
} =

*λ*^{
q
}(1 −

*λ*)

^{(K − q)}. In the

*q* occupied frequency slots,

*r*_{
j
} = (

*σ*_{0}^{2} +

*σ*_{
X
}^{2})

^{−1}, while in the (

*K – q*) vacant frequency slots,

*r*_{
k
} =

*σ*_{0}^{−2}. Therefore,

$\sum}_{j=1}^{K}{r}_{j}=\frac{q}{\left({\sigma}_{0}^{2}+{\sigma}_{X}^{2}\right)}+\frac{\left(K-q\right)}{{\sigma}_{0}^{2}$. It follows that, for a given situation with

*q* occupied frequency slots and (

*K – q*)

*vacant* frequency slots, the

*conditional* probability that the minimum power sample is in an

*occupied* frequency slot is

${P}_{\mathit{oq}}={\left({\sigma}_{0}^{2}+{\sigma}_{X}^{2}\right)}^{-1}{\left[\frac{q}{\left({\sigma}_{0}^{2}+{\sigma}_{X}^{2}\right)}+\frac{\left(K-q\right)}{{\sigma}_{0}^{2}}\right]}^{-1}$, and the

*conditional* probability that the minimum power sample is in a

*vacant* frequency slot is

${P}_{\mathit{vq}}={\sigma}_{0}^{-2}{\left[\frac{q}{\left({\sigma}_{0}^{2}+{\sigma}_{X}^{2}\right)}+\frac{\left(K-q\right)}{{\sigma}_{0}^{2}}\right]}^{-1}$. Subsequently, the probability that the minimum power sample is in the area of

*occupied* frequency slots can be expressed as

$\begin{array}{l}{P}_{o}={\displaystyle {\sum}_{q=0}^{K}\left(\begin{array}{c}\hfill K\hfill \\ \hfill q\hfill \end{array}\right){P}_{q}q{P}_{\mathit{oq}}}\\ \phantom{\rule{1.5em}{0ex}}{\displaystyle ={\sum}_{q=0}^{K}\left(\begin{array}{c}\hfill K\hfill \\ \hfill q\hfill \end{array}\right)\frac{{\lambda}^{q}{\left(1-\lambda \right)}^{\left(K-q\right)}q}{q+\left(K-q\right)\left(1+{\sigma}_{X}^{2}/{\sigma}_{0}^{2}\right)}}\cdot \end{array}$

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Similarly, the probability that the minimum power sample is in the area of

*vacant* frequency slots can be expressed as

$\begin{array}{l}{P}_{v}={\displaystyle {\sum}_{q=0}^{K}\left(\begin{array}{c}\hfill K\hfill \\ \hfill q\hfill \end{array}\right){P}_{q}\left(K-q\right){P}_{\mathit{vq}}}\\ {\displaystyle \phantom{\rule{1.5em}{0ex}}={\displaystyle {\sum}_{q=0}^{K}\left(\begin{array}{c}\hfill K\hfill \\ \hfill q\hfill \end{array}\right)\frac{{\lambda}^{q}{\left(1-\lambda \right)}^{\left(K-q\right)}\left(K-q\right)\left(1+{\sigma}_{X}^{2}/{\sigma}_{0}^{2}\right)}{q+\left(K-q\right)\left(1+{\sigma}_{X}^{2}/{\sigma}_{0}^{2}\right)}}}\text{.}\end{array}$

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It is interesting to verify that *P*_{
o
} + *P*_{
v
} = 1, and for *K* = 1, *P*_{
o
} = *λ*, *P*_{
v
} = 1 − *λ*.

When miss detection occurs, the SU transmits and hence introduces interference to the PU in only

*one occupied* frequency slot with the probability of

*P*_{
m
}(

*γ*)

*P*_{
o
}/

*K*. In this event, the PU rate in the presence of SU interference is

${log}_{2}\left[1+\frac{{L}_{\mathit{PK}}}{{I}_{S}+{\sigma}_{0}^{2}}\right]$ (in b/s/Hz) where

${L}_{\mathit{PK}}={S}_{\mathit{PK}}{\left|\frac{1}{\sqrt{N}}{\displaystyle {\sum}_{l=0}^{L-1}{h}_{\mathit{FPl}}{e}^{-\frac{j2\pi lk*}{N}}}\right|}^{2}$ is the PU signal power received at the PR with

