Anne Girault

Marie Ferrua

Benoît Lalloué

Claude Sicotte

Aude Fourcade

Fatima Yatim

Guillaume Hébert

Mario Di Palma

Etienne Minvielle

Internet-based technologies to improve cancer care coordination: Current use and attitudes among cancer patients

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Today, a number of important changes are altering cancer care delivery. We observe an increase in outpatient care services coupled with a decrease in inpatient care. Clinical advances have improved the survival rates for most cancers, leading health professionals to treat cancer as a chronic disease. With oral therapies, more cancer patients, even during active treatment, can also be cared for from home. These changes, which could save important costs in a context of high financial constraints, are, however, challenging current cancer care coordination practices [START_REF]ASCO Report : The state of Cancer Care in America[END_REF], [START_REF] Kelly | Delivering maximum clinical benefit at an affordable price: engaging stakeholders in cancer care[END_REF]. Better coordination between patients and providers and among providers, is utterly needed to ensure adequate follow-up of patients.

Internet-based technologies (IBT) such as patient portals, websites and applications, managed by healthcare institutions, have therefore been recognized as a significant lever to improve cancer care coordination practices [START_REF] Clauser | Improving modern cancer care through information technology[END_REF].

A recent systematic review conducted in the US, in 2013, highlighted the four recurrent components of cancer coordination: 1/ roles and models for communication and transfer of care between primary care physicians and oncologists during active treatment and survivorship; 2/ care navigation through designated personnel or telecommunication processes among care team members; 3/ treatment summaries and survivorship care plans; and 4/ multidisciplinary communication accompanying patient and practice management within the framework of the Chronic Care Model [START_REF] Gorin | Cancer care coordination systematic review and meta-analysis: Twenty-two years of empirical studies[END_REF], [START_REF] Bodenheimer | Improving primary care for patients with chronic illness[END_REF].

In light of this, IBT can bring valuable opportunities to improve cancer care coordination by enhancing patient-provider communication, by monitoring adverse events and by providing better patient follow-up at distance [START_REF] Lewis | Patients' and healthcare professionals' views of cancer follow-up: systematic review[END_REF], [START_REF] Atherton | Email for clinical communication between patients/caregivers and healthcare professionals (Review)[END_REF].

However, more evidence is needed regarding cancer patient's current use and willingness to use IBT to monitor their health. First, it is important to know more about their physical connectivity to Internet. In the general population, although physical connectivity to Internet remains a concern, the "digital divide" is narrowing. For instance, in the EU28, 79% of households had access to the internet in 2013 and 76% had a broadband internet connection, compared with 55% and 42% respectively in 2007 [START_REF] Eurostat | Accès et Utilisation d'internet en 2013[END_REF]. Nevertheless, cancer patients can have specific characteristics compared to the general population, especially as cancer patients tend to be older [START_REF] Chou | Health-related Internet use among cancer survivors: data from the Health Information National Trends Survey, 2003-2008[END_REF]. Secondly, it is required to understand the attitudes regarding computers, internet and applications as they may play an important role in the willingness to use them for their health [START_REF] Baker | Use of the Internet and E-mail for Health Care Information[END_REF], [START_REF] Davis | Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology[END_REF]. Thirdly, the question of the influence of social inequalities has to be addressed. In the literature, the most frequent sociodemographic factors found to be predictors of IBT use were age, education, socioeconomic status, gender, place of living and social isolation [START_REF] Eysenbach | The impact of the Internet on cancer outcomes[END_REF]- [START_REF] Or | A systematic review of patient acceptance of consumer health information technology[END_REF].

Scope of the study

Patients and Methods

A questionnaire-based survey was conducted in June 2013 in Gustave Roussy within seven outpatient departments (medical oncology for prostate, breast, skin, head and neck, endocrine, gastric and cervical cancers, radiotherapy, radiology, anesthesia, hematology).

Setting

Gustave Roussy

Gustave Roussy is the largest comprehensive cancer center in Europe, and is located in the suburbs of Paris. The hospital cares for about 50,000 cancer patients annually. Patients come from the full range of socioeconomic backgrounds.

Project Scope

This study was conducted in response to the needs of the CAPRI (Cancer, Parcours de Soins, Région Ile de France) project that is in the process of being implemented at Gustave Roussy (2013-2016). Its objective is to improve the quality of care and coordination for cancer patients treated in the hospital through the implementation of a coordination platform based on two nurse navigators and a dedicated online portal in order to improve the coordination of patients across the care continuum.

Questionnaire design

The questionnaire was built upon a preliminary literature review conducted between January and June 2013 on Google Scholar™, Web of knowledge™ and Pubmed™. It consisted of a total of 38 multiple-choice questions and one open question. The questionnaire comprised three parts: (i) Use of internet through computers, mobile phones and tablets (ii) Willingness to use information technologies for their health, (iii) Socio-demographics.

