Theoretical expression of link performance in OFDM cellular networks with MIMO compared to simulation and measurements

Abstract

The objective of this paper is to establish a theoretical expression of the link performance in the downlink of a multiple input multiple output (MIMO) cellular network and compare it to the real Long-Term Evolution (LTE ) performance. In order to account for the interference, we prove that the worst additive noise process in the MIMO context is the white Gaussian one. Based on this theoretical result, we build an analytic expression of the link performance in LTE cellular networks with MIMO. We study also the minimum mean square error (MMSE) scheme currently implemented in the field, as well as its improvement MMSE-SIC (successive interference cancellation) known to achieve the MIMO capacity. Comparison to simulation results as well as to measurements in the field shows that the theoretical expression predicts well practical link performance of LTE cellular networks. This theoretical expression of link performance is the basis of a global analytic approach to the evaluation of the quality of service perceived by the users in the long run of their arrivals and departures.

Appendix A: MIMO flat fading channel with additive noise

We shall establish in the present appendix a useful lower bound of the capacity of a general additive noise channel, where the noise is not necessarily Gaussian nor white. Our motivation is that interference in wireless networks does not have necessarily these properties.

1.1 A.1 Model

Consider a discrete-time model of a multiple input and multiple output (MIMO) channel with t transmitting and r receiving antennas such that, at each time n = 1,2,…, the channel output $Y_{n}\in \mathbb {C}^{r}$ is related to the channel input $X_{n}\in \mathbb {C}^{t}$ by

$$ Y_{n}=HX_{n}+B_{n},\quad n=1,2,\ldots $$

(A.1)

where H is a complex matrix of dimension r×t modelling the fading, and $B_{1},B_{2},\ldots \in \mathbb {C}^{r}$ is the noise process. We assume that the fading matrix H is deterministic. The channel input is subject to a power constraint of the form

$$\frac{1}{n}\sum\limits_{k=1}^{n}X_{k}^{\ast}X_{k}\leq\mathcal{P},\quad n=1,2,\ldots $$

where $\mathcal {P}$ is a given positive constant and $X_{k}^{\ast }$ designates the transpose complex conjugate of X _k . Note that the above constraint concerns the total power aggregated over all the t transmitters and averaged over n channel uses. The channel (A.1) is called MIMO additive noise channel with deterministic fading.

1.2 A.2 Capacity lower bound

We are interested in the capacity of the channel (A.1) when the noise samples B ₁, B ₂,… are not necessarily Gaussian nor independent. We shall in fact establish an explicit lower bound for this capacity.

We begin by some definitions and notation. The identity matrix of dimension r×r is denoted by I _r . The covariance matrix of a centred random vector $X\in \mathbb {C}^{t}$ is denoted by

$$\Gamma_{X}=E\left[ XX^{\ast}\right] $$

The covariance matrix of two centred random vectors $X\in \mathbb {C}^{t}$ and $Y\in \mathbb {C}^{r}$ is denoted by

$$\Gamma_{XY}=E\left[ XY^{\ast}\right] $$

A random vector $X\in \mathbb {C}^{n}$ is called circularly symmetric if e ^iϕ X has the same distribution as X for all $\phi \in \mathbb {R}$ which implies that X is centred.

From now on, all the considered random vectors are assumed to have well-defined entropies [31, §1.3]. For example, if the random vector $X\in \mathbb {C}^{n}$ has a density p _X with respect to the Lebesgue measure on $\mathbb {C}^{n}$, then its entropy is defined by $h(X)=-\int _{\mathbb {C}^{n}}p_{X}(x)\log p_{X}(x)dx$ provided the Lebesgue integral is well defined. We denote by $I\left (X;Y\right )$ the mutual information between two random vectors X and Y which is related to the entropy by [31, Theorem 1.6.2]

$$ I\left( X;Y\right) =h\left( X\right) -h\left( X|Y\right) $$

(A.2)

where $h\left (X|Y\right )$ is the entropy of X conditionally to Y.

We give now two preliminary lemmas.

Lemma A.1

Let X ₁ ,X ₂ ,…,X _n be random vectors in $\mathbb {C}^{t}$ and Y ₁ ,Y ₂ ,…,Y _n be random vectors in $\mathbb {C}^{r}$ . Denote $X^{\left (n\right )} =\left (X_{1},X_{2} ,\ldots ,X_{n}\right )$ and $Y^{\left (n\right )} =\left (Y_{1},Y_{2} ,\ldots ,Y_{n}\right )$ . If X ₁ ,X ₂ ,…,X _n are independent, then

$$I\left( X^{\left( n\right)} ;Y^{\left( n\right)} \right) \geq\sum\limits _{k=1}^{n}I\left( X_{k};Y_{k}\right) $$

