On the energy efficiency of base station cooperation under limited backhaul capacity

Abstract

Recently, energy-efficient (EE) communications have received increasing interest specially in cellular networks. Promising techniques, such as multiple input multiple output (MIMO) and base station (BS) cooperation schemes, have been widely studied in the past to improve the spectral efficiency and the reliability. Nowadays, the purpose is to investigate how these techniques can reduce the energy consumption of the systems. In this paper, we address for a single-user scenario, the energy efficiency of two BSs cooperation under limited backhaul capacity. In order to evaluate the EE metric, we provide first an information-theoretical analysis based on the outage probability, for a quantization model over the backhaul. Then, we extend this EE analysis to a more practical approach with data transmission over the backhaul. For both approaches, we identify by numerical/simulation results the cooperation scenarios that can save energy depending on the backhaul capacity.

Appendices

Appendix: A Proof of Proposition 1: rate-distortion theory

In rate-distortion theory, the quantization problem consists of representing a source X by an estimate X ̂ subject to some distortion measure or quantization error such that

$$ D=E\left[|X -\hat{X}|^{2}\right]. $$

(A.1)

For complex Gaussian source, the rate-distortion R(D) metric is defined by

$$ R(D)=\left\{\begin{array}{ll} \log_{2}\frac{\sigma_{X}^{2}}{D}, & 0\leq D\leq\sigma_{X}^{2}\\ 0, & D>\sigma_{X}^2. \end{array} \right. $$

(A.2)

It is modeled by the backward test channel

$$ X=\hat{X}+Z_{q}, $$

(A.3)

where X ̂ and Z _q are independent, $X\sim \mathcal {N}_{c}\left (0,\sigma _{X}^{2}\right )$, $Z_{q} \sim \mathcal {N}_{c}\left (0,D=\sigma _{Z_{q}}^{2}\right )$ and $\sigma _{\hat {X}}^2=\sigma _{X}^2-D$.

Following this model, a forward test channel can be equivalently constructed as

$$ \hat{X}=c(X+\eta), $$

(A.4)

where X and η are independent, η being the equivalent quantization noise $\eta \sim \mathcal {N}_{c}\left (0,\sigma _{\eta }^{2}\right )$ and c a scaling constant such that

$$ \sigma_{\eta}^2=\frac{\sigma_{X}^2D}{\sigma_{X}^2-D}\quad\text{and}\quad c=\frac{\sigma_{X}^2-D}{\sigma_{X}^{2}}. $$

(A.5)

It can be readily verified that according to this model, we have,

$$\begin{array}{@{}rcl@{}} E\left[|X-\hat{X}|^{2}\right]&=& E\left[\left|\frac{D}{\sigma_{X}^{2}}X-\frac{\sigma_{X}^2-D}{\sigma_{X}^{2}}\eta\right|^{2}\right]=D \\ E\left[|\hat{X}|^{2}\right] &=& \left(\frac{\sigma_{X}^2D}{\sigma_{X}^2-D}\right)^{2} E\left[|X+\eta|^{2}\right] \\ &=&\sigma_{X}^2- D. \end{array} $$

(A.6)

For our system model, BS1 quantizes X and forwards it to BS2 over the backhaul link with capacity C _b . BS2 decodes then an estimate X ̂ of X with some distortion entailed by the backhaul. This distortion can be expressed in terms of the backhaul rate according to (A.2)

$$ D(C_b)=\sigma_{X}^{2} 2^{-C_{b}}. $$

(A.7)

then we can define the forward test channel from Eqs. (A.4), (A.5) with

$$\begin{array}{@{}rcl@{}} \sigma_{\eta}^{2} =\frac{\sigma_{X}^{2} 2^{-C_{b}}}{\sigma_{X}^2-2^{-C_{b}}}\quad\text{and}\quad c=1-\frac{2^{-C_{b}}}{\sigma_{X}^{2}}. \end{array} $$

(A.8)

This concludes the Proof of Proposition 1.

B Proof of Proposition 2: difference of correlated gamma variables

As presented in [12], Type II McKay Distribution is defined as follows

Definition 3 Type II McKay Distribution

A random variable Δ follows Type II McKay distribution with parameters a > − (1 / 2), b > 0 and | c | < 1 when the PDF of Δ is

$$ f_{\Delta}(\delta)=\frac{\left(1-c^{2}\right)^{a+\frac{1}{2}}|\delta|^{a}}{\sqrt{\pi}2^ab^{a+1}\Gamma\big(a+\frac{1}{2}\big)}e^{-\delta\frac{c}{b}}K_{a}\left(\frac{|\delta|}{b}\right), $$

(A.9)

where K _a (.) is the modified Bessel function of the second kind and of order a.

