Annemie Somers
email: annemie.somers@uzgent.be

Hugo Robays

Kurt Audenaert

Georges Van Maele

Marc Bogaert

Mirko Petrovic

The use of hypnosedative drugs in a university hospital: has anything changed in ten years time?

Keywords: Hypnosedatives (HSs), prescribing patterns, formulary guidelines, risk factors

Aim:

To investigate the use of hypnosedatives (HSs) before and during hospitalisation; to explore the relationship between their use and various demographic and clinical variables; to compare the results with data from a similar study performed in 2000 with particular interest for the adherence to the hospital formulary guidelines. Methods: Cross-sectional observational survey of 326 hospitalised patients recruited from 10 wards of the Ghent University Hospital, with a patient interview and by evaluation of the medical and nursing files. Results: In 30.7% of the patients the use of a hypnosedative (HS) before admission was reported. According to the patient interview, 33.1% of the patients used a HS during hospitalisation. But, according to the medical and nursing files, the use of HSs in the hospital was 10% higher (43.3%). In 19.4% of the patients who took HSs before admission, their use was discontinued in the hospital. In 15.6% of the patients who took no HS before admission, a HS was started in the hospital, according to the formulary guidelines (data from files). There was a positive correlation between use of HSs in the hospital and older age, longer hospitalisation, not coming from home, a higher number of HSs taken before hospitalisation, sleeping problems emerging during hospitalisation and CNS disorders respectively. In comparison with the year 2000, we registered a slight decrease in the use of HSs during hospitalisation and a decrease in the number of newly started patients.

Conclusions: The prevalence of HS use in our university hospital is high, mostly as a result of continuation of HSs that were started before admission, as there seems to be no general policy of active cessation. Compared to the survey performed ten years ago, fewer patients are newly started on HSs, and when this is the case, the formulary guidelines are followed.

INTRODUCTION

Hypnosedatives (HSs) are widely prescribed for treatment of insomnia and anxiety, but their long-term use is a reason for concern, because of dependency, side effects, and cost [START_REF] Buscemi | The efficacy and safety of drug treatments for chronic insomnia in adults: a meta-analysis of RCTs[END_REF][START_REF] Panneman | The costs of benzodiazepine-associated hospital-treated fall Injuries in the EU: a Pharmo study[END_REF][START_REF] Holbrook | The diagnosis and management of insomnia in clinical practice: a practical evidence-based approach[END_REF]. Older patients are especially at risk for impaired cognitive and movement function, with an increased risk of inappropriate sedation, falls and fractures [START_REF] Tamblyn | A 5-year prospective assessment of the risk associated with individual benzodiazepines and doses in new elderly users[END_REF][START_REF] Petrovic | Is there a rationale for prescription of benzodiazepines in the elderly? Review of the literature[END_REF][START_REF] Glass | Sedative hypnotics in older people with insomnia: meta-analysis of risks and benefits[END_REF].

It is well known that the worldwide use of HSs is high, both inside and outside the hospital [START_REF] Pelissolo | Anxiolytic and hypnotic use by general hospital inpatients. The impact of psychopathology and general medical conditions[END_REF][START_REF] Wancata | Use of anxiolytics and hypnotics in gynecological, surgical and medical departments of general hospitals[END_REF][START_REF] Bell | Initiation of benzodiazepines in the elderly after hospitalization[END_REF][START_REF] Stuffken | The impact of hospitalisation on the initiation and long-term use of benzodiazepines[END_REF]. In Belgium, there is a high prevalence of HSs prescription: surveys conducted in 2001, 2004 and 2008 [11] demonstrated an increasing use of HSs (reported use during the last 2 weeks was 5%, 8% and 9% respectively), and found that the use of HSs increases with age (2004: 17% in patients aged 75 years or more). The Belgian Centre for Pharmacotherapeutic Information in its formulary gives guidance for treatment of acute and chronic insomnia, and advises short term use of benzodiazepines in case non-pharmacological approach is not helpful. However, the choice and dose of benzodiazepines that could be used are not mentioned, and no distinction is made between alprazolam or lorazepam and other benzodiazepines.

In 2000, we registered the use of HSs before and during hospitalisation for 493 patients admitted to ten wards of our hospital. HSs were defined as benzodiazepines and benzodiazepine-like hypnotics (the so called Z-drugs, i.e. zolpidem, zaleplon and zopiclone); we also included two tricyclic antidepressants, i.e. mianserine (ATC N06AX03) and trazodone (ATC N06AX05), as in our hospital, these are only used for HS purposes, as well as the antihistaminic dimetindene (ATC R06AB03). We found that 29% of these patients took a HS at home and that in 14% of these patients the HS was discontinued during their hospital stay. Furthermore, a HS was prescribed in the hospital in 45.2% of the patients in total and in 28.6% of the 493 patients a HS was started [START_REF] Warie | The use of hypnosedative drugs in a university hospital setting[END_REF]. In view of this high consumption, the hospital formulary guidelines (available upon request) were adapted and distributed throughout the hospital. Briefly, in case of sleeping problems, HSs are not recommended; in case of persisting and debilitating inner tension or adjustment problems, an intermediate-acting hypnosedative is recommended (e.g. lormetazepam or lorazepam), in a low dose and for a short period of time (preferably only once, maximum 1 or 2 weeks). These guidelines were published in the hospital formulary in 2003 and in 2005 as a printed version, which was sent to the medical staff and all head nurses. In 2007, an electronic drug bulletin was distributed with precise recommendations in case a HS is necessary, namely to prescribe lormetazepam 1 mg for a short period of time, and to stop newly started treatment before discharge. It was also stipulated in the drug bulletin that gradually discontinuation of HSs started before hospitalisation should be considered.

