## 3. Theoretical Analysis of Joint PAPR Reduction and Predistortion

Among the large variety of linearization techniques, digital predistortion is by far the one the most popular [Fra16, Din04, Was04]. The main principle of this technique is to apply the inverse function of the PA transfer function to cancel the nonlinearities at the PA output [INM89, SC92]. Therefore, the cascade of the predistorter and the PA gives a linear response. Letting $H_{PA}$ and $H_{PD}$ be the PA and predistortion baseband transfer functions, ideal computation of $H_{PD}$ through digital predistortion theoretically leads to

$$H_{PA} \circ H_{PD} = I_{d}, \qquad (6.36)$$

where $\circ$ is the composite function and the $I_d$ is the identity. The advantage of predistortion is that it is relatively easy to implement at a low cost. As such, predistortion is frequently applied, not also in research, but also in commercially available products [SSS03].

Even though the predistortion modeling and PA modeling are two different applications, the working principles of both are almost identical to each other. In fact, the PA modeling aims at finding the exact PA transfer function, while the intention of the predistortion is to estimate the inverse transfer function of PA characteristics. Nonetheless, both the PA characteristics and its inverse transfer function can be based on the same mathematical models. That is why most of PA models previously mentioned can be used in the predistortion modeling such as the Rapp model or the polynomial model for example. Hence, in the case of Rapp model, the transfert function of the predistortion module can be expressed as,

$$H_{PD}(r) = \frac{r}{\left(1 + \left(\frac{r}{A}\right)^{2a}\right)^{\frac{1}{2a}}}, \qquad (6.37)$$

which has exactly the same form as (6.37), except the use of a predistortion knee factor $a$ instead of the exact knee factor $b$ of the amplifier<fnref n="9" />.

On the other hand, it is also possible to adopt the polynomial model to model the predistortion function which is a good compromise between compensation performance and complexity. Thus, the predistorted signal can be expressed as follows

$$x_{PD}(t) = \sum_{k=0}^{K_p-1} a_{2k+1}x(t)^{2k+1}, \qquad (6.38)$$

with $a_{2k+1}$ the nonlinear polynomial coefficients and $K_p$ the parameter which determine the nonlinear model order. Note that the objective of the conventional DPD modeling is to find the inverse transfer characteristics of PA to compensate the static nonlinearity of PAs without consideration of memory effects. In practice, one is able to estimate the inverse PA characteristic using a special training signal. Consequently, the resulted DPD transfer function does not need to be adapted since the PA characteristics are assumed to be unchanged during the operating time. Many solutions exists to handle the estimation process and get the polynomial parameters $a_{2k+1}$, with various pros and cons in terms of stability, bandwidth capacities and complexity [MMK+06].

## 3.3 ANALYTICAL DERIVATIONS OF THE EVM

Various closed-forms of the EVM have been derived during the PhD of A. Cheaito and published namely in [R.1][R.2][C.12], taking into account the activation or not of the different modules of the system (PAPR reduction through clipping and linearization through predistortion) and the model of the power amplifier (Rapp model, polynomial model, memory polynomial model). All these contributions have consisted in leading the mathematical derivations of the EVM from the following definition,

$$\text{EVM} = \sqrt{\frac{\mathbb{E}\{|\epsilon(t)|^2\}}{\mathbb{E}\{|x_1(t)|^2\}}} \qquad (6.39)$$

$$\text{with,} \qquad \epsilon(t) = z(t) - x_1(t) \qquad (6.40)$$

<footnote n="9">9. Perfect linearization would be performed with $a = b$.</footnote>
# Chapitre 6. Optimization of Wireless Broadcast Transmissions

While it is obvious that the denominator of the EVM is simply representing the input signal power $P_{x_1}$, we also understand that the computation of the numerator needs some efforts. Given that $\epsilon(t)$ can be seen as a complex random variable $\epsilon(r, \theta)$ with uniformly distributed phase $\theta$, the derivation that should be led can be stated as :

$$ \mathbb{E}\{|\epsilon|^2\} = \int_{0}^{2\pi} \int_{0}^{\infty} |\epsilon(r, \theta)|^2 \frac{1}{2\pi} f_R(r) dr d\theta \quad (6.41) $$

with $f_R(r)$ the PDF of the amplitude of $\epsilon$ which results from the activation of the clipping function or not (according to (6.31)) but also on the power amplifier and predistortion models. The computation can then take various forms. The results obtained in the cases of the Rapp model and the polynomial model are briefly summarized in the following paragraphs.

## ► Rapp model amplifier case

In the Rapp model case, let us recall that the non linearities only affect the amplitude of the input signal. In the most general case when considering a clipping threshold of $A_{max}$ and combining the transfer functions of the predistortion and the power amplifier, Eq. (6.41) can be restated as,

$$ \mathbb{E}\{|\epsilon|^2\} = \frac{\int_{0}^{A_{max}} |r - H_{EQ}(r)|^2 f_r dr}{I_1} + \frac{\int_{A_{max}}^{\infty} |r - H_{EQ}(A_{max})|^2 f_r dr}{I_2} \quad (6.42) $$

$$ \text{where, } \quad H_{EQ}(r) = \frac{r}{\left( \left(1 - \left(\frac{r}{\Lambda}\right)^{2a}\right)^{\frac{b}{a}} + \left(\frac{r}{\Lambda}\right)^{2b} \right)^{\frac{1}{2b}}} $$

The first integral form $I_1$ corresponds to the distortion term due to the amplification non-linear effects which remains when imperfect predistortion is used (i.e. $b \neq a$). The second term is essentially coming from the clipping effect. After some manipulations, these two integral forms can be derived and an analytic expression of the EVM is obtained as stated below. The proof yielding this general result is given in [R.2].

## Contribution II.7 (Analytic Expression)

### EVM – Rapp model
The EVM of a clipped and amplified multicarrier signal assuming a Rapp model for the power amplifier and the predistortion module is

$$ \text{EVM} = \left[ 1 + G_{\text{clip}} \left( 1 - \frac{2\Gamma(\frac{3}{2}, \Lambda)}{\sqrt{G_{\text{clip}}}} \right) \Lambda e^{-\Lambda} + \sum_{i=0}^{+\infty} \left[ \left( \frac{-\frac{1}{b}}{i} - 2 \left( \frac{-\frac{1}{2b}}{i} \right) \right] \sum_{j=0}^{+\infty} \left( \frac{-\frac{b}{a}}{\frac{1}{2b} + i} \right) \times (-1)^j \frac{\gamma(aj + bi + 2, \Lambda)}{((1 - e^{-\Lambda})\text{IBO})^{aj+bi}} \right]^{1/2} \quad (6.43) $$

with IBO the power back-off applied to the input signal, $\Lambda$ the clipping ratio defined in (6.32) and $G_{\text{clip}}$ the gain of the predistorted amplifier at the clipping threshold,

$$ G_{\text{clip}} = \left| H_{\text{EQ}} \left( \sqrt{\frac{\Lambda}{1 - e^{-\Gamma}} \text{IBO}} \right) \right|^2 \quad (6.44) $$

The obtained EVM expression only depends on the IBO, clipping ratio $\Lambda$ and the power amplifier and predistortion parameters (knee factors $a$ and $b$). Also, the form can be particularized in the case when no clipping is operated, taking $\Lambda \to \infty$. In that case, the first term of the expression tends to unity and the incomplete lower Gamma function $\gamma(x, \Lambda)$ tends to the complete Gamma function $\Gamma(x)$ in the second term. Also, if no predistortion is used, it can be proven that the same expression is obtained simply discarding the terms involving the $a$ knee factor [R.1].
## 3. Theoretical Analysis of Joint PAPR Reduction and Predistortion

<figure><img src="image_34.png" /><figcaption>FIGURE 6.22– Theoretical and simulated EVM versus IBO without predistortion. Knee factor is $b = 1.6$</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>FIGURE 6.23– Theoretical and simulated EVM versus IBO with predistortion. Knee factors are $a = 1.5$ and $b = 1.6$.</figcaption></figure>

Even if the obtained formula can not been considered as a closed-form due to the infinite series expansion, we verified that an evaluation at a reasonable order is sufficient to get a very good accuracy of the EVM. This can be observed in Fig. 6.22 and Fig. 6.23 which gives some EVM curves obtained in simulation and applying the theoretical expressions at finite order, namely $O = 4$ without predistortion and $O = 6$ with predistortion. The fact that a higher order is needed in the predistortion case is due to the two non-linear functions involved in the computation (amplifier and predistortion gains) instead of a single one when no predistortion is used.

Above all, these curves reflect the two main expected effects owing to clipping and predistortion. On one hand, when clipping is activated, we observe that the threshold $\Lambda$ directly affects the EVM floor. As commonly known, the lower the clipping threshold, the larger the EVM. However, the proposed analytical expressions provide numerical prediction of the degradation. On the other hand, when predistortion is used, the EVM values at low IBO are significantly reduced while there floor values are not changed. This reflects the linearity improvement of the predistorted power amplifier which can directly be foreseen through the proposed expressions.

### ► Polynomial model case

In the polynomial model case, $|e|$ is obtained using the polynomial expression given in Eq. (6.34) for $z(t)$. In the general case, the polynomial coefficients are complex and the non linearities now affect the amplitude and the phase of the input samples, which means that the complete computation of Eq. (6.41) has to be led. After some manipulations detailed in [R.2], it is possible to extract from Eq. (6.41) four integral forms leading to complete and incomplete Gamma functions. The final obtained result is given in Contrib. II.8.

The EVM expression in (6.45) only depends on the input back-off, clipping ratio $\Lambda$ and on the polynomial coefficients. Note that the predistortion effect is simply taken into account through these coefficients, since a predistorted non-linear amplifier can be represented by an equivalent polynomial with new coefficients. Indeed, the residual non linearity effects are reflected by the two summations over $k$ and $k'$. In particular, it is verified that EVM equals zero when $b_1 = 1$ and $\forall k > 1$, $b_k = 0$, i.e. for a perfectly linear normalized amplifier. This will be namely obtained with a perfect predistortion. As in the case of the Rapp model, it is also possible to get the EVM expression when no clipping is applied, by letting $\Lambda \to \infty$. In that case, the exponential terms disappears and the incomplete Gamma functions simply tend to factorial functions (since their arguments are integers).

Interestingly, contrary to the Rapp model, the obtained expression here is a closed-form. As most of power amplifiers can be accurately approximated with polynomial representation of sufficient order, this gives a high importance to the proposed result. Note that even a Rapp amplifier can be modeled using polynomial formulation. In Fig. 6.24 for example, we give the EVM versus IBO curves obtained with our closed form, but modeling the Rapp amplifier with a polynomial form of order $K = 6$. The very good match between the simulated and theoretical results confirm the validity and accuracy of the closed-form. In Fig. 6.25, the same kind of results are given but now using a true amplifier used for DVB-T2 applications [NXP10] and for which a polynomial form of order $K = 6$ has been used. Again we conclude that our closed-form is valid.
<figure><img src="image_36.png" /><figcaption>FIGURE 6.24– Theoretical and simulated EVM versus IBO for a Rapp amplifier without predistortion. Polynomial representation of order K = 6 is used, for an original knee factor of the Rapp model of b = 2.</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>FIGURE 6.25– Theoretical and simulated EVM versus IBO for a DVB-T amplifier [NXP10] without predistortion. Polynomial representation of order K = 6 is used.</figcaption></figure>

## Contribution II.8 (Analytic Expression)

### EVM - Polynomial model

The EVM of a clipped and amplified multicarrier signal assuming a polynomial model for the power amplifier and the predistortion module is

$$ \text{EVM} = \left[ 1 + \sum_{k=1}^{K} \sum_{k'=1}^{K} v_{kk'} (\gamma(k+k', \Lambda) + \Lambda^{k+k'-1} e^{-\Lambda}) - u_k (\gamma(k+1, \Lambda) + \Lambda^{k-\frac{1}{2}} \Gamma(\frac{3}{2}, \Lambda)) \right]^{\frac{1}{2}} \quad (6.45) $$

where $u_k$ and $v_{kk'}$ are defined as,

$$ u_k = 2 \frac{\mathfrak{R}\{b_k\}}{((1 - e^{-\Lambda})\text{IBO})^{k-1}} \quad \text{and,} \quad v_{kk'} = \frac{b_k b_{k'}^*}{((1 - e^{-\Lambda})\text{IBO})^{k+k'-2}} \quad (6.46) $$

with {$b_k$}$_{1 \le k \le K}$ the coefficients of the polynomial model of order K.

### ► Memory Polynomial model case

The last case of our study was to take into consideration memory effects which are known to have a strong influence on signal distortion. This case can be seen as a generalization of the polynomial case discussed before. Derivations are quite similar as in the non-memory polynomial case, but a bit trickier because of the refined model of Eq. 6.35. The complete derivations have been published in [R.1] and the final result is provided in Contrib. II.9 hereafter.

Eventually, when memory effects are taken into consideration, the EVM closed-form is similar to that obtained with the memoryless polynomial model except in the definition of parameter $v_{kk'}$ which now integrates the contribution of memory coefficients, i.e. with index $q > 0$. From this closed-form, it is then possible to analyze the impact of memory on the EVM. In Fig. 6.26, we give the EVM curves using a power amplifier with fifth odd-order nonlinearity (K = 5) and memory depth of Q = 0 or Q = 2. The values of the polynomial coefficients $b_{kq}$, which are taken from [D+04], were extracted from an actual Class AB PA.
## 3. Theoretical Analysis of Joint PAPR Reduction and Predistortion

<figure><img src="image_38.png" /><figcaption>FIGURE 6.26- Theoretical and simulated EVM as a function of the IBO, with and without clipping, with and without memory effects (Q = 2, K = 5).</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>FIGURE 6.27- EVM as a function of the memory depth at IBO=10 dB.</figcaption></figure>

### Contribution II.9 (Analytic Expression)

EVM - Memory Polynomial model The EVM of a clipped and amplified multicarrier signal assuming a memory polynomial model for the power amplifier and the predistortion module is given by Eq. (6.45) where $u_k$ and $v_{kk'}$ are restated as,

$$u_k = 2 \frac{\mathcal{R}\{b_{k0}\}}{((1 - e^{-\Lambda})\text{IBO})^{k-1}} \quad \text{and,} \quad v_{kk'} = \sum_{q=0}^{Q} \frac{b_{kq} b_{k'q}^*}{((1 - e^{-\Lambda})\text{IBO})^{k+k'-2}} \quad (6.47)$$

with {$b_k$}$_{1 \le k \le K}$, $0 \le q \le Q$ the coefficients of the polynomial model of order K and memory depth Q.

From the comparison between the theoretical (star markers) and simulated (continuous lines) results, we first conclude that our analytical evaluations perfectly match the simulated EVM. Besides, these results allow us to highlight the significant difference between the EVM of the amplified signal using memory and memoryless PA models. In particular, EVM curves reach a floor at high IBO when memory effects are taken into account. Indeed, a strong performance loss is obtained at large IBO values, however corresponding to expected linear regions of the PA characteristics. This proves that the memory effects can not be ignored in practice and may even become predominant compared to non-linearity effects. In addition, it appears that clipping leads to smaller performance degradations than the memory effect, once again proving the necessity of introducing the memory effects in analytical models.

To further quantify the weight of the memory effects on EVM results, Fig. 6.27 depicts the EVM as a function of the memory depth, for an IBO of 10 dB and different clipping thresholds. These curves are obtained from the proposed closed-form. As expected, the deeper the memory of the amplifier, the stronger the EVM. Hence the memoryless polynomial model (Q = 0) is not sufficient for modeling the power amplifier behavior when the memory effects are present. For example, when clipping is deactivated, the EVM of amplified signals using memoryless polynomial model is 2.5%. However, using memory polynomial model with memory depth of 2, the EVM increases to 6.8% which is quite significant.
<figure><img src="image_40.png" /><figcaption>FIGURE 6.28– EVM chart for the Rapp model. EVM is given versus predistortion accuracy for variable IBO. Clipping ratio is 10 dB.</figcaption></figure>

## 3.4 APPLICATION TO OPTIMAL SETTING OF THE TRANSMITTER

From the obtained analytical expressions of the EVM, it is now possible to carry out thorough analyses of the global behavior of a transmitter depending on the various identified parameters. The aim is to find optimality behavior regimes of the transmitter regarding some signal quality constraints. For instance, given a particular power amplifier, a major question that arises in practice is how to control both the clipping ratio and the predistortion accuracy to respect some target EVM according to standard specifications, while maximizing the power efficiency of the amplifier.

In the case of the Rapp model for example, it is possible to build some EVM chart gathering several EVM curves versus the quality of the predistortion function for various IBO values. The predistortion accuracy can be simply represented by the ratio between the knee factors of the amplifier and the predistortion functions, i.e. *a/b*. An example of such chart is given Fig. 6.28, for a clipping ratio of 10 dB. The optimal operating point is when *a* = *b* but means high complexity is spent to estimate the amplifier response and have tight predistortion. But we note that some degree of freedom can be exploited considering some target EVM, (for instance 2% represented here which corresponds to the average EVM in the DVB-T2 requirements). Namely, the chart shows how some less accurate predistortion can be tolerated if some IBO merging is taken. Conversely, maximizing the power efficiency and thus reducing the IBO as much as possible, imposes a minimum accuracy of the predistortion function. Hence, such chart can be useful in the perspective of finding some adequate trade-off between power efficiency and complexity.

In the case of the polynomial model, it is even possible to go a bit further in the analysis. Let us consider the chart of Fig. 6.29 consisting of EVM curves versus IBO for various clipping ratios $\Lambda$. The power amplifier considered here is of order $K = 6$, with coefficients extracted from [NXP10]. The predistortion complexity is represented by the order of the polynomial used for the linearization ($K_p = 5$). The optimal operating regime of the transmitter is easily identified as being the curve corresponding to the lower limit of the network of curves, i.e. for which EVM is minimized at each IBO. Then plotting such optimal curves for various orders of the predistortion polynomial yields the results of Fig. 6.30. This final chart allows to find the adequate IBO to use for a given predistortion complexity for a given target EVM. Thanks to the proposed closed-forms, these chart are easy to build and only need the knowledge of the polynomial coefficients of the power amplifier.
## 3. Theoretical Analysis of Joint PAPR Reduction and Predistortion

<figure><img src="image_41.png" /><figcaption>**FIGURE 6.29**– EVM chart for the Polynomial model. EVM is given versus IBO for variable clipping ratios. Power amplifier if of order K = 5 and predistortion of order K<sub>p</sub> = 5.</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>**FIGURE 6.30**– Final EVM chart for the polynomial model. EVM is given versus IBO for variable predistortion orders. Power amplifier is of order K = 5.</figcaption></figure>

## 3.5 CONCLUSION

This study comes at a complement to the algorithmic studies on PAPR reduction detailed before, with a more theoretical approach and wide scope. Our aim was to provide new analytical tools for the overall optimization of transmitters. On this subject, only upper-bounds or approximated expressions were available in the literature, which was not bringing sufficient insight into the performance and interactions between the various modules of transmitters. The derived analytic forms are a first attempt towards the elaboration of a complete set of analytic laws that are expected to help designing and properly setting the different modules of a transmitter. The first provided charts already show how it is possible to play with many degrees of freedom, address some performance/efficiency/complexity trade-off and evaluate their impact. However, as it will be detailed in my research perspectives, additional efforts are still needed to have a complete set of tools.

### Summary (PhD/Dissemination/Projects)

- 1 PhD (A. Cheaito) and 1 Master Internship (S.M. Fahad)

- 2 international journal papers [R.1][R.2]

- 5 communications in international conferences and workshops [C.2][C.11][C.12][C.23][C.25]

- 2 communications in national conferences [N.1][N.3]

- 1 Best paper award [C.12]

- 1 invited talk at IEEE Globecom 2017, Special session on selected Wireless Communications Letters

- 1 project LabEx project (CominLab TEPN)
<figure><img src="image_43.png" /></figure>
# Optimization of the Wireless Broadband Access

## Chap. 7

> There is no science in this world like physics.
> Nothing comes close to the precision with which physics
> enables you to understand the world around you.

> - Neil de Grasse Tyson -

### Selected Topics

In this chapter, I will present the main contributions of my research at the level of wireless broadband communication systems, with a special focus on the general idea of spatial filtering. During the last years, spatial filtering concepts, also commonly referred to as beamforming or spatial precoding, have progressively been introduced in many communication models hereby revealing some new paradigm we have been trying to appropriate and exploit for new system optimizations. To highlight our contributions on this topic, I have selected the three following aspects of the work :

1. Time Reversal Based Communications : Performance Analysis for Single-User Transmission

2. Spatial Filtering for MU-MIMO : Spectral Resource Optimization in Moderate-Scale Antenna regime

3. Practical Spatial filtering Strategies for massive MIMO mmWave Communications

## 1 ANALYSIS OF TIME-REVERSAL BASED COMMUNICATIONS

### 1.1 MOTIVATIONS

As already reminded in section 6, multiple antenna concepts have brought about a real revolution in the field of wireless communications. Looking back to the late 1990s, the seminal works of Foschini, Telatar, Calderbank, Tarokh and others [Fos96, Tel99, TSC98] were promising impressive capacity gains behind the ever-considered single link Shannon limit. MIMO has thus rapidly been viewed as a goldmine of researches and developments, not solely for academics but also for system designers and network operators who were already facing exponential bitrate demands. Therefrom, MIMO has become a key technology in the evolution of wireless systems.

In the early years if its existence, MIMO was essentially considered as a new degree of freedom upon which some additional channel diversity (spatial) could be reaped to fight against selectivity and multipath effects. MIMO integration in system models were thus carried out following the same general approach as that used when dealing with frequency and time diversities. The main first contributions on MIMO systems were hence at the level of space-time code design (in addition to the essential synchronization and channel estimation issues) [TJC99, ZG02, GH03, LV07].
# Chapitre 7. Optimization of the Wireless Broadband Access

Around 2005, a new vision of MIMO attracted interest through the multiuser dimension of many communication system models. Multiuser downlink scenarios were considered with the idea of using a set of single antenna users as a virtual antenna array [SSH04, CM04]. Spatial multiuser diversity could then be exploited to send data streams in parallel targeting different users. Hence was introduced the concept of MU-MIMO, and accordingly the idea of spatial division multiple access (SDMA). In 2003 Caire and Shamai had already shown in [CS03] that Dirty Paper Coding strategies could succeed in achieving the capacity region of the downlink broadcast channel<fnref n="1" />. However, the design of optimal Dirty Paper Coding turns out to be a hard non-linear problem, and, more importantly, the fundamental assumption that channel state information should be available at the transmitter (CSIT) was not convincing practical system designers. CSIT was historically an advantage reserved to static transmission systems, as for example wired communications.