Survey items asked for patients' use of information technologies (computer, mobile phones, tablets) for accessing to the Internet and their intention to use these devices for health care.

To investigate whether different groups within the population had different patterns of use, we selected five socio-demographic variables including age, gender, socioeconomic status (based on employment status), number of people in the household, and the type of locality they live in (rural/urban). Intention to use IBT for different services in health was measured using a 5-point Likert scale ranging from 1 (definitely not useful) to 5 (definitely useful). The services were: provision of information about disease and treatment, provision of information about care and support, peer communication in support groups, patient-provider communication by e-consultation, autonomous symptom monitoring, and symptom monitoring with e-monitoring of the care provider.

To ensure validity in this method, we tested the questionnaire through face validity, using two complementary approaches. First, we gathered a panel of experts (2 physicians, 2 pharmacists, 2 nurses and 2 senior researchers) to evaluate the questionnaire. Following their suggestions, minor modifications were made. Second, content validity was then checked by passing the questionnaire to a group of patients (n=20) within Gustave Roussy to ensure the questions were relevant and properly answered by patients. Different adjustments were made after analysis of the comments. Eventually, the final draft of the questionnaire was reviewed in consultation with a statistician to ensure that the questions could be coded appropriately for data analysis.

Data collection

Patient eligibility

Patients over the age of 18 were asked by front desk staff if they were willing to complete an anonymous questionnaire regarding their current use and attitudes toward new information technologies while waiting for their appointment. The medical staff of Gustave Roussy was asked to give his consent before and patient consent was necessary to be enrolled in the study.

Survey administration

Copies of a self-administered questionnaire were distributed by front desk outpatient clinic staff to patients checking in for appointments with their physician at Gustave Roussy. A box was displayed to return the questionnaires, once completed. A member of staff was available to answer questions the patients may have. The staff was well informed about the objectives of the study. The questionnaire was distributed between June 6 th and June 14 th , 2013, during seven non-consecutive days.

Ethical consent

The study received a certificate, delivered by CNIL (The French national board which enforces law on data protection) on June 4 th , 2013 to guarantee personal data protection.

Consent to participate was implied by the completion and return of the questionnaire. The questionnaires were fully anonymous.

Analysis

Survey data were analyzed with an optical scanner. Statistical analyses were performed using R. Results were considered significant at α= 0.01.

We first computed descriptive statistics based on survey responses. Then, an analysis based on spearman coefficients and Fisher's exact tests was conducted to investigate correlations between characteristics of patients' IBT usage and attitudes, and their age, gender, socioeconomic status, social isolation (number of people in the household), and place of living (urban/rural). For the multivariate analysis, multinomial logistic regressions were done including socioeconomic status and age as independent variables. The outcome variables for the first set of regressions was "frequency of use of mobile phone", "frequency of use of smartphone", and "frequency of use of computer". The outcome variable of the last regression was the perceived ease of use of IT devices by patients ("I feel able to use a computer, a smartphone or a tablet"). Likelihood ratio tests were conducted to ensure for the goodness of fit of the models. In the models, socioeconomic status was coded as a categorical variable. The manager category served as the reference group.

Results

Sample characteristics

Of the 1609 questionnaires distributed, 1371 were returned. The participation level was 85%.

Questionnaire with more than five missing answers were excluded from the survey. Finally, 1072 questionnaires were selected (final response rate = 67%).

The study sample consisted of 1072 questionnaires. Respondents to the survey were 53 years old in average. 70% of them were women. Participants represented a broad spectrum of professional categories. (see Table 1) "Insert Table 1 here"

The sample was consistent with the hospital population. As a comparison, patients coming to Gustave Roussy in 2012, had a median age of 56 years and 62% of them were women.

Use

71% of patients used a mobile phone every day. Computers were also used on a daily basis by a majority of the study sample (63%), or a few times a week (21%). Tablets and smartphones were still minority but the proportion of people using it is supposed to grow quickly for a year to another. Smartphone users accounted for about 40% of the population.

Most patients reported never using tablets (see Figure 1). "Insert Figure 1 here" 93% of our diverse population accessed the Internet from home. Among them, 68% used Internet every day. Only 7% of them didn't have access at home.

Influence of socio-demographics

Age and employment status were significantly associated with the frequency of use of mobile phones, smartphones and computers (p<0.05). The respective correlation coefficients were negative and moderate (coefficients between -0.25 et -0.49).

Age and employment status were then included in the logistic models.