We repeated the registration in 2009, to evaluate whether the prevalence and dynamics of HS use was changed. Moreover, we wanted to collect more clinical data, in order to search for risk factors for HSs use which could orient us to more targeted actions.

METHODS

Setting and study design

This prospective, observational, monocentric study was initiated by the Pharmacy and Therapeutic Committee of the Ghent University Hospital, Belgium and performed on ten hospital wards after approval of the hospital Ethics Committee. The same definition of HSs was used as in the previous study. The ten wards were those where the registration in the year 2000 was performed. The wards included were cardiology, pneumology, nephrology, psychiatry, physical rehabilitation, maternity, head and neck surgery, thoracic and vascular surgery, plastic surgery and abdominal surgery. Five of these wards had the highest and five wards had the lowest consumption of HSs in the year 2000. The method was identical, namely patient interview (after informed consent) by means of a questionnaire, and examination of the medical and nursing files. The interview used in 2009 was almost identical with the one used in 2000. The examination of the medical and nursing files was somewhat altered due to the implementation of electronic prescribing on half of the ten wards. The patients' health problems were also recorded.

We included only patients who stayed at least two nights in the hospital, and interviewed all consecutively admitted patients to the ten hospital wards, during a six weeks period per ward.

Data collection

The use of HSs before and during hospitalisation was measured by means of a patient interview and by examination of the nursing and medical files. The interview took place on a random day during the patient's hospital stay, mostly about a week after admission but this was different depending on the length of stay. Before the interview, the patient was informed about the purpose of the study and was asked for written informed consent.

The patient interview (available upon request) consisted of 18 questions concerning sleeping problems and HS use before and during hospital stay, and additionally questioned the intended HS use after discharge. The information on HS use was completed with information from the medical and nursing files, done at the moment of patient interview and at the moment of the discharge. In addition, the reason for hospitalisation and the health problems, based on a predefined list of 36 items, were registered. These 36 items were grouped into 9 categories namely cardiovascular, respiratory, gastro-intestinal, metabolic disorders, blood related disorders, infection, malignancy, central nervous system pathology, and orthopaedic and gait disorders. Metabolic disorders mainly concerned diabetes and furthermore acute and chronic renal insufficiency, gout and osteoporosis; central nervous system pathology included amongst others dementia, depression, epilepsy, Parkinson's disease and peripheral neuropathy. Demographic data that were collected concerned age, gender, origin (home, nursing home, other hospital), and length of stay. The data collection was performed by a hospital pharmacist.

Data analysis

The data were processed in Microsoft Access and transposed to SPSS

RESULTS

Demographics and health problems

During the 2.5 months of the survey, 330 patients were asked to participate; four of them did not accept. The demographic data of the 326 included patients are shown in Table 1. This population consisted of 155 females (47.5%) and 171 males (52.5%). Their mean age was 53.5 years (SD 18.6 years). Twelve patients were transferred from a nursing home, 27 from another hospital, and 287 were living at home before admission. The mean length of hospital stay was 19.1 days (SD 31.8 days, median 9 days). Four of the patients died during their hospital stay.

The distribution of the patients over the ten wards as well as the length of stay per ward is presented in Table 2. During the 6 week survey, around 40 patients were included per ward, except for the rehabilitation and the psychiatry ward, because of longer length of stay and less turn-over of patients as a consequence. In addition, the study duration was shorter for the abdominal surgery ward because of a delay in agreement for participation in the study.

Most of the patients included in the study suffered from cardiovascular disorders (31.0%), followed by metabolic disorders (28.5%), malignancy (15.0%), respiratory disorders (14.1%), gastro-intestinal disorders (9.5%), central nervous system disorders (9.2.%), orthopaedic or gait disorders (7.7%), infection (4.9%), or blood related disorders (2.8%).

Sleeping problems and hypnosedative use before admission

The patient interview revealed that 131 of the patients (40.2%) suffered from sleeping problems before hospitalisation; out of these 131 patients, 80 (61.1%) took a HS on a regular basis. Out of the 195 patients who did not report sleeping problems before admission, 20 (10.3%) did use a HS, resulting in a total of 100 (30.7%) patients using HSs before admission (Table 3).