Hence, in the late 2010s, one major question that arose in the optimization of MIMO systems (and that has not found a definite answer untill today) was mainly related to the practical design of MU-MIMO schemes, using linear spatial treatments at the transmitter and that could accomodate limited channel feedback from the users. Such spatial domain linear operations can be called **spatial filters** as they act on the spatial branches and modify there complex gains. Such filters have to be continuously adapted to the propagation conditions to make the communication to multiple users well estalibshed, and possibly be non-interferent. With this spatial filtering perspective, time-reversal techniques appeared to us as a potentially well suited candidate as it will be further explained. We therefore started some preliminary studies in 2008 and launched a PhD program (T. Dubois) on that subject in 2009 in parallel with an ANR project (TRIMARAN). Hereby, we first focused on single user scenarios (SU-MIMO). In 2011, a second PhD started (H. Fu) to work on complementary analyses in the SU-MIMO case and extend them to the MU-MIMO context. In parallel to this, important hardware developments were led within the research team with other collegues to provide proof of concepts of our studies [C.32]. These latter aspects will be further discussed in the proposed perspective works in section 9.

## 1.2 TIME REVERSAL FILTER CONCEPT

### ▶ Origin

The time-reversal concept takes its origin from the research works of Fink et al. on ultra-sound and underwater acoustic waves in the 1990s [Fin92, DTF99]. Based on the acoustic wave propagation equation, Fink demonstrated that it was possible to re-create an acoustic impulse at a precise source location, by generating adequate waves from an infinite number of transducers surrounding that source in a closed contour. The waves to generate for this should exactly correspond to the time-reversed versions of those measured by the transducers during a first step, called sounding phase, when the source emits an impulse. Fink based his proof of the time-reversal concept on the Green function of the non-dissipating propagation environment,

$$G(r_1, r_2, w) = G(r_2, r_1, w) \qquad (7.1)$$

which represents the generalization of the channel impulse response taking into account of the overall propagation environment and in particular the location $r_1$ and $r_2$ in space of the reference input/output. The symmetry of the Green function with respect to space variables yields the fundamental reciprocity property which essentially says that the roles of the transmitter and receiver can be interchanged in harmonic propagation regime. Fink realized many experiments on acoustics wave as the one depicted in Fig. 7.1 to prove the concept of time-reversal and particularly the back-focusing effects of propagation waves in the form of focal spots at the target location. Fink introduced the idea of **time-reversal** filters used at the transducers which make the overall generated wave propagates back through the same environment in a reverse way, hereby retracing its former paths and eventually focusing its power in space at the receiver. This phenomenon is called spatial focusing and is observed in practice in the form of a focal spot. Other experiments tended to show that efficient focusing effects could also be obtained with a limited number of transducers not necessarily sourrounding the target location, assuming a sufficiently dispersive propagation environment. This is the idea of time-reversal mirror as illustrated in Fig. 7.2.

<footnote n="1">1. Such channel is referred to as Broadcast Channel using the information theory terminology, which has not to be confused with the broadcast transmission mode we have been considering in this document.</footnote>
# 1. Analysis of Time-Reversal Based Communications

<figure><img src="image_44.png" /><figcaption>**FIGURE 7.1–** Time Reversal experiment using a Time Reversal Cavity (non-dissipating medium).</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>**FIGURE 7.2–** Time Reversal experiment using a Time Reversal Mirror (inhomogeneous medium).</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>**FIGURE 7.3–** Time Reversal experiment with electromagnetic waves in a reverberrant chamber</figcaption></figure>

During the 2000s, time-reversal was further applied to electromagnetic waves for many system configurations [LdRT$^{+04}$, OHE$^{+04}$, LdT$^{+06}$]. Time-reversal refocusing properties are today well understood mathematically. Many works have in particular analyzed subwavelength spatial focusing effects in highly dispersive media [OKPP05, LdTF07].

## ► Application to Wireless Communications

The time-reversal concepts theoretically studied and demonstrated by physicians can actually be translated into communication terminology. Embedding time-reversal concepts in a communication chain amounts to equip a transmitter with a time-domain pre-filter built from the knowledge of the CSI. Provided sufficient richness of the scattering environment hence, time-reversal filters are capable of confining electromagnetic energy to narrow beams. This strong property makes it a promising solution for energy efficient transmissions [WWH$^{+11}$], allowing for spatial discrimination between multiple users [HYW$^{+12}$] but also leading to low-complexity receivers [EVH$^{+04b}$].

In the field of wireless communications, TR was initially studied for Ultra WideBand (UWB) systems, i.e. considering signals spreading over large bandwidths [NBEZ09, NZZK09]. In such a case, efficient focusing effect is expected from the highly time-dispersive nature of the channel response resulting from the high temporal resolution used to observe the propagation environment. Indeed, the larger the bandwidth, the better the resolution for resolving multipaths, and the thinner the focal spot. Applying TR to IR-UWB signals is straightforward, by directly using the reversed channel impulse response as the pulse-shaping function at the transmitter. First studied as a pre-RAKE transmission solution leading to low-complexity UWB receivers [SEH$^{+04}$, PFDB09], TR
# Chapitre 7. Optimization of the Wireless Broadband Access

was also proposed for multiple-user UWB communications [NKE07].

More recently, TR was applied to multiple antenna transmitters with the idea of enriching the propagation channel through the exploitation of the spatial dimension, hereby improving the focusing effects without increasing the bandwidth [NZZK09]. In that case, one TR filter is used for each antenna element of the transmitter according to the spatial link between this antenna and the receiver antenna. Thus, the transmit antenna array plays the role of the so-called time-reversal mirror (TRM) as experienced with acoustic waves and optics [DTF99]. The larger the antenna array, the more accurate the time reversal effect regarding the overall propagation environment, and the better the spatial focusing.

We actually started our work on time-reversal by leading proof of concept studies from realistic measurements or embedded in electromagnetic reverberation chambers to characterize the focusing effects obtained with TR-based multiple-input-multiple-output (MIMO) wireless systems [C.54]. Some examples of obtained focusing effects is depicted in Fig. 7.3. The wave focusing is clearly obtained in space around the target, but also in time. From this starting point, our aim was to apply the concept to OFDM multiple antenna systems and to lead some system design and theoretical studies for such systems.

## ▶ Generic TR-based System Model

Let us consider a general system model as depicted in Fig. 7.4 made of one transmitter equipped with M antennas and K single antenna receivers. Denoting $h_{m,k} \in \mathbb{C}^{L \times 1}$ the sampled L-tapped impulse response of the baseband channel from antenna m to receiver k, the TR filter associated to this channel consists in the following:

$$g_{m,k} = \frac{1}{\sqrt{M}} \cdot [h_{m,k}(L-1), \dots, h_{m,k}(2), h_{m,k}(0)]^H \quad (7.2)$$

with $h_{m,k}(i)$ being the $i$th entry of $h_{m,k}$. Hence, $g_{m,k}$ is a FIR filter of the same size as $h_{m,k}$, but whose tap values are conjugated and arranged in a reverse chronological order compared to $h_{m,k}$. Clearly, $g_{m,k}$ is the temporal matched filter of $h_{m,k}$. Note that factor $\frac{1}{\sqrt{M}}$ is used for power normalization purpose along the antenna dimension. As evident from Fig. 7.4, there are as many TR filters as the number of antennas and targeted users, i.e. a total of M×K filters. Note that the knowledge of the channel state is theoretically needed at the transmitter side to feed these filters, which means that TR is applicable to closed loop systems. However, the idea behind time-reversal filters is that they can be refreshed periodically by simply storing the impinging signals coming from simple short impulses sent by the users. This is expected to be a quite quick and low-complex process.

Let $s_k \in \mathbb{C}^{N \times 1}$ be a frame of N complex symbols intended to user k. By modeling the convolution operation by Toeplitz matrix multiplication, the signal transmitted by each antenna can then be expressed as,

$$x_m = \frac{1}{\sqrt{K}} \cdot \sum_{k=1}^{K} T(g_{m,k}) \cdot s_k \quad (7.3)$$

where $\frac{1}{\sqrt{K}}$ guarantees a unity power factor whatever the number of users. In this equation, $T(v)$ means that vector v is converted into a Toeplitz matrix<fnref n="2" /> which first column is $[v^T, 0, \dots, 0]^T$ and first row is $[v(1), 0, \dots, 0]$. From Eq. (7.3) and Fig. 7.4, it is important to see that in a TR-MIMO system, each antenna branch sends the same data symbols $s_k$, the spatial dimension being exploited for beamforming purpose rather than for space diversity and multiplexing as in conventional MIMO systems. However, multiuser multiplexing is obtained if $K > 1$ since data symbols of various users k are accumulated after TR filtering and then sent simultaneously.

In the single user case, a unique data frame $s_k$ is transmitted at a time. After classical manipulations, we can express the received signal as,

$$y_k = \frac{1}{\sqrt{K}} \cdot T(\tilde{W}_k) \cdot s_k + n_k \quad \text{with, } \tilde{W}_k = \sum_{m=1}^{M} w_{m,k} \quad (7.4)$$

and defining $w_{m,k}$ as

$$w_{m,k}(l) = \frac{1}{\sqrt{M}} \cdot \sum_{i=-L+1}^{L-1} h_{m,k}^*(i) h_{m,k}(i+l-L) \quad (7.5)$$

<footnote n="2">2. Note that the size of T is determined by that of the vector used for right multiplication. In the present case, $T(g_{m,k})$ is of size $(L+N-1) \times N$.</footnote>
# 1. Analysis of Time-Reversal Based Communications

<figure><img src="image_47.png" /><figcaption>**FIGURE 7.4**– Generic multiple user multiple antenna system model using Time Reversal filters.</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>**FIGURE 7.5**– Equivalent propagation channels obtain with TR filtering with $M \in [1, 8, 32, 64]$. Indoor environment, central frequency is 2.4 GHz, bandwidth is 20 MHz.</figcaption></figure>

the equivalent MISO channel response experienced by the data symbols, including TR precoding. Vector $w_{m,k}$ turns out to be the autocorrelation of channel vector $h_{m,k}$, which is at the fundamental basis of TR concept. Due to TR precoding thus, symbols $s_k$ pass through an equivalent channel which impulse response is the autocorrelation of the initial channel. More precisely, $\tilde{W}_k$ is a sum of autocorrelation functions and represents the baseband equivalent model for the radio propagation channel, including TR precoding, from the M-element transmit array to the receive antenna $k$. In the time domain, these autocorrelation functions consist of a central peak and some side lobes. Some examples of these autocorrelation functions measured in an Indoor office environment are given Fig. 7.5. Owing to the TR effect, the peaks add up constructively at the receive antenna location, whereas the side lobes are likely to recombine destructively by virtue of random phases. With sufficiently rich channel responses then, equivalent channel $\tilde{W}_k$ may converge to a dirac delta function revealing a Gaussian channel response. Such an effect is expected to be obtained in case of a large frequency bandwidth, but also with a high number M of decorrelated channel responses as observed in Fig. 7.5. One may understand that the transmission becomes then easy to handle using simple threshold detectors at the receiver side. However, if autocorrelation functions are not sufficiently thin due to a lack of scattering richness in the MISO channel, the receiver will have to treat residual ISI and compensate for channel distortions by means of equalization.

Let us now extend the received signal expression in a multiple user scenario, that is when various symbol vectors are transmitted simultaneously. In that case, let $k = j$ be the user of interest, Eq. (7.4) rewrites as :

$$y_j = \frac{1}{\sqrt{K}} \cdot \underbrace{T(\tilde{W}_j) \cdot s_j}_{\text{useful term}} + \frac{1}{\sqrt{K}} \cdot \underbrace{\sum_{k \neq j} T(\tilde{\omega}_{kj}) \cdot s_k + n_j}_{\text{Interference term}} \quad (7.6)$$

where we now have to introduce $\tilde{W}_{kj} = \sum_{m=1}^{M} w_{m,kj}$ as,

$$w_{m,kj} = \frac{1}{\sqrt{M}} \cdot \sum_{m=1}^{M} \sum_{i=-L+1}^{L-1} h_{m,k}^{*}(i) h_{m,j}(i+n-L) \quad (7.7)$$

These latter functions are the MISO equivalent cross-correlation functions due to the combined effect of TR filtering for user $k$ and channel response from the transmit station to user $j$. These functions are responsible for the mutiuser interference (MUI) term and models the recombination of waves intended to competing users. In the ideal case of fully decorrelated channel responses between users, cross-correlation functions $w_{m,kj}$ are cancelled and Eq. (7.6) becomes equivalent to Eq. (7.4). This can be interpreted as a consequence of the spatial focusing effect. As in the SU-MIMO case, decorrelation properties are strongly related to the bandwidth and the number of antennas. The larger the bandwidth and the higher the number of antennas, the weaker the cross-correlation functions.
## 1.3 TR-OFDM SYSTEM DESIGN

### ► Frequency domain Time Reversal

In practice, as the combination of MIMO and OFDM is often considered in many systems, it is interesting to pose the problem of TR precoding in MIMO-OFDM. Namely, combination with OFDM is an simple and efficient solution to get rid of residual ISI as a consequence of imperfect autocorrelation functions. One simple approach for this is the use of the famous guard interval (GI) as a cyclic extension in preamble of the time-domain OFDM symbols. Taking this mechanism as reference, it is well known that every linear convolution function occurring between the OFDM modulator and demodulator is translated into circular convolution. This also means that the related Toeplitz matrices in Eq. (7.6) become circulant. Hence, it is very easy to get the equivalent expression of Eq. (7.6) but for OFDM waveform as,

$$Y_j = \frac{1}{\sqrt{K}} \cdot \underbrace{\tilde{W}_j \cdot S_j}_{\text{useful term}} + \frac{1}{\sqrt{K}} \cdot \underbrace{\sum_{k \neq j} \tilde{W}_{kj} \cdot S_k}_{\text{Interference term}} + N_j \quad (7.8)$$

in which $Y_j = F_N y_j$, $S_k = F_N s_k$ are the frequency domain symbol vectors with $F_N \in \mathbb{C}^{N \times N}$ the N-point Digital Fourier Transform (DFT) matrix. The equivalent channel and interferent diagonal matrices $W_j$ and $W_{kj}$ such that

$$\left\{ \begin{array}{l} \tilde{W}_j = \frac{1}{\sqrt{M}} \cdot \sum_{m=1}^{M} \text{Diag}\{H_{m,j}\} \text{Diag}\{G_{m,j}\} \\ \tilde{W}_{kj} = \frac{1}{\sqrt{M}} \cdot \sum_{m=1}^{M} \text{Diag}\{H_{m,j}\} \text{Diag}\{G_{m,k}\} \end{array} \right. \quad (7.9)$$

with $H_{m,k} = F_N h_{m,k}$. This result simply confirms that the time-domain TR FIR filter can be implemented as a point-wise multiplication with the data symbols before OFDM modulation by using the frequency domain coefficients $G_{m,k} = F_N g_{m,k}$ of the TR filter obtained through DFT. From the TR filter definition in Eq. (7.2), is is straightforward to write

$$G_{m,k} = \frac{1}{\sqrt{M}} \cdot H_{m,k}^* \quad (7.10)$$

which indicates that the frequency domain TR filter essentially amounts to applying to each subcarrier $n$ of the OFDM spectrum a complex weighting factors $H_{m,k}^*(n)$ being the conjugate of the frequency domain channel gain. Eq. (7.10) can be viewed as a matched filter, applied to each subcarrier $n$ and each antenna $m$, i.e. to the overall space-time channel. The corresponding transmitter is depicted in Fig. 7.6, highlighting the frequency domain TR precoding stage.

Some interesting interpretations can be done from this result, in comparison to the expression obtained in Eq. (7.6). First, the frequency domain equivalent channel $\tilde{W}_j$ appears as a summation of real coefficients. More precisely, the $n$th entry of $\tilde{W}_j$, corresponding to the $n$th subcarrier of the OFDM signal, is $\tilde{W}_j(n) = \frac{1}{\sqrt{M}} \sum_m |H_{m,j}(n)|^2$. This pure real value has to be understood as the effect of the constructive recombinations of waves as already discussed in the previous section. Thus, the larger the M, the better the equivalent channel gain $\tilde{W}_j(n)$ on each subcarrier $n$ and the stronger the resulting useful term. Equivalently, the better the time focusing. On the other hand, the interfering component $k$ on subcarrier $n$ experiences the following equivalent channel $\tilde{W}_{kj}(n) = \sum_m H_{m,j}(n) H_{m,k}^*(n)$. Hence, the interfering symbols are weighted by a sum of complex numbers with independent amplitudes and phases. As such, the interfering power tends to be naturally reduced, namely with large M, which consists in another interpretation of the spatial focusing concept. Eventually, this comes in favor of the MU multiplexing ability of the TR-based precoded system. TR filters can thus be used for SDMA purpose.

Such important interpretations of time-reversal filtering behaviour with large M coincides with the initial postulate relying on wave propagation and which was telling that wave focusing could be theoretically obtained with a sufficiently large number of transducers. Interestingly, this point of view which is known for many years from the studies of Fink is another interpretation of the famous massive MIMO paradigm introduced by Marzetta in 2010 [Mar10a]. Some connections with massive MIMO and MU-MIMO literature will thus be regularly established in the sequel.
# 1. Analysis of Time-Reversal Based Communications

<figure><img src="image_49.png" /><figcaption>FIGURE 7.6– TR-OFDM system model with Time Reversal applied in frequency domain.</figcaption></figure>

<figure><img src="image_50.png" /><figcaption>FIGURE 7.7– Optimal guard interval setting in TR-OFDM with variable number of antennas.</figcaption></figure>

## ▶ Guard Interval Settings

One first parameter to correctly set when carrying out OFDM communications is the guard interval. Yet, as time-reversal is originally processed through a time domain FIR filter, it leads to a straightforward time dispersion of the symbols which comes in addition to the dispersion produced by the channel itself. It other words, the equivalent channel experienced by the filtered OFDM symbols is of theoretical length $2L - 1$, if L is the original channel length. However, when implementing the time reversal filter in frequency domain, linear convolution translates into circular one, and the additional time dispersion disappears.

Nonetheless, in both cases, the resulting channel impulse response will benefit from the properties of autocorrelation functions. In particular, the equivalent channel will have a symmetric power delay profil, with the highest tap in the middle of the response. Hence, it can be shown that a judicious choice to ease the synchronization stage at the receiver is to circularly extend the OFDM symbols with a prefix and a suffix, as summarized in Contrib. III.1 and shown in [C.44].

## Contribution III.1 (System Design)

### CPCS Guard Interval Setting for a TR-OFDM system

Consider an TR-OFDM system of order N. A conventional OFDM receiver correctly recovers the OFDM frame synchronization if each OFDM symbol is prepended and appended with a cyclic extension (Cyclic Prefix Cyclic Suffix - CPCS) of $v = \frac{\Delta}{2}$ samples. Hence, the lth OFDM symbol $s_l$ is,

$$s_l = G_{CPCS} F_N^H X_l \quad \text{with, } G_{CPCS} = \begin{bmatrix} O_{v \times (N-v)} | I_v \\ \hline I_N \\ I_v | O_{v \times (N-v)} \end{bmatrix} \quad (7.11)$$

where 0 and I are zero-entry and identity matrices, $F_N$ is the square Fourier matrix and $X_l$ is the vector of data symbols.

In addition, depending on the time-domain (TD) or frequency-domain (FD) implementation of the time reversal filter, we have,

$$\Delta_{FD} = \frac{\Delta_{TD}}{2} \quad (7.12)$$
# Chapitre 7. Optimization of the Wireless Broadband Access

Complementary to this analysis, owing to the time focusing effects, guard interval durations are expected to be less using a TR filter than in common OFDM. We led simulations using measured channel responses in order to identify the reduction of the guard interval when increasing the number of antennas. Our results, as presented in Fig. 7.7 confirm the expected behaviour.

## ► Receiver Synchronization

As above mentioned, using TR filtering results in an equivalent channel response with linear phase. We showed that this particular feature can be advantageously exploited to derive a very simple ML estimation of the OFDM symbol synchronisation error at the receiver. Hence we proved that time reversal filtering can also be leveraged to ease the OFDM symbol timing recovery and lead to a very simple algorithm which can be processed with the help of few pilots, and even blindly if the number of transmit antennas and subcarriers high sufficiently high. The main principle of this simple algorithm is based on a least-square estimation, as summarized hereafter and argued in [C.44].