"Insert Table 2 here"

As shown in table 2, age and employment status were predictors of the frequency of use of mobile phones, smartphones and computers. The odds of using a mobile phone or a smartphone every day rather than less than once a week decreased by 5% for one-unit increase in age. The odds of using a mobile phone every day compared to less than once a week decreased by 68% for a retired compared to a manager. The odds of using a computer every day compared to less than once a week decreased by 77% for a worker or a retired compared to a manager and by 71% for an unemployed.

Intention to use IBT for health care

Patient attitudes toward e-health activities were mainly positive. A significant majority valued the use of IBT in health care, and especially, the possibility to enhance communication with providers (see Table 3).

"Insert Table 3 here"

Communication with physicians via video was not perceived as useful by a majority of respondents (44%). The opportunity to chat with peer patients was not seen as really important to most of them (only 44%). Otherwise, no significant gap among attitudes between the different patient groups could be identified.

Perceived ease of use

Above all, patients were not reluctant to use IBT. 84% of the population studied declared to be able to use a computer, a tablet or a smartphone. Only 8% disagreed.

Influence of socio-demographics

Age and employment status were significantly associated with the perceived ease of use of mobile phones, smartphones and tablets (p<0.05). The respective correlation coefficients were negative and moderate (coefficients between -0.25 et -0.49). Age and employment status were then included in the logistic models.

"Insert Table 4 here"

As shown in table 4, Perceptions regarding the ability to use IBT devices were negatively associated with age and employment status. The odds of agreeing with the sentence "I feel able to use such devices" decreased by 7% for one-unit increase in age. The odds of feeling able to use the devices decreased by 76% for a worker, 75% for a retired, and 82% for an unemployed compared to a manager.

Except that, the only issue raised by respondents was the question of data confidentiality. A third of the population (32%) reported being worried about this matter.

Overall, a majority of the patients included in our study sample were willing to use IBT for their health care.

Discussion

The first findings indicated that access and use of IBT were widespread in the population.

These proportions align with the projections that had been made for developed countries. It is estimated that there were six billion mobile phones in 2013, with over 85% of the world's population having access to a mobile signal [START_REF] Kwok | Phoning in data[END_REF]. It is less evident for tablets and smartphones (46%) which are still used by a minority, even if the figures could change rapidly (e.g. the proportion of people owning a smartphone has doubled between 2012 and 2013 in the general population in France [START_REF] Credoc | La diffusion des technologies de l'information et de la communication dans la société française[END_REF]).

The second findings were related to patient willingness to use IBT for their health. 80% of respondents considered the possibility to get an improved access to their medical records as a priority. It is something observed for other clinical conditions over the last decade [START_REF] Ross | The Effects of Promoting Patient Access to Medical Records : A Review[END_REF], yet far fewer (7%) had experience doing so [START_REF] Turley | Association between personal health record enrollment and patient loyalty[END_REF]. We can highlight that chatting with peer patients was not necessary according to most patients (54%) even if some blogs have been developed with success over the last decade [START_REF] Coiera | Social networks, social media, and social diseases[END_REF].

The third findings were related to the influence of age and socioeconomic status in both access to and perceived ease of use of IBT. This measure of influence should be considered in a dynamic way (for instance, the percentage of social network users aged fifty-five to sixtyfour rose from 9 percent at the end of 2008 to 43 percent by mid-2010) [START_REF] Topol | The Creative Destruction of Medicine: How the Digital Revolution Will Create Better Health Care[END_REF].

Challenges

Implementation of such IBT required health care organizations to comply with standards to ensure patients a sufficient level of privacy and security when using internet-based technologies [START_REF] Hall | For telehealth to succeed, privacy and security risks must be identified and addressed[END_REF].

The differences in attitudes and beliefs according to patient characteristics advocate for a customized approach. This study shows that older patients are less likely to use web-based tools. As old patients are also less likely to search for information about their cancer [START_REF] Protière | Heterogeneity of cancer patient information-seeking behaviors[END_REF],customization is necessary in order to adapt IBT to their specific needs [START_REF] Minvielle | Managing customization in health care: A framework derived from the services sector literature[END_REF].Moreover, other services can be developed to address patient preferences and provide more customized care such as the opportunities to schedule appointments online and to receive health-related reminders according to their specific needs.

Another challenge could consist in providers fearing that a consequence of IBT could be the cuts in hospital staffing considering the restraint number of inpatients visits. At the same time, the predictions of shortage of oncologists may alleviate the effect [START_REF]ASCO Report : The state of Cancer Care in America[END_REF],

Last, the increase in the use of IBT can alter the doctor-patient relationship. Patients will expect providers to be much more responsive as they would need instant responses. At the same time, they will also need maintain face-to-face hospital contact [START_REF] Bartlett | Developing a useful, user-friendly website for cancer patient follow-up: users' perspectives on ease of access and usefulness[END_REF] so the balance will be hard to find.