The main reasons for HS use mentioned by the patients were a sleep initiation problem (n=68) or a sleep maintenance problem (n=36), as well as nervousness (n=30), fear (n=7) and pain (n=4). Seventeen percent of the patients that took HSs before admission had been given information on future discontinuation of HS use by the treating physician.

According to the medical and nursing files, HS were used before admission by 108 patients, with a total of 153 HSs. Thirty-three patients used more than 1 HS before admission (22 patients took 2 HSs, 10 patients took 3 HSs and 1 patient took 4 HSs). The most frequently used HSs were alprazolam (21.3%), lormetazepam (18.7%), zolpidem (14.8%) and lorazepam (11.0%). A Fisher's exact test revealed that females reported more sleeping problems before hospitalisation than males (females 47.1% versus males 33.9%, p = 0.017).

Suffering from a metabolic disorder was positively correlated with sleeping problems before hospitalisation (metabolic disorder 52.7% versus no metabolic disorder 32.2%, p = 0.002). A Mann-Withney test showed that patients dealing with sleeping problems before admission were significantly older (p = 0.001).

Sleeping problems and hypnosedative use during hospitalisation

Concerning sleeping problems in the hospital, 84 patients mentioned a combination of sleep initiation and sleep maintenance troubles (25.8%), 57 patients mentioned sleep maintenance problems (17.5%) and 38 patients mentioned sleep initiation problems (11.7%). In total, 179 patients (or 54.9 % of the study population) had difficulties to sleep in the hospital.

According to the patient interview, 80 patients with sleeping problems, and 28 patients without sleeping problems, took a HS during hospitalisation, resulting in a total of 108 patients or 33.1% of the total study population (Table 3). Ninety of these 108 patients were aware of the brand name.

The main reasons for HS use in the hospital mentioned by the patients, were continuation of HS use before admission (n=51), sleeping problems emerging in the hospital (n=36), worsening of already existing sleeping problems in the hospital (n=7), fear (n=19) and pain (n=6). According to the patient interview, 35 patients were newly started on HSs during hospitalisation (i.e. 15.5% of the patients who did not take a HS before admission). Information about discontinuation of HSs in the future was given by the treating physician to 2 patients that were newly started on HSs during hospitalisation (5.7%).

According to the medical and nursing files, HS use during hospitalisation was found for 141 patients (43.3%), with a total of 235 HSs, although not all HSs were used at the same time (Table 3). For 66 of the 141 patients who took a HS, more than one HS was prescribed (46.8%): 45 patients took 2 HSs, 14 took 3 HSs and for 7 patients 4 HSs were prescribed. Use of more than one HS was more frequent on the psychiatry, thoracic and vascular surgery, rehabilitation, plastic surgery and abdominal surgery wards. For 49 HSs in total, it was not clear whether the drug was effectively administered, and for 18 patients, the use was only upon request. Only for 83 HSs, a starting and stopping date was found.

According to the patient files, HSs were newly started in a higher percentage of patients were newly started in HSs in comparison to what was reported in the patient interviews, namely 49 (22.5%), but after exclusion of upon request orders without real HS use and after exclusion of a single administration (e.g. before endoscopic procedures), we found that HSs were newly started in 34 patients were newly started on HSs during hospitalisation (15.6%). Most of these patients were admitted at a surgical ward (53.3%), or at the rehabilitation ward (26.7%).

The most frequently HSs prescribed during hospitalisation were lormetazepam (32.3%), alprazolam (23.4%), and lorazepam (15.3%). For 18 patients (12.8%), the HS was prescribed upon request and for 83 HSs (35.3%) a starting and stopping date was found.

The linear-by-linear association showed a strong correlation between sleeping problems at home, and the number of used HSs before, during, and after hospitalisation. Sleeping problems emerging in the hospital were correlated with the number of HSs used in the hospital (p = 0.002), but not with the number of HSs used before admission (p = 0.053). The Spearman correlation test revealed a positive correlation between HS use (before, during and after hospitalisation) and older age (p < 0.001), and between HS use during hospitalisation and length of stay (p < 0.001). Furthermore, the linear-by-linear association showed a positive correlation between use of HSs in the hospital and not coming from home (p = 0.032), a higher number of HSs taken before hospitalisation (p < 0.001) and CNS disorders (p < 0.001).

The highest prevalence of HS use was found on the rehabilitation, psychiatry, plastic surgery, thoracic and vascular surgery and cardiology wards. When the patients of the maternity ward are excluded (since there was no use of HSs found), we come to a prevalence of nearly 50% of the patients taking a HS during hospitalisation.

Hypnosedative use after discharge

The patient interviews learned that for 27 patients from the 100 patients who declared to take a HS at home, the HS was stopped during hospitalisation (27.0%).

Two of these patients told they wanted to restart the HS, one patient hesitated. For the 73 patients who continued to take the HS in the hospital, 57 patients (76%) told they wanted to continue after discharge, 8 patients (11%) said they wanted to stop, and 8 patients (11%) had not yet decided.