If X ₁, X ₂are two correlated Gamma variables i.e., Bivariate Gamma variables, X ₁, X ₂ ∼ Γ(k, θ ₁, θ ₂, ρ), the authors of [12] proved that their difference Δ = X ₁ − X ₂follows Type II McKay distribution with the parameters

$$\begin{array}{@{}rcl@{}} a &=& k-\frac{1}{2} \\ b&=&\frac{2\theta_{1}\theta_{2}(1-\rho)}{\sqrt{(\theta_1-\theta_2)^2+4\theta_{1}\theta_{2}(1-\rho)}}\\ c&=&-\frac{\theta_1-\theta_{2}}{\sqrt{(\theta_1-\theta_2)^2-4\theta_{1}\theta_{2}(1-\rho)}} \end{array} $$

(A.10)

where the conditions b > 0 and |c| < 1 are met as long as ρ < 1.

In the case of k = 1, $a=\frac {1}{2}$ and this McKay distribution reduces to

$$ f_{\Delta}(\delta)= \frac{1-c^{2}}{2b}e^{-\frac{\delta c+ |\delta|}{b}}. $$

(A.11)

Thus, it is possible to find a closed-form expression for the CDF of the McKay Type II distribution. It can be written as following

$$ F_{\Delta}(\delta)=\left\{\begin{array}{ll} \frac{1+c}{2}\exp\left(\frac{1-c}{b}\delta\right), & \delta<0 \\ 1-\frac{1-c}{2}\exp\left(-\frac{1+c}{b}\delta\right), & \delta\geq 0. \end{array} \right. $$

(A.12)

In our system model, we have Rayleigh fading channel, thus k = 1. We have also δ = λ σ ² ≥ 0, so we can apply the second line of the CDF formula to calculate the outage probability.

$$ F_{\Delta}\left(\lambda\sigma^{2}\right)= 1-\frac{1-c}{2}\exp\left(-\frac{1+c}{b}\lambda\sigma^{2}\right). $$

(A.13)

This concludes the Proof of Proposition 2.

C Proof of Proposition 3: mutual information of finite constellations

The mutual information of finite constellations over Gaussian channel $Y=\sqrt {\text {SNR}}X+Z$ is defined by

$$ I(X,Y)=H(Y)-H(Y|X)= H(Y)- H(Z) $$

(A.14)

with X ∈ M-QAM, and Z the Gaussian noise Z ∼ 𝒩 _c (0, N ₀ with entropy given by

$$ H(Z)= \log_{2} (\pi e N_0) $$

(A.15)

On the other hand, the entropy of Y is computed as follows

$$\begin{array}{ll} H(Y)&=\mathrm{E}_{X,Y}\{-\log_2(p(y))\} \notag\\ &=\sum_{x_i} p({x_i}) \mathrm{E}_{Y|X=x_i}\{-\log_2(p(y))\} \notag\\ &=\sum_{x_i} \frac{1}{M}\mathrm{E}_{Y|X=x_i}\{-\log_2(p(y))\} \notag\\ &=\mathrm{E}_{Y|X}\{-\log_2(p(y))\} \notag\\ &=\mathrm{E}_{Y|X}\left\{-\log_2 \left[\sum_x p(y|x)p(x) \right] \right\} \notag\\ &=\mathrm{E}_{Y|X}\left\{-\log_2 \left[\frac{1}{M}\sum_x p(y|x) \right] \right\} \\ &=\mathrm{E}_{Y|X}\!\left\{\!-\!\log_2 \!\left[\! \frac{1}{M}\!\sum_x \!\frac{1}{\pi N_0}\! \exp\left(\!- \frac{|Y-\sqrt{\mathrm{SNR}}X|^ 2}{N_0} \!\right) \!\right] \!\right\} \end{array} $$

(A.16)

Note that the third and the sixth lines follow from the fact that we have uniform probability $p(x_i)=\frac {1}{M}$, for all symbols x _i ∈ M-QAM.

Therefore, this mutual information can be represented in Fig. 12 for several M-QAM modulations, with M = 4, 16 and 64.

Fig. 12

Mutual information of QAM constellations for the Gaussian channel

In order to approximate the Shannon capacity C = log₂(1 + SNR), it can be noticed that for a M = 2^m-QAM constellation with a rate of R = m bits/cu, a lower rate corresponding to a lower order modulation ∼ (m − 1) should be used.

For instance, we can choose for

• 4-QAM, R _QAM ≤ 1. 5

• 16-QAM, R _QAM ≤ 3

• 64-QAM, R _QAM ≤ 5

This concludes the Proof of Proposition 3.