### Contribution III.2 (Algorithm)

#### Multiple Antenna RT-OFDM Synchronization

Given a MIMO TR-OFDM system, ML symbol offset estimation $\epsilon_j$ at receiver j can be obtained using a subset $\mathcal{P}$ of pilot subcarriers as,

$$ \hat{\epsilon}_j = \left| \frac{1}{2\pi} \cdot \frac{\sum_{n \in \mathcal{P}} n |Y_j(n)| \arctan \left( \frac{\mathrm{Im}\{Y_j(n)X^*(n)\}}{\mathrm{Re}\{Y_j(n)X^*(n)\}} \right)}{\sum_{n \in \mathcal{P}} n^2} \right| \quad (7.13) $$

Also, when $M \gg 1$, and assuming a sufficient number N of subcarriers a blind algorithm can be processed as,

$$ \hat{\epsilon}_j = \left| \frac{1}{2\pi} \cdot \frac{3}{N^3} \cdot \sum_{n=0}^{N-1} n \arctan \left( \frac{\mathrm{Im}\{Y_j(n)\}}{\mathrm{Re}\{Y_j(n)\}} \right) \right| \quad (7.14) $$

## ► Comparison with other spatial filters

From the previous considerations, it appears that time-reversal filtering can be viewed as a particular multiantenna linear precoding function. Hence, some commonalities with classical MU-MIMO precoding schemes may be identified. Having a look back to the literature of MU-MIMO systems, and in especially regarding linear precoders, matched-filter-like solutions have already been proposed for the design of MIMO systems using CSI at the transmitter. The main two ones are Maximum Ratio Transmission (MRT) [Lo99] and Equal Gain Transmission (EGT) [LH03]. Considering the same OFDM-MIMO system model as previously introduced, we can restate such precoders for a particular user k as

$$ \text{EGT:} \quad G_{m,k} = \frac{1}{\sqrt{M}} \Phi_{m,k}^* \quad \text{with, } \Phi_{m,k}(n) = \frac{H_{m,k}}{|H_{m,k}|} \quad (7.15) $$

$$ \text{MRT:} \quad G_{m,k} = \mathrm{Diag}\{\Lambda_k\} H_{m,k}^* \quad \text{with, } \Lambda_k(n) = \frac{1}{\sqrt{\sum_{m=1}^{M} |H_{m,k}(n)|^2}} \quad (7.16) $$

Compared to time-reversal, the EGT precoding scheme yields a transmission with a unitary gain on every link, in frequency and space domains. It consists in a simple phase rotation and thus avoids any power imbalance between the signals propagated by each antenna. This can be a positive aspect regarding the power amplification efficiency as discussed in 6.
# 1. Analysis of Time-Reversal Based Communications

On the other hand, comparing Eq. (7.16) with Eq. (7.10), it is concluded that TR and MRT precoding follow the same overall approach by applying the conjugate of the frequency domain channel responses, thus leading to similar (but not strictly equivalent) conclusion in terms of time and spatial focusing properties. The difference between them merely comes from power normalization. Indeed, TR precoding is based on a simple equal power allocation across the antennas, whilst being strictly identical whatever the subcarrier index and the user. With MRT, the power normalization can be rather interpreted as a spatial normalization which may differ from one subcarrier to another as well as between users. It can easily be shown that the power factor $\Lambda_k$ simply corresponds to the singular value of the MISO channel on subcarrier n [R.11].

We will analyze the difference between these two schemes in terms of capacity and perfomance the next two sections. Let us first try to identify for each technique the focusing gain we obtained, i.e. the average useful power that is transfered to the receiver. Focusing gain can be understood as the array gain obtained at the antenna level when a particular precoding scheme is used. The obtained results, published in [R.11] are reported in the following contribution.

## Contribution III.3 (Analytic Expression)

### Focusing gain of matched filter like spatial filters

Assuming independent normalized complex Gaussian fading per antenna, the ergodic power array gain obtained for the TR, MRT and EGT spatial filtering schemes in an MIMO-OFDM system with M antennas are,

$$ \text{TR} \quad \mathcal{G}_{\text{TR}} = M + 1 \qquad (7.17) $$

$$ \text{MRT} \quad \mathcal{G}_{\text{MRT}} = M \qquad (7.18) $$

$$ \text{EGT} \quad \mathcal{G}_{\text{EGT}} = 1 + \frac{\pi}{4}(M - 1) \qquad (7.19) $$

This results already show that MRT and TR become equivalent when $M \gg 1$. On the other hand, EGT gives the lowest focusing gain, since such spatial filter only plays with the phases of the channel complex gains, hereby losing one degree of freedom. We conclude that TR is the best choice in terms of average SNR maximization, which makes it the best strategy for green communications.

Last but not least, as it has been noticed that the MRT precoding vector applied on each subcarrier n is taking into account the gains of the spatial links for all the antennas, it turns out that MRT precoding has to be computed in a centralized way. Indeed, the channel responses of different transmit antenna pairs need to be gathered to compute the normalization factor $\lambda_k(n)$, which is required for each subcarrier of each user. With TR or EGT filtering, each precoding vector can rather be processed independently at each transmit antenna. Hence, TR and EGT could easily be envisaged in systems with distributed antennas. This is not the case of MRT.

After the system level analysis discussed so far, the aim of our work during the first years of studies on time-reversal (i.e. 2009–2011), was to establish new results from a green communication point of view. We then targeted single user downlink scenarios, considering the spatial filters as promising solutions for link budget enhancement rather than for multiuser management. In the previous section, we have already classified the matched-like filters in terms of focusing gains. In the following ones, we give the results of our investigations in terms of theoretical ergodic capacity and BER performance of the SU-MIMO precoded systems.

## 1.4 ACHIEVABLE RATE IN SINGLE USER SCENARIOS

From the introduced system model in 1.3, and focusing on a particular subcarrier n, getting the achievable rate of the system in the SU case amounts to developing the following expression depending on the used precoding scheme,

$$ \mathcal{R} = \mathbb{E} \left\{ \frac{1}{N} \sum_{n=0}^{N-1} \log_2 \left( 1 + |\tilde{W}_k(n)|^2 \gamma \right) \right\} \qquad (7.20) $$
where $\gamma$ is the reference ratio between data symbol and noise variances, i.e. $\gamma = \sigma_s^2/\sigma_n^2$, and where $\tilde{W}_n$ is the equivalent channel for one particular subcarrier embedding the normalization factor per antenna. The problem is then a derivation of achievable rate for SISO link, but involving a dedicated statistical fading model. Boosted by the introduction of the Massive MIMO context around 2010, numerous results about achievable rates have been produced regarding precoded MIMO schemes during our own studies on time-reversal. Most of the contributions were however focusing on the asymptotical results with infinite number of antennas M. As we were more interested in practical system design, we first tried to provide contributions in a moderate-scale antenna regime.

► Equivalent Channel Statistical Analysis

A first step towards the derivation of closed-forms of the achievable capacity was to identify the statistical law that could properly model the equivalent channel fading. Without loss of generality, we assume independent and identically distributed variables across the subcarriers. In [R.11], we provided an analysis on this subject which can be summarized as follows.

Contribution III.4 (Analytic Expression)

Statistical Model of the Spatially Filtered MISO Channel

Assuming independent normalized complex Gaussian fading per antenna, the equivalent fading channel models of a precoded MISO system with M antennas are,

$$ \text{TR} \quad \tilde{W}_{\text{TR}} \sim \text{Gamma}\left(k = M, \theta = \frac{1}{\sqrt{M}}\right) \qquad (7.21) $$

$$ \text{MRT} \quad \tilde{W}_{\text{MRT}} \sim \text{Nakagami}(\mu = M, w = M) \qquad (7.22) $$

$$ \text{EGT} \quad \tilde{W}_{\text{EGT}} \sim \text{Nakagami}\left(\mu = M, w = 1 + \frac{\pi}{4}(M-1)\right) \qquad (7.23) $$

where $k$ and $\theta$ are the shape and scale parameters of the Gamma distribution, and $\mu$ and $w$ are the shape and spread parameters of the Nakagami distribution.

This first analysis tells us that the EGT and MRT schemes yield the same statistical behaviour of their equivalent channel. They namely present the same shape parameter and only differs on their spread values. As the shape parameter corresponds to the number of degrees of freedom of the law, we conclude that EGT and MRT can theoretically exploit the same channel diversity order. As for the spread parameter, it is directly equal to the average power of the distribution, hence indicating that MRT will benefit from highest average SNR than EGT. This latter coincides with the previous analysis on the focusing gain in Eq. (7.17). On the other hand, the TR equivalent channel is governed by a Gamma distribution with also M degrees of freedom. By comparing the Gamma and Nakagami CDF with identical shape parameters, it turns that the Gamma distribution is much more spread but exhibits a higher average value. This means that a higher average SNR can be expected however with a higher probability of deep fades. As the log is a concave function, this will positively influence the expectation in Eq. 7.20 for low SNR values, but negatively for high SNR values.

These considerations, published in [R.11] can be easily verified through simulations as proposed in Fig. 7.8. As expected, the capacity curves of MRT and EGT follow exactly the same trend with some outperformance of the EGT scheme. However, the TR curve crosses the MRT curve for some SNR, hence outperforming the MRT for low SNR range while suffering from degradation at high SNR. Hence we conclude that the TR filter is the optimal solution at low SNR. Instead, MRT is optimal at high SNR. The former follows the matched filter criterion, hence maximizing the SNR. The latter relies on the singular value $\Lambda_k(n)$ of each MISO channel, hence maximizing the rate at high SNR.

Overall, since the number of degrees of freedom is the same for the three cases, the capacity curve exhibit the same slope at high SNR. Fig. 7.8 also gives the achievable rates when combining spatial filtering and OSTBC. We indeed studied this combination since matched-type prefilters yield an equivalent channel with real values (i.e. fading coefficients are real numbers), which enables easy combination with OSTBC of rate one for an arbitrary number of antennas [WC09a]. However, leading the same approach in terms of statistical analysis leads to the
# 1. Analysis of Time-Reversal Based Communications

<figure><img src="image_51.png" /><figcaption>FIGURE 7.8- Achievable Rate comparison bewteen various spatial filtering methods with M = 2 antennas.</figcaption></figure>

conclusion that achievable rates are dramatically reduced in that case. More insights on this aspect have been given in [R.11]

## Closed-forms and bounds

To go a step beyond, we also worked on the theoretical derivation of the expectation in Eq. (7.20), aiming at providing new closed-forms for a moderate number of antennas. We succeeded for the MRT and EGT cases for which the integral forms are the same and are tractable identifying some special functions as stated in the following closed-form expression.

### Contribution III.5 (Analytic Expression)

#### Closed-Form of the achievable rate in SU MRT/EGT MISO

Assuming independent normalized complex Gaussian fading per antenna, a precoded MISO system with M antennas yields an achievable rate expressed as,

$$ \mathcal{R}_{\text{MRT,EGT}} = \frac{e^{\kappa^{-1}}}{(M-1)! \kappa^M \log 2} \sum_{m=0}^{M-1} \binom{M-1}{m} (-1)^{M-1-m} \left[ \kappa^{m+1} m! (\Psi(m+1) + \log \kappa) + \frac{1}{(m+1)^2} {}_2F_2\left(m+1, m+1; m+2, m+2; -\frac{1}{\kappa}\right) \right] \quad (7.24) $$

where $\kappa = \frac{G_{\text{MRT,EGT}}}{M} \times \gamma$ is a so-called normalized SNR taking into account the focusing gains as given in Contrib. III.3.

The obtained closed-form involves the Euler Digamma function $\Psi(x)$ and the generalized hypergeometric function ${}_pF_q(\cdot; \cdot; x)$. Note that this closed-form is valid whatever the SNR and whatever the number M of antennas. Similar closed-forms can actually be found in the literature for the MRT case, but following another approach to lead the derivation [NMDL13]. Our approach allows to generalize the result to the EGT case. We partially published this result in an invited paper [C.20]. As far as time-reversal filtering is concerned, the integral form becomes untractable. Instead we thus derived some capacity lower bounds for the high SNR regime, but still without any assumption on the number of antennas.
### Contribution III.6 (Analytic Expression)

#### Achievable rate lower bounds (high SNR) for TR, MRT and EGT

Assuming independent normalized complex Gaussian fading per antenna, achievable rates at high SNR for TR/MRT/EGT precoded MISO system with M antennas are,

$$ \text{TR} \quad R_{\text{TR}}^{\text{low}} = 2 \frac{\Psi(M)}{\log 2} + \log_2 \left( \frac{1}{M} \gamma \right) \qquad (7.25) $$

$$ \text{MRT} \quad R_{\text{MRT}}^{\text{low}} = \frac{\Psi(M)}{\log 2} + \log_2 \left( \frac{G_{\text{MRT}}}{M} \gamma \right) \qquad (7.26) $$

$$ \text{EGT} \quad R_{\text{EGT}}^{\text{low}} = \frac{\Psi(M)}{\log 2} + \log_2 \left( \frac{G_{\text{EGT}}}{M} \gamma \right) \qquad (7.27) $$

On the other hand classical Jensens's inequality can be used to obtain some upper-bounds. Interestingly, the obtained bounds are equivalent to those obtained in the massive MIMO asymptotic regime. This can be understood from the so-called channel hardening effect which occurs when M increases to infinity. From 2010 to 2015, many publications have provided such kind of upper-bounds using the theory of large random matrices. In the case of TR precoding, following either the Jensen's inequality or the large matrix assumption, we find the following upper-bound. It can be proved that this result holds for MRT and EGT precoding using the appropriate focusing gain $G$.

### Contribution III.7 (Analytic Expression)

#### Achievable rate upper bounds (large M) for TR

Assuming independent normalized complex Gaussian fading per antenna, achievable rates at high SNR for a TR precoded MISO system with M antennas is,

$$ R_{\text{TR}}^{\text{up}} = \log_2 (1 + G_{\text{TR}} \gamma) \qquad (7.28) $$

### ► Simulations

In order to show the accuracy of these bounds and closed forms, we give in Fig. 7.9 and 7.10 two examples of achievable rate plots in the TR and EGT cases. For the EGT case, the closed-form is given in comparison to simulated results for validation. For EGT and TR, lower and upper-bounds can be compared to the simulations. We conclude that the provided lower-bounds are more tight in a moderate-scale antenna regime that the upper-bounds, as soon as the SNR is of at least 5 dB. In addition, they turn out to be also very accurate if M is large, which can be explained by the fact that at large M the focusing gain makes the SNR substantially increase. Hereby, the convergence to the high SNR regime is faster. Hence, the proposed lower-bounds can be preferred than other existing bounds to approach the achievable rate of precoded MISO systems without any assumption upon the size of the antenna array.

## 1.5 BIT ERROR RATE PERFORMANCE ANALYSIS

In the same way we had been working on the derivation of achievable rates, we also tried to obtain the closed form expressions of the BER performance in the case of TR filtering, but also for the other precoders. Out main results have been published in [R.11]. The derivation merely consists in averaging the Q-function $Q(.)$ over the PDF of the equivalent channels, as identified in Eq. (7.21). For QPSK modulation for example, the theoretical
# 1. Analysis of Time-Reversal Based Communications

<figure><img src="image_52.png" /><figcaption>FIGURE 7.9– Achievable rate of the TR-OFDM system (Simulation and lower bound at high SNR).</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>FIGURE 7.10– Achievable rate of the EGT-OFDM system (simulation, closed-form and lower bound at high SNR).</figcaption></figure>

average BER in the TR case is obtained by deriving the following integral,

$$ \mathcal{P}_e(\gamma, M) = \int_0^\infty \mathcal{Q}(r\sqrt{\gamma}) \times \underbrace{\frac{(M^2 + M)^{\frac{M}{2}}}{(M-1)!} r^{M-1} e^{-(M^2+M)^{\frac{1}{2}r}} dr}_{f_T(r,M)} \quad (7.29) $$

in which one may recognize the PDF of the Gamma distribution which models the equivalent channel in the TR case. Similar forms are obtained with EGT and MRT precoding, simply modifying the involved PDF. Using particular integral results of the literature, namely [GRJ+65], it is possible to get a closed-form for each precoding case. For example, we give hereafter the obtained result for the TR case which is the most challenging case in terms of mathematical derivations. Note that this result is valid whatever the number of antennas and for all positive SNR in dB. For the MRT and EGT cases, computations need slightly less effort as they can be viewed as a particular case of BER derivations over Nakagami channels for which one can find already published material as for example in [HB06].

## Contribution III.8 (Analytic Expression)

### BER Closed-form for TR filtering

Assuming independent normalized complex Gaussian fading per antenna and QPSK constellations, the BER performance of a TR precoded MISO system with M antennas is,

$$ \mathcal{P}_e = \frac{1}{2} - \sqrt{\frac{1}{2\pi}} \sum_{m=0}^{M-1} \kappa^{\frac{m+1-M}{2}} e^{\frac{1}{4\kappa}} \mathcal{D}_{m-M} \left( \frac{1}{\sqrt{\rho}} \right) \quad (7.30) $$

with $\kappa = \frac{G_{TR}}{M} \gamma$ the normalized SNR, and $\mathcal{D}_p$ the pth order parabolic cylinder function.

This closed-form will be further exploited in section 2 for MU-MIMO system optimization with the concept of partial frequency re-use. However, additional insights can be brought from their series development, namely getting the asymptotic behaviour at high SNR. Doing this for each scheme leads us to the following results given here as an example for QPSK modulation [R.11].
Contribution III.9 (Analytic Expression)

Asymptotic (high SNR) QPSK performance in TR/MRT/EGT SU-MISO

Assuming independent normalized complex Gaussian fading per antenna and QPSK constellations, the asymptotic BER performances at high SNR of a precoded MISO system with M antennas are,

$$
\text{TR} \quad \lim_{\gamma \to \infty} \mathcal{P}_e^{\text{TR}} = \frac{1}{2^{M+1}} \frac{(M+1)^{\frac{M}{2}}}{\left(\frac{M}{2}\right)!} \left(\frac{1}{\kappa}\right)^{\frac{M}{2}} \qquad (7.31)
$$

$$
\text{MRT} \quad \lim_{\gamma \to \infty} \mathcal{P}_e^{\text{MRT}} = \frac{1}{2^{M+1}} \frac{(2M)!}{(M!)^2} \left(\frac{1}{\kappa}\right)^M \qquad (7.32)
$$

$$
\text{EGT} \quad \lim_{\gamma \to \infty} \mathcal{P}_e^{\text{EGT}} = \frac{1}{2^{M+1}} \frac{(2M)!}{(M!)^2} \left(\frac{1}{\kappa}\right)^M \qquad (7.33)
$$

with $\kappa = \frac{G_{TR,MRT,EGT}}{M}$ $\gamma$ the normalized SNR.

As already expected from the study of the equivalent channel statistics, we notice that MRT and EGT have the same asymptotic behaviour, the only difference being at the level of the normalized SNR, once again resulting from the weaker focusing gain of EGT compared to MRT. As far as TR is concerned, the closed-form exhibits some differences compared to the other two. The most important think we can learn from these forms is the fact that MRT and EGT exploit the full diversity of the channel, i.e. the number M of spatial branches. Instead, TR exploits half of this diversity, which could appear as a bit surprising. The simulated results given in Fig. 7.11 and 7.12 for QPSK modulation confirm these theoretical analysis. MRT and EGT BER curves follow the same slope straightforwardly given by the number M of antennas, whereas the TR results suffer a loss of half the diversity order.

This particular trend however results from the simple fact that a Gamma distribution is roughly equivalent to the square of a Nakagami. This indeed yields that the QPSK symbols transmitted through a TR filter experience deep fades with higher probability than in the case of EGT or MRT precoding. Conversely, they also experience favorable channel gains with higher probability. These latter can however not compensate the former since the concavity of the Q-function. Eventually, we can interpret that the TR filter boosts the spatial links already exhibiting favorable gains to the detriment of the links with poor gains. This leads us to the conclusion that TR filtering can be viewed as a spatial matched filter but also including some implicit power allocation mechanism across the frequency spectrum. Contrary to that, EGT does not imply any amplitude modification, and hence does not produce any kind of power allocation. Also, MRT involves power factor corrections (namely matrix Λ in Eq. (7.16)) to compensate for the natural power boost of its time-reversal filter part.

1.6 CONCLUSION

This first part of our studies on spatial filters has been dedicated to time-reversal filtering. This has made us learn many important points about the characteritics of time-reversal in comparison to its EGT and MRT counterparts. A major difference is that TR can be implemented in time or frequency domain, whereas EGT and MRT are applicable only in frequency (or at least by means of two DFTs). A first important corollary to this property is that time-reversal is waveform-agnostic. A second corollory is its ability to be implemented in a pure distributed antenna scenario since no data exchange is needed to compute the filter applied to each antenna. Interestingly, this latter property is also true for EGT since no centralized gain control has to be computed with this scheme.

Then, we have identified the fact that time-reversal have to be viewed as a space-time matched filter to the propagation channel, whereas MRT is a matched filter in the frequency domain only. This means that time-reversal maximizes the power transfer through the space-time channel hence maximizing the receive SNR globally over the considered frequency band. As a pure flat fading precoding approach instead, MRT maximizes the power transfer along the spatial axis only, independently from one narrow band to another, i.e. without
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

<figure><img src="image_54.png" /><figcaption>**FIGURE 7.11**– Bit error rate performance of SU OFDM-MISO systems using various precoding schemes with $M = 2$ transmit antennas.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>**FIGURE 7.12**– Bit error rate performance of SU OFDM-MISO systems using various precoding schemes with $M = 4$ transmit antennas.</figcaption></figure>

accounting for the channel characteristics along the frequency axis. As a result, time-reversal performance both depend on the bandwidth and the number of antennas, whereas MRT is solely depending on the antenna scale.

From a theoretical point of view, time-reversal is the optimal solution at low SNR, by virtue of the matched filter criterion. Instead, MRT uses a power adjustment based on the singular value of the channel. Thus, it represents the optimal solution at high SNR, by virtue of the maximum capacity criterion. As for EGT, it can be viewed as an optimal precoding scheme at high SNR and low SNR, but under the constraint of strict equal power allocation across the antennas.

Eventually, it is worth remembering that the time-reversal concept comes propagation wave considerations and naturally leads to the concept of large-scale antenna systems. Massive MIMO systems have been introduced from a different perspective, judiciously applying random matrix theory to MIMO systems described by channel matrices of high dimension. This brilliant idea came in 2010 thanks to T. Marzetta, whereas the concept of time-reversal mirrors and cavity which embodies all the main concepts of spatial focusing was already existing since 1990 with the works of M. Fink.

### Summary (PhD/Dissemination/Projects)

- 1 PhD (T. Dubois)

- 1 international journal paper [R.11]

- 4 communications in international conferences and workshops [C.26][C.43][C.44][C.54]

- 1 GDR workshop

- 1 Brevet [P.1]

- 1 ANR project (TRIMARAN)

# 2 SPATIAL FILTERING FOR MU-MIMO IN MODERATE-SCALE ANTENNA REGIME

## 2.1 MOTIVATIONS

The first part of our studies on spatial filters has somehow allowed to bridge the two concepts of time-reversal and massive MIMO. Many issues have however not been addressed in this first part, among which the spatial
focusing properties of the precoding filter in a multiuser context, i.e. their ability to efficiently separate the users in spatial domain (SDMA). These aspects are investigated in the present section under the constraint of a raisonnable number of antennas. Indeed, as we started our studies on time-reversal, we were ourselves not that aware of being working on massive MIMO which was just being emerging at this period. Hence, our first feeling around 2011 was to investigate spatial filtering concepts when using not so large antenna arrays, what we call here the moderate-scale antenna regime.