According to the discharge documents, for 21 of the 108 patients who took a HS before admission, the use of HSs was to be discontinued (19.4%), and for 58 patients (17.8%), use of a HS after discharge was envisaged (Table 3). Out of the 21 patients in whom the HS use was stopped, 9 were admitted at a surgical ward and 12 at an internal medicine ward. A linear-by-linear association showed no correlation between sleeping problems in the hospital and intended use of HSs after discharge (p = 0.73).

Comparison between 2000 and 2009

The mean and median age of the patients of 2009 was almost the same as in The percentage of patients using HSs before hospitalisation was 29.0% in the year 2000 and 33.1% in 2009, but this difference was non-significant (Chi-square p = 0.21). There was a significant decrease in newly started patients (28.6% in 2000 versus 15.6% in 2009; p<0.001), and a borderline significant increase in discontinued patients (14.0% in 2000 versus 19.4% in 2009; p = 0.055). Nevertheless, there was no significant decrease in percentage of patients using HSs during hospitalisation (45.2% in 2000 versus 43.3% in 2009; p = 0.58), probably due to the higher percentage of patients using HSs before admission, that continued these drugs in the hospital.

DISCUSSION

Strengths and weaknesses

The strength of this study is that the use of HSs before and during, and the intended use after hospitalisation, were recorded accurately, using different sources namely a combination of patient interview, and evaluation of the medical and nursing files. This gave us the opportunity to report highly reliable data about HS use. Furthermore, we searched for risk factors (medical and demographic data), to identify the patients that had the greatest risk of prolonged HS use.

Another strength is the possibility of comparison with the study performed in the year 2000, by using the same patient interview and by including the same hospital wards, which allowed us to evaluate the evolution of HS use.

Our study has several limitations. First of all, the formulary guideline was introduced only passively (by mail and by publication in the formulary), and was perhaps not enough known by the prescribing physicians. It could have been more effective if it would have been implemented in a more proactive way, e.g. by teaching or by prospective advice. Secondly, the wards were selected according to the HS use in the year 2000, namely five wards with the highest and five wards with the lowest consumption. This means that some wards were excluded, either with routine active cessation or with inappropriate HS therapy (e.g. the geriatric and pneumology ward), which could have biased our results. Therefore, in the future, a hospital wide survey would be more suitable, e.g. when electronic prescribing has been introduced in all hospital wards. Another limitation concerns the data at discharge, since use of HSs after discharge was extracted from the discharge documents, and follow-up of the patient (e.g. by telephone call) was not performed.

Use of HSs before and during hospitalisation

We found a discrepancy between the patient interviews and the medical and nursing files. The lower incidence of HS use reported by the patients could be due to fact that use of a HS as an anxiolytic was not perceived as sleep medication; furthermore some patients, e.g. the severely ill, could not remember properly their medication use, and some HSs were started between the patient interview and the consultation of the patient files. One would expect however that the nursing file would list every administration of drugs during hospitalisation correctly and would therefore be considered as the golden standard considering drug use. However, we noticed that in the nursing and medical files the drugs taken before admission were not always listed correctly (e.g. 'home medications to be continued') and sometimes home medications were kept by the patients themselves (although this is against the agreed procedures). This resulted in loss of information about stopping dates in the patient files, which made correct interpretation of combined HS use very difficult. For instance only for three of the seven patients for whom four HSs were prescribed, we can be sure that the four HSs were combined at the same time. It is therefore likely that the real use of HSs was somewhere in between 33.1% (reported by the patients) and 43.3% (out of the medical and nursing files).

Notwithstanding the difference between patient interview and file registration, the prevalence of HS use before and during hospitalisation to our hospital is high. In some published studies performed in hospitals, a large variability in HS use, from 15.7% to 45%, was reported [7;8;13-17]. The variability of the results could be due to the characteristics of the population studied, the method for data collection and the duration of the studies. Furthermore, some of these studies were not performed recently, and HS use has probably changed over time.

The comparison of the results from almost ten years ago to those now in the same hospital is probably more meaningful, and learns us that HS use during hospitalisation has not decreased, mostly as a result of continuation of HSs started before admission, as there seems to be no general policy of active cessation. The positive news is that in comparison to ten years ago, a smaller number of patients were newly started on HSs in the hospital, and when this was the case, the formulary guidelines were followed.

Risk factors

We explored the risk factors for HS use during hospitalisation. Since we found that information about discontinuation was provided for only 2 patients who were newly started, we can presume that the risk factors are predictive for prolonged use after discharge.

-Concerning demographic characteristics, HS use in the hospital was significantly higher in older patients, in patients who were not coming from home, and in patients who stayed longer in the hospital. The influence of age, length of stay and co-morbidities has also been found in previous studies [9;10;15;16].

