Depending on the SU transmitter properties and the PU quality-of-service requirements, there may be different power constraints in a spectrum sharing network. Here, we first study the case where no power allocation is done by the SU transmitter, and the codewords are sent with a fixed power. Later, we will relax this constraint, permitting adaptive power allocation by the SU transmitter.

### PU interference power constraint and nonadaptive SU transmission power

Limiting the PU average received interference power, which for fixed SU transmission power is found as *T*_{s}*EG*_{sp}, to be less than a threshold *μ*, we have {T}_{\mathsf{\text{s}}}\le \frac{\mu}{E{G}_{\mathsf{\text{sp}}}}. Then, as the transmission rate of Gaussian channels is an increasing function of the SIR [37, 38], the optimal case is obtained by equality. On the other hand, we can instead consider the case where the instantaneous received interference power is with probability *ξ* less than a threshold *η*. In this case, as we have

Pr\left\{\mathsf{\text{In}}{\mathsf{\text{t}}}_{\mathsf{\text{p}}}\le \eta \right\}=Pr\left\{{T}_{\mathsf{\text{s}}}{G}_{\mathsf{\text{sp}}}\le \eta \right\}={F}_{{G}_{\mathsf{\text{sp}}}}\left(\frac{\eta}{{T}_{\mathsf{\text{s}}}}\right)

the optimal transmission power satisfying the instantaneous interference power constraint Pr{Int_{p} ≤ *η*} ≥ *ξ* is obtained by

{T}_{\mathsf{\text{s}}}=\frac{\eta}{{F}_{{G}_{\mathsf{\text{sp}}}}^{-1}\left(\xi \right)}.

(8)

Here, {F}_{{G}_{\mathsf{\text{sp}}}}^{-1}\left(.\right) is the inverse function of the SU-PU fading cdf which for Rayleigh-fading channels simplifies to {F}_{{G}_{\mathsf{\text{sp}}}}^{-1}\left(x\right)=\frac{-1}{\lambda}log\left(1-x\right).

### PU outage probability constraint and nonadaptive SU transmission power

Assuming that the PU message is transmitted at a fixed rate *R*_{p}, the PU outage probability^{b} is found as

\begin{array}{lll}\hfill Pr{\left\{\mathsf{\text{outage}}\right\}}_{\mathsf{\text{p}}}& =Pr\left\{log\left(1+\frac{{T}_{\mathsf{\text{p}}}{G}_{\mathsf{\text{pp}}}}{{T}_{\mathsf{\text{s}}}{G}_{\mathsf{\text{sp}}}}\right)\le {R}_{\mathsf{\text{p}}}\right\}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =Pr\left\{\frac{{G}_{\mathsf{\text{pp}}}}{{T}_{\mathsf{\text{s}}}{G}_{\mathsf{\text{sp}}}}\le \frac{{e}^{{R}_{\mathsf{\text{p}}}}-1}{{T}_{\mathsf{\text{p}}}}\right\}={F}_{{\Omega}_{\mathsf{\text{p}}}}\left(\frac{{e}^{{R}_{\mathsf{\text{p}}}}-1}{{T}_{\mathsf{\text{p}}}}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}

(9)

in which {F}_{{\Omega}_{\mathsf{\text{p}}}}\left(.\right) is the cdf of the variable {\Omega}_{\mathsf{\text{p}}}=\frac{{G}_{\mathsf{\text{pp}}}}{{T}_{\mathsf{\text{s}}}{G}_{\mathsf{\text{sp}}}} obtained with the same procedure as in (7). In this way, constraining the PU outage probability to be less than *π*_{p}, i.e., Pr {outage}_{p} ≤ *π*_{p}, the SU transmission power is obtained as

\begin{array}{c}{T}_{\mathsf{\text{s}}}={\u2308\frac{{T}_{\mathsf{\text{p}}}}{{e}^{{R}_{\mathsf{\text{p}}}}-1}\frac{\gamma -\sqrt{{\gamma}^{2}-\frac{\phi}{2}}}{\phi}\u2309}^{+}\\ \phi =2{\pi}_{\mathsf{\text{p}}}\left(1-{\pi}_{\mathsf{\text{p}}}\right),\\ \gamma =\phi +1-{\beta}_{\mathsf{\text{ppsp}}}^{2}\end{array}

(10)

where ⌈*x*⌉^{+} ≐ max{0, *x*}. Finally, it is worth noting that, with appropriate scalings, (10) can be mapped to the case where, with some probability, the PU received SIR is constrained to be higher than some threshold.

### Power adaptation at the SU transmitter

Intuitively, the SU-SU achievable rates can be increased by adaptive power allocation at the SU transmitter. For instance, we can consider the case where the SU transmission powers are determined such that, while the SU transmission rates are maximized, the PU average received interference power does not exceed a given threshold. However, under perfect CSI assumption, the SU average transmission power is directly related to the PU received interference power. Therefore, we can consider the SU average transmission power instead. In this way, since the average SU transmission power is obtained by {\stackrel{\u0304}{T}}_{\mathsf{\text{s}}}={\int}_{0}^{\infty}{T}_{\mathsf{\text{s}}}\left(\omega \right){f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)\mathsf{\text{d}}\omega, the optimal transmission power for every SU received SIR realization is found by the Lagrangian objective function

\Upsilon =\underset{0}{\overset{\infty}{\int}}{f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)log\left(1+\omega {T}_{\mathsf{\text{s}}}\left(\omega \right)\right)\mathsf{\text{d}}\omega -\rho \underset{0}{\overset{\infty}{\int}}{T}_{\mathsf{\text{s}}}\left(\omega \right){f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)\mathsf{\text{d}}\omega

(11)

in which *ρ* is the Lagrange multiplier satisfying the SU power constraint {\stackrel{\u0304}{T}}_{\mathsf{\text{s}}}\le T (or equivalently, the PU average received interference power constraint) [19]. Setting \frac{\partial \Upsilon}{\partial {T}_{\mathsf{\text{s}}}\left(\omega \right)}=0, the optimal transmission powers maximizing (6) are obtained by the water-filling equation

{T}_{\mathsf{\text{s}}}\left(\omega \right)={\u2308\frac{1}{\rho}-\frac{1}{\omega}\u2309}^{+}.