To validate the Massive MIMO concept, theoretical lower bounds when $M \to \infty$ had already been derived in the early 2010s and show a promising increase of the network capacity of order of $M$ [JAMV11]. For a constrained $M$ however, spatial separation between multiple signals may become unperfect or be obtained at the cost of decreased useful power, hereby degrading the global system performance. This is commonly reflected in most of the works through the evaluation of multiuser interference (MUI) and signal to interference and noise ratio (SINR). Based on this, it is possible to find the critical $M$ needed to approach the capacity bounds as done by Caire or Hoydis around 2012 [HCPR12b, HTBD+13a]. At the time of our studies hence, we could find various contributions dealing with SINR maximization through resource allocation mechanisms taking benefit of the beamforming gains [BSHD13a, ZDPC13], or also looking into user clustering strategies and playing with spatial, frequency or even time dimensions (e.g. [JCS14]). Our complementary idea was to tackle the problem of inter-user interference following two aspects :

- Study the interference term when $M$ is moderately large, not to use strong assumption on channel hardening as done in many papers

- Investigate partial frequency or sub-band reuse between users to control the MUI term, i.e. allowing some adaptive frequency overlap between several user signals

Even if there is an abundant literature regarding SINR-based investigations, no analytical study had been provided on these aspects. As we were working on time-reversal, we proposed to combine time-reversal filtering with the partial frequency reuse concept as as subject of a new PhD program (Hua FU), that started in 2011 in a complementary way with the studies previously presented. This program was partially included in the framework of the ANR TRIMARAN project and with contractual studies with Orange Labs.

## 2.2 MU-MIMO ACHIEVABLE RATE IN THE MODERATE-SCALE ANTENNA REGIME

### ► Matched filters vs min-Interference filters

A first step in our studies was to extend the theoretical achievable rate results from the SU-MIMO case to the MU-MIMO case. When dealing with MUI, instead of using the matched filtering approaches studied before (TR, MRT, EGT), one may prefer null-steering strategies such as zero-forcing (ZF) [YM13b] or Regularized ZF [NE08]. ZF spatial filtering is a simple method to guarantee strict absence of the MUI terms by exciting orthogonal propagation modes of the channel. Using a classical flat fading matrix representation of the MU-MIMO system, the vector of the received symbols at each of the $K$ receivers $Y(n)$ for a particular subcarrier index $n$ can be stated as,

$$Y(n) = \underbrace{H(n)G(n)S(n) + N(n)}_{W(n)} \quad (7.34)$$

where $G(n)$ is the $M \times K$ spatial precoding matrix and $H(n)$ the MU-MIMO channel matrix of size $K \times M$. Using ZF precoding means forcing the equivalent channel matrix $W(n)$ to be an identity matrix. This is simply obtained imposing

$$G = \frac{1}{\eta} H^H (H^H)^{-1} \quad (7.35)$$

where $\eta$ is the power normalization factor of the filter. We understand that such a power normalization factor is directly influenced by the structure of the channel matrix, depending on the number of spatial links. Especially, if the channel matrix is ill-conditioned, the inversion will lead to a high value, and hence a dramatic degradation of the SNR. If the entries of channel matrix $H$ follow unit variance i.i.d complex Gaussian processes, it can be shown that the focusing gain $G_{ZF}$ of the ZF scheme tends to $\frac{M-K}{K}$ [WCDS12a]. This means that ZF precoding can not accommodate a two important increase of the number of users, unless the number of transmit antennas is sufficient, as in the large-scale antenna regime. This behaviour is illustrated in Fig. 7.13.
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

<figure><img src="image_56.png" /><figcaption>FIGURE 7.13– Comparative achievable rates MU-MIMO systems using ZF and MRT precoding schemes with M = 8 transmit antennas.</figcaption></figure>

From these considerations, and adding the complexity fact of matrix inversion, we chose to pursue our studies preferring matched-filtering approaches to deal with multi-user configurations. Then, the key aspect of the study was the evaluation of the impact of the MUI terms on the performance. We first led the analysis through the derivation of the achievable rate as explained in the sequel.

### ► Achievable rate analysis

The generalization of Eq. (7.20) to the case of a MU-MIMO system leads to the following expression,

$$ \mathcal{R}_{\text{MU-MIMO}} = \mathbb{E} \left\{ \frac{1}{N} \sum_{n=0}^{N-1} \log_2 \left( 1 + \frac{|\tilde{W}_k(n)|^2 \gamma}{\sum_{j \neq k} |\tilde{W}_{kj}(n)|^2 \gamma + 1} \right) \right\} \quad (7.36) $$

with the definitions of $\tilde{W}$ taken from the system model introduced in section 1.3. The derivation of this equation is not straightforward since the useful and interference terms may not be independent. Indeed equivalent channels $\tilde{W}_{kj}$ embed the channel fading coefficients of the spatial links also involved in $\tilde{W}_k$. A capacity lower bound can be derived as proposed in [JAMV11], based on the common worst-case of uncorrelated additive interference modeled as a complex Gaussian distribution of the same variance [HH03]. Even if such assumption can be accepted for asymptotically high M, it turns out to be very inaccurate in a moderate-scale antenna regime. The other common approach used by many authors since the introduction of the Massive MIMO concept is to invoke the channel hardening effect which occurs at large M [YM13b]. This roughly consists in stating that the interference and useful terms become somehow deterministic if $M \to \infty$. In that case, we get a much simpler expression for the achievable rate,

$$ \mathcal{R}_{\text{MU-MIMO}} \underset{M \gg 1}{\longrightarrow} \frac{1}{N} \sum_{n=0}^{N-1} \log_2 \left( 1 + \frac{\mathbb{E}\{\tilde{W}_k^2(n)\}\gamma}{\sum_{j \neq k} |\mathbb{E}\{\tilde{W}_{kj}^2(n)\}|\gamma + 1} \right) \quad (7.37) $$

which can also be almost viewed as the convergence of the achievable rate to its upper-bound based on Jensen's inequality. However, at moderate scale antenna regime, this strong assumption does not hold. One of our contributions was then to work out the initial achievable rate derivation of Eq. (7.37), i.e. without assuming channel hardening considerations. Our first investigations led us to study in details the distributions of the useful and interference terms up to the conclusion that these terms can eventually be modeled by two independent random processes. In other words, although the useful term and the interference term are both function of a
common random process, they can be considered as independent. This result essentially holds because of the uniformly distributed phase of the common random process. From this fact, the useful term can be modeled as proposed in Contrib. III.4. As for the interference terms, they do not follow a Gaussian distribution but rather a Gamma distribution. This first result is formally stated in the following contribution.

## Contribution III.10 (Analytic Expression)

### Approximated Average Achievable Rate of TR/MRT/EGT MU-MIMO Systems

Assuming independent normalized complex Gaussian fading per antenna link, the achievable rate of a MU-MIMO system with K users based on TR/MRT/EGT precoding is accurately approximated as

$$ \tilde{R} \approx \int_{X \in \mathbb{R}^+} \int_{Y \in \mathbb{R}^+} \log_2 \left( 1 + \frac{x^2}{y+1} \right) p_X(x) p_Y(y) dx dy \quad (7.38) $$

where X and Y are two independent random processes with X following the laws given in Contrib. III.4 for each spatial filter, and $Y \sim \text{Gamma}(1, K - 1)$.

It is interesting to note that the distribution of the interference term in Contrib. III.10 is not changing from one spatial filter scheme to another, whereas the distribution of the useful term does (according to already established results in Contribution II.8). As an example in Fig. 7.14, it is verified that the proposed random processes provide a tight model for the achievable rate computation in the TR precoding case. The model is valid whatever the number of antennas and for all SNR range. From the above contribution, it is even possible to go a step further in the EGT and MRT cases, for which it is possible to gather the two random processes X and Y into a single one which directly models the SINR, as proposed hereafter.

## Contribution III.11 (Analytic Expression)

### Approximated Average Achievable Rate of MRT/EGT MU-MIMO Systems

Assuming independent normalized complex Gaussian fading per antenna link, the achievable rate of a MU-MIMO system with K users based on MRT and EGT precoding is accurately approximated as

$$ \tilde{R} \approx \int_{Z \in \mathbb{R}^+} \log_2(1+z) p_Z(z) dz \quad (7.39) $$

where Z is a random processes with PDF,

$$ p_Z(z) = \frac{\theta e^{\frac{1}{(K-1)\gamma}}}{(K-1)(M-1)!} \frac{z^{M-1}}{\left(z + \frac{\theta}{K-1}\right)^{M+1}} \Gamma\left(M+1, \frac{1}{\gamma}\left(\frac{1}{\theta} + \frac{1}{(K-1)}\right)\right) \quad (7.40) $$

with $\theta = 1$ for MRT and $\theta = \frac{1+\frac{\pi}{4}(M-1)}{M}$ for EGT, and $\Gamma(x, s)$ the upper incomplete Gamma function.

However, it turns out that no closed-form solutions to the ergodic capacity can be established due to the high complexity of the statistical law. In the literature, one can find closed-forms only in the massive MIMO regime [HCPR12b]. Nonetheless it is possible to get some lower and upper bounds as in the SU-MIMO case, without invoking the asymptotic regime. These bounds have partly been proposed in [C.20] and are still being worked out for future publications. As an example, the lower bounds are summarized in the following contribution.
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

<figure><img src="image_57.png" /><figcaption>FIGURE 7.14– Exact and approximated achievable rate of the TR MU-MIMO system. Approximation is obtained from the random processes proposed in Contrib. III.10.</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>FIGURE 7.15– Achievable rate of the MRT MU-MIMO OFDM system (Simulation, Approximation, Lower-bound at high SNR, and Upper-bound at high SINR).</figcaption></figure>

### Contribution III.12 (Analytic Expression)

#### Lower Bounds of the Average Achievable Rate of MRT/EGT MU-MIMO Systems

Assuming independent normalized complex Gaussian fading per antenna link, the lower bounds of the achievable rate at high SINR regime of a MU-MIMO system with K users based on TR/MRT/EGT precoding are

$$R_{\text{MU-MIMO}}^{\text{low}} = R_{\text{SU-MIMO}}^{\text{low}} + \frac{e^{\frac{1}{(K-1)\gamma}}}{\log 2} \text{Ei}\left(-\frac{1}{(K-1)\gamma}\right) \quad (7.41)$$

with $R_{\text{MU-MIMO}}^{\text{low}}$ given in Contrib. III.6 for TR/EGT/MRT, and $\text{Ei}(x)$ the exponential integral function.

Corresponding simulation results are given in Fig. 7.15 in the case of the MRT filters. It appears that the obtained bounds are less accurate in the MU case than in the SU case, unless M or SNR is sufficiently large. Hence, the derivation of closed-forms or approximated closed-forms for the theoretical achievable rate in MU-MIMO systems without particular assumption on the antenna scale is still an open problem. Since the end of our work in 2015 on this subject, massive MIMO concepts have been so largely accepted that the new contributions about average achievable rate have always been considering the channel hardening effect as a basic assumption, thereby not considering the case when M remains moderate.

## 2.3 BER PERFORMANCE WITH PARTIAL BANDWIDTH OVERLAP

After our work on average achievable rate, we have been interested in practical performance using conventional modulation constellations. As already discussed at the beginning of section 2.2, using MRT/TR/EGT filters is expected to lead to potentially strong MAI terms. Instead of reducing such MAI level by means of orthogonalization methods as with ZF and thus losing at the level of the focusing gain, we have proposed some partial frequency reuse mechanisms to control the MUI as detailed hereafter. The reason for excluding ZF of the scope of our study was also because we were focusing on system designs adapted to decentralized scenarios, i.e.
<figure><img src="image_59.png" /><figcaption>FIGURE 7.16– Simple illustration of the two-user partial frequency band overlapping (PFBO) scheme.</figcaption></figure>

in the case of cooperative spatial filtering between distant stations. For the same reason, MRT precoding could not be considered in our scenario. Hence only EGT and TR are the selected filters for the rest of the study.

## Partial frequency bandwith overlapping system model

From the system model introduced in 1.3, spatial matched filters such as TR and EGT are not initially designed to cancel the inter-user interference in MU-MIMO systems. The interference term is only asymptotically zeroed or can at least be neglected compared to noise thanks to the tight focusing effect at large M. In a moderate-scale antenna regime however, the system performance is expected to be directly resulting from the SINR level due to imperfect spatial focusing. The idea behind the proposed partial frequency reuse scheme is to control such SINR level by constraining the inter-user interference to appear in some frequency subband only. Accordingly, instead of reusing the whole available bandwidth for each user as in the classical MU-MIMO approach, two particular users would only share a common subband, or a common subset of subcarriers in the multicarrier case. The principle is depicted in Fig. 7.16 for a two-user scenario. Applying the sub-band reuse concept to the system model in 1.3 leads to the following received signal for one particular user j considered as the user of interest,

$$Y_j = \frac{1}{\sqrt{K}} \cdot \underbrace{\tilde{W}_j \cdot S_j}_{\text{useful term}} + \frac{1}{\sqrt{K}} \cdot \underbrace{\sum_{k \neq j} \tilde{W}_{kj} \cdot B_k \cdot S_k}_{\text{Interference term}} + N_j \quad (7.42)$$

in which we introduce the diagonal sub-band assignment matrices $B_k$ which diagonal entries are $B_k(n) = 1$ if the nth subcarrier is assigned to user k, and 0 otherwise. Accordingly, we may define the frequency overlap ratio of the system as,

$$\tau = \frac{1}{K} \sum_{k=1}^{K} \tau_k \quad \text{with, } \tau_k = \frac{1}{N} \text{Tr} \left( \sum_{j \neq k} B_j \right) \quad (7.43)$$

with $\tau_k$ is the overlap ratio of user k. An overlap ratio $\tau_k$ means that user k shares $\tau_k$ % of its subcarriers with others users$^3$. If all assignment matrices $B_k$ are identity, the overlap ratio reaches K × 100% and the MAI term is maximized. This latter case is the typical situation of conventional MU-MIMO systems.

From a general perspective, the optimal setting of such a system is highly complex, since it can be viewed as a mixed integer programming problem, albeit some greedy or iterative solutions can be found. However, our aim was rather to evaluate the system BER degradation regarding the overlap ratio and then understand how such a degree of freedom can be exploited regarding some quality of service constraints. Indeed, any sub-band assignment algorithm would need some evaluation of the performance degradation to decide how to share the spectrum between several users. SINR based measurements could be used in a first approach, but we will see that more precise performance estimation can be obtained.

Eventually, the underlying issue to address is the evaluation of the BER of a MU-MIMO system with a variable number of users, a variable number of antennas and using a finite symbol alphabet according to a particular constellation. Under the massive MIMO assumption again, the answer is almost straightforward since

<footnote n="3">3. Note that random assignment matrices could also be used to target a particular overlap ratio without changing the rest of the study.</footnote>
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

the interference terms can be rationally approximated as extra additive Gaussian noise [R.4]. Yet, in the moderate-scale antenna regime, further investigations are needed as developed during the work of H. Fu and summarized in the sequel.

### ► BSPK Constellation with two competing users

The simplest model to start with is the case of Binary Phase Shift Keying (BPSK) modulation with only two users assigned to a same sub-band. We derived the theoretical BER of such a situation, when EGT filtering is used<fnref n="4" />. Beyond the obtained BER closed-form, the developed analysis gives new insights into how fading effects and MUI terms behave in a MU-MIMO system with not so large M values. Indeed, on the subject of interfering systems, one can find some problems with close formulations in the literature, for example about cochannel interference in cellular systems [BC04] or in jamming situations [LAZ07]. However, in the papers dealing with theoretical derivations (and not simulations) interfering signals are mostly considered as Gaussian process. In multiuser MIMO systems however, the MUI term experiences the same channel fading as the useful signal, which introduces correlation in the received signal and makes the Gaussian approximation inappropriate. During the PhD of H. Fu, we experimented a new way of modeling the interfering signal as directly taking part of the fading effect of the channel. We proposed our method and published our results in [R.4]. The main points are summarized below.

Let us first give the simplified expression of the received signal $y^{(1)}$ (for the user of interest) as,

$$y^{(1)} = \frac{1}{\sqrt{M}} \sum_{m=1}^{M} |H_{m}^{(1)}| x^{(1)} + \frac{1}{\sqrt{M}} \sum_{m=1}^{M} |H_{m}^{(1)}| e^{j\theta_{m}} X^{(2)} + n \quad (7.44)$$

with $\theta_m$ a phase rotation due to EGT. This model involves the desired symbol $x^{(1)}$ and the interfering symbol $x^{(2)}$. Since $x$ belong to a finite alphabet of reduced dimension common models based on the classical Gaussian approximation of the interfering term or on other RV assumed independent of the useful term are likely to yield poor BER estimates [JAMV11]. One may rather try to get the joint p.d.f. for the useful and interference terms. To circumvent such a hard task, we rather proposed in our study to treat the MUI term as an additional distortion factor applied to the useful term. This is obtained letting $x^{(2)} = \rho e^{j\phi} x^{(1)}$, with $\rho$ and $\phi$ depending on the constellation point coordinates. This leads to the modified model,

$$y^{(1)} = \frac{1}{\sqrt{M}} \sum_{m=1}^{M} |H_{m}^{(1)}| (1 + \rho e^{j\psi_{m}}) x^{(1)} + n \quad (7.45)$$

where $\psi_m = \theta_m + \phi$ such that $\psi_m \sim \text{Unif}(0, 2\pi)$. This expression highlights the fact that the system can be viewed as a particular fading model governed by the complex weighting factor $\xi$. Our idea was to find some statistical model to $\xi$ and then derive the average BER. In the simple case of BPSK constellation, finding the convenient PDF of $\xi$ is tractable under some reasonable assumptions and approximations. The first result we obtained, considering BSPK and two users, is stated in the following Contrib. III.14 and detailed in [R.4].

It is noticed that the statistic law that is obtained in Eq. (7.46) of Contrib. III.14 is the same as that of the EGT SU-MIMO system (see Contrib. III.4), with however different parameter expressions. The proposed model is controlled by parameters $\tilde{m}$ and $\tilde{\omega}$ which can be simplified if M is sufficiently high, in practice as soon as $M > 8$. The former reveals the intrinsic (inverse) fading amplitude of the MISO Rayleigh channel, while the latter represents a kind of “overfading” effect due to interference. The equivalent channel can hence be viewed as a global fading model for the interfered Rayleigh MISO channel.

In Fig. 7.17, the typical scatter plot obtained at the receiver for one of the two possible BPSK symbols is represented. As evident from the shape of the cloud of points, the interference could not been considered as Gaussian additive. In Fig. 7.18, the comparison between the empirical PDF and the theoretical Nakagami PDF is given, confirming the accuracy of the proposed model for various M.

<footnote n="4">4. It turned out that TR precoding led to much more complex derivations which were in most of the case even not tractable.</footnote>
<figure><img src="image_60.png" /><figcaption>FIGURE 7.17– Received BPSK scatter plot without noise with EGT precoding and two competing two users.</figcaption></figure>

<figure><img src="image_61.png" /><figcaption>FIGURE 7.18– PDF of the amplitude of the received In-phase component with BPSK, EGT precoding and two competing users.</figcaption></figure>

### Contribution III.13 (Analytic Expression)

#### Nakagami equivalent fading model of a two-user BPSK EGT MU-MIMO System

Assuming independent normalized complex Gaussian fading per antenna link, the equivalent fading model of a two-user BPSK EGT MU-MIMO transmission can be approximated as,

$$y = \xi x + n \quad \text{where, } \xi \sim \text{Nakagami}(\tilde{\mu}, \tilde{\omega}) \qquad (7.46)$$

with $n$ the Gaussian noise sample, and $\xi$ an equivalent Nakagami fading model such that,

$$\tilde{\mu} \left[ \frac{\Gamma(\tilde{\mu})}{\Gamma(\tilde{\mu} + \frac{1}{2})} \right]^2 = 1 + \frac{\frac{6}{\pi} - 1}{M} \quad \text{and,} \quad \tilde{\mu} \xrightarrow{M \gg 1} \tilde{\mu} = \frac{\pi}{24 - 4\pi} M + \frac{1}{8} \qquad (7.47)$$

$$\tilde{\omega} = \frac{\pi(M-1) + 6}{\pi(M-1) + 4} \quad \text{and,} \quad \tilde{\omega} \xrightarrow{M \gg 1} \tilde{\omega} = 1 \qquad (7.48)$$

### ► Extension to QSPK Constellation and to multiple users

From Eq. (7.45), it is possible to apply the proposed method to the QPSK case and it can be shown that the Nakagami distribution can be used as a good equivalent fading model again. However, as illustrated by the scatter plot of Fig. 7.19, the cloud of points is crossing the real and imaginary axes, which translates into a PDF with non-zero values for $x < 0$. Yet, the Nakagami distribution is strictly positive definite. As the global shape of the PDF seems convenient, we proposed to modify the Nakagami distribution into a so-called translated Nakagami distribution. The shifted version of the PDF is such that it can be defined for positive values only. Many values of translation $\Delta$ may be used in practice. A convenient choice, obtained through extensive simulations, is to take $\Delta = 1$. Finally, extension to higher order modulations, e.g. 16QAM, becomes harder to study, albeit possible. Hopefully, the need to treat such cases is questionable since the Gaussian approximation becomes progressively acceptable as the order the constellation increases.