-As for diagnosis, there was a positive correlation between use of HSs in the hospital and CNS disorders, as well as the intake of a HS before admission.

-With regard to doctors' characteristics, the HS use before admission could not be studied, but we explored more in detail the changes during hospitalisation, and found that relatively more HSs were newly started, and fewer were actively stopped in patients at surgical wards, in comparison with internal medicine wards. However, this should be interpreted with caution since this was a monocentric study in which not all hospital wards were included.

Actions for the future

Our results indicate that older patients who took HSs before admission and who suffer from co-morbidities (especially psychological disorders) are at risk of prolonged HS use and adverse events as a consequence. Future preventive actions must take this population into consideration. We found that 143 of the 162 HS users (88.3%) were not informed by the treating physician about risk of dependence and were not stimulated to reduce HS use. The distribution of formulary guidelines for HS use has been useful for reducing initiation of HSs in the hospital, but another approach is needed for reduction of the high HS use, over the boundaries of the hospital setting.

As mentioned before, there is no general policy of active cessation of HS use in our hospital, and this is probably the same in many other hospitals. The finding that HSs are not often discontinued in the hospital can perhaps be explained by the fact that hospitalisation in itself leads to anxiety and insomnia. Moreover, abrupt discontinuation is not recommended since this could lead to withdrawal symptoms. Hospitalisation could however be the good moment for drug therapy optimisation, especially for older patients with several co-morbidities [START_REF] Saltvedt | Patterns of drug prescription in a geriatric evaluation and management unit as compared with the general medical wards: a randomised study[END_REF]. Combined HS use should be discouraged to prevent adverse drug events, and clinicians should focus on short-term use in newly started patients and should inform patients about the need to restrict the use in time. HS use could also be decreased by computer based reminders regarding appropriate use [START_REF] Agostini | Improving sedative-hypnotic prescribing in older hospitalized patients: provider-perceived benefits and barriers of a computer-based reminder[END_REF]. Furthermore, a tapering scheme could be proposed to the patient, and implemented after the acute phase of illness, through electronic communication between the hospital and the community-based physicians. This could especially be the case for patients admitted to surgical wards, were immediate discontinuation is perhaps not suitable, but where evaluation of appropriate pharmacotherapy is not routinely applied. This action should be facilitated by the government through dissemination of guidelines for HS use and by setting up pilot projects in hospitals.

Several actions have already been taken in our hospital, including automatic stop orders in the electronic prescribing system for newly started HS use, clinical pharmacists' advice to lower HS doses in case patients are sleeping well and to change long-acting substances to intermediate acting HSs (e.g. a clinical pharmacist has been attached to the ward of thoracic vascular and plastic surgery, and more surgical wards should be foreseen for clinical pharmacy in the future). Furthermore, a protocol for active cessation is being set up, using a fast withdrawal scheme over one week, for patients aged more than 55 years suffering from different pathologies [20;21].

TABLES

  version 17 for statistical analysis. The influence of age, residence before hospitalisation and previous HS use on sleeping problems and on the number of used HSs before and during hospitalisation was assessed by a Chi-Square Test (Fisher's exact test and linear-by linear association) and by logistic regression. The difference in HS use before, during, and after hospitalisation was analysed by the Friedman test and by the Wilcoxon signed-ranks test. The McNemar test was used for determining the difference between sleeping problems before and during hospitalisation. The influence of age and length of stay on sleeping problems was assessed by the Mann-Whitney U-test. The Spearman coefficient was used to analyse the correlation between age and length of stay on the one hand and use of HSs on the other hand. Finally, the difference between HS use in the year 2000 and the year 2009 was assessed by a Chi-square test.

  2000 (mean age: 53.5 years in 2009 and 53.7 in 2000, median age was the same namely 55 years).

Table 1 :

 1 Demographic data of the participating patients (n=326)

	Females	Males	Total
	(n = 155)	(n = 171)	(n = 326)

Table 2 :

 2 Number of patients, mean age, median length of stay (LOS) and percentage of HS users per ward during admission (n=326)

		Number of	Mean age	Median	% of HS
		patients	(years)	LOS (days)	users
	Abdominal surgery	17	62.1	13	52.9%
	Cardiology	49	67.3	9	53.1%
	Head and neck surgery	29	59.8	5	31.0%
	Maternity	42	29.9	4	0.0%
	Nephrology	42	58.0	9	42.9%
	Plastic surgery	28	44.3	13.5	60.7%
	Pneumology	43	54.4	8	30.2%
	Psychiatry	15	48.5	39	73.3%
	Rehabilitation	20	39.6	108	75.0%
	Thoracic and vascular surgery	41	62.2	6	56.1%

Table 3 :

 3 Incidence of HS use: comparison between the year 2000 and 2009

	2000	2009

N.A.: not available (*) percentage calculated on the number of patients who took no HS before admission (**) percentage calculated on the number of patients who took a HS before admission

Table 4 :