Proof

The mutual information may be expressed in terms of the entropy as follows [31, Theorem 1.6.2]

$$I\left( X^{\left( n\right)} ;Y^{\left( n\right)} \right) =h\left( X^{\left( n\right)} \right) -h\left( X^{\left( n\right)} |Y^{\left( n\right)} \right) $$

Since X ₁, X ₂,…, X _n are independent, the entropy $h\left (X^{\left (n\right )} \right )$ may be decomposed as the sum of the individual entropies [31, Theorem 1.3.2 (h.6)] $h\left (X^{\left (n\right )} \right ) =\sum \limits _{k=1}^{n}h\left (X_{k}\right )$. On the other hand, the conditional entropy $h\left (X^{\left (n\right )} |Y^{\left (n\right )} \right )$ may be bounded as follows

$$\begin{array}{@{}rcl@{}} h\left( X^{\left( n\right)} |Y^{\left( n\right)} \right) & =&\sum\limits _{k=1}^{n}h\left( X_{k}|Y^{\left( n\right)} ,X_{1},\ldots,X_{k-1}\right) \\ & \leq&\sum\limits_{k=1}^{n}h\left( X_{k}|Y^{\left( n\right)} \right) \leq \sum\limits_{k=1}^{n}h\left( X_{k}|Y_{k}\right) \end{array} $$

where for the first equality, we use [31, Theorem 1.3.2 (h.7)], and for the two above inequalities, we use [31, Theorem 1.3.2 (h.7)] and [31, Theorem 1.3.2 (h.5)], respectively. Combining the above three equations, we get the desired result.

The following lemma may be seen as an extension of [31, Theorem 1.8.6] or [19, Lemma II.2] to the complex case. Our proof is inspired by [32] and [33].

Lemma A.2

Consider three random vectors $X\in \mathbb {C} ^{t}$ , $Y,\tilde {Y}\in \mathbb {C}^{r}$ . Assume that the random vector $\left (X,\tilde {Y}\right )$ is circularly symmetric Gaussian with the same covariance matrix as $\left (X,Y\right )$ and that Γ _Y is invertible. Then,

$$I(X;Y)\geq I(X;\tilde{Y}) $$

Proof

For any deterministic matrix $A\in \mathbb {C}^{t\times r}$,

$$ h\left( X|Y\right) =h\left( X-AY|Y\right) \leq h\left( X-AY\right) $$

(A.3)

where for the above inequality, we use [31, Theorem 1.3.2]. In particular, taking $A=\Gamma _{XY}\Gamma _{Y}^{-1}$ in which case A Y is the best quadratic approximation of X by a linear function of Y, and letting U = X−A Y, we get

$$ h\left( X-AY\right) =h(U)\leq\log\left[ \det\left( \pi e\Gamma_{U}\right) \right] $$

(A.4)

Combining the above two inequalities, we get $h\left (X|Y\right ) \leq \log \left [ \det \left (\pi e\Gamma _{U}\right ) \right ]$. Then, Eq. (A.2) implies

$$ I\left( X;Y\right) \geq h\left( X\right) -\log\det\left( \pi e\Gamma _{U}\right) $$

(A.5)

Apply now the above arguments with $\tilde {Y}$ in the role of Y. Observe that $\tilde {U}=X-A\tilde {Y}$ is circularly symmetric Gaussian, thus equality holds in (A.4). Moreover, $\tilde {U}$ is independent from $\tilde {Y}$ since $\Gamma _{\tilde {U}\tilde {Y}}=E\left [ \tilde {U}\tilde {Y}^{\ast }\right ] =\Gamma _{XY}-A\Gamma _{Y}=0$, and decorrelation implies independence for circularly symmetric Gaussian random vectors. Thus, equality holds also in (A.3) which shows that

$$I\left( X;\tilde{Y}\right) =h\left( X\right) -\log\det\left( \pi e\Gamma_{\tilde{U}}\right) $$

which combined with the observation that $\Gamma _{U}=\Gamma _{X}-\Gamma _{XY}\Gamma _{Y}^{-1}\Gamma _{YX}=\Gamma _{\tilde {U}}$ and (A.2) finishes the proof of the desired inequality.

We show now that the above lemmas permit to deduce a lower bound for the capacity of the channel (A.1). The considered channel has memory, i.e. different channels uses are not independent because of the noise samples might be correlated, thus its information capacity C is defined as follows

$$C=\liminf_{n\rightarrow\infty}\frac{1}{n}C^{\left( n\right)} $$

where

$$C^{\left( n\right)} =\sup_{X^{\left( n\right)} }\left\{ I\left( X^{\left( n\right)} ;Y^{\left( n\right)} \right) ;\frac{1}{n}\sum\limits _{k=1}^{n}E\left[ X_{k}^{\ast}X_{k}\right] \leq\mathcal{P}\right\} $$

where $X^{\left (n\right )} =\left (X_{1},\ldots ,X_{n}\right )$ is a random object with values in $\left (\mathbb {C}^{t}\right ) ^{n}$; and $Y^{\left (n\right )}$ is the output of the channel associated to the input $X^{\left (n\right )}$.