It is also possible to extend our model to the multiple user case, i.e. for more than two users. In such a case, the translated Nakagami solution turns out to be a good approximation to the equivalent fading channel as illustrated in Fig. 7.20 where a translation factor of $\Delta = 1$ is used to make the Nakagami PDF be defined over $[-1, +\infty)$. The proposed translated Nakagami model is summarized in the following contribution. A journal paper is in preparation for the unified approach exposed here.
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

<figure><img src="image_62.png" /><figcaption>FIGURE 7.19– Received QPSK scatter plot with EGT precoding with two competing two users, and empirical PDF of the amplitude on the real and imaginary parts.</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>FIGURE 7.20– PDF of the amplitude of the received In-phase component with BPSK, EGT precoding and 3 competing users.</figcaption></figure>

### Contribution III.14 (Analytic Expression)

#### Translated-Nakagami equivalent fading model of a K-user BPSK EGT MU-MIMO System

Assuming independent normalized complex Gaussian fading per antenna link, the equivalent fading model of a K-user QPSK EGT MU-MIMO transmission can be approximated as,

$$y = v_{I}x_{I} + jv_{Q}x_{Q} + \Delta + n \quad \text{where, } v_{I,Q} \sim \text{Nakagami}(\tilde{\mu}, \tilde{\omega}) \quad (7.49)$$

with n the Gaussian noise sample, and $v_{I,Q}$ two independent Nakagami processes defined as,

$$\tilde{\mu} = \frac{\alpha \pi M + \Delta^2 (4 - \pi)}{8(K + 1) - 4\pi} + \frac{1}{8} \quad \text{with, } \alpha = 1 + 2\Delta^2 + 2\Delta \sqrt{\frac{\pi(M - 1) + 4}{\pi M}} \quad (7.50)$$

$$\tilde{\omega} = \frac{\pi(M - 1) + 2(K + 1)}{\pi(M - 1) + 4} + 2\Delta \sqrt{\frac{\pi M}{\pi(M - 1) + 4}} + \Delta^2 \quad (7.51)$$

in which $\Delta$ is a translation factor adjusted through simulations.
# Chapitre 7. Optimization of the Wireless Broadband Access

<figure><img src="image_64.png" /><figcaption>**FIGURE 7.21-** Average BER of the EGT MU-MIMO system with two competing users using BPSK modulation and 100% bandwidth overlap.</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>**FIGURE 7.22-** Average BER of the EGT MU-MIMO system with two competing users using BPSK modulation and variable overlap ratios $\tau$.</figcaption></figure>

## Simulation results

From the developed statistical models, it is then possible to derive some closed-forms of the average BER. As an example, using the parameters $\tilde{\mu}$ and $\tilde{\omega}$ of the equivalent Nakagami fading channel, the closed-form obtained with BPSK/QPSK writes,

$$P_e(\gamma_b) = \frac{\Gamma_B(\tilde{\mu} + \frac{1}{2})}{\Gamma_B(\tilde{\mu} + 1)} \cdot \frac{[\frac{\tilde{\mu}}{\tilde{\omega}}]^{\tilde{\mu}} \gamma_b^{\frac{1}{2}}}{[\gamma_b + \frac{\tilde{\mu}}{\tilde{\omega}}]^{\tilde{\mu} + \frac{1}{2}}} \times {}_2F_1\left(1, \tilde{\mu} + \frac{1}{2}, \tilde{\mu} + 1, \frac{\tilde{\mu}}{\tilde{\omega}}\right) \quad (7.52)$$

where $\gamma_b = \frac{\sigma_e^2}{N_0}$. This last result gives us a formula to predict the BER of a MU-MIMO system with an arbitrary number of antennas and an arbitrary number of users utilizing the same sub-band. Fig. 7.21 gives an example of the obtained results with BPSK modulation for the two-user case. Comparison is made between the proposed Nakagami model and the common Gaussian approximation. It is clearly concluded that the proposed model is much more accurate in predicting the BER than the Gaussian approach.

Finally, it is possible to run performance simulations of the proposed partial frequency overlap system. Depending on the overlap ratio $\tau$, the global BER is obtained through a weighted average between the BER in the single user case for the non-overlapped sub-bands and in the multi-user case when the sub-band is used by more than one user. Hence, the system BER is computed using both the theoretical BER formula obtained in SU-MIMO case (see Eq. 7.30) and the above formula for the MU-MIMO case (Eq. (7.52)). In Fig. 7.22, an example of BPSK BER curves is given for various overlap ratios, in the case of two competing users with M = 4 antennas. It is noticed that the Gaussian approximation becomes acceptable from M = 32 antennas and almost equivalent to our model for M = 64 antennas. This is due to the Nakagami distribution which tends to a Gaussian one as the scale parameter $\mu \to \infty$. Hence, the proposed Nakagami model can be viewed as a general statistical model for MU-MIMO interfering channels, also valid for low M.

Finally, in Fig. 7.23, a synthetic performance chart is given allowing to identify which overlap ratio can be tolerated for a target BER performance of $5 \times 10^{-3}$ depending on the number of available antennas. We clearly see the progressive beneficial effect of increasing the number of antennas. Using our closed-forms, it is then possible to obtain such kind of system level analysis for any number of antennas and not solely in the massive MIMO regime.
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

<figure><img src="image_66.png" /><figcaption>FIGURE 7.23– Maximal overlap ratio $\tau$ for a target BER of $5 \times 10^{-3}$ and for variable number M of antennas.</figcaption></figure>

## 2.4 EXTENSION TO SS-OFDM WAVEFORM

In the same general study on MU-MIMO systems with partial sub-band overlap, we also try to study the influence of a spread-spectrum (SS) component added to the OFDM waveform. As SS-OFDM is known to be robust to narrow-band interference, the MU-MIMO SS-OFDM system with sub-band overlap is expected to outperform the OFDM system, at least in some situations that have to be identified. Following the same approach as previously, we analyzed the BER performance of the SS-OFDM system based on an inter-channel interference model and then compared the robustness of the SS-OFDM and OFDM waveforms to the overlap ratio, depending on the number of antennas.

### ► SS-OFDM system model

The combination of SS with OFDM, yielding so-called linear-precoded OFDM (LP-OFDM) [Kai02], has been widely studied and demonstrated to offer an additional multipath diversity gain to conventional OFDM [DJ98, Pop99]. In addition, spread-spectrum techniques are well-known to increase the resistance of a system to the interference and jamming. Hence, in our context of sub-band reuse, a competing user can be viewed as a possibly narrow-band interference which negative impact could be advantageously mitigated using spreading sequences. SS can easily be associated to OFDM by applying the spreading sequences to the symbols conveyed by the OFDM subcarriers, in the form of a linear precoding function applied to the frequency domain symbol vector before OFDM modulation through DFT. To accommodate various transmission constraints, many spreading code families have been proposed in the literature [Gol67]. As we are in a synchronous communication system, we have been using the classical orthogonal Walsh-Hadamard spreading sequences [TZ06]. Formally, the transmitted signal, including spatial filtering can be stated as,

$$Y_j = \frac{1}{\sqrt{K}} \underbrace{\tilde{W}_j PS_j}_{\text{useful term}} + \frac{1}{\sqrt{K}} \underbrace{\sum_{k \neq j} \tilde{W}_{kj} B_k PS_k}_{\text{Interference term}} + N_j \quad (7.53)$$

with the spreading code matrix P introduced to model the SS-OFDM waveform in the form of a linear precoding. Note that the same spreading matrix is used for all the users. However, looking at the interference term, it is observed that the sub-band allocation matrix $B_k$ acts as a puncturing process applied to the spreading matrix P of the interfering users. Hence, unless a 100% overlap ratio is used, the only user who receives the totality of the code chips is the user of interest $j$. Therefore, a partial frequency overlap translates into a partial sequence overlap, as simply illustrated in Fig. 7.24 in a two-user scenario. It is finally noticed that each user in a full-load
# Chapitre 7. Optimization of the Wireless Broadband Access

<figure><img src="image_67.png" /><figcaption>FIGURE 7.24– Illustration of a two-user partial frequency overlap model with SS-OFDM waveform.</figcaption></figure>

SS-OFDM situation, i.e. each user makes use of all the available sequences to spread its own data symbols over the spectrum. This means that the spreading sequences are not used for code division multiple access (CDMA) but rather to whiten the interference terms.

At the receiver side, despreading has to be process to sum up all the code chips of all sequences hence retrieving the data symbols. Due to the fading channel effects, the orthogonality (or pseudo-orthogonality) between the spreading sequences is lost. In practice, it is then convenient to equalize the receive frequency domain signals before despreading. It is well-known that the MMSE equalizer gives the best results in terms of BER performance [HLGHB01]. In such a case, the output symbol vector at the receiver side, after equalization and despreading can be expressed and decomposed as,

$$ 
\begin{aligned}
\hat{S}_j &= \frac{1}{\sqrt{K}} P^H E_j \tilde{W}_j P S_j + \frac{1}{\sqrt{K}} \sum_{k \neq j} P^H E_j \tilde{W}_{kj} B_k P S_k + \tilde{N}_j \\
&= Z_j S_j + \underbrace{I}_{\text{ICI}} + \underbrace{J(\tau) + \tilde{N}_j}_{\text{MAI}}
\end{aligned}
\quad (7.54) $$

where matrix $E_j$ accounts for equalization at the jth receiver. Compared to the OFDM case, an additional Inter-code Interference term (ICI) has to be considered owing to the imperfect orthogonality among the spreading sequences after MMSE equalization.

## ► Approximate BER

The general study of each of these terms have to be led in the perspective of some BER theoretical derivation. In the literature, some coarse estimation of the performance degradation in case of partial sub-band reuse can be found, defining the so-called jammer-to-signal ratio [PKH+03, LAZ07]. As it will be highlighted later on, this approach is far from being accurate in the BER prediction. Paying particular attention to our system model, it is noticed that each signal component results from an averaging over all the subcarriers owing to the spreading function. In this way, the coloured narrow band interference responsible of the MAI term turns out to be transformed into a white noise over the whole bandwidth. This common assumption can be verified in our case and is shown to be a sufficient approximation at high SNR range, (typically above 10 dB), and for a high, albeit moderated, number of antennas (typically $M > 8$). Similar conclusion can be drawn for the ICI term, hence leading to an homogeneous expression of the received symbols, whatever the subcarrier index $n$, as

$$ \hat{S}(n) = \rho S(n) + \omega_{\text{ICI}} + \omega_{\text{MAI}} + \omega_{\text{N}} \quad (7.55) $$

where $\omega_{\text{ICI}}$, $\omega_{\text{MAI}}$ and $\omega_{\text{N}}$ follow independent complex Gaussian processes, and $\rho$ is a real random variable driving the average fading due to the combined effects of spreading/despreading, spatial filtering, channel propagation and equalization. From this fact, we have finally showed that a good approximation of the BER could be obtained simply taking into account second order statistics of the random processes, as summarized in the following contribution. Note that the result remains valid for other fading models than the theoretical complex Gaussian model. This means that the proposed approximate BER formula can be used for any channel model, as soon as the second order statistics of the involved random variable can be extracted. Some further results at this level have been published in [C.27].
## 2. Spatial Filtering for MU-MIMO in Moderate-Scale Antenna regime

### Contribution III.15 (Analytic Expression)

#### Approximated Average BER for a MU-MIMO SS-OFDM System

Assuming independent fading per antenna link, the average BER performance of a MU-MIMO SS-OFDM system using EGT/MRT/TR precoding can be approximated through,

$$ BER \approx Q \left( \sqrt{\frac{\mathbb{E}[|\rho|^2] \sigma_s^2}{\mathbb{E}[|\omega_{ICI}|^2] + \mathbb{E}[|\omega_{MAI}|^2] + \mathbb{E}[|\omega_N|^2]}} \right) \quad (7.56) $$

with the variance of $\rho$, $\omega_{ICI}$, $\omega_{MAI}$ and $\omega_N$ specifically computed from the system model of Eq. (7.54) taking into account the precoding scheme and the channel model.

### ► Performance results

Using the proposed approximate BER formula, we give in Fig. 7.25 the BER curves obtained with SS-OFDM over BRAN-A channels [Med98]. A comparison is made between the simulated results, the theoretical results extracted from our approximation and the results obtained applying the partial-jamming approach found in literature, e.g. in [LAZ07]. The given curves represent various settings of the overlap ratio, from $\tau = 1\%$ to 100%, with a number of antennas set to $M = 4$. It is clearly concluded that the proposed strategy gives the best prediction of the BER, namely concerning the level of the error floor at high SNR.

In addition, using the BER expressions obtained with OFDM and SS-OFDM systems, it is also possible to lead comparative simulations in the context of MU-MIMO transmissions with partial sub-band reuse. In particular, it is interesting to study in what extend SS-OFDM can be more robust to MAI and tolerate a higher overlap ratio compared to OFDM. As already discussed, SS-OFDM is expected to offer better immunity in a moderate-scale antenna regime. First results showing the superiority of the SS-OFDM waveform are given in Fig. 7.26 in the case of $M = 10$ when the overlap ratio $\tau$ varies from 1% to 12%. The robustness of the SS-OFDM waveform over the OFDM waveform is clearly observed, since the former can better accommodate an increasing overlap ratio than the latter. However, it can be understood that the resistance of SS-OFDM to the interference starts failing from a given overlap ratio value. In addition, as SS-OFDM also suffers some ICI, its performance may even become worse than that offered by OFDM. To highlight this, Fig. 7.27 gives the maximum overlap ratio tolerated by each waveform for a target BER depending on the number of antennas. Interestingly, the SS-OFDM waveform maintains its advantage if $M < 10$, that is in the moderate-scale antenna regime.

<figure><img src="image_68.png" /><figcaption>FIGURE 7.25– Average BER of the EGT MU-MIMO system based on SS-OFDM. QPSK modulation, $M = 4$ antennas, two competing users with variable overlap ratio $\tau$.</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>FIGURE 7.26– Comparison of average BER between OFDM and SS-OFDM in a EGT MU-MIMO system. QPSK modulation, $M = 8$ antennas, two competing users with variable overlap ratio $\tau$.</figcaption></figure>
<figure><img src="image_70.png" /><figcaption>FIGURE 7.27– Achievable bandwidth reuse rate versus the number M of antennas at target BER=$10^{-3}$.</figcaption></figure>

## 2.5 CONCLUSION

In this work, we have chosen to investigate the MU-MIMO system performance under the constraint of a moderate number of antennas at the central station, typically between 8 and 16. We did definitely not assume in our work any asymptotic regime of the MU-MIMO system, hence trying to tackle the problem of achievable rate and BER theoretical derivations with more general perspective. This has allowed us to bring some complementary insights in the very fertile period that had started since the introduction of the massive MIMO paradigm.

Our approach has been based on a thorough statistical analysis and modeling of the so-called equivalent channel resulting from the combined effect of spatial filtering and wireless propagation. As a result, we have been able to highlight the progressive focusing effects of matched-type spatial filters when increasing the number of cooperative antennas, providing some new analytical expressions of the achievable rate and BER. However, most of the closed-forms were obtained with MRT and EGT filters, the TR filter generally leading to much more complex expressions. Therefore, closed-form expressions for TR-based MU-MIMO systems are still to be solved.

On one other hand, most of our derivations have been based upon i.i.d complex Gaussian MIMO channel models. Even if this strong assumption allows for general system behaviour prediction, it may be far from describing the reality of the propagation conditions for practical systems. Nonetheless, the developed methodology based on statistical equivalent channel models may be easily extended to any propagation characteristics. Such analyses may even become essential since large-scale MIMO systems are expected to be deployed in mmWave bands, where i.i.d complex Gaussian models lack consistency.

## Summary (PhD/Dissemination/Projects)

- 1 PhD (H. Fu)

- 1 international journal paper [R.4]

- 3 communications in international conferences and workshops [C.20][C.27][C.37]

- 1 GDR workshop

- 2 Tutorial in conferences (APWiMob'15, Softcom'15)
# 3. Practical Beamforming Strategy for mmWave Communications

## 3

### PRACTICAL BEAMFORMING STRATEGY FOR MMWAVE COMMUNICATIONS

#### 3.1

##### MOTIVATIONS

The studies led during the PhD of Hua FU and summarized in the previous section were essentially focused on theoretical aspects based upon the common i.i.d. complex Gaussian model of the spatial links. The obtained results have a general scope and allows to understand the main trends in terms of MU-MIMO system performance, namely without particular assumption on the antenna array size. Yet, the translation of such results in practical propagation environments remains questionable, which encouraged us to start complementary studies with a more system-level view.

Four years after our first investigations about time-reversal and spatial filters, the concept of massive MIMO was already initiating deep changes in the way of envisaging the evolution of future communication systems [LETM13]. Firstly, the scientific community enthusiastically took up this new subject which was offering numerous new optimization opportunities. In parallel, industrials involved in the strategic reflexions of the future generation of wireless networks (5G) were starting considering Massive MIMO techniques as a key enabler to support higher achievable data rates [KPR12] [NNB<sup>+13</sup>]. A general trend that was also progressing at that period was the natural evolution towards higher frequency bands, namely the so-called mmWave usage. People were hence understanding that a joint use of millimeter waves and massive MIMO schemes could be viewed as a win-win scenario [EAHAS<sup>+12</sup>]. Indeed, MmWave regime enabled antenna arrays to be deployed with reasonable array form factors at such frequencies, which makes the large-scale antenna assumption become more conceivable.

Last but not least, network densification was a general idea which was also starting to be seriously considered as an additional degree of freedom for future wireless communication networks to achieve better coverage and higher data rates [BLM<sup>+14a</sup>] [JMZ<sup>+14</sup>]. Densely deploying small cells with high data rate requirements and small per cell coverage area encourages using new solutions adapted to Line-of-Sight (LOS) channels which are indeed favourable well adapted to mmWave frequencies [KPR12].

At that period started the IRT (Institute for Research and Technology) initiatives in France, with namely in Rennes IRT Bcom which was launched in 2013. I was involved in this new research structure to take part in new prospective works, and in particular in Massive MIMO studies. We then had the opportunity to launch a new PhD program on the topic of large-scale antenna systems in mmWaves. Our aim was to investigate the specificities of mmWave propagation conditions in the perpespective of designing adequate spatial filters. The results discussed hereafter have been provided in the framework of the RS-COM project within Bcom, under ANR and regional financial supports.

#### 3.2

##### SYSTEM MODEL WITH GEOMETRIC-BASED CHANNEL DESCRIPTION

► Channel angular spectrum

The basic system model considered in this work is similar to that of the previous sections, except that the used channel matrix is of particular structure resulting from the specificities of the propagation channel in mmW. More precisely, as the carrier frequency increases, it can be shown that the propagation channel becomes sparser in the sense that less propagation rays can survive than in lower frequency bands due to higher losses when waves interact with obstacles. As a consequence, the channel matrix is mainly governed by the specular components of the propagation environment, corresponding to the most significative propagation rays. Adding the fact that off course the higher the operating frequency, the more accurate the ray-optics approximations, the MU-MIMO channel matrix can accurately be obtained from a pure geometrical representation of the propagation environment. From such a geometrical approach, a straightforward correlation between the antenna links is naturally obtained, giving more consistency to the MIMO channel model than using i.i.d. random fading processes.

To go further with this idea it becomes possible to provide an angular visualization of the channel as proposed in Fig. 7.28. The propagation channel can indeed be described through its angular spectrum, i.e. the collection
# Chapitre 7. Optimization of the Wireless Broadband Access

<figure><img src="image_71.png" /><figcaption>FIGURE 7.28- Representation of the MIMO channel from its angular spectrum.</figcaption></figure>

of complex components associated to angles of arrival (or departure) of the impinging rays, in the horizontal (azimuth) and vertical (elevation) planes. Using an infinite size antenna array, each ray can be perfectly isolated and discriminated and the channel can be represented by a sum of specular components (angle-based Dirac delta functions). Hence, the specular angular spectrum can be stated as,

$$S(\theta, \varphi) = \sum_{i} \sum_{j} \sqrt{\beta_{ij}} e^{j\Phi_{ij}} \delta(\theta - \theta_{i}, \varphi - \varphi_{j}) \quad (7.57)$$

with $\theta$ and $\varphi$ the azimuth and elevation angles, respectively.

Yet, observing the propagation environment through an antenna array of limited size translates into some spatial filtering effect of the angular spectrum. Each impinging ray is then viewed through the antenna array angular response, commonly referred to as array factor $F(\theta, \varphi)$. Therefore, the angular spectrum taking into account the finite antenna array size writes,

$$S(\theta, \varphi) = \sum_{i} \sum_{j} \sqrt{\beta_{ij}} e^{j\Phi_{ij}} F(\theta - \theta_{i}, \varphi - \varphi_{j}) \quad (7.58)$$

The well-known examples of linear or planar arrays yield Dirichlet functions for the array factor expressions [Chr00].

## ► Geometrical MIMO Channel Representation

From the above considerations, it is then possible to imagine a ray-based MIMO channel matrix description taking into account the angles of arrival of the propagation paths and the geometrical structure of the antenna array. Indeed, from the antenna array geometry it is possible to define the so-called steering vectors representing the phase deviations which occurs at each radiating element of the antenna array when a plane wave impinges with a particular angle of arrival. For a planar antenna array made of $M = M_H \times M_V$ horizontal and vertical elements as represented in Fig. 7.30 for example, such steering vector states as,

$$\mathbf{a}_{\theta, \varphi} = [1, e^{j\nu}, \dots, e^{j(M_H - 1)\nu}]^T \otimes [1, e^{j\nu}, \dots, e^{j(M_V - 1)\nu}]^T \quad (7.59)$$

with $\otimes$ the kronecker product, and

$$u = \frac{2\pi}{\lambda} d_V \sin(\varphi) \quad v = \frac{2\pi}{\lambda} d_H \sin(\theta) \cos(\varphi) \quad (7.60)$$

where $\theta$ and $\varphi$ are the azimuth and elevation angles of the considered impinging ray (see Fig. 7.30), while $d_V$ and $d_H$ represent the distance between two vertical antennas and between two horizontal antennas of the planar antenna array. These expressions are easily found from the phase shifts at each antenna of the array, as illustrated in Fig. 7.29.