 4 Incidence of HS use during hospitalisation, per ward

	Ward	patients	HS users	%	>1 HS
	Abdominal surgery	17	9	52.9	4
	Cardiology	49	26	53.1	7
	Head and neck surgery	29	9	31.0	3
	Maternity	42	0	0	0
	Nephrology	42	18	42.9	6
	Plastic surgery	28	17	60.7	9
	Pneumology	43	13	30.2	5
	Psychiatry	15	11	73.3	9
	Rehabilitation	20	15	75.0	9
	Thoracic and vascular surgery	41	23	56.1	14
	Total	326	141	43.3	66

ACKNOWLEDGEMENTS

The authors are indebted to pharmacist F. Lavreau, for her valuable help with the data collection, to the participating heads of the medical departments, the head nurses and to the participating patients. This study was not funded and the authors have no conflicts of interest.

CONCLUSION

The prevalence of HS use in our university hospital is high, mostly as a result of continuation of HSs that were started before admission. In comparison to ten years ago, HSs are started in fewer patients in the hospital, and when this is the case, the formulary guidelines are followed. There is still a long way to go to reduce the high consumption of HS, and strategies such as warnings through the electronic prescribing system, clinical pharmacy advice and active tapering and discontinuation schemes for targeted patients should be explored.


Abdoulaye Bagayoko
email: abdoulaye.bagayoko@orange-ftgroup.com

Inbar Fijalkow
email: inbar.fijalkow@ensea.fr

Patrick Tortelier
email: patrick.tortelier@orange-ftgroup.com

Power control of spectrum-sharing in fading environment with partial channel state information

This paper addresses the spectrum-sharing for wireless communication where a cognitive or secondary user shares a spectrum with an existing primary user (and interferes with it). We propose two lower bounds, for the primary user mean rate, depending on the channel state information available for the secondary-user power control and on the type of constraint for spectrum access. Several power control policies are investigated and the achieved primary-user mean rates are compared with the lower bounds.

Specially, assuming all pairs of transmitter-receiver are achieving real-time delay-sensitive applications, we propose a novel secondary-user power control policy to ensure for both users, at a given occurrence, predefined minimum instantaneous rates. This power control uses only the secondary-user direct links gains estimations (secondary-to-secondary link and secondary-to-primary link). We take into account all links in the network.

I. INTRODUCTION

When looking at the radio frequency spectrum, all frequencies below 3GHz have been allocated to specific uses [START_REF] Sahai | Fundamental design tradeoffs in cognitive radio systems[END_REF]. However, regulatory bodies in various countries found that most of the radio frequency spectrum is inefficiently utilized. The 2002 report of the Federal Communications Commission (FCC)'s Spectrum Policy Task Force made the recommendation that FCC develops a spectrum policy that allows more flexible access to spectrum [START_REF] Tortelier | Spectrum Policy Task Force Report[END_REF]. Spectrum-sharing, for unlicensed and licensed bands, and cognitive September 9, 2010 DRAFT radio have been proposed as promising solutions for improving the spectrum efficiency. Therefore, these topics have received a lot of attention in the technical papers where it is often a concern of designing spectrum-sharing rules and protocols which allow the systems to share the bandwidth in a way that is efficient and compatible with the incentives of the individual systems, [START_REF] Tortelier | Spectrum Policy Task Force Report[END_REF] to [START_REF] Peha | Approaches to spectrum sharing[END_REF].

Power control for spectrum-sharing users has been widely studied. In particular, [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] investigated the maximum ergodic capacity of a secondary user under joint peak and average interference power constraints at the primary receiver. The optimal power control derived in [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] to achieve the secondary maximum ergodic capacity is function of the channel state information (CSI) of the secondary user and of the link between the secondary transmitter and the primary receiver. However, this optimal power allocation does not take into account the interference of the primary user to the secondary user. Moreover, in non-outage states, the secondary's received power could be weak, providing bad quality to the secondary service. [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF] presents a criterion to design the secondary transmit power control by introducing a primary-capacity-loss constraint (PCLC). This method is shown to be superior over the previous ones in terms of achievable ergodic capacities of both the primary and the secondary links. It protects the primary transmission by ensuring that the maximum ergodic capacity loss of the primary link, due to the secondary transmission, is no greater than some predefined value [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF]. However, to enable the PCLC-based power control, [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF] assumes that not only the CSI of the secondary fading channel and the fading channel from the secondary transmitter to the primary receiver (noted g 22 and g 12 in Fig. 1) are known to the secondary transmitter, but also the CSI of the primary direct links (g 11 and g 21 ). [START_REF] Etkin | Spectrum Sharing in Unlicensed Bands[END_REF] investigates cooperative and non-cooperative scenarios of spectrum-sharing for unlicensed bands. The cooperative assumption may be realistic when the different systems are jointly designed with a common goal. They can be complying with some standard or regulation, or they can be as transmitter-receiver pairs of a single global system. Assuming a selfish behavior (non-cooperative scenario) may be more realistic 1 when systems are competing with one another to gain access to the common medium. However, one can imagine spectrum-sharing for systems that carry out real-time delay-sensitive applications, e.g. voice and video. It is then crucial to guarantee, for a given occurrence, predefined minimum instantaneous rates for both the users.