Proposition A.1

Assume that the covariance matrix $E[B_{k}B_{k}^{\ast }]$ of the noise B _k is finite for all $k\in \mathbb {N}$ and denote

$$\mathcal{N}_{n}=\frac{1}{n}\sum\limits_{k=1}^{n}E[B_{k}B_{k}^{\ast}] $$

Then, the information capacity of the channel (A.1), given the fading matrix H, is lower bounded by

$$ C\geq\liminf_{n\rightarrow\infty}\left[ \log_{2}\det\left( I_{r} +\frac{\mathcal{P}}{t}HH^{\ast}\mathcal{N}_{n}^{-1}\right) \right] $$

(A.6)

The above inequality remains true under the additional constraint that the signals emitted by the transmitting antennas are independent and have equal powers.

Proof

Consider independent inputs X ₁, X ₂,…, then by Lemma A.1

$$I\left( X^{\left( n\right)} ;Y^{\left( n\right)} \right) \geq {\displaystyle\sum\limits\limits_{k=1}^{n}} I\left( X_{k};Y_{k}\right) $$

Assume now that each X _k is circularly symmetric Gaussian, independent of B _k , and with covariance matrix $E[X_{k}X_{k}^{\ast }]=\frac {\mathcal {P}} {t}I_{t}$. Note that

$$E\left[ X_{k}Y_{k}^{\ast}\right] =E\left[ X_{k}X_{k}^{\ast}H^{\ast}\right] +E[X_{k}B_{k}^{\ast}]=\frac{\mathcal{P}}{t}H^{\ast} $$

and

$$\begin{array}{@{}rcl@{}} E\left[ Y_{k}Y_{k}^{\ast}\right] &=&E\left[ HX_{k}X_{k}^{\ast}H^{\ast} \right] +E[B_{k}B_{k}^{\ast}]\\ &=&\frac{\mathcal{P}}{t}HH^{\ast}+E[B_{k} B_{k}^{\ast}] \end{array} $$

Consider $\tilde {B}_{k}$ circularly symmetric Gaussian independent from X _k and with the same covariance matrix as B _k . Denoting $\tilde {Y}_{k}=HX_{k}+\tilde {B}_{k}$, then $\left (X_{k},\tilde {Y}_{k}\right )$ is circularly symmetric Gaussian with the same covariance matrix as $\left (X_{k},Y_{k}\right )$. We deduce from Lemma A.2 that

$$\begin{array}{@{}rcl@{}} I(X_{k};Y_{k}) & \geq& I(X_{k};\tilde{Y}_{k})\\ & =&h(\tilde{Y}_{k})-h(\tilde{Y}_{k}|X_{k})\\ & =&h(\tilde{Y}_{k})-h(\tilde{B}_{k})\\ & =&\log_{2}\left[ \det\left( \pi e\left( \frac{\mathcal{P}}{t}HH^{\ast} +E[B_{k}B_{k}^{\ast}]\right) \right) \right] \\ && -\log_{2}\left[ \det\left( \pi eE[B_{k}B_{k}^{\ast}]\right) \right] \\ & =&\log_{2}\det\left( I_{r}+\frac{\mathcal{P}}{t}HH^{\ast}E[B_{k}B_{k} ^{\ast}]^{-1}\right) \end{array} $$

Using Jensen’s inequality and convexity of the function $A\mapsto \log _{2} \det \left (I_{r}+\frac {\mathcal {P}}{t}HH^{\ast }A^{-1}\right )$ on the set of positive definite matrices of $\mathbb {C}^{r\times r}$, cf. [19, Lemma II.3], we obtain

$$\frac{1}{n} {\displaystyle\sum\limits\limits_{k=1}^{n}} I\left( X_{k};Y_{k}\right) \geq\log_{2}\det\left( I_{r}+\frac{\mathcal{P}} {t}HH^{\ast}\mathcal{N}_{n}^{-1}\right) $$

which concludes the proof of (A.6). The last statement in the Proposition follows from the fact that the above inequality is proved for inputs X ₁, X ₂,… such that each X _k is circularly symmetric Gaussian with covariance matrix $E[X_{k}X_{k}^{\ast }]=\frac {\mathcal {P}} {t}I_{t}$, which implies that, for each $k\in \mathbb {N}^{\ast }$, the components of the vector X _k are independent from each other.

We make an observation and give a corollary.

Remark A.1

Assume that B ₁, B ₂,… have the same covariance matrix $E[B_{k} B_{k}^{\ast }]=\mathcal {N}$, then, the right-hand side of (A.6) equals

$$\log_{2}\det\left( I_{r}+\frac{\mathcal{P}}{t}HH^{\ast}\mathcal{N} ^{-1}\right) $$

Note that the above formula gives also the capacity of a MIMO channel with additive circularly symmetric Gaussian noise process with independent samples and equi-partition of power between the transmitting antennas.

