On the Spectral Efficiency of Energy Constrained Short-Range Communicating Systems

Abstract

Recently the energy efficiency (EE) in wireless communication is becoming one of the key performance metric for wireless communicating systems. For battery driven system like wireless sensor networks and ad-hoc networks, the energy conservation is a critical factor for node life. In addition, the spectral efficiency (SE) has been traditionally used as a performance index for wireless transmission. This paper investigates the optimum spectral to energy efficiency tradeoff especially for short range communications. The analysis starts with the Shannon case in additive white gaussian noise channel where some theoretic results are developed when considering both transmit and circuit energy. Then, point to point communication system with uncoded M-ary quadrature amplitude modulation (MQAM) is considered. At the device layer, a system level power consumption model is proposed. At the physical layer, a novel and accurate approximation of the bit error rate (BER) function of the signal to noise ratio is made. The total energy per bit is formulated and the link between the Shannon limit capacity and MQAM based communication is established. The impact of distance, bandwidth, power consumption and BER as a quality-of-service parameter on EE–SE tradeoff is analyzed. It is shown that, varying distance, bandwidth and circuit power consumption induce more impact in the low SE regime whereas, the BER has more impact on the high SE regime. Moreover, the energy optimal spectral efficiency for MQAM is obtained in closed-form and confirmed by numerical results.

Appendix

1.1 Proof of Lemma 1

The objective function $E_\mathrm{bt}$ can be bounded as follow: $\forall t \in ]0,\theta ]$ we have $1\le e^{t}\le e^{\theta }$ so

$$\begin{aligned}&\theta \le \int _{0}^{\theta } {e}^{t} \, \mathrm {d}t\le \theta e^{\theta }\end{aligned}$$

(32)

$$\begin{aligned}&\theta \le e^{\theta }-1\le \theta e^{\theta }\end{aligned}$$

(33)

$$\begin{aligned}&1\le \frac{e^{\theta }-1}{\theta } \le e^{\theta } \end{aligned}$$

(34)

$$\begin{aligned} {\text { Hence,}}\quad \quad&A\ln 2+\frac{C}{\theta } \le E_\mathrm{bt}(\theta )\le A2^\theta \ln 2 \end{aligned}$$

(35)

Finally, $E_\mathrm{bt}$ can be approximated to the right hand expression for low $\theta$ values and to the left hand expression for high $\theta$ values.

1.2 Proof of Lemma 2

First we express a general form of (4) as follow

$$\begin{aligned} E_\mathrm{bt}(\theta )=A\frac{2^{\theta }-1}{\theta }+C\frac{1}{\theta } \end{aligned}$$

(36)

To prove the quasi-convexity of (4), we recall that for $\theta \ge 0$ a function is strictly quasi-convex if

$$\begin{aligned} S_{\beta }=\left\{ \theta \ge 0 \mid E_\mathrm{bt}(\theta )\le \beta \right\} \end{aligned}$$

(37)

is a strictly convex set for any real number $\beta$. Note that when $\beta \le 0$, $S_{\beta }$ is empty since $E_\mathrm{bt}(\theta )>0$. Hence $S_{\beta }$ is strictly convex as no point exist on the contour. Now we investigate the case when $\beta > 0$. In such case, $S_{\beta }$ is equivalent to $S_{\beta }=\left\{ \theta \ge 0 \mid g(\beta ,\,\theta )=A\left( 2^{\theta }-1\right) -\beta \theta \le 0 \right\}$. Since $g(\beta ,\,\theta )$ is strictly convex in $\theta$, if we take $\theta _{1}$ and $\theta _{2}$ two points of the contour of $S_{\beta }$ ( $\theta _{1}>0$ and $\theta _{2}>0$) then we have $\forall \theta \in \left( \theta _{1},\,\theta _{2} \right)$, $g(\beta ,\,\theta )< \max \left\{ g(\beta ,\,\theta _{1}),\,g(\beta ,\,\theta _{2}) \right\} \le 0$. The first inequality comes from the strict quasi-convexity of $g(\beta ,\,\theta )$ w.r.t $\theta$ [34]. Hence, $S_{\beta }$ is strictly quasi-convex set when $\beta > 0$. Thus we have the strict quasi-convexity of $E_\mathrm{bt}(\theta )$.