Then, considering the MISO channel between the antenna array and a single antenna terminal, each geometrical path between both ends yields one impinging ray to the antenna array which itself translates into one
## 3. Practical Beamforming Strategy for mmWave Communications

<figure><img src="image_72.png" /><figcaption>FIGURE 7.29– Relative phase shifts experienced at each antenna of a linear array (vertical and horizontal).</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>FIGURE 7.30– Plannar antenna array with azimuth and elevation angles of arrival.</figcaption></figure>

steering vector with the corresponding angle of arrival. Gathering all the geometrical paths for one terminal user $k$ thus leads to the following channel vector,

$$h_k = \sum_{l=1}^{L_k} \sqrt{\beta_{k,l}} e^{j\phi_{k,l}} a_{k,l} \quad (7.61)$$

with $a_{k,l}$ the steering vector corresponding to angle of arrival $\theta_{k,l}$, $\phi_{k,l}$. From such a model, the MU-MIMO channel matrix with K users and M antennas at the central station finally writes,

$$H = \sum_{l=1}^{L} D_{l}^{1/2} \Phi_{l} A_{l} \quad (7.62)$$

where $A_l = [a_{1,l}, a_{2,l}, ..., a_{K,l}]^T \in \mathbb{C}^{K \times M}$ is the steering matrix of path $l$ and $D_l \in \mathbb{C}^{K \times K}$ (resp. $\Psi_l$) is the diagonal complex matrix describing path attenuation (resp. phase shifts) with their $k$th diagonal entries calculated as :

$$(D_l^{1/2})_k = \sqrt{\beta_{k,l}} \quad (\Psi_l)_k = e^{j\phi_{k,l}} \quad \forall k \in [1, K] \quad (7.63)$$

As in a mm-wave scenario, the propagation environment can be described from its macro-structure by a few geometrical rays, it is essential to make use of such channel model based on steering vectors. Differenciating between the small-scale and large-scale propagation effects (typically the path-losses), the overall geometrical based channel model at the level of a whole coverage area can be written as,

$$H = \underline{D}^{1/2} \sum_{l=1}^{L} \tilde{D}_{l}^{1/2} \Phi_{l} A_{l} \quad (7.64)$$

with $\tilde{D}_l^{1/2} = \underline{D}^{1/2} \tilde{D}_l^{1/2}$ meaning that the large-scale path-loss associated to each user is facotrized. This model can be applied for each subcarrier of an OFDM signal.

### ► MU-MIMO system model

The MU-MIMO system model introduced in the previous sections can then integrate the geometrical ray-based model. Considering a downlink Massive MIMO system with M transmit antennas at the central station and a set of K single antenna users, the received signal vector for one particular flat fading channel writes,

$$Y = \underline{D}^{1/2} \sum_{l=1}^{L} \tilde{D}_{l}^{1/2} \Phi_{l} A_{l} \cdot \underbrace{GS}_{X} + N \quad (7.65)$$

with $X$ the vector of precoded symbols and $G$ is the normalized multiuser spatial filtering matrix, i.e. $E[\text{Tr}(G^H G)]$. In the following, we will consider that spatial filtering is applied at the transmit side in a pure SDMA mode allowing for transmitting different data to each user at the same time and in the same bandwidth.
► Beamsteering approach of conventional spatial filters

From the aforementioned system model, one can re-express the most common spatial filters of the literature already introduced in the previous sections, namely MRT, EGT, TR and ZF. In the case of matched-filter like schemes, it is even possible to find a direct expression of the filters from the steering matrices $A_l$, hence obtaining filters in the form of a sum of beamsteering precoders. As described in more details in [C.19], this leads to the following result.

Contribution III.16 (Analytic Expression)

Beamsteering based MU-MIMO spatial matched filters

Assuming the geometrical model of the MU-MIMO channel of Eq. (7.65), the TR and EGT spatial filter matrices can be expressed as a combination of digital beamsteering matrices as,

$$ \text{EGT : } G = \frac{1}{\sqrt{M}} \sum_{l=1}^{L} B_l \Phi_l^H \qquad (7.66) $$

$$ \text{TR : } G = \frac{1}{\sqrt{M}} \sum_{l=1}^{L} B_l \Phi_l^H D_l^{1/2} \qquad (7.67) $$

$$ \text{MRT : } G = \frac{1}{\sqrt{M}} \sum_{l=1}^{L} B_l \Phi_l^H D_l^{1/2} \Lambda \quad \text{where, } (\Lambda)_k = \frac{1}{\sum_{l=1}^{L} \beta_{k,l}} \qquad (7.68) $$

with the digital beamsteering matrix $B_l$ enclosing the hermitian transpose of the steering vectors of the K users for their lth path as,

$$ B = A_l^H \qquad (7.69) $$

This contribution essentially tells us that conventional spatial matched filters can be interpreted as linear combinations of unitary beamsteering beamformers associated to the angle of arrival of the geometrical paths of each user. As an example, one unitary beamsteering vector of, i.e. one column of the beamsteering matrix $B_l$, applies the adequate phase shifts to the input signal such that the antenna array exhibits a radiation pattern with a maximum gain steered in the direction of the angle of arrival of the treated path. This idea, inherited from the traditionnal analog beamforming theory, is illustrated in Fig. 7.31 for a single steered beam. Eventually, can be simply interpreted as the formation of multiple beams so as to produce the waves propagating along the geometrical rays that will reach and recombine constructively at each intended user. MRT and TR does form the same beams but modify their amplitude to maximize the SNR of each user. Hence, the latters can be viewed as spatial Rake pre-filters expressed in the angular domain.

An interesting consequence of such geometrical decomposition of the channel is the reduced number of parameters to estimate to build the spatial filters. Hence, instead of estimating the $M \times K$ entries of the initial channel matrix, as commonly defined in MU-MIMO systems, only $L \times K$ angles of arrival and the related complex coefficients have to be estimated. In the context of massive MIMO systems in which $M \to \infty$, we understand that the geometrical description of the channel becomes highly advantageous. In addition, another interest of the geometrical basis representation is its inherent reciprocity, whatever the sub-band used. More precisely, the purely geometrical structure of the channel is not varying with the subcarrier index or between two adjacent sub-bands. Hence, putting apart the phase terms in matrices $\text{Phi}_i$, all the other components remains unchanged in uplink or dowlink, even in an FDD duplexing mode. This characteristic is of high interest according to channel state information feedback mechanisms in FDD mode, which is identified of one of the major issues in massive MIMO systems.
## 3. Practical Beamforming Strategy for mmWave Communications

<figure><img src="image_74.png" /><figcaption>FIGURE 7.31– Steerable array beam through phase shifters.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>FIGURE 7.32– Example of multi-user digital Beamforming based on the LOS component of three users.</figcaption></figure>

### ► Digital Beamsteering for LOS environments

From a general point of view, the received signal carried by the $\tilde{W} = HG$ equivalent channel can be decomposed in order to separate the direct path components from the reflected ones. For a particular user $k$, the equivalent channel vector can be written as,

$$ \tilde{w}_k = \underbrace{\sqrt{\beta_{k,1}} e^{j\phi_{k,1}}} \over \text{Direct path} \mathbf{a}_{k,1}^\mathrm{T} \mathbf{g}_k + \underbrace{\sum_{l=2}^{L_k} \sqrt{\beta_{k,l}} e^{j\phi_{k,l}} \mathbf{a}_{k,l}^\mathrm{T} \mathbf{g}_k}_{\text{Reflections}} \quad (7.70) $$

where $g_k$ is the $k$th row of the spatial precoding matrix $G$. Due to the strong path-losses in a mm-waves, it is expected that the propagation channel is mainly governed by the line-of-sight components for each user link and that the reflected paths remain of low amplitude. In addition, at mm-wave frequencies, it is appropriate to use small cells within which the LOS connectivity between transmitter and receiver is very probable. It is then possible to envisage suboptimal spatial filters by relaxing the constraint on the complete knowledge of the channel. Hence, we proposed to build beamsteering filters only based on the main path component of each user. If we assume a purely LOS channel, i.e. only considering the direct path from each user to the central station, the channel matrix simplifies to,

$$ H_{\mathrm{LOS}} = D_{1}^{\frac{1}{2}} \Phi_{1} A_{1} \quad (7.71) $$

Hence a very simple multi-user digital beamsteering (DBS) scheme can be used, as proposed in [C.19], consisting of forming only one beam per user steered in the direction of the main path of that user as illustrated in Fig. 7.32. In that case, no particular attention has to be paid to the phase component of the unique propagation ray of each user since no constructive combination between multiple beams has to be obtained. Hence, the only pieces of information about the channel that have to be known by the transmitter to build the DBS filter are reduced to the angles of arrival of the LOS rays for each user. Interestingly, this channel parameters are intrinsically reciprocal, in TDD and FDD modes. This important result, summarized in Contrib. III.18 on top of next page, has been detailed in [C.21].

The proposed beamformer leads to maximizing the emitted power only in the main direction of each user. It can be visualized as a filter adapted to the channel described by its main steering vector (angles $\theta_{k,1}$ and $\phi_{k,1}$).

### ► Simulation results and analysis

The proposed DBS filter has been experimented and compared to other approaches through simulations. A practical propagation environment as illustrated in Fig. 7.33 has been created and simulated using ray-tracing methods to produce the channel matrices of the MU-MIMO system. Two types of channel responses can be obtained : the purely LOS channel, only retaining the direct path between the central station and each user, and the complete multipath channel made of the LOS and NLOS components.
Contribution III.17 (System Design)

TDD/FDD suboptimal Multiuser Digital Beamsteering (DBS)

In a MU-MIMO system with K users which channels each exhibit one strong propagation ray with an angle of arrival ($\theta_k, \varphi_k$) at the central station, can be multiplexed over the space dimension using the following digital beamsteering (DBS) matrix,

$$G = \frac{1}{\sqrt{M}} A^{\mathcal{H}} \qquad (7.72)$$

with the digital beamsteering matrix A enclosing the steering vectors of the K users for their strongest path as,

$$A = [a_{\theta_1, \varphi_1}, \dots, a_{\theta_K, \varphi_K}]^T \qquad (7.73)$$

where $a_{\theta_k, \varphi_k}$ is defined in Eq. (7.59).

In Fig. 7.34, we give the synthetic results obtained for the proposed simple DBS solution opposed to the ZF and MRT filters, in the case of the LOS and LOS+NLOS channels at 60 GHz and using a square 6 antenna array. It is first verified that the DBS and MRT results converge in the purely LOS channel, as expected from the previous analysis on the geometrical channel representation. Then, the ZF filter exhibits contrasted performance depending on K. Indeed, the ZF filter hardly succeeds in accomodating more than 10 users and even fails compared to MRT and DBS schemes when K becomes to large. As already discussed, ZF relies on channel matrix inversion which can properly be achieved only if the matrix is well conditioned. Practically, with K users, the channel matrix shall be of rank K at least to be invertible. Yet, with K users in the street area, we showed in [C.21] and proved that the channel rank is essentially of the order of the horizontal size $M_H$ of the antenna array in purely LOS channels, and slightly more if some multipaths occur. Hence, in our example, as $M_H = 6$, the optimum number of users to address in parallel with ZF is $K \approx 6$, otherwise experiencing some performance drop. For the two other filters, the increase of the number of users yields higher interbeam/interuser interference and hence a sum capacity saturation. The interference terms are of higher amplitude when more propagation modes are present, i.e. when multipaths are activated, which explains the better results in LOS compared to LOS+NLOS. However, this effect remains of limited impact on performance since multipaths components vanish rapidly in mm-waves. Finally, since DBS only exploits the knowledge of the LOS component, its performance is slightly degraded compared to the MRT filter. However, as it requires only two angles per user to be produced, the DBS filter can be viewed as a interesting performance/complexity trade-off.

## 3.4 LEACKAGE BASED POWER ALLOCATION

In the following of the work on spatial filtering for mm-waves based on DoA, we investigated power allocation techniques to enhance the previously presented results on achievable rate. It is well known from the literature that power allocation in Multi User (MU) MIMO [FSY11, WWGY10, WYZD13, JLL03] is an additional processing layer that can be utilized separately or jointly with other blocks in order to optimize the system spectral or energy efficiency. However most of the contributions available for MU-MIMO systems are based on iterative procedures than can hardly be scaled to massive MIMO system topologies. As for massive MIMO systems, is was already shown around 2015 that low complexity linear precoders such as MRT can achieve considerable gains when coupled together with an interference wise power allocation [ZZZ13, CL14]. However, no practical low-complexity algorithms were available at that time. After the proposal of the previously introduced DoA-based spatial filters, our idea was to extend such filters with power allocation solutions of low complexity. Due to the fact that MRT/EGT/TR or DBS strategies only aims at maximizing the received signal and doesn't consider the interference, we focused on adapting these kinds of spatial filters taking into account the inter-beam interference, or more exactly what we will call the beam leakage in the sequel. This can be done with extra processing layer
## 3. Practical Beamforming Strategy for mmWave Communications

<figure><img src="image_76.png" /><figcaption>FIGURE 7.33– Simulation scenario as a street canyon area with a planar antenna array at the central station. Channel response is obtained through LOS and reflected rays.</figcaption></figure>

<figure><img src="image_77.png" /><figcaption>FIGURE 7.34– Total achievable rate using ZF, MRT and DBS spatial filters in purely LOS and LOS+NLOS channels. Planar antenna array with M = 6 × 6 elements.</figcaption></figure>

referred to as LB-PA (Leakage-based Power Allocation) aiming at mitigating the interference power produced by competing users, but not following the strict nulling strategy of ZF. Contrary to all the proposed algorithms in the literature, the LB-PA solution is a closed-form non iterative one as long as partial or complete channel information is known at the central station.

### ► Beam leakage

Let us introduce the beam leakage metric which will be considered as the main building block of our power allocation algorithm. The leakage has to be understood from a central station perspective as a measure of how the formation of a beam to a given user will harm its neighbours. In other words the leakage effect, translates the propensity to impact neighbouring transmissions, as simply illustrated in Fig. 7.35. Note that leakage differs from interference. The latter is a measure for one given user k of the signal power perceived from the signals transmitted to other users, while the former measures the nuisance level that one transmission to a user k creates upon other users. The leakage L_k created by the transmission to user k is expressed as,

$$L_k = \sum_{j \neq k=1}^{K} |\mathbf{h}_j^{\mathrm{T}} \mathbf{g}_k|^2 \qquad (7.74)$$

where g_k is the precoding vector constructed to target user k.

Since the leakage is an evaluation of the power emitted in the direction of the other users, it is possible to assess its level exploiting the angular model of the channel introduced in Eq. (7.65). The leakage can hence be restated using the channel steering vectors as,

$$L_k = \sum_{j \neq k}^{K} \sum_{l=1}^{L_j} \beta_{j,l} |\mathbf{a}_{j,l}^{\mathrm{T}} \mathbf{g}_k|^2 \qquad (7.75)$$

where β_{j,l} and a_{j,l} are the path-losses and steering vectors describing the propagation channel for user j as reminded in Fig. 7.36. Note that both expressions in Eq. (7.74) and (7.75) are valid for a given flat fading channel. The leakage evaluation is thus depending on the subcarrier in case of an OFDM waveform.

### ► Leakage with reduced channel knowledge

From the above expression, it is clear that the calculation of the leakage level requires a complete knowledge of the channel response, however reduced to some particular angles of arrival if using the geometrical channel
<figure><img src="image_78.png" /><figcaption>FIGURE 7.35– Illustrating the effect of the leakage</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>FIGURE 7.36– Angle-based channel knowledge for one user.</figcaption></figure>

description. In the case of a propagation scenario strongly conditioned by the LOS component of the channel, as is the case in mmWave and/or in the small-cell environment, it is favorable to implement a spatial filters based on the main angles of arrival of the propagated rays such as those introduced in Contrib. III.18). In that case, a simpler expression of the leakage can be formulated depending only on the steering vectors of the users. In addition, the aforementioned properties of mm-waves make it possible to provide an approximate expression of the leakage focusing on the main path of each user, leading to

$$ \tilde{L}_k = \sum_{j=1 \neq k}^{K} \beta_{j,1} |\mathbf{a}_{j,1}^{\mathcal{H}} \mathbf{a}_{k,1}|^2 \qquad (7.76) $$

This expressions assumes that the leakage level is mainly governed by the emitted power in the main direction of each user, i.e. by the LOS components of the MU-MIMO channel. Interestingly, in contrast to the former expressions given by Eq. (7.74) or Eq. (7.75), the expression we propose here does not depend on the frequency response H, impulse response of the channel, nor the number of subcarriers N but only the geometric structure of the channel. It is even possible to push further the simplication of the leakage evaluation by removing this path-loss parameter β, hence only measuring the amount of interfering energy that is radiated toward the other users from an angular point of view, regardless of their distance to the central station. Then a basic, albeit consistent, leakage definition can be provided as,

$$ \tilde{L}_k = \sum_{j=1 \neq k}^{K} |\mathbf{a}_{j,1}^{\mathcal{H}} \mathbf{a}_{k,1}|^2 \qquad (7.77) $$

which means that only the angles of arrival of the LOS rays are exploited to evaluate the leakage level.

The use of both reduced channel knowledge solutions aforementioned in a realistic environment allows the BS to approximately assess the interference that each transmission produces on each user in a common coverage area. The leakage can be evaluated in any system with a channel estimation process, but the simple definitions given here allows for a very economic parameter estimation for each user, since only 2K values have to be obtained (2 angles per user). In addition to the extremely simple expression of the leakage in that case (low complexity and low overhead), such a definition allows for greater fairness of power allocation between users as it will not hinder cell edge users more than other users.

## 3.5 LEAKAGE BASED POWER ALLOCATION

From the latter definitions of leakage, it is now possible to envisage leakage minimization power allocation strategies and then act on the individual power weights per user in order to minimize the interbeam interference effects. In case the power distribution is uniform among all users, $P_k = \frac{P_T}{K}$ with $P_T$ the total power emitted and K the number of users. On this basis, a situation of equity between users in terms of leakage level is to guarantee
## 3. Practical Beamforming Strategy for mmWave Communications

that any user does not produces more leakage than another. Hence, the optimization problem can be stated as,

$$ \mathcal{P}: \quad \min_{P_k, \forall k} \max_{L_k} P_k L_k \qquad (7.78) $$

$$ \text{w.r.t.} \quad \sum_{k=1}^{K} P_k = P_T $$

with $P_T$ the total transmit power to be shared between users. The resolution of such a problem leads to the following power allocation algorithm, proposed in [C.9].

### Contribution III.18 (Algorithm)

#### Leakage-Based Power Allocation for MRT/DBS spatial filters

In a MU-MIMO system with K users and a total transmit power $P_T$, MRT or DBS spatial filters yield uniform leakage if the power allocated to each user is such that,

$$ P_k = \frac{P_T}{1 + \sum_{j \neq k} \frac{L_k}{L_j}} \qquad (7.79) $$

with $L_k$ the leakage level of user $k$ evaluated from Eq. (7.75) (perfect Leakage estimation), from Eq. (7.76) (LOS component approximation) or from Eq. (7.77) (LOS angle-based approximation).

### ▶ Simulation results

The various simplifications proposed for the leakage level evaluation will naturally lead to a sub-optimality of the precoding and power-weighting processes. However, the simulation results provided in Fig. 7.37 show that the performance degradation remains minimal in practical situations. It is indeed observed that the accuracy of the leakage evaluation yields negligible effect on the performance results, hence concluding that the underlying complexity reduction is worth interest. Interestingly, it is also noticed that using the proposed power allocation algorithm makes the simple digital beamsteering and the MRT filters perform similarly, whereas MRT is out-performing DBS with uniform power allocation. Last but not least, the power allocation algorithm succeeds in decreasing the gap significantly between MRT/DBS and ZF. These results unleash the main advantage of the proposed algorithm, which is reducing the benefit of using a high complexity-high overhead signal processing techniques.

## 3.6 CONCLUSION

Between the beginning and the end of this study, massive MIMO mmWave systems have clearly evolved as key enablers for the future generation of wireless networks (5G) [TWA+16, XMH+17] to support higher achievable data rates. Due to the wide bandwidth and high antenna gains, it is today recognized that massive MIMO mmWave systems are able to achieve very high per user peak data rates of tens of Gigabits per second (Gbps) [SAHC14a].

In our study, we have started investigating spatial filter concepts in the context of mm-waves propagation environments exploiting the sparse nature of the channel mainly governed by the its geometrical structure as suggested by the works of Rappaport [RRE14]. On this basis, we have shown that angle based precoders could provide almost optimal performance, however using a limited number of information parameters about the channel state, namely as many azimuth and elevation angles as there are users. This contributes to the growing conviction that DoA estimation and angle based precoders are simple and effective solutions for high frequency small cell systems.

However, working at such high frequencies imposes a lot of hardware and channel implications that need to be tackled in order to unleash the high potentials of Massive MIMO. Such questions will be part of the perspective works detailed in 9.
<figure><img src="image_80.png" /><figcaption>FIGURE 7.37– Total achievable rate of a MU-MIMO system using MRT/DBS/ZF spatial filters with and without leakage-based power allocation. Planar antenna array of 16 elements.</figcaption></figure>

## Summary (PhD/Dissemination/Projects)

- 1 PhD (A. Rozé)

- 1 submitted international journal paper

- 3 communications in international conferences and workshops [C.9][C.19][C.21]

- 1 IRT-Bcom Project (RS-COM)
> Great discoveries and improvements
> invariably involve
> the cooperation of many minds.
> 
> – Alexander Graham Bell –

Selected Topics

This chapter introduces the last field of research I have been developping during these last years on the hybridization between wireless broadcast and broadband networks. Contrary to the previous two topics which were mainly concentrating on physical layer aspects, this research field takes place at an upper-layer stage aiming at assesssing the leverage of making two different networks cooperate and interwork. The selected aspects of our work are :

1. Semi-analytic analysis of dual mode broadcast/unicast networks

2. Wireless hybrid broadcast/broadband network optimization

# 1 SEMIANALYTIC ANALYSIS OF DUAL MODE BROADCAST/UNICAST NETWORKS

## 1.1 MOTIVATION

From 2000 to 2010, the progressive disappearance of the analogue television throughout the world have released large amounts of spectrum in the UHF and VHF bands. This so-called digital dividend have coincided with the beginning of the tremendous growth in the area of mobile telecommunications following the arrival of smart phones, and has motivated the assignment of the newly available frequencies to mobile oriented services, namely the 791-821 MHz and 832-862 MHz bandwidths for the deployment of the LTE standard [Rep14]. Even if this new spectrum opportunity has been a windfall for mobile network operators, it can not be considered as the unique long term solution to the data tsunami. Indeed, new trends of data consumption over wireless broadband networks have started to be witnessed around 2010 with the generalization of the mobile Internet access. On the one hand, consumer needs have become more and more user-specific because of the growing demand for surfing the Internet and making use of social networks on mobile terminals. On the other hand, the number of downloaded/streamed popular multimedia contents and the usage of handheld terminals for watching TV programms have known a constant inscrease. As a matter of fact, users are more and more watching linear services (such as live TV) on their tablets and smartphones which are connected to a mobile broadband network relying on a unicast transmission system.