*S*_{
PK
} =

*KS*_{
P
},

${h}_{\mathit{FPl}}={h}_{0}\cdot {d}_{P}^{-\frac{\alpha}{2}}{h}_{\mathit{Fl}}$,

*I*_{
S
} =

*S*_{
S
}*h*_{0}^{2}*d*_{
SP
}^{−α}|

*h*_{
FSP
}|

^{2} represents the interference from SU to PR, and

*σ*_{0}^{2} is the thermal noise power in a certain frequency slot. Otherwise, in the absence of SU interference, with the probability of [

*λ* −

*P*_{
m
}(

*γ*)

*P*_{
o
}/

*K*], the PU rate in the presence of SU interference is

${\text{log}}_{2}\left[1+\frac{{L}_{\mathit{PK}}}{{\sigma}_{0}^{2}}\right]$ (in b/s/Hz). In other words, the PU link rate of a certain frequency slot can be represented as

$\begin{array}{l}{C}_{\mathit{PU}}=E\left\{\left(\lambda -\frac{{P}_{m}\left(\gamma \right){P}_{o}}{K}\right){\text{log}}_{2}\left[1+\frac{{L}_{\mathit{PK}}}{{\sigma}_{0}^{2}}\right]\right.\\ \phantom{\rule{3.5em}{0ex}}\left.+\frac{{P}_{m}\left(\gamma \right){P}_{o}}{K}{\text{log}}_{2}\left[1+\frac{{L}_{\mathit{PK}}}{{I}_{S}+{\sigma}_{0}^{2}}\right]\right\}\left(\text{in\hspace{0.17em}b/s/Hz}\right)\text{,}\end{array}$

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where *E*{.} represents the expectation over fading and all PU and SU radio locations in the network.

When the false alarm occurs, SU will not transmit (while the frequency slot is actually vacant), effectively introducing capacity loss. This event occurs with the probability of

*P*_{
f
}(

*γ*)

*P*_{
v
}. In other words, the event that frequency slot is actually vacant and the SU correctly detects the absence of the PU with the probability of [1 −

*P*_{
f
}(

*γ*)]

*P*_{
v
}, and, in this case, the SU can transmit in the vacant frequency slot with the rate of

${\text{log}}_{2}\left[1+\frac{{L}_{S}}{{\sigma}_{0}^{2}}\right]$ (in b/s/Hz) where

*L*_{
S
} =

*S*_{
S
}*h*_{0}^{2}*d*_{
S
}^{−α}|

*h*_{
FS
}|

^{2} is the SU signal power received at the SR. On the other hand, in the event of miss detection, SU transmits in an

*occupied* frequency slot in the presence of interference from the PU with the probability of

*P*_{
m
}(

*γ*)

*P*_{
o
} and the rate of

${\text{log}}_{2}\left[1+\frac{{L}_{S}}{{I}_{\mathit{PK}}+{\sigma}_{0}^{2}}\right]$ (in b/s/Hz) where

${I}_{\mathit{PK}}={S}_{\mathit{PK}}{\left|\frac{1}{\sqrt{N}}{\displaystyle {\sum}_{l=0}^{L-1}\phantom{\rule{0.36em}{0ex}}{h}_{\mathit{FPSl}}{e}^{-\frac{j2\pi l{k}^{*}}{N}}}\right|}^{2}$ represents the interference from PU to SR with

*S*_{
PK
} =

*KS*_{
P
}, and

${h}_{\mathit{FPSl}}={h}_{0}{d}_{\mathit{PS}}^{-\frac{\alpha}{2}}{h}_{\mathit{Fl}}$. Therefore, the SU link rate can be expressed as

$\begin{array}{l}{C}_{\mathit{SU}}=E\left\{\left[1-{P}_{f}\left(\gamma \right)\right]{P}_{v}{\text{log}}_{2}\left[1+\frac{{L}_{S}}{{\sigma}_{0}^{2}}\right]\right.\\ \phantom{\rule{3.5em}{0ex}}\left.+{P}_{m}\left(\gamma \right){P}_{o}{\text{log}}_{2}\left[1+\frac{{L}_{S}}{{I}_{\mathit{PK}}+{\sigma}_{0}^{2}}\right]\right\}\left(\text{in}\phantom{\rule{0.12em}{0ex}}\text{b/s/Hz}\right)\text{.}\end{array}$

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