1.3 Proof of Lemma 3

Setting the derivative of $E_\mathrm{bt}(\theta )$ with respect to $\theta$ to zero we try to find the optimum spectral efficiency $\theta ^{\star }$. After rearranging the terms, we take the following steps

$$\begin{aligned} A\frac{\ln (2)e^{\ln (2)\theta }-e^{\ln (2)\theta }+1}{\theta ^{2}}&=\frac{C}{\theta ^{2}}\end{aligned}$$

(38)

$$\begin{aligned} e^{\ln (2)\theta -1}\left( \ln (2)\theta -1\right)&=\frac{C-A}{eA} \end{aligned}$$

(39)

by using the Lambert function $W_{L}$ since it is the solution to $x=W_{L}(x)\exp \left( W_{L}(x)\right)$ we have

$$\begin{aligned}&\quad \quad \ln (2)\theta -1=W_{L}\left( \frac{C-A}{eA}\right) \end{aligned}$$

(40)

$$\begin{aligned} {\text {therefore}}&\quad \quad \theta ^{\star }=\frac{1}{\ln (2)}\left( 1+ W_{L}\left( \frac{C-A}{eA}\right) \right) \end{aligned}$$

(41)

1.4 Proof of Lemma 5

We begin by establishing the first inequality. Suppose that $P_\mathrm{pr}$ is independent of d then

$$\begin{aligned} E_\mathrm{bt}\left( \theta ^{\star }(d_{2}),\,d_{2}\right)&=\alpha N_{0}G_{1}d_{2}^{k}\frac{2^{\theta ^{\star }(d_{2})}-1}{\theta ^{\star }(d_{2})}+\frac{P_\mathrm{pr}}{\theta ^{\star }(d_{2}) B} \nonumber \\&=\left( \frac{d_{2}}{d_{1}} \right) ^{k} \alpha N_{0}G_{1}d_{1}^{k}\frac{2^{\theta ^{\star }(d_{2})}-1}{\theta ^{\star }(d_{2})}+\frac{P_\mathrm{pr}}{\theta ^{\star }(d_{2}) B} \nonumber \\&>\alpha N_{0}G_{1}d_{1}^{k}\frac{2^{\theta ^{\star }(d_{2})}-1}{\theta ^{\star }(d_{2})}+\frac{P_\mathrm{pr}}{\theta ^{\star }(d_{2}) B} > E_\mathrm{bt}\left( \theta ^{\star }(d_{2}),\,d_{1}\right) \ge E_\mathrm{bt}\left( \theta ^{\star }(d_{1}),\,d_{1}\right) \end{aligned}$$

(42)

The last inequality is due to the fact that $\theta ^{\star }(d_{1})$ minimizes $E_\mathrm{bt}\left( \theta (d),\,d_{1}\right)$. To prove the rest of lemma we set the derivative of (4) w.r.t $\theta$ and let $q(\theta (d))=\frac{2^{\theta (d)}-1}{\theta (d)}$. By exploiting the fact that $\theta ^{\star }(d_{1})$ is the solution of $E_\mathrm{bt}^{\prime }\left( \theta (d),\,d_{1}\right) =0$, the following equality hold

$$\begin{aligned} \alpha N_{0}G_{1}d_{1}^{k}q^{\prime }(\theta ^{\star }(d_{1}))=\frac{P_\mathrm{pr}}{\theta ^{\star ^{2}}(d_{1})B} \end{aligned}$$

(43)

By setting the derivative of $E_\mathrm{bt}\left( \theta ,\,d\right)$ w.r.t $\theta$ and using (43) it can be shown that

$$\begin{aligned} E_\mathrm{bt}^{\prime }\left( \theta ^{\star }(d_{1}),\,d_{2}\right)&=\alpha N_{0}G_{1}d_{2}^{k}q^{\prime }(\theta ^{\star }(d_{1}))-\frac{P_\mathrm{pr}}{\theta ^{\star ^{2}}(d_{1})B}\nonumber \\&=\left( \frac{d_{2}}{d_{1}}\right) ^{k}\frac{P_\mathrm{pr}}{\theta ^{\star ^{2}}(d_{1})B}-\frac{P_\mathrm{pr}}{\theta ^{\star ^{2}}(d_{1})B}=\frac{P_\mathrm{pr}}{\theta ^{\star ^{2}}(d_{1})B}\left( \left( \frac{d_{2}}{d_{1}}\right) ^{k}-1\right) >0 \end{aligned}$$

(44)

As $E_\mathrm{bt}\left( \theta ,\,d\right)$ is strictly quasi-convex so $E_\mathrm{bt}^{\prime }\left( \theta ,\,d\right) >0$ when $\theta >\theta ^{\star }(d)$ and $E_\mathrm{bt}^{\prime }\left( \theta ,\,d\right) <0$ when $\theta <\theta ^{\star }(d)$. Therefore $\theta ^{\star }(d_{1}) >\theta ^{\star }(d_{2})$.