Hence, mobile network operators have to accomodate these two antagonist phenomena : content individuali-
zation and massive consumption of popular contents among which an increased proportion of linear services. The
former issue can partially be solved using larger bandwidths, reducing the size of the cells in dense environments
(femtocells), or even taking advantage of local area networks, namely WiFi, as soon as they are available to each
user. To address the latter yet, multicast or broadcast instead of the by-default unicast transmission mode may be more adapted owing to their user factorization nature. Indeed, delivering identical pieces of data to different users in a unicast manner duplicates the bandwidth consumption and becomes a greedy strategy when the number of users increases. Meanwhile the broadcast approach provides service access to multiple users with a constant spectrum utilization. The complementarity between unicast and broadcast modes has already been addressed by the last 3GPP-LTE specifications that define a broadcast component called E-MBMS (evolved multimedia broadcasting multicast service). E-MBMS can be activated in some situations to optimize the spectrum use in cellular networks [1].

Nevertheless, broadcast transmission mode is expected to provide more benefit in case of large coverage areas in which the factorization effect is higher, which is typically the case with the high-power high-tower (HPHT) network topology of conventionnal TV broadcast networks. Hence, one new idea which emerged around 2010 was consisting in a cooperation between cellular network and terrestrial broadcast network infrastructures for the exploitation of the VHF/UHF frequencies in a more efficient way. The cooperation would rely on a complementary exploitation of terrestrial digital TV transmitters to broadcast selected multimedia contents within a large coverage area. The objective is to reduce the multimedia unicast traffic on the cellular networks by enabling the off-load of some contents from one unicast network to a broadcast one. The dual unicast/broadcast modes of the 3GPP-LTE can be viewed as a cooperation between two virtual networks, however based on the same cellular infrastructure. In the case of the broadcast/broadband network hybridization, the cooperation concept is pushed a step further.

We decided to put some efforts on this topic as we were working in parallel with mobile networks operators and broadcast networks operators, trying to find some new approach to face the famous data tsunami. Our study, which has been led in the framework of the ANR MCube project, dealt both on the definition of a common physical layer suited for the overall hybrid network, and on the performance evaluation methodology of the hybrid network depending on various application scenarios. The work on the common physical layer are not detailed in this manuscript but has been published in [R.8] and [B.2] as a collaborative study with other academic and industrial partners involved in the ANR MCube project. In the following, we rather focus on our contributions concerning the cooperative aspects between unicast and broadcast networks. Such a global hybrid network approach raises the question of how to define some unified model and to how decide which sub-network to exploit between the cellular unicast one and the terrestrial broadcast one, to deliver a service to a given number of end users. These aspects have been addressed in the PhD work of Nicolas Cornillet summarized in the sequel.

## 1.2 SYSTEM LEVEL HYBRID BROADBAND/BROADCAST WIRELESS NETWORK MODEL

### ► Hybrid Network System Model

The hybrid network considered in our study is the combination of two components. The first component is a classical cellular network made of a great number of relatively small hexagonal cells of side $c$ with low power base stations providing unicast broadband signals according to a mobile communication standard such as LTE. The second component is a broadcast network of coverage radius $R_B$ based upon one single high-power transmitter emitting a broadcast signal such as DVB-T2 for instance. Each component is assumed to use distinct frequency bands so that no inter-component interference can occur. Typically, the broadcast component uses a VHF carrier frequency, e.g. around 800 MHz, while the unicast component is operated around 2-3 GHz. The coverage area of the broadcast network is circular and completely overlaps the cellular network as depicted in Fig. 8.1. The overall coverage area of the hybrid network corresponds to that of the broadcast network and the broadcast transmitter is used as the coordinate reference, i.e. it is considered as the center of the hybrid network. Typical value for the broadcast network coverage radius is $R_B = 100$ km, while hexagonal cells have a side length of about $c = 1$ km.

The users are supposed to have dual-system receiver capabilities such that they can receive the proposed service from either the broadcasting system or the unicast system. According to such a hybrid network, various user location patterns can be considered and have strong influence on the overall system performance. In our study, we used a uniform distribution of the users within the overall coverage area, but also a two-dimensional Gaussian distribution (centered or not) for instance representing the situation of a coverage area at the scale of a whole city which center coincides with the center of the Gaussian distribution.
# 1. Semi-analytic analysis of dual mode Broadcast/Unicast Networks

<figure><img src="image_81.png" /><figcaption>**FIGURE 8.1– Proposed hybrid unicast/broadcast network**</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>**FIGURE 8.2– PDF of the distance within a unicast cell : comparison between the simulation and theoretical results assuming a Gaussian user distribution.**</figcaption></figure>

Such topology fixes the spatial pattern of the hybrid network from which it will then be possible to compute some performance metrics for example in terms of average spectral efficiency, achievable rate, energy consumption or service success rate. Even if they can be computed building a system level simulator, we were mostly interested in deriving some analytical or semi-anaytical expressions for such metrics, as detailed further. In that perspective, some statistical functions have to be defined such as the probability density function (PDF) of the distance between users and transmit stations, and then the probability density function of the achievable rate per user for each component of the network. Classical derivations based on the geometrical properties of the hybrid coverage model are used in the sequel to obtain the desired PDF.

## ▶ PDF of the user distance to transmit stations

A key step to derive statistical performance metrics of the hybrid network is to compute the PDF of the user distance to the nearest transmit station, in the broadcast and unicast cases. Such PDF is needed to further derive the statistical properties of the spectral efficiency attained within the region of interest, depending on the used component (unicast or broadcast). Our fist contributions have then been to provide the PDF of the user distance to the cellular and broadcast stations for various spatial user distributions, e.g. uniform or Gaussian. We give in Contrib. IV.1 hereafter the result for a two-dimensional Gaussian distribution of the user, in the general case when the Gaussian function is not necessarily centered regarding the hybrid network.

## Contribution IV.1 (Analytic Expression)

### PDF of the user distance to transmitter in the hybrid network with Gaussian user distribution

Given a hybrid network made of a single transmitter broadcast network of coverage radius $R_B$ and a cellular network with hexagonal cells of side length $c$, the probability density function $p_R(r)$ of the distance $r$ between a user and a transmit station of each component assuming a Gaussian distribution of the users across the overall coverage area are

$$ \text{Broadcast} \qquad p_R(r) = \frac{1}{1 - Q_1\left(\frac{\Delta}{\sigma}, \frac{R_B}{\sigma}\right) \sigma^2} e^{-\frac{\Delta^2+r^2}{2\sigma^2}} I_0\left(\frac{\Delta r}{\sigma^2}\right) \quad 0 \le r \le R_B \qquad (8.1) $$

$$ \text{Unicast} \qquad p_R(r) = \begin{cases} \frac{2\pi r}{3A} & 0 \le r \le c \\ \frac{2r}{A} \arcsin\left(\frac{c\sqrt{3}}{2r}\right) & c < r \le c\sqrt{3} \\ \frac{2r}{A} \arcsin\left(\frac{c\sqrt{3}}{2r}\right) - \frac{2\pi r}{3A} & c\sqrt{3} < r \le 2c \end{cases} \quad \text{with, } A = \frac{3\sqrt{3}}{2}c^2 \qquad (8.2) $$

where $\sigma$ is the standard deviation of the Gaussian distribution such that $\sigma \gg c$ and $\Delta$ is the distance between the center of the Gaussian distribution and the center of the hybrid network. (i.e. broadcast transmitter location). $Q_1(\cdot, \cdot)$ and $I_0(\cdot)$ are the Marcum Q-function and the modified Bessel function of the first kind and order 0, respectively.
The proof yielding the expressions of (8.1) and (8.2) have been published in [C.45]. With this first result, one may observe that the distribution of the distance between the users and the nearest unicast station does not depend on the Gaussian function parameter $\sigma$ governing the spatial distribution of the users in the hybrid network. This fact is valid under the mentioned assumption that $\sigma \gg c$, indicating that the spread of the Gaussian distribution is sufficiently wide compared to the hexagonal cell dimension. Indeed, it can be shown in that case that the user distribution within a cell is almost uniform [C.36]. This is for example verified through Monte-Carlo simulation as illustrated in Fig. 8.2. The histogram of the simulation results is obtained considering a Gaussian distribution of the users at the scale of the broadcast coverage area, while the theoretical plot is obtained from Eq. (8.2). Hence, from one unicast cell to another, the expression of the PDF is valid and will remain the same. Then, the only difference between cells rather comes from the number of users per cell depending on the cell location within the hybrid network. This can be reflected by the probability that a particular user falls within the coverage area of a particular cell in the hybrid network. Such result summarized in the Contrib. IV.2 was also part of the study published in [C.36].

## Contribution IV.2 (Analytic Expression)

### Probability of a user falling within a particular unicast cell coverage

Given the above defined hybrid network with Gaussian distribution of the users with standard deviation $\sigma$, the probability $P_c(u \in C_i)$ that one user $u$ of the network falls within the coverage are of one particular cell $C_i$ of dimension $c$ is approximated as

$$P_c(u \in C_i) \approx \frac{1 - Q_1\left(\frac{\delta_i}{\sigma}, \frac{\tilde{R}_c}{\sigma}\right)}{1 - Q_1\left(\frac{\Delta}{\sigma}, \frac{R_B}{\sigma}\right)} \quad \text{with} \quad \tilde{R}_c = \sqrt{\frac{3\sqrt{3}}{2\pi}} c \quad (8.3)$$

where $\delta_i$ is the distance between the base station of the $i$th cell and the center of the Gaussian distribution, $\Delta$ is the distance between the center of the Gaussian distribution and the broadcast transmitter location.

## PDF of the attainable spectral efficiencies

The previously derived PDF of the distance between users and transmit stations can be translated into a PDF of the achievable rates of the users for each component of the hybrid network. To that end, one has to establish the link between the distance and the performance of the system. This is commonly obtained integrating some link budget relation into the model, and some tabulated physical layer performance results.

As far as the link budget is considered, each component of the hybrid network is generally associated with an adequate propagation attenuation model providing the path-loss $L(r)$ as a function of the distance of the receiver to the transmit station. In the case of broadcasting system from high-power transmitters, the Hata model is generally used [Hat80]. For cellular systems, the LTE specifications recommend the use of a dedicated model given in [2]. In both cases, the path-loss depends not only on the distance and carrier frequency, but also on the height of the transmit and receive antennas. From such models, it is easy to express the receive signal to noise ratio (SNR) $\Gamma(r)$ for each component of the hybrid network as,

$$\Gamma(r) = \Gamma_0 \left( \frac{R_{max}}{r} \right)^{\alpha} \quad (8.4)$$

with $\alpha$ the path-loss attenuation coefficient depending on the propagation model, $R_{max}$ the maximum coverage distance of each component, i.e. $R_{max} = R_B$ for broadcast and $R_{max} = 2c$ for unicast, $\Gamma_0$ some minimal reference SNR. This latter can for instance correspond to the minimal required SNR at the boundary of the coverage area (broadcast or broadband) allowing for a transmission at a given QoS.

From the path-loss model, the next step is then to integrate the tabulated SNR results obtained from pure physical layer simulations over various channel models. For example, Table 8.1 gives the list of the spectral efficiencies related to various modulation and coding schemes of the DVB-T2 standard [v1.12a] at different levels of SNR assuming a binary error rate of $10^{-11}$.
# 1. Semi-analytic analysis of dual mode Broadcast/Unicast Networks

<table><caption>TABLE 8.1– Spectral efficiencies versus SNR for the DVB-T2 system. Target BER=10⁻¹¹.</caption><thead><tr><th>index i</th><th>Γ<sub>i</sub> (dB)</th><th>ν<sub>i</sub> (bit/s/Hz)</th></tr></thead><tbody><tr><td>1</td><td>2.0</td><td>0.87</td></tr><tr><td>2</td><td>4.1</td><td>1.18</td></tr><tr><td>3</td><td>5.3</td><td>1.31</td></tr><tr><td>4</td><td>6.6</td><td>1.45</td></tr><tr><td>5</td><td>6.9</td><td>1.74</td></tr><tr><td>6</td><td>9.6</td><td>2.36</td></tr><tr><td>7</td><td>11.0</td><td>2.60</td></tr><tr><td>8</td><td>12.8</td><td>2.89</td></tr><tr><td>9</td><td>13.9</td><td>3.07</td></tr><tr><td>10</td><td>14.4</td><td>3.54</td></tr><tr><td>11</td><td>16.1</td><td>3.94</td></tr><tr><td>12</td><td>18.2</td><td>4.34</td></tr><tr><td>13</td><td>19.0</td><td>4.72</td></tr><tr><td>14</td><td>20.5</td><td>5.25</td></tr><tr><td>15</td><td>22.9</td><td>5.78</td></tr><tr><td>16</td><td>24.5</td><td>6.14</td></tr><tr><td>17</td><td>25.8</td><td>6.49</td></tr></tbody></table>

<figure><img src="image_83.png" /><figcaption>FIGURE 8.4– Effective spectral efficiency v<sub>eff</sub> as a function of SNR for a LTE signal (AWGN channels)</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>FIGURE 8.3– Probability density function of spectral efficiencies for the broadcast component</figcaption></figure>

<figure><img src="image_85.png" /><figcaption>FIGURE 8.5– PDF of spectral efficiencies for the unicast component with HARQ mechanisms.</figcaption></figure>

From such SNR tables, one can then compute the PDF $p_Y(v)$ of the spectral efficiencies which can be expected or which can be effectively granted to each user of the network. This is obtained through simple integration of the PDF of the distance over rings of coverage associated to the different levels of spectral efficiency of each standard assuming a predefined target bit or packet error rate. Such computation has been proposed and detailed in [C.36] and leads to the following expression.

## Contribution IV.3 (Analytic Expression)

### PDF of the achievable spectral efficiencies

Given the defined hybrid network, the probability density function $p_Y(v)$ of the spectral efficiency $v$ which can be expected within the coverage area of each component of the hybrid network is

$$p_Y(v) = \sum_{i=0}^{N_i} \int_{r_i-1}^{r_i} p_R(r) \delta(v - v_i) dr \quad \text{with, } r_i = e^{\frac{1}{\alpha} \log\left(\frac{\Gamma_i R_{max}}{r_0}\right)} \quad (8.5)$$

where $p_R(r)$ is the PDF of the distance defined in Contrib. IV.1, $\Gamma_i$ is the minimum required SNR for spectral efficiency $v_i$ as tabulated for a particular PHY and QoS.

This last result allows us to compute the probability to which a given user has the chance to benefit from a spectral efficiency $v_i$ depending on the used sub-network. To illustrate this the PDF of the broadcast component using $\alpha = 2.5$, $F_p = 800$MHz, and $R_{max} = 100$km has been computed and is depicted in Fig. 8.3. In the same way, the PDF of the unicast component can be exemplified on the basis of 3GPP-LTE specifications as
provided in [C.36]. In this latter case however, the system level model has to take into account the retransmission mechanisms used to improve the QoS of the system, namely the HARQ (Hybrid Automatic Repeat reQuest). The spectral efficiency values then translate into effective spectral efficiencies, integrating the statistical number of occuring repetitions. Hence, one can build a bijective function $\mathcal{G}$ that links the effective spectral efficiency with the SNR as shown in Fig. 8.4. Such a function can be tabulated after PHY layer simulations and then be used to derive the PDF of the spectral efficiency. In that case, the PDF takes the form of a continuous function of $v$ instead of a discrete form as previously stated in Contrib. IV.3. The new PDF form is introduced in the contribution below and is exemplified in Fig. 8.5 in the LTE system case over Gaussian channels [C.36].

## Contribution IV.4 (Analytic Expression)

### PDF of the achievable spectral efficiencies with HARQ

Given the defined hybrid network, the probability density function $p_Y(v)$ of the spectral efficiency $v$ of the unicast component using HARQ is

$$p_Y(v) = p_R(r) \frac{2c}{\alpha\Gamma} \left(\frac{\mathcal{G}^{-1}(v)}{\Gamma}\right)^{\frac{1}{\alpha}-1} \mathcal{G}'(v) \quad (8.6)$$

where $p_R(r)$ is the PDF of the distance defined in Contrib. IV.1 for the unicast component and $\mathcal{G}(v)$ is the tabulated function of the effective spectral efficiency versus SNR involving HARQ mechanisms.

## 1.3 SEMI-ANALYTIC HYBRID NETWORK PERFORMANCE EVALUATION

From the derived PDFs, it is then possible to compute some statistical performance metrics of a network in which a broadcast and a unicast components are available. In such a hybrid network, indeed, a given service $S$ can be transmitted through either the unicast component or the broadcast component. The choice of the component to use can be driven from various criteria according to objective criteria such as capacity, QoS, delay or energy considerations. In [C.36], we for instance defined two switching criteria based on spectral efficiency and energy efficiency gains denoted as $G_{U \to B}$ when selecting the broadcast component instead of the unicast one :

$$G_{spectral}^{U \to B} = \frac{v^{(B)}}{v^{(U)}} \quad \text{and,} \quad G_{energy}^{U \to B} = \frac{E^{(U)}}{E^{(B)}} \quad (8.7)$$

### ► Spectral efficiency switching criterion

A fundamental difference between the unicast and broadcast transmission modes is the way to adapt the spectral efficiency used to deliver a common service to multiple users. In the unicast mode, a so-called link adaptation is carried out to adapt the modulation and coding scheme regarding the quality of the channel (SNR). Hence, the resulting spectral efficiency at the scale of a cell corresponds to the average of the user spectral efficiencies. Contrary to that, in the broadcast case, the service is delivered using the same spectral resource for all the users of the coverage area. Hence, the choice of the modulation and coding scheme is based on the worst user channel conditions, that is of the users located at the boundary of the cell. On that basis, the gain in terms of spectral efficiency can be viewed as a random variable resulting from the fact that spectral resource is exploited with the latter two strategies. We showed that such a gain can from Eq. (8.7) be formally expressed as,

$$\lambda = G_{spectral}^{U \to B} = \min_k v_k^{(B)} \times \sum_{k=1}^{N} \frac{1}{v_k^{(U)}} \quad (8.8)$$

This expression already indicates that the potential gain obtained switching from a unicast to a broadcast transmission mode is depending on the number of users. If $N$ is large, the right-hand side of the above expression corresponding to the cost in spectral efficiency of the unicast component dramatically increases, which comes
# 1. Semi-analytic analysis of dual mode Broadcast/Unicast Networks

in favor of the broadcast component hereby yielding a potentially high spectral gain. To go further, we provided the analytical expression of the PDF of the spectral gain as a function of the number of user [C.45]. This key result is stated in the following Contrib. IV.5. It can be analyzed that the obtained PDF depends both on the choice of the broadcast spectral efficiency $v_0^{(B)}$ and on the number N of users. If $v_0^{(B)} < 1$ and N is low, it can be shown that the switching gain is statistically negative. If $v_0^{(B)}$ is increased or N becomes sufficiently large, the gain can reach potentially high positive values. This has to be viewed as the user factorization gain naturally provided by the broadcast transmission mode. Further analyses namely given in [C.45] show that the spectral gain follows a modified Nakagami distribution which tends to be almost deterministic when N is large enough.

## Contribution IV.5 (Analytic Expression)

### PDF of the unicast-to-broadcast spectral gain

Given the defined hybrid network, the probability density function $p_{\Lambda}(\lambda)$ of the spectral gain $\lambda$ obtained when switching from the unicast to the broadcast component to deliver a common service S is,

$$p_{\Lambda}(\lambda) = \frac{2m^m}{\Gamma(m)\omega^m} v_0^{(B)} (\lambda v_0^{(B)})^{2m-1} e^{-\frac{m}{\omega}(\lambda v_0^{(B)})^2} \quad (8.9)$$

where $v_0^{(B)}$ is the spectral efficiency selected for the broadcast component, and parameters $\omega$ and $m$ defined as a,

$$\omega = N^2 (\mathbb{E}[v])^2 + N \mathbb{V}[v] \quad m = \frac{\omega^2}{4(\mathbb{E}[v])^2 \mathbb{V}[v] N^3}$$

with N the number of users demanding service S in the network and v the random variable modeling the spectral efficiency of the unicast component which PDF is given by Contrib. IV.3 or Contrib. IV.4.

## ► Energy efficiency switching criterion

The energy efficiency gain is similar to the spectral efficiency gain but integrates some complementary correction term in its expression regarding the transmit power level used for each component of the network. In the previously derived spectral gain, the transmit power of one component could indeed be arbitrarily augmented to positively influence that component, which lacks fairness. The energy efficiency gain takes the potential power imbalance between the two components into consideration as,

$$G_{\text{energy}}^{U \to B} = G_{\text{spectral}}^{(U \to B)} \times \frac{L^{(U)}(2c)}{L^{(B)}(R_B)} \frac{\Gamma_0^{(B)}}{\Gamma_0(U)} \quad (8.10)$$

where $L(.)$ are the maximum power attenuations experienced at the coverage boundaries for each component of the network (i.e. $2c$ or $R_B$) and computing from the adequate path-loss models as already discussed. $\Gamma_0$ are the minimum value of SNR that is required to guarantee a reliable reception for a signal which transmission is carried out at the minimal spectral efficiency among the available ones for each component.

From this expression, it is understood that the maximal coverage distance of the broadcast component comes as a penalty compared to the cellular case since $L^{(B)}(R_B)$ is expected to be far higher than $L^{(U)}(2c)$. This penalty has to be compared to the user factorization gain discussed above. The simulation results presented in the following show how these two features jointly behave when N varies.

## 1.4 HYBRID ACCESS PERFORMANCE ANALYSIS

Many aspects can enter in consideration in the decision process between the activation of either the unicast or the broadcast transmission mode in a hybrid network in which both are available. Most of the already

<footnote>a. with $\mathbb{E}[.]$ and $\mathbb{V}[.]$ denoting the statistical expectation and variance, respectively.</footnote>
conducted studies focus on the transmission of live mobile TV services [CHH08a, Bri08, UK06]. In that case, audience measurements provide a good criterion to decide which service has to be transmitted through the broadcast component. For other types of services, there is a lack of such a criterion. In the semi-analytical study summarized above, we have provided some statistical metrics allowing to easily evaluate the gain obtained when switching from the unicast to the broadcast component. Some of our results, published in [C.45] and [C.36], are reported below.