Perspectives

As patients are open to use them, IBT could play a significant role in cancer care coordination in the near future.

At the hospital level, IBT could offer opportunities to improve quality of care and save costs by enhancing coordination between professionals throughout the care pathway [START_REF] Zhou | Improved quality at Kaiser Permanente through e-mail between physicians and patients[END_REF]. Focal points for the hospital are the possibility to reduce mails and telephone communications and to limit the number of unjustified readmissions and missed appointments (i.e. patients would not show up for a planned appointment) that are an increasing waste of time and money for both the hospital and the whole healthcare system.

At the patient level, IBT can limit the need for routine in-person outpatient visits, and monitor adverse events associated with chemotherapies. IBT can also provide reassurance by maintaining contact with the clinical team, and by providing useful information to the patients [START_REF] Lewis | Patients' and healthcare professionals' views of cancer follow-up: systematic review[END_REF], [START_REF] Basch | Use of information resources by patients with cancer and their companions[END_REF]- [START_REF] Cappiello | Breast cancer survivors: information and support after treatment[END_REF]. They may offer additional services such as medication refills, appointment scheduling, or access to general medical information [START_REF] Turley | Association between personal health record enrollment and patient loyalty[END_REF].

Another aspect of cancer care redesign could be its transition toward a patient-centered service [START_REF] Schwamm | Telehealth: seven strategies to successfully implement disruptive technology and transform health care[END_REF]. Indeed, IBTs are effective tools to support care customization through online needs assessments and day-to-day patient follow-up via a web-based platform. It is important to notice that these web-based tools can allow a patient to access the information independently and repeatedly; the information is better readable; and the user can link the information to other sources of medical information, available on the Internet. Also, they can be adapted to the patient's wishes and knowledge level by sending e-mails through secure messaging when they want.

Last, IBTs could also be used to facilitate real-time data collection of patients' health status.

Patient-driven data are becoming of paramount importance as they can provide useful information to health professionals [START_REF] Banerjee | Web-Based Patient-Reported Outcomes in Drug Safety and Risk Management[END_REF].

Limitations

Limitations of this study include its sampling from a single center in a metropolitan area, so the results may have limited generalizability, although overall access and intentions were consistent with the results found in other studies made on the general population. In addition, respondents may be influenced by social desirability and be tempted to conform their answers to what is expected from them. This could affect the numbers we got in terms of willingness to use IBT in healthcare [START_REF] Paulhus | Measurement and Control of Response Bias[END_REF].

Conclusion

This study confirmed the possibility for cancer patients to use internet-based technologies for their health. No major obstacles to the development of a cancer care coordination program based on IBT could be identified. The effects of age and socioeconomic status have to be addressed.

  Based on a patient survey, the three objectives of the study were: -To understand the current level of use of IBT (computers, tablets, mobile phones and smartphones) -To assess the intention to use IBT for their health -To determine what socio-demographic criteria could be predictors of the use and willingness to use new IBT in healthcare

Table 1 . Description of respondents Table 1

 1

	Characteristics	Participants
	Age	53.37
	Gender (% women)
	Women	70%
	Men	30%
	# of people in the household (%)
	1	16%
	2	36%
	3 et +	47%
	n/a	1%
	Professionnal categories (%)
	Retired	26.4%
	Manager	22.2%
	Worker	21.8%
	Unemployed	7.4%
	Other	13.6%
	n/a	4.3%
	Localisation (%)
	Urban	78%
	Rural	18%
	n/a	4%

Table 2 . Coefficients of the multinomial logistic regression on frequency of use Frequency of use of a mobile phone Frequency of use of a smartphone

 2

	Table 2

Table 3 . Perceived usefulness of IBT applications Table 3

 3

	Applications	I
		Agree vs disagree	Neutral vs disagree
	Age	-0.07**	-0.03
	Employment
	manager	1	1
	worker	-1.43*	0.33
	retired	-1.37*	0.27
	unemployed	-1.7*	0.28
		*<0.01
		**<0.001

am able to use a computer, a tablet, or a smartphoneTable 4 . Coefficients of the multinomial logistic regression on perceived ease of use

 4

Table 4 Figure 1. Frequency of use per IT device.

 4

	Figure 1
		Mobile Phones	6%	4%	15%				71%
		Smartphones				44%		3%	43%
			Computers	7%	6%	21%		63%
			Tablets					54%		7%	7%	19%
				0%		10%	20%	30%	40%	50%	60%	70%	80%	90% 100%
		Never	Less than once a week	A few times per week	Every day	Missing data
		Never	Less than once a week		A few times per week	Every Day Missing
											data
	Tablets	54,00%	7,00%						7,00%	19,00%	13,00%
	Computers	7,28%	6,44%						21,18%	62,50%	2,61%
	Smartphones 44,00%	1,00%						3,00%	43,00%	10,00%
	Mobile	5,88%	4,29%						14,55%	71,18%	4,10%
	phones

Acknowledgements

The study was supported by the Foundation Philanthropia Lombard-Odier and the French National Agency for Research (ANR-MMO).

