A Closed Form Estimate of TVWS Capacity Under the Impact of an Aggregate Interference

Abstract

Radio spectrum is a necessary barrier for flourishing of economic activities through provision of wireless services. The radio spectrum suitable for the propagation of wireless signals is a limited resource and hence requires optimal allocation as collectively dictated by regulatory, technical and market domains. The current global move to switch from analogue to digital TV has opened up an opportunity for the re-allocation of this valuable resource. In one way, spectrum bands once used for analogue TV broadcasting will be completely cleared, leaving a space for deploying new licensed wireless services, and in another way, digital television technology geographically interleaves spectrum bands to avoid interference between neighboring stations—leaving a space for deploying new unlicensed wireless services. The focus of the paper is to assess the availability of geographically interleaved spectrum, also known as television spectrum white spaces (TVWS). The focus of this paper is to asses the potential of deploying white space devices to meet broadband needs in rural India. The main contribution of this paper is a closed form estimate of the average white capacity under the influence of an aggregate interference from other secondary devices. Also, simulations over a representative terrain considering Federal Communications Committee rules and aggregate interference, conclude that, when Digital Television is introduced there could still be approximately 6.14 Mbps of average TVWS personal capacity available.

Appendix

1.1 Case I

Without considering fading on CR transmit channels and considering the interference from CRs spree over large area i.e. $r_{\textit{max}}\rightarrow \infty$ in the model. Power control at CR node is not employed and nominal transmit power is $P_{c(nom)}=1$

$$\begin{aligned} {\mathcal {L}}_{I_{0}}(s)= {\mathbb {E}}_{\varPhi }\prod _{x\in \varPhi \setminus A_{\textit{max}}} {\mathbb {E}}_{h}\bigg [{\textit{exp}}(- sh_{c} P_{c(nom)}\parallel x\parallel ^{-\alpha }) \bigg ] \end{aligned}$$

(41)

for non fading case expectation over fading process is not considered.

$$\begin{aligned} {\mathcal {L}}_{I_{0}}(s)= {\mathbb {E}}\prod _{x\in \varPhi \setminus A_{\textit{max}}} {\mathbb {E}} {\textit{exp}}(-sx^{-\alpha })= {\textit{exp}}\left\{ - \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-sx^{-\alpha })\lambda '(x)dx\right\} \end{aligned}$$

(42)

where, $\lambda '(x)= \lambda A$.

• A is area of unit circle: $A=\pi$

• $= {\textit{exp}}\bigg \{-\pi \lambda \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-sx^{-\alpha })dx\bigg \}$

• Substituting $x\rightarrow x^{1/2}$

• $= {\textit{exp}}\bigg \{-\pi \lambda \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-sx^{-\alpha /2})dx\bigg \}$

• $={\textit{exp}}\bigg \{-\pi \lambda \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-s(x^{-1})^{\alpha /2})dx\bigg \}$

• Substituting $x^{\alpha /2}\longrightarrow r$

• $x=r^{2/\alpha }$

• $dx=dr\cdot 2/\alpha r^{2/\alpha -1}$

  $$\begin{aligned} {\mathcal {L}}_{I_{0}}(s)= {\textit{exp}}\left\{ -\pi \lambda \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-s/r)^{2/\alpha }r^{(2/\alpha ) -1})dr\right\} \end{aligned}$$

  (43)

• Let $\eta =2/\alpha$

  $$\begin{aligned} {\mathcal {L}}_{I_{0}}(s)= {\textit{exp}}\underbrace{\bigg \{-\pi \lambda \int ^{r_{\textit{max}}}_{0} 1-{\textit{exp}}(-s/r)\eta r^{\eta -1}dr\bigg \}}_{{\textit{Term}} T_{I}} \end{aligned}$$

  (44)

Solving for the integral Term $T_{I}$, $T_{I}=\int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r)\eta r^{\eta -1}dr$, for the interference from large CR network: Let $r_{\textit{max}}\rightarrow \infty$ $T_{I} = \lim _{r_{\textit{max}}\rightarrow \infty } \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r)\eta r^{\eta -1}dr$

If R is random variable, It can be notted that the above integral is expected value ${\mathbb {E}}[((R/s)^{-1})^{n}]$ of an exponential random variable R with mean 1.

Since ${\mathbb {E}}[R^{P}]=\varGamma (1+P)$ from the definition of Gamma Function.

$$\begin{aligned} \varGamma (P) \triangleq \int _{0}^{\infty }x^{P-1}e^{-x}dx \end{aligned}$$

(45)

Then,

$$\begin{aligned} {\mathbb {E}}[((R/s)^{-1})^{n}T_{I}=\pi \lambda \eta s^{n}\varGamma (1-\eta ) \end{aligned}$$

(46)

It follows that, ${\mathcal {L}} _{I_{0}}={\textit{exp}}[-\pi \lambda \eta s^{n}\varGamma (1-\eta )]$ is the Laplace agreegate interference from large CR network to $PR_{x}$. When fading is not considers on CR channels.

1.2 Case II

The fading on CR transmit channels is considered power control at CR transmitters is not considered.