## ► Energy gain

Let us first investigate the gain obtained regarding the energy efficiency criterion as defined in (8.10). Our goal is to find which one of the components or sub-network can deliver a service to the users with the smallest overall power consumption, taking into account the spectral efficiency granted to the users. Fig. 8.6 gives the evolution of the global gain $G_{U→B}$, which correspond to the broadcast gain over unicast, when the total number N of users changes within the proposed hybrid network. The given results are obtained using system and channel attenuation parameters specified in [v1.12a, 2]. Simulations results are obtained by direct numerical computation of the global gain $G_{U→B}$ of (8.10) through uniform random locations of the N users. From these computations, curves corresponding to “average”, “minimum” and “maximum” cases are reported in the figure. The theoretical results are the average gains obtained with the proposed closed-form of Contrib. IV.5 which is verified to be very accurate as soon as $N > 10$. Then, it turns out that an average gain of 0dB is obtained when $5.0 × 10^3$ users are active. This value represents the threshold that could drive the off-load of traffic from the unicast to the broadcast component of the hybrid network. This value can seem quite high, but recall that in the studied system, the area of the broadcast cell is $12.1 × 10^3$ times greater than the area of a unicast cell. This means that the threshold eventually represent an average of 0.42 user per unicast cell, which correspond to less than 1 user every 2 cells. In other words, from energy considerations, the use of the large scale broadcast component is more favourable than the small scale unicast one as soon as one half of the unicast cells are requested for the same service delivery. This tends to prove that an hybrid unicast/broadcast coverage approach would allow global energy consumption reduction. In comparison, LTE-EMBMS-like systems are more flexible, allowing the switch from unicast to broadcast component cell by cell, but need that at least 2 users in the same cell use the same service for this switch to have an interest.

## ► Mixed broadcast/broadband strategy

In our study, further simulation have been led when changing the minimal spectral efficiency of the broadcast component. More precisely, instead of selecting the minimal spectral efficiency $v_1$ and ensure a maximal coverage area, a higher $v_i$ is chosen to increase the throughput at the level of the broadcast component. If the transmit power of the broadcast station is not changed, the related coverage area is reduced. As such, we may switch from a pure unicast component to a mixed broadcast/unicast coverage : the users falling into the coverage area of the broadcast station may be served through the broadcast component and the other users will have no other possibility that the unicast component. In Fig. 8.7, we give how such a mixed strategy leads to some additional gain in terms of energy efficiency. Various “Mixed” modes according to switching spectral efficiency values ranging from $v_2$ to $v_7$ are tested, and the “Mixed Opt.” corresponds to the optimal mixed mode among all the possibilities.

## ► Unicast traffic off-load

The idea of data off-load can be applied to evaluate the reduction of the traffic load over the unicast component in some practical cases such as the periodic update of a given application. In that case, a large amount of terminals may be requested to download a common amount of data which could actually be pushed in advance in the memory of the terminals using a broadcast strategy. Based on some public information available in [Fra13], we took the example of the update of the Youtube application for Android terminals. In France for example, the average density of Android terminals is of about 26,4 per square km, hence leading to about $8,3^5$ terminals in the considered hybrid network of radius 100 km. Considering an average file weight of 6,6 Mo, the total traffic load to transmit such file to the whole terminals of the coverage area reaches 2,7 To in unicast mode, whereas it remains limited to 6,6 Mo in the broadcast mode if this mode is selected for the whole coverage area.

As previously used, we can also operate the hybrid network in a mixed broadcast/unicast strategy by reducing the broadcast coverage area while increasing the spectral efficiency. Fig. 8.8 gives the simulation results in such
# 1. Semi-analytic analysis of dual mode Broadcast/Unicast Networks

<figure><img src="image_86.png" /><figcaption>**FIGURE 8.6**– Analytical results and simulations of the energy efficiency gain versus the number of active users in a broadcast/unicast hybrid network with $R_B = 100$ km and $c = 1$ km.</figcaption></figure>

<figure><img src="image_87.png" /><figcaption>**FIGURE 8.7**– Energy efficiency gain versus the number of active users for the mixed broadcast/broadband strategy.</figcaption></figure>

<figure><img src="image_88.png" /><figcaption>**FIGURE 8.8**– Distribution of the unicast traffic load per cell in a mixed broadcast/unicast transmission strategy. Left figure : unicast mode only. Right figure : broadcast mode activated $v = 4,34$ b/s/Hz.</figcaption></figure>

a configuration for a Gaussian distribution of the users in the hybrid network and when the spectral efficiency of the broadcast component is set to 4,34 b/s/Hz. By this example, it is highlighted that a joint optimization of the broadcast and unicast components of a hybrid network can also be used to reduce the load of the unicast cells.

## 1.5 CONCLUSION

Network cooperation and convergence is a research topic that has taken an increased importance these last years since it opens new degrees of freedom in the optimization of communication systems. Most of the studies consider cooperation between cellular and local area networks as a major driver for increased system efficiency. The originality of our work has been to rather investigate cooperation aspects through the prism of the broadcast/unicast complementarity, hence addressing cooperation scenarios involving the conventional broadcast high-power high-tower infrastructure.

In this first PhD program conducted in the framework of the ANR MCube project, we have defined a new hybrid network topology and derived the related theoretical performance expressions based on a statistic model of the network metrics such as spectral and energy efficiency. On this basis, we have proposed switching criteria helping for the operation of the envisaged hybrid network. This led us to demonstrate the potential advantage
having a dual mode broadcast/unicast network.

During the work of the MCube project, we therefore promoted for a unified vision of the broadcast tech-
nologies, embedded in cellular networks to enable such cooperative scenarios. For instance, we contributed
collectively with other partners to the organization of a dedicated Open Event at Orange Labs Paris gathering
many industrials taking part in the DVB consortium and the 3GPP group. From an implementation perspective,
we demonstrated in [R.8] how to specify a universal physical layer using a flexible signal to provide a common
broadcasting system for the mobile broadband and TV broadcast networks. In the meantime, other initiatives
such as the so-called dynamic broadcast [QNR11, NR12, QR13] or tower-overlay strategies [JR13, RIJ14] have
emerged, with roughly the same idea of making cellular and broadcast network cooperate. In [RIJ14] in particular,
the authors propose a cooperation scenario in which the LTE data traffic is offloaded to a DVB-T2 network, by
using the DVB-T2 Future Extension Frame (FEF) concept together with the LTE carrier aggregation functiona-
lity, with the same philosophy as we introduced in [R.8]. All these ideas confirm that there is an advantage in
such interworking between a broadcast and a unicast component, at least from a theoretical point of view.

Summary (PhD/Dissemination/Projects)

- 1 PhD (N. Cornillet)

- 1 International journal paper and 1 book chapter

- 3 communications in international conferences and workshops [C.36][C.45][C.51]

- 1 ANR Project (M-Cube)

- 1 presentation to an Open Event at Orange Labs

2

WIRELESS HYBRID BROADCAST/BROADBAND NETWORK OPTIMIZATION

2.1 MOTIVATION

Convergent network scenarios involving broadcast and broadband wireless networks have known a growing
interest during the early 2010s. At the end of the PhD program summarized in the previous section, broad-
cast/unicast hybridization was even considered as a potential key technology in the perspective of the definition
of the future 5th Generation mobile networks. The 5G vision of some industrials was consisting in a global
standard unifying already existing network infrastructures and technologies, potentially also including broadcast
networks. In addition, the definition in 2013 of the new MBMS operation on demand (MooD) mode in the
3GPP Release 12 [3GP15b] was a crucial step in the introduction of broadcast strategies in wireless broadband
networks, since MooD allowed users to dynamically switch between unicast and broadcast transmission modes.
This idea turned out to be very close to that we had investigated in our hybrid broadcast/broadband network
studies and encouraged us to pursue our theoretical works on the optimization of hybrid operating modes and
on the definition of efficient switching criteria.

In parallel to such an evolution, the upcoming 2015 Wireless Regulation Congress (WRC) putting again
after the 2012 edition the UHF broadcasting band under high pressure for being partly attributed to mobile
broadband operators [MGL+15, IR13, BR14], broadcast network operators were very interested in new strategies
to demonstrate the advantage of operating such frequency bands using their high-power high-tower infrastructure.
One promising approach for them was to enable cooperation strategies with wireless broadband networks.

In 2013, we then commonly defined with TDF and IRT-Bcom a new PhD program aiming at leading further
investigations on broadcast/broadband hybrid network optimization. Some other works than ours, for instance
[RECEH13c, RECEH13a], had shown that a broadcast/broadband network cooperation could be a promising and
efficient solution, namely in the so-called broadcast coverage extension scenario. Broadcast coverage extension
can be viewed as the dual scenario of the broadband off-load scenario previously introduced. As detailed later, a
major common aspect of such scenarios relies on the coverage adaptation of the broadcast tower. In the PhD of
P.A. FAM, started in 2013, we hence went on the hybrid network works initiated during the PhD of N. Cornillet
## 2. Wireless hybrid broadcast/broadband network optimization

<figure><img src="image_89.png" /><figcaption>FIGURE 8.9- Hybrid network model with partial coverage of the broadcast station.</figcaption></figure>

by integrating broadcast coverage adaptation. Our aim was to provide new theoretical results in term of hybrid network capacity and energy consumption and show in what extend a joint broadband/broadcast operation could lead to a better service coverage.

## 2.2 HYDRID NETWORK WITH ADAPTIVE BROADCAST COVERAGE

### ► Network model

As in the previous study, the hybrid network consists of two Orthogonal Frequency Division Multiplexing (OFDM) systems : a broadcasting system composed of a single High Power High Tower (HPHT) site located at the center of the service area and a mobile broadband system composed of $N_c$ Low Power Low Tower (LPLT) sites. Based on the last discussions about the second digital dividend on the UHF broadcasting band around 2013, we will consider in this study that both the mobile broadband and broadcast systems operate in the UHF 700 MHz band which may be shared between the broadcast and the broadband networks. However, the two systems are supposed to be operated in two different frequency subbands, without any mutual interference. The broadcast system will benefit from a bandwidth $B^{(B)}$, while the each unicast cell will distribute between the users a number of $N_{RB}$ resource blocks each having a bandwidth $B_{RB}^{(U)}$.

Contrary to the hybrid network model studied in the previous section, we consider in this work that the area of interest of radius $R_{max}$ is jointly covered using the broadcast and broadband networks as represented Fig. 8.9. The broadcast station is assumed to have a coverage area of radius $R_B \le R_{max}$. As such, the overall coverage is ensured for a common set of services to M users, some $M^{(B)}$ users being served by the central broadcast station, the other $M^{(U)}$ users located at outside the broadcast range being attached to the cellular base stations upon a unicast transmission mode. It is assumed that the service is always available and that the M considered users are demanding the service. A minimum capacity $C_{req}$ is required for each user to properly receive the proposed service. This threshold capacity will hence govern the coverage area of the broadcast station, but also the spectral resource sharing for each unicast cell. In the latter case namely, a total

We are then interested in the study of the optimal planning parameters of the broadcast network i.e. the optimal broadcast coverage radius that maximizes a given utility function. We can for instance aim at maximizing the service capacity or the energy efficiency of the hybrid network given a targeted service rate. The useful metrics are introduced in the following paragraphs.
## ► Average Success Service Rate

Let us first define the average service capacity in the hybrid network. Such a definition was not used in the previous study. The service success rate $η_s$ is defined as the number of users that have access to the proposed service over the total number M of users in the service area. Since the broadcast system is planned such that all users in the broadcast coverage area can access the service, therefore all users attached to the HPHT site have access to the service. However, for the unicast system, when the cells are overloaded, the access to the service is denied for some users. This is taken into account through the service success rate given at the level of the hybrid network by

$$ \mathbb{E}[\eta_s] = \frac{1}{M} \left( \mathbb{E}[M^{(B)}] + \sum_{i=1}^{N_c} \mathbb{E}[\eta_i] \right). \quad (8.11) $$

in which $η_i$ is the success rate for the ith cell. In practice, the number of users associated to each network component changes over time, hence the expectation. Considering a uniform distribution of the users in the network, a first contribution has been to derive a closed-form for the average service success rate as discussed later.

## ► Average Service Capacity

From the M users demanding the set of services in the hybrid network, the average service capacity is defined as the total average achievable rate gathering the contributive part of both components as,

$$ \mathbb{E}[C_s^{(H)}] = \mathbb{E}[C^{(B)}] + \sum_{i=1}^{N_c} \mathbb{E}[C_i^{(U)}] \quad (8.12) $$

where $C^{(B)}$ is the total achieved capacity of the broadcast mode and $C_i^{(U)}$ is the capacity of the unicast mode in the ith cell of the wireless broadband network. The achieved rate of the broadcast mode will be directly influenced by the selected reference SNR $\Gamma_0^{(B)}$ yielding the coverage radius and the targeted spectral efficiency. As for the broadband network, the achieved rate will depend on the spectral resource sharing between users in each cell of the network, that is on the service success rate defined above.

## ► Energy Efficiency

Complementary to our previous study on hybrid network optimization, we have integrated here an accurate power consumption model extracted from the results of the EARTH project [RECEH13a]. According to [RECEH13a, DDG+12], base stations represent a dominant share of the total power consumption in unicast and broadcast networks. The proposed power model is based on a combination of the power consumption of base station components and sub-components such as analog Radio Frequency (RF), baseband (BB) processing, power amplifier and the power system (cooling). Furthermore, the power model uses a linear function that maps the consumed input power P of a base station to achieve a certain RF output power Pmax at the antenna. Therefore, the power consumption Pin of a base station can be obtained from the following function,

$$ P_{in} = \begin{cases} P_0 + \Delta_p P_{max} \rho & \text{if station in ON} \\ P_{sleep} & \text{otherwise} \end{cases} \quad (8.13) $$

where $P_0$ is a linear model parameter that represents the power consumption at the zero RF output power, $P_{max}$ is the maximum RF output power at the antenna, $P_{sleep}$ is the power consumption in sleep mode, $\Delta_p$ is the slope of the load-dependent power model and depends on the type of the base station, and $\rho$ is the ratio of the number of carriers used to the total number of carriers, also called as the resource usage ratio. As an example, 8.2 shows power consumption model parameters for the different types of base stations used in this study, i.e. LPLT and HPHT transmitters.

<table><caption>TABLE 8.2- Power consumption parameters for the LPLT and HPHT transmitters.</caption><thead><tr><th></th><th>P<sub>max</sub> [W]</th><th>Δ<sub>p</sub></th><th>P<sub>0</sub> [W]</th><th>P<sub>sleep</sub> [W]</th></tr></thead><tbody><tr><td>LPLT<sup>1</sup></td><td>40</td><td>4.75</td><td>260</td><td>150</td></tr><tr><td>HPHT<sup>2</sup></td><td>5000</td><td>3</td><td>260</td><td>150</td></tr></tbody></table>
## 2. Wireless hybrid broadcast/broadband network optimization

HPHT transmitters. Note that the resource usage ratio $\rho$ should be set to 1, since all available subcarriers are used for data transmission in a broadcasting system. From such a model, the average power consumption of the hybrid network for the service delivery can be expressed as,

$$P_s^{(H)} = P_{in}^{(B)} + \sum_{i=1}^{N_c} P_{in,i}^{(U)} \qquad (8.14)$$

## 2.3 SERVICE CAPACITY MAXIMIZATION

### ► Optimization Problem

We have first investigated the hybrid network optimization with adaptive broadcast range in the perspective of service capacity maximization. In other words, we have tried to show in what extend the cooperation between the unicast and broadcast components leverages the overall achievable rate of the network. Some intuitive reasoning can already be applied to understand the origin of the potential capacity gain when cooperation is processed. As stated in the previous section, the broadcasting system is usually planned to cover the entire service area using the worst case scenario, i.e. taking basis on the boundary SNR value. This planning strategy leads to a low service capacity with a high transmission power of the HPHT site. Furthermore, from (8.11) it can be seen that when the number of users in the unicast system increases, the service success rate decreases since the number of RBs available in a LPLT cell is limited. Based on these observations, we understand that finding an adequate coverage of the HPHT site should lead to an increase of the service capacity of the hybrid network subject to a targeted service success rate $\eta_{ref}$. Therefore the service capacity maximization problem we have treated in our work can be stated as

$$ \begin{array}{ll} \mathcal{P}1: & \displaystyle \max_{0 \le r_B \le 1} \mathbb{E}[C_s^H] \\ \text{subject to} & \mathbb{E}[\eta_s] \ge \eta_{ref}, \end{array} \qquad (8.15) $$

where $r_B = R_B/R_{max}$ is the normalized broadcast coverage radius with respect to the entire area of interest. The maximization of the service capacity in Problem $\mathcal{P}1$ can be viewed as the optimization of the offloading capacity of the hybrid network from its unicast to its broadcast component. Reversely, it can be understood as a maximization of the coverage area of the hybrid network, using the broadband component as a coverage extension means.

### ► Analytic formulation

To solve Problem $\mathcal{P}1$, we need to seek a more convenient analytical expression of the hybrid service capacity and service rate. As far as the broadcast component is considered, a straightforward expression of the service rate can be given assuming a uniform distribution of the users in the area of interest,

$$ \mathbb{E}[C_s^{(B)}] = Mr_B^2 B^{(B)} \log_2 \left( 1 + \Gamma_0^{(B)} r_B^{-\alpha_B} \right) \qquad (8.16) $$

where $\alpha_B$ is the path-loss exponent associated to the broadcast network. In the cellular system, the computation needs more effort since we need to express the average capacity $C_i^{(U)}$ of a cell in the network (see (8.12)). As detailed in [R.3], this can be achieved by first deriving the average achievable rate for one resource block RB and then getting the expression of the service success rate per cell. The main results we obtained are given in the following Contrib. IV.6 for a uniform distribution of the users. This result shows that the achievable rate corresponds to the minimal rate obtained at the boundary of the cells (the left term in (8.17)) augmented by an average rate given by the Beta function (the right term). This latter term from the uniform user distribution assumption. A different distribution would lead to a modified expression of such a term.

The expression given in Contrib. IV.6 is at the basis of any further statistical performance analysis at the level of the unicast cells. In particular, we have then derived the average service success rate per unicast cell, as given in the Contrib. IV.7. However, getting such a result theoretically depends on the resource allocation strategy carried out by the unicast station to serve multiple users in the cell, namely distributing the available
resource blocks among them. In a first attempt, a uniform allocation of the resource blocks among the users can be assumed. This approach can be seen as non realistic compared to practical allocation mechanisms, but we have shown through simulations that the impact of the average service rate is negligible [C.17], at least when the number of users is sufficiently high. Uniform RB allocation means that each user is assigned a fixed number of resource blocks based on the average capacity per resource block computed in Contrib. IV.6 and the total number of available resource block $N_{RB}^{(U)}$. Upon such assumption, the derivation of the average success rate per cell becomes tractable and yields the expression given in Contrib. IV.7 and detailed in [R.3].

## Contribution IV.6 (Analytic Expression)

### Average capacity of resource blocks with uniform user distribution

Given a uniform distribution of the users in the service area, and considering a minimum reference SNR $\Gamma_0^{(U)}$ at the boundary of the cells, the average capacity of a resource block in unicast mode is,

$$ \mathbb{E}[C_{RB}^{(U)}] = B_{RB}^{(U)} \log_2(1 + \Gamma_0^{(U)}) + (\Gamma_0^{(U)})^{\alpha_U} \frac{B_{RB}^{U}}{\log 2} B\left(\frac{1}{1 + \Gamma_0^{(U)}}, \frac{2}{\alpha_U}, 1 - \frac{2}{\alpha_U}\right) \quad (8.17) $$

with $\alpha_U$ is the path-loss exponent in unicast mode, $B_{RB}^{(U)}$ the RB bandwidth and $B(.,..,.)$ the incomplete Beta function.

## Contribution IV.7 (Analytic Expression)

### Average service success rate per cell with uniform user distribution and resource allocation

Given a uniform distribution of M users in the service area and considering a unicast transmission mode based on a uniform resource allocation of the $N_{RB}^{(U)}$ resource blocks in each cell of radius $R_U$, the average service success rate per cell based upon a unicast transmission mode is,

$$ \mathbb{E}[\eta_i^{(U)}] = \min \left( 1, \frac{N_{RB} \mathbb{E}[C_{RB}^{(U)}]}{M r_U^2 C_{req}} \right) \quad (8.18) $$

where $r_U = R_U/R_{max}$ is the normalized cell coverage radius with respect to the entire area of interest, $C_{req}$ is the required service capacity, $N_{RB}$ is the total number of resource block per cell and $\mathbb{E}[C_{RB}^{(U)}]$ is given by Contrib. IV.6.

This expression of the average success rate yields a simple interpretation. If the number of users M is sufficiently low, the number $N_{RB}$ of available resource blocks is higher than the number of required resource blocks to serve the users regarding the average rate given by Contrib. IV.6. Consequently, the ratio involved in the min function is greater than 1, hence leading to a 100% success rate. Otherwise, the ratio is less than 1 and directly gives the effective success rate.

From the latter two contributions, we have shown in [R.3] that Problem P1 can be restated into the following analytic form,

$$ \begin{array}{ll} \mathcal{P}1: & \displaystyle \max_{0 \le r_B \le 1} M \left[ r_B^2 B^{(B)} \log_2 \left( 1 + \Gamma_0^{(B)} r_B^{-\alpha_B} \right) + C_{req} (1 - r_B^2) \mathbb{E}[\eta_i^{(U)}] \right] \quad (8.19) \\ & \text{subject to} \\ & r_B^2 + \mathbb{E}[\eta_i^{(U)}] (1 - r_B^2) \ge \eta_{ref}, \end{array} $$

### ► Problem solution

A major contribution of our work on broadband/broadcast hybrid network optimization has been to derive the closed-form solution to the above Problem P1. A first important result was to prove the existence of an