Conflict of interest statement

The authors declare that they have no competing interests.


Chao Zhang

Samson Lasaulce

Vineeth Varma

Using Continuous Power Modulation for Exchanging Local Channel State Information

Keywords: Distributed power control, interference channel, iterative water-filling algorithm, global channel state information, power domain channel estimation

Using continuous power modulation for exchanging local channel state information

I. INTRODUCTION

To perform operations such as power control or interference coordination in distributed interference networks, a transmitter requires knowledge about the channel gains or qualities of the different links in presence. Aside from limitations such as complexity, one limitation in the implementation of globally efficient distributed power control policies, is the problem of channel state information (CSI) availability at the transmitter. Typically, global or network performance metrics depend on global CSI, i.e., all the channel gains in presence. While many estimation techniques or protocols to acquire local CSI (i.e., the channel gains of links from all transmitters to a given receiver) at a transmitter are available in literature (see e.g., [START_REF] Ozdemir | Channel estimation for wireless OFDM systems[END_REF] [START_REF] Boudreau | Interference coordination and cancellation for 4G networks[END_REF]), there are not many works providing techniques to acquire global CSI. The existing techniques to acquire global CSI, typically rely on the existence of inter-transmitter signaling channels (see [START_REF] Gjendemsjø | Binary power control for sum rate maximization over multiple interfering links[END_REF] [4] [START_REF] Deghel | Queueing stability and CSI probing of a TDD wireless network with interference alignment[END_REF]), which may be unavailable in practice. Remarkably, it has been shown recently in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] that global CSI1 can be acquired from the sole knowledge of the received signal strength indicator (RSSI) or signal-tointerference plus noise ratio (SINR), and therefore making dedicated feedback and inter-transmitter channels unnecessary. The motivation for assuming such a feedback is given by power control schemes such as the iterative water-filling algorithms (IWFA) [START_REF] Yu | Distributed multiuser power control for digital subscriber lines[END_REF] [START_REF] Scutari | The MIMO iterative waterfilling algorithm[END_REF] which assume a transient or exploration period during which the power control algorithm updates the transmit power. For this, each transmitter updates its power by relying on individual SINR feedback. As shown in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF], SINR feedback is sufficient to allow every transmitter to recover local CSI and exchange it by using the novel idea of power modulation, namely, the transmit power itself is used to embed information about the channel.

In this letter, we present an alternative to the solution given in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] to exchange local CSI. The latter solution involves quantizing local channel gains, power modulating the corresponding bits with discrete power levels, and using a lattice-type decoding scheme at each transmitter, which has to recover the power levels of the others. The CSI exchange technique in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] has a least two advantages: it can be used in wireless systems where only discrete power levels are allowed (see e.g., [START_REF] Sesia | The UMTS Long Term Evolution, From Theory to Practice[END_REF]); it can be made robust to local CSI estimation noise and imperfections on the knowledge of the RSSI (or SINR). However, if continuous power levels are allowed, local CSI is well estimated, and the RSSI quality is good, a much simpler solution can be used namely, to use continuous power modulation (CPM), which is the purpose of this letter. Technically, the technique proposed here distinguishes from [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] by the fact that it uses continuous power modulation (in a very specific way which allows local CSI exchange), and relies on time-sharing (which is especially relevant for power domain channel estimation). Additionally, the proposed technique has the advantage of having very low complexity.

Important notations

II. PROPOSED TECHNIQUE

We consider an interference channel with K transmitterreceiver pairs which are assumed to operate on the same band [START_REF] Gjendemsjø | Binary power control for sum rate maximization over multiple interfering links[END_REF]; the multi-band scenario readily follows and is considered in the section dedicated to simulations (Sec. II). The channel gain of the link between Transmitter i P t1, ..., Ku and Receiver j P t1, ..., Ku is denoted by g ij ě 0 and is assumed to lie in the interval rg min ij , g

p i ptq " K ÿ j"1 a ji ptqr g ji (1)
where

a ji ptq " a ji ptq P max g max ji , a ji ptq ě 0, K ÿ j"1 a ji ptq " 1. (2)
Therefore, the power levels used during the exploration phase 2 conveys information about local CSI. It turns out that local CSI information can be recovered, provided the interference is observed either through RSSI or SINR feedback. Here we consider RSSI feedback, which has the advantage of directly leading to linear estimators. In general, the RS power at Receiver i on time-slot t writes as follows:

ω i ptq " K ÿ j"1 g ji p j ptq `σ2 (3)
where σ 2 is the receive noise variance. But in a real wireless system the RS power is quantized and transmitted through a noisy feedback channel (the corresponding quantizer and channel will be specified in Sec. II). Thus Transmitter i has only access to an observed or a noisy version of ω i ptq, which is denoted by r ω i ptq. To facilitate and make more accurate the local CSI exchange procedure, the used power levels during the exploration phase are imposed to follow a time-sharing rule. This means that during the exploration phase the power level of Transmitter i, is chosen either to follow [START_REF] Ozdemir | Channel estimation for wireless OFDM systems[END_REF] or to be zero. Assuming time-sharing for the exploration phase, Transmitter j is only active when t P tt j `1, t j `2, . . . , t j `Ku with t j :" jpK ´1q . The observed RS power at Transmitter i when Transmitter j is active expresses as: r ω j i ptq " ω j i ptq `zj i,1 ptq " g ji p j ptq `σ2 `zj i,1 ptq " r g ji p j ptq `σ2 `zj i,1 ptq `zj i,2 ptq " r g ji p j ptq `σ2 `zj i ptq (4) 2 Just as for the IWFA, tuning the power levels in the proposed way may induce some optimality loss, but in the scenarios of interest the influence of the exploration phase on the performance is negligible.

where z j i,1 ptq " r ω j i ptq ´ωj i ptq and z j i,2 ptq " pg ji ´r g ji q p j ptq. By substituting p j ptq in (4) by its expanded version (1), it follows that the sequence of RS powers observed by Transmitter i when Transmitter j is active, expresses as:

r ω j i " P j r g j r g ji `zj i `σ2 1 (5)
where ω j i " pω j i pt j `1q, ..., ω j i pt j `Kqq T , r g j " pr g 1j , ..., r g Kj q T , z j i " pz j i pt j `1q, ..., z j i pt j `Kqq T , 1 " p1, 1, ..., 1q T , and

P j " P max ¨a1j pt j `1q g max 1j . . . a Kj pt j `1q g max Kj . . . . . . . . . a 1j pt j `Kq g max 1j . . . a Kj pt j `Kq g max Kj ‹ ‹ ‹ ‹ ‹ ' . (6)
This means that when j is broadcasting from time t j `1 to t j `K, the sequence of transmit powers it uses (as a column vector), over the K time slots, is given by P j r g j . Finally, the local CSI estimate r g j is estimated at Transmitter i as:

p g i j " P j ´1 r g ji ´r ω j i ´σ2 1 ¯. ( 7
)
where p g i kj denotes the estimate of the channel g kj by transmitter i. Of course, writing the above implicitly assumes that the coefficients a ji are chosen properly. Note that more advanced estimators such as the minimum mean square error (MMSE) or maximum likelihood (ML) estimators might be used but here, low complexity is prioritized. Normal requirements in terms of local CSI and RSSI qualities are the targeted operating conditions for the proposed technique. Once a global CSI estimate is available, it becomes possible for Transmitter i, i P t1, ..., Ku, to maximize a common network performance criterion under the form

upp 1 , ..., p K ; p G i q, (8) p
G i " " p g i 1 ¨¨¨p g i K ı
being the global channel matrix estimate (this setup has been coined for distributed CSI and studied in [START_REF] Kerret | Degrees of freedom of the network MIMO channel with distributed CSI[END_REF]).

III. COMPARISON OF THE PROPOSED TECHNIQUE WITH [6]

A. Technical comparison Compared to the local CSI exchange technique provided in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF], the proposed technique has two distinguishing technical features. First, the transmit power during the exploration phase is continuous. Thus, the proposed technique can be seen as a complementary technique to [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] for scenarios in which discrete power levels are not allowed. Additionally, the continuous power is chosen in a very specific manner, namely to be the linear combination of the channel gains. Second, only one transmitter is active at a time during the local CSI exchange phase, which makes the estimation procedure simple, but it is observed to be very efficient via simulations. To understand the underlying problem let us consider a special case K " 3 and Transmitter 1. When three users are active at a time and g 31 ąą g 21 it becomes difficult to recover p 2 from ω 1 ptq " g 11 p 1 ptq `g21 p 2 ptq `g31 p 3 ptq `σ2 . One drawback for only activating one transmitter at a time appears if only SINR feedback is available instead of RSSI feedback. In the presence of SINR feedback, at least two users have to be active at a time to allow information exchange in the power domain.