(12)

The Lagrange multiplier *ρ* is determined based on

\rho =\underset{z}{arg}\left\{\underset{z}{\overset{\infty}{\int}}\left(\frac{1}{z}-\frac{1}{\omega}\right){f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)\mathsf{\text{d}}\omega =T\right\}

(13)

where we have

\begin{array}{lll}\hfill \underset{z}{\overset{\infty}{\int}}\left(\frac{1}{z}-\frac{1}{\omega}\right){f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)\mathsf{\text{d}}\omega & \stackrel{\left(b\right)}{=}\underset{z}{\overset{\infty}{\int}}\frac{1-{F}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)}{{\omega}^{2}}\mathsf{\text{d}}\omega \phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \stackrel{\left(c\right)}{=}{T}_{\mathsf{\text{p}}}\left\{\frac{1}{z{T}_{\mathsf{\text{p}}}}+\frac{1}{2}\right.\sqrt{{\left(1+\frac{1}{z{T}_{\mathsf{\text{p}}}}\right)}^{2}}-\frac{4{\beta}_{\mathsf{\text{ssps}}}^{2}}{z{T}_{\mathsf{\text{p}}}}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \phantom{\rule{1em}{0ex}}+\left({\beta}_{\mathsf{\text{ssps}}}^{2}-1\right)log\left(1-2{\beta}_{\mathsf{\text{ssps}}}^{2}+\frac{1}{z{T}_{\mathsf{\text{p}}}}+\sqrt{{\left(1+\frac{1}{z{T}_{\mathsf{\text{p}}}}\right)}^{2}-\frac{4{\beta}_{\mathsf{\text{ssps}}}^{2}}{z{T}_{\mathsf{\text{p}}}}}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \phantom{\rule{1em}{0ex}}-\left(\right)close="\}">\frac{1}{2}+\left(1-{\beta}_{\mathsf{\text{ssps}}}^{2}\right)log\left(2-2{\beta}_{\mathsf{\text{ssps}}}^{2}\right)& .\phantom{\rule{2em}{0ex}}\\ \hfill \text{(4)}\end{array}\n \n \n \n (5)\n \n \n \n

(14)

Again, (*b*) is obtained by partial integration, and (*c*) is based on (7) and some calculations. Finally, replacing (12) in (6), the SU-SU channel capacity with average SU transmission power (or equivalently, PU average interference power) constraint is obtained by

\begin{array}{lll}\hfill {C}_{\mathsf{\text{SU-SU}}}& =\underset{\rho}{\overset{\infty}{\int}}{f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)log\left(1+\omega {T}_{\mathsf{\text{s}}}\left(\omega \right)\right)\mathsf{\text{d}}\omega \phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\underset{\rho}{\overset{\infty}{\int}}{f}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)log\left(\frac{\omega}{\rho}\right)\mathsf{\text{d}}\omega \phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\underset{\rho}{\overset{\infty}{\int}}\frac{1-{F}_{{\Omega}_{\mathsf{\text{s}}}}\left(\omega \right)}{\omega}\mathsf{\text{d}}\omega \phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\frac{1}{2}\left\{log\left(1-2{\beta}_{\mathsf{\text{ssps}}}^{2}+\frac{1}{\rho {T}_{\mathsf{\text{p}}}}+\sqrt{{\left(1+\frac{1}{\rho {T}_{\mathsf{\text{p}}}}\right)}^{2}-\frac{4{\beta}_{\mathsf{\text{ssps}}}^{2}}{\rho {T}_{\mathsf{\text{p}}}}}\right)\right.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \phantom{\rule{1em}{0ex}}+\mathsf{\text{arctanh}}\left(\frac{1+\frac{1}{\rho {T}_{\mathsf{\text{p}}}}-\frac{2{\beta}_{\mathsf{\text{ssps}}}^{2}}{\rho {T}_{\mathsf{\text{p}}}}}{\sqrt{{\left(1+\frac{1}{\rho {T}_{\mathsf{\text{p}}}}\right)}^{2}-\frac{4{\beta}_{\mathsf{\text{ssps}}}^{2}}{\rho {T}_{\mathsf{\text{p}}}}}}\right)+log\left(\right)close="\}">\left(\frac{1}{\rho {T}_{\mathsf{\text{p}}}}\right)-log\left(\frac{2-2{\beta}_{\mathsf{\text{ssps}}}^{2}}{{\beta}_{\mathsf{\text{ssps}}}^{2}}\right)& \phantom{\rule{2em}{0ex}}\\ \hfill \text{(5)}\end{array}\n \n \n \n (6)\n \n \n \n

(15)

where the second (the third) equality comes from partial integration (partial integration and some manipulations). Finally, it is interesting to note that (12) implies no spectrum sharing at *weak* channels realizations, i.e., *T*_{s}(*ω*) = 0, *ω* ≤ *ρ*, where the saved power is spent on *good* fading realizations, i.e., *T*_{s}(*ω*) *>* 0, *ω > ρ*. Then, based on (13), it is easy to show that the threshold *ρ* is a decreasing function of the average transmission power *T*. That is, more realizations of the variable Ω_{s} receive powers as the average transmission power increases (water-filling).