In this paper, we consider the spectrum-sharing scheme of Fig. 1 where a secondary or cognitive user (CR) shares a spectrum first licensed to a primary one (PR). We investigate the lower bounds of the primary mean rate according to the channel state information available for the secondary power control and to the type of constraint for spectrum access. Theses lower bounds allow us to evaluate the protection performance of different types of power control at the secondary transmitter by comparing the achieved primary mean rate with its lower bounds. In particular, we propose a novel secondary power control policy to ensure for both users, at a given occurrence, predefined minimum instantaneous rates. Contrary to the optimal power controls, derived in [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] and [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF], and the non-cooperative games in [START_REF] Etkin | Spectrum Sharing in Unlicensed Bands[END_REF], the goal of the new allocation strategy is neither to achieve, in any case, maximum possible rate, nor to maximize selfish utilities. The particularity of the new suboptimal allocation strategy is to achieve, in a same frequency band, applications that require a given minimum instantaneous rates. Furthermore, this power control uses only the secondary direct links gains estimations (estimations of g 22 and g 12 ).

The remainder of this paper is organized as follows. In the next section, we describe the system and signals model, our main assumptions and the problem we tackle. We investigate two lower bounds for the primary user mean rate, in section III. Power control for secondary user is considered in section IV.

Finally, conclusions are discussed in section V.

II. PROBLEM FORMULATION

A. System and channel model

We consider the network depicted in figure 1 with two users transmitting in the same frequency band and interfering with each other. The first user (PR) is assumed to be the licensee of the spectrum and is called primary user. The second user (CR) is the secondary user. We assume the fading channels are flat and timediscrete. We define the power gains of direct links by g 11 and g 22 . The power gains of transverse links are noted g 12 and g 21 as depicted in Fig. 1. The estimations of g 11 , g 22 , g 12 and g 21 are respectively noted by ĝ11 , ĝ22 , ĝ12 and ĝ21 . The channels power gains are assumed to be independent and identically distributed according to exponential distribution with parameters λ ij , i, j ∈ {1, 2}. Moreover they are supposed to be stationary, ergodic and mutually independent from the noise. The noise power spectral density is denoted by σ 2 . We assume very simple receivers in which all interfering signals are processed as noise.

Thus, with Gaussian signalling, the instantaneous rates (expressed in nats/s/Hz) of the primary and the secondary users may be expressed as C 1 = log

  1 + p 1 g 11 σ 2 + p 2 g 12   and C 2 = log   1 + p 2 g 22 σ 2 + p 1 g 21   ,
where p 1 and p 2 denote, respectively, the primary user transmit power and the secondary user transmit power. This assumption is somewhat pessimistic, and our results thus form a conservative lower bound.

In practice, some form of multi-user detection allowing for interference suppression or mitigation may be used to enhance the rates achieved. The mean rates are defined as

C 1 E [C 1 ] and C 2 E [C 2 ],
where E [x] denotes the mean of the random variable x.

B. Main goal

We consider a secondary user trying to access a licensed spectrum. We study the impact of its transmission on the reception quality of the primary user. In contrast, the primary user does not care about its interference to the secondary user. We aim to investigate lower bounds for the primary mean rate according to the CSI available for the secondary power control and to the type of constraint for spectrum access. We then compare these bounds to the primary achievable mean rates when the secondary user is performing different power control policies. In particular, we propose a novel power control policy, for the secondary user, when all pairs of transmitter-receiver are achieving real-time delay-sensitive applications.

For simplicity, in the sequel, we assume the primary user performs a constant power control. Therefore, we have p 1 = P1 , where P1 denotes the mean transmit power of the primary user.

1) Lower bounds for the primary user mean rate: the lower bound for the primary user mean rate is investigated in two different spectrum-sharing scenarios:

• the first scenario is called in this paper unconstrained spectrum-sharing. It consists in a theoretical spectrum-sharing where the secondary user is subject to no constraint from the primary user other than the limited-mean-transmit-power constraint. A lower bound for the primary mean rate is derived when secondary user performs a {ĝ 22 , ĝ21 }-dependent power control/scheduling,

• the other scenario is called constrained spectrum-sharing. Secondary transmission is subject to some interference constraints from the primary user. To meet the interference constraints, we assume that the secondary-to-primary link gain estimation is available at the secondary transmitter. A lower bound for the primary mean rate is derived in a more general case when secondary user performs a {ĝ 22 , ĝ21 , ĝ12 }-dependent power control/scheduling.

September 9, 2010 DRAFT

2) Secondary power control: we investigate different power control schemes and compare the primary achievable mean rate to its lower bounds. In particular, we propose an original secondary power control policy with the following requirements:

• the secondary user can only estimate the channel gains g 22 (secondary-to-secondary link) and g 12 (secondary-to-primary link),

• each spectrum user needs given outage performance to achieve its service.