B. Comparison of CSI exchange phase

If one assumes that the number of required observations has to be equal to the number of unknowns to estimate, the local CSI exchange technique of [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] requires KpK ´1q timeslots. During this CSI exchange phase, regular communication occurs in parallel, but with potentially high interference. Under the same assumption, the technique proposed here requires T " K 2 time-slots for the exchange phase since each transmitter has K channel gains to be exchanged and only one user is active at a time (regular communication is effectively time-division multiple access). For K being respectively equal to 2, 3, and 4, this corresponds to an additional cost in terms of time-slots of 100%, 50%, and 33%. Indeed, the number of effectively interfering users using the same channel (meaning operating on the same frequency band, at the same period of time, in the same geographical area) will typically be small in practice and not exceed 3 or 4, which makes the number of time-slots of the exploration phase reasonable.

C. Simulations

As mentioned in Sec. II, we consider the multi-band case. Here, s will stand for the band index, and S for the number of bands. We assume, for ease of exposition, that the channel gain statistics (path losses) are symmetric over all the bands; which is why in some places, the band index is removed. To be able to compare the technique provided in this letter to the state-ofthe-art technique given in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF], we consider two performance metrics, namely, the estimation signal-to-noise ratio (ESNR) and the sum-rate, which are defined as follows:

ESNR i " Er}G} 2 s Er}G ´p G i } 2 s (9)
where }.} 2 stands for the Frobenius norm and G " rG 1 ¨¨¨G S s is the global channel matrix and the entries of G s are the channel gains for "band s" g s ij , i and j being respectively the row and column indices. The matrix p G i corresponds to the estimate which is available to Transmitter i. The sum-rate is defined as: ˙. The quantizer used by Receiver i produces labels which are sent to Transmitter i through a feedback channel and correctly received with probability 1 ´ε with 0 ď ε ď 1. We also need to define the signal-to-interference ratio (SIR) for the case of K " 2 transmitters:

upp 1 , ..., p K ; Gq " S ÿ s"1 K ÿ i"1 log 2 p1 `SINR s i pp s 1 , ...,
SIRpdBq " 10 log 10 ˆEpg 11 q Epg 21 q ˙" 10 log 10 ˆEpg 22 q Epg 12 q ˙(11)

which can be written when the channel statistics are symmetric (which we assume to be true in our simulation settings). We chose the parameters g min ij " 0.01Epg ij q, g max ij " 5Epg ij q (when i ‰ j) and Epg ii q " 1 for all the simulations in our setting, with Epg ij q determined by the SIR indicated. The channel gain dynamics g max ij g min ij is thus equal to 27 dB, which is a quite typical value in real systems. The ESNR is obtained by averaging over 10 4 realizations for the channel matrix, the channel gains being chosen independently and according to

an exponential law φ ij pg ij q " 1 Epg ij q exp ˆ´g ij Epg ij q ˙(that
is, Rayleigh fading is assumed). The local CSI estimates are assumed to be perfectly known.

Influence of the choice of the parameters a ij ptq. First, we study the impact of the choice of the matrix P j on the ESNR. Denote by A j the matrix whose entries are the coefficients a ij ptq for all t P tt j `1, . . . , t j `Ku, i.e.,

Interestingly, for typical scenarios, the particular choice A j " I only induces a quite small performance loss with respect to the optimal choice e.g., measured in terms of ESNR. In this respect, we perform simulations to compare the ENSR obtained by choosing the best possible A j over that of choosing A j " I, for all j. For S " 1, K " 2, N " 8, ε " 1%, SNRpdBq " 30, and SIRpdBq " 10 our comparison has shown that choosing the best matrix only provides marginal improvements. Indeed, for typical values for the SIR (say above 5 dB), estimating the cross-channel gains reliably lead to matrices which are quite similar to the identity matrix; otherwise, the influence of the cross-channel gains in the sum [START_REF] Ozdemir | Channel estimation for wireless OFDM systems[END_REF] or in the RS power might be dominated by that of the direct channel. Therefore, for the rest of this letter, we will choose A j " I for all j, as this choice results in a very low complexity technique and guarantees the invertibility of the power matrix in [START_REF] Yu | Distributed multiuser power control for digital subscriber lines[END_REF].