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)= {\mathbb {E}}\prod _{x\in \varPhi /A_{\textit{max}}} {\textit{exp}}(-s h_{c} \parallel x\parallel ^{-\lambda }) \end{aligned}$$

(47)

Since $x$ and $h_{c}$ are independent random variable due to independence of fading and point process. and the fact that,

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)&= {\textit{exp}}\bigg \{-{\mathbb {E}}_{h} \underbrace{\int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s h_{c} x^{-\lambda })\lambda '(x)dx}_{T_{\textit{II}}}\bigg \}\end{aligned}$$

(48)

$$\begin{aligned} T_{\textit{II}}&= \pi \lambda \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s h_{c} x^{-\lambda })\lambda '(x)dx \end{aligned}$$

(49)

From above equation, $T_{\textit{II}}$ and $r_{\textit{max}} \rightarrow \infty$ $T_{\textit{II}} \lim _{r_{\textit{max}}} \rightarrow \infty \pi \lambda \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-sh_{c}/r)\eta r^{\eta -1}dr$

The above integral’s expected value is

$$\begin{aligned} T_{\textit{II}}[((R/sh_{c})^{-1})^{\eta }]={\mathbb {E}}[(sh_{c})^{\eta }\varGamma (1-\eta )] \end{aligned}$$

(50)

then

$$\begin{aligned} T_{\textit{II}}&= \pi \lambda {\mathbb {E}}(sh)^{\eta } \varGamma [1-\eta ]\end{aligned}$$

(51)

$$\begin{aligned}&= \pi \lambda {\mathbb {E}}[h^{\eta }] \varGamma [1-\eta ]s^{\eta } \end{aligned}$$

(52)

If Rayleigh fading is considered ${\mathbb {E}}[h^{\eta }] \varGamma [1-\eta ]$

$$\begin{aligned} \therefore {\mathcal {L}} _{I_{0}}(s)={\textit{exp}}\bigg \{-\pi \lambda \varGamma (1-\eta )s^{\eta }\varGamma (1+\eta ) \bigg \} \end{aligned}$$

(53)

From the property of the Gamma function

$$\begin{aligned} \varGamma (1-\eta )\varGamma (1+\eta )&= \frac{\pi \eta }{\sin (\pi \eta )}\end{aligned}$$

(54)

$$\begin{aligned} \therefore {\mathcal {L}} _{I_{0}}(s)={\textit{exp}}\bigg \{-\pi \lambda \frac{\pi \eta s^{\eta }}{\sin (\pi \eta )}\bigg \}&= {\textit{exp}}\bigg (\frac{-\lambda \pi ^{2}\eta s^{0.5}}{\sin (\pi \eta )}\bigg ) \end{aligned}$$

(55)

1.3 Case III

For non fading case and finite CR network i.e. of radius $r_{\textit{max}}$.

From $T_{I}$

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)&= {\textit{exp}}\bigg \{-\pi \lambda \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r)\eta r^{\eta -1}dr\bigg \}\end{aligned}$$

(56)

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)&= {\textit{exp}}\bigg \{-\pi \lambda \eta \underbrace{\int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r) r^{\eta -1}dr}_{T_{\textit{III}}} \bigg \}\end{aligned}$$

(57)

$$\begin{aligned} T_{\textit{III}}&= \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r) r^{\eta -1}dr\end{aligned}$$

(58)

$$\begin{aligned}&= r_{\textit{max}}-\varGamma [(1-\eta ),s/r_{\textit{max}}]s^{(\eta -1)} \end{aligned}$$

(59)

then,

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)={\textit{exp}}\bigg \{-\pi \lambda \eta r_{\textit{max}}-\varGamma [(1-\eta ),s/r_{\textit{max}}]s^{(\eta -1)}\bigg \} \end{aligned}$$

(60)

1.4 Case IV

For fading case and finite CR network.

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)&= {\mathbb {E}}_{h}\bigg [{\textit{exp}}\bigg \{-\pi \lambda \int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r)\eta r^{\eta -1}dr\bigg \}\bigg ]\end{aligned}$$

(61)

$$\begin{aligned} {\mathcal {L}} _{I_{0}}(s)&= {\textit{exp}}\bigg \{-\pi \lambda \eta \underbrace{\int _{0}^{r_{\textit{max}}}1-{\textit{exp}}(-s/r) r^{\eta -1}dr}_{T_{IV}}\bigg \}\end{aligned}$$

(62)

$$\begin{aligned} T_{IV}&= {\mathbb {E}}_{h}\bigg [\int _{0}^{r_{\textit{max}}} [1-{\textit{exp}}(-sh/r)]r^{-\eta -1}dr\bigg ] \end{aligned}$$

(63)

The above integral gives,

$$\begin{aligned} T_{IV}&= {\mathbb {E}}h_{c}^{\eta }\int _{0}^{r_{\textit{max}}}r^{\eta -1}\cdot e^{-r}dr\end{aligned}$$

(64)

$$\begin{aligned}&= \int _{0}^{r_{\textit{max}}}\bigg \{-\varGamma [(1-\eta )\cdot s/r_{\textit{max}}]s^{\eta -1}\bigg \}\end{aligned}$$

(65)

$$\begin{aligned} \therefore {\mathcal {L}} _{I_{0}}(s)&= {\textit{exp}}\bigg \{-\pi \lambda \eta {\mathbb {E}}_{h_{c}^{\eta }}\bigg \{r_{\textit{max}} -\varGamma [(1-\eta ), s/r_{\textit{max}}]s^{(\eta -1)}\bigg \} \end{aligned}$$

(66)

$\varGamma [(1-\eta ),s/r_{\textit{max}}]s^{(\eta -1)}$—incomplete Gamma function. For Rayleigh fading ${\mathbb {E}}_{h_{c}^{\eta }}=\varGamma (\eta )$,

$$\begin{aligned} {\mathcal {L}}_{I_{0}}(s)={\textit{exp}}\bigg \{-\pi \lambda \eta \varGamma (\eta ) r_{\textit{max}}-\varGamma [(1-\eta ),s/r_{\textit{max}}]s^{\eta -1}\bigg \} \end{aligned}$$

(67)