The following Corollary of Proposition A.1 states that, for a single input and single output (SISO) channel t = r = 1, the worst additive noise process distribution (not necessarily white nor Gaussian) for capacity with given second moment is the additive white Gaussian noise (AWGN). This result may be seen as an extension of Gallager’s result [34, Theorem 7.4.3] for memoryless channels to the channels with memory. It may also be deduced from Shannon’s result [17, Theorem 18]—proved there by the entropy power inequality—and from the fact that the entropy power is not larger than the average power.

Corollary A.1

Consider a SISO channel whose input and output, at time n, represented by $X_{n}\in \mathbb {C}$ , $Y_{n}\in \mathbb {C}$ , respectively, are related by

$$Y_{n}=X_{n}+B_{n},\quad n=1,2,\ldots $$

where the noise process $B_{1},B_{2},\ldots \in \mathbb {C}$ is assumed stationary and satisfies $E\left [ \left \vert B_{n}\right \vert ^{2}\right ] =N$ . Assume that the channel has a power constraint in the form $\frac {1} {n}\sum \limits _{k=1}^{n}\left \vert X_{k}\right \vert ^{2}\leq \mathcal {P}$ . Then, the information capacity C of the channel is lower bounded by

$$ C\geq\log_{2}\left( 1+\frac{\mathcal{P}}{N}\right) $$

(A.7)

1.3 A.3 Asymptotic analysis

Consider a particular case where the noise samples B ₁, B ₂,… are i.i.d. each being circularly symmetric Gaussian with covariance matrix $E[B_{n}B_{n}^{\ast }]=NI_{r}$ where N is a given positive constant. In this case, the right-hand side of (A.6 ) equals

$$ \log_{2}\det\left( I_{r}+\frac{\mathcal{P}}{tN}HH^{\ast}\right) =\log _{2}\det\left( I_{r}+\frac{P}{t}HH^{\ast}\right) $$

(A.8)

where $P=\frac {\mathcal {P}}{N}$ is the signal to noise power ratio (SNR). For given t and r, the capacity (A.8) depends on H.

Frequently, one is interested in the ergodic capacity, that is the expectation of the capacity with respect to the fading matrix H assumed random with a given distribution. Assume for example that H has i.i.d. components each being circularly symmetric Gaussian with variance 1. In this case, the ergodic capacity may be calculated with the help of the analytical result given by Telatar [5, Theorem 2].

Alternatively, the capacity (A.8) may be approximated with the help of the following asymptotic result saying that when the number of transmitting and receiving antennas go to infinity, the capacity per receiving antenna converges to a deterministic limit.

Lemma A.3

[12, Equations (9), (38)] , [35, Appendix] Assume that the fading matrix $H\in \mathbb {C}^{r\times t}$ has i.i.d. components, centred and with variance 1. Assume that $t,r\rightarrow \infty$ such that $\frac {t}{r}\rightarrow \beta \in \mathbb {R}_{+}^{\ast }$ , then

$$ \frac{1}{r}\log\det\left( I_{r}+\frac{P}{t}HH^{\ast}\right) \rightarrow \mathcal{C}\left( P,\beta\right) $$

(A.9)

almost surely, where

$$\begin{array}{@{}rcl@{}} \mathcal{C}\left( P,\beta\right) & :=&\beta\log\left( 1+\frac{P}{\beta} -\frac{1}{4}\mathcal{F}\left( \frac{P}{\beta},\beta\right) \right) \\ && +\log\left( 1+P-\frac{1}{4}\mathcal{F}\left( \frac{P}{\beta} ,\beta\right) \right)\\ &&-\frac{\beta}{4P}\mathcal{F}\left( \frac{P}{\beta} ,\beta\right)\notag\\ \end{array} $$

(A.10)

where

$$\mathcal{F}\left( \xi,\beta\right) =\left( \sqrt{\xi\left( 1+\sqrt{\beta} \right) ^{2}+1}-\sqrt{\xi\left( 1-\sqrt{\beta}\right) ^{2}+1}\right) ^{2} $$

The above asymptotic result gives a good approximation of the expectation $E\left [ \frac {1}{r}\log \det \left (I_{r}+\frac {P}{t}HH^{\ast }\right ) \right ]$ even for a small number of antennas as already observed in [22, Table 1] for the value SNR=10 dB and confirmed for the whole typical range of SNR values in wireless cellular networks in Fig. 6 surprisingly even for a single antenna at the emitter and at the receiver.

Fig. 6

Comparison of the analytic capacity and the asymptotic formula for MIMO t×r (t transmitting antennas and r receiving antennas)