Proposed scheme versus state-of-the-art. We compare the performance in terms of sum-rate of our proposed technique, with the state-of-the-art techniques in Fig. 1. We consider the parameters K " 4 (number of users), S " 2 bands, perfect local CSI estimate and SIRpdBq " 10. The RSSI feedback is assumed to be with N " 8 bits and ε " 0.01. We compare the performance measured by the sum-rate using CSI exchange using our proposed scheme, of CPM, with that of: 1) perfect global CSI (ideal case); 2)local CSI exchange using a power modulation based on Lloyd-Max quantization (as in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF]) with 2 or 16 quantization levels; 3) the iterative water-filling algorithm (IWFA). When global CSI is available (perfect or otherwise), we implement a team best response dynamics (BRD)3 to select the power control, where each transmitter uses the CSI available for the BRD. In the case of IWFA, no exchange of local CSI is required (no phase 2), but there is a time taken for the algorithm to converge. This figure demonstrates the performance improvement offered by our proposed modulation technique in terms of sum-rate, and we can observe that the team-BRD with CPM achieves a sumrate that is very close to that with perfect global CSI (which is the ideal case). Team BRD with perfect global CSI Team BRD with CPM Team BRD with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] of 16 power levels Team BRD with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] of 2 power levels IWFA with perfect local CSI Fig. 1: Comparison of average sum-rate obtained by using the proposed technique (continuous power modulation) with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF]. We observe that our technique results in an average sum-rate that is very close to the ideal case of perfect global CSI. N=8, ε=1%, with CPM N=4, ε=5%, with CPM N=1, ε=5%, with CPM N=8, ε=1%, with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] N=4, ε=5%, with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] N=1, ε=5%, with [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] Fig. 2: ESNR against SIR. Using continuous power modulation to exchange local CSI appears to be a relevant choice when the RSSI quality is good or even medium. Under severe conditions (e.g., when only an ACK/NACK-type feedback is available for estimating the channel), quantizing the channel gains and power modulating the corresponding labels as in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] is more appropriate. Fig. 2 represents the ENSR in dB against the SIR in dB for which represents the sum-EE as a function of the SNR and assumes a similar setting to Fig. 1 except that here, SNRpdBq " 30, K " 2 users and S " 1 bands. Three scenarios are considered: pN, εq " p8, 1%q, pN, εq " p4, 5%q, and pN, εq " p1, 5%q. The first scenario corresponds to typical conditions in terms of quality for the RSSI, while the two others correspond to quite severe conditions. The scenario with only one RS power quantization bit can be seen as a scenario with an ACK/NACK feedback. For each scenario, two schemes are compared: the scheme proposed in this letter and the one used in [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] which relies on channel gain quantization (here with 1 or 4 bits), discrete power modulation with L levels and lattice decoding. In Fig. 2 we compare with the case of L " 2.

Fig. 2 clearly shows that the proposed technique provides a performance in terms of ESNR, which is independent of the SIR level; this is one of the effects of using time-sharing in the exploration phase. When the quality of the RSSI is good, it is seen that the proposed technique provides a very significant gain in terms of ESNR; the gain ranges from 10 dB to 25 dB, depending on the SIR level. It is only when the RSSI quality is severely degraded (namely, when pN, εq " p1, 5%q) that the proposed technique does not perform well when compared to the technique of [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] when using L " 2 (as seen in Fig. 2).

IV. CONCLUSION

The two main features of the proposed technique are using continuous power modulation in a very specific manner, and to activate one transmitter at a time. Although continuous power modulation may be thought to be sensitive to imperfect RSSI feedback and imperfect local CSI, it is shown to perform very well under normal conditions in terms of RSSI quality.

  A j " ¨a1j pt j `1q . . . a Kj pt j `1q . . . . . . . . . a 1j pt j `Kq . . . a Kj pt j `Kq ‹ '.

  Ki the local channel gain estimates available to Transmitter i. As mentioned in Sec. I, local CSI can be acquired by using classical estimation techniques which, for example, assume that each transmitter sends to the a pilot or training sequence in the signal domain. One of the key features of the proposed technique is to adjust the power level of Transmitter i on time-slot t P t1, ..., T u, as follows:

	to be constant over each transmitted data frame, a frame being
	composed of several time-slots. Each frame comprises two
	phases. During the first T ě 1 time-slots, transmit power levels
	p 1 ,...,p K vary from time-slot to time-slot and corresponding
	received signal (RS) power measurements are used to estimate
	global CSI (exploration phase), p i being subject to a classical
	power limitation p i P r0, P max s. Once the transmitters have
	acquired a global CSI estimate, they can find the fixed power
	level at which each of them will operate over all the remaining
	time-slots (exploitation phase).
	Let us denote by r g 1i , ..., r g
	max ij s. Channel gains are assumed

It is essential to note that the estimation techniques proposed in the present paper, and in[START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF], differ from classical estimation techniques as they are performed in the power domain, and do not require training or pilot signals. While channel acquisition may take some time, note that regular communication is uninterrupted and occurs in parallel.

Team BRD for sum-rate with power control implies that each transmitter iteratively updates its best transmit power given the other transmit powers until all the powers converge. Since global CSI is available, each transmitter can do this offline by assuming an arbitrary initial power vector and with perfect global CSI, all transmitters will converge to the same equilibrium.