More precisely, we ensure that the secondary transmission meets the following constraints:

Prob g11, g21 (C 1 ≤ C 0 ) ≤ ϵ (1) Prob g11, g21 ( C 2 ≤ C ′ 0 ) ≤ ϵ ′ , ( 2
)
where Prob g11, g21 (x) denotes the probability of event "x" over the distributions of g 11 and g 21 . The given rates C 0 and C ′ 0 are the minimum necessary rates for the services of, respectively, the primary and the secondary users. In general, (1) and (2) ensure that primary and secondary instantaneous rates are greater than C 0 and C ′ 0 most of the time, the occurrence is determined by the maximum outage probabilities ϵ and ϵ ′ .

3) Channel and parameters estimation: For more real channels, one can include path-loss to the means, 1/λ ij , of the channels power gains g ij . The channels gains estimations ĝij and the means values 1/λ ij can be brought to the transmitters thanks to the following protocol. First, transmitter i, for i ∈ {1, 2}, sends a pilot signal of normalized power, then, receivers i and j (j ̸ = i) estimate simultaneously the values of λ ii , λ ji , ĝii and ĝji . Moreover, one can imagine the existence of a low rate control channel that the receivers can use to feed back λ ii , λ ji , ĝii and ĝji , [START_REF] Ghasemi | Fundamental limits of spectrum-sharing in fading environments[END_REF]. Finally, one can also imagine a coordination channel between transmitters that they can use to communicate to each other. So, to perform the proposed power control, as shown farther, secondary user needs to know P1 , λ 11 , λ 21 , ϵ, ϵ ′ , C 0 and C ′ 0 . We assume that P1 , λ 11 , ϵ and C 0 are sent to the secondary user via the coordination channel or by a band manager which mediates between the two parties.

III. LOWER BOUNDS OF THE PRIMARY USER MEAN RATE

In this section, we investigate two lower bounds for the primary user mean rate according to spectrum access constraints and available channel state information at the secondary user transmitter.

A. Unconstrained spectrum-sharing

In this part, we are interested in a scenario of spectrum-sharing where there is neither collaboration between the two users, nor interference or capacity loss constraint. The result will be that, contrary to [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] DRAFT what we could imagine, the optimal power control for the secondary link, does not cause the most harmful interference to the primary transmission. We assume that

E [p 2 ] ≤ P2 , (3)
where P2 denotes the maximum mean transmit power of the secondary user.

Since the secondary user rate C 2 is function of g 22 and g 21 only, we assume that to achieve a desired rate, without an interference constraint, the secondary user performs a power scheduling/control scheme such that the transmit power p 2 can be expressed as:

p 2 = ψ (1) (ĝ 22 , ĝ21 ), (4)
thanks to appropriate techniques to estimate g 22 and g 21 . ψ (1) is a {ĝ 22 , ĝ21 }-dependent function or operator. It includes all power control schemes which depend either on ĝ22 only, or on ĝ21 only, or both ĝ22 and ĝ21 , and constant power control scheme. The primary mean rate can be expressed as:

C 1 = E        log        1 + P1 g 11 g 12 σ 2 g 12 + p 2               .
Thanks to the independence of g 11 , g 12 , g 22 and g 21 , it follows that

C 1 = E g11, g12        E {g22, g21}/{g11, g12}        log        1 + P1 g 11 g 12 σ 2 g 12 + p 2                     
, where E a, b [x] denotes the expectation of the random variable x over the joint distribution of the random variables a and b, while E a/b [x] denotes the expectation of the random variable x over the conditional distribution of a given b.

Moreover, we have:

E {g22, g21}/{g11, g12}        log        1 + P1 g 11 g 12 σ 2 g 12 + p 2               ≥ log        1 + P1 g 11 g 12 σ 2 g 12 + E [p 2 ]        ≥ log   1 + P1 g 11 σ 2 + P2 g 12   ,
where the first inequality is due to Jensen inequality 2 . The second inequality is due to the power constraint 2 Because of the convexity of the x-dependent function log

  1 + A B + x   with A ≥ 0, B ≥ 0 and x ≥ 0.
(3). Finally, we obtain:

C 1 ≥ C (1) 1,min E   log   1 + P1 g 11 σ 2 + P2 g 12     .
The mean rate C

(1)

1,min is achieved for a constant power control from the secondary user: p 2 = P2 . Therefore, in this unconstrained spectrum-sharing, constant power control of the secondary user, p 2 = P2 , achieves the lower bound of the primary mean rate. C

(1) 1,min can be expressed (appendix A) as:

C (1) 1,min = P1 P1 -λ11 λ12 P2 [ exp ( σ 2 λ 11 P1 ) E 1 ( σ 2 λ 11 P1 ) -exp ( σ 2 λ 12 P2 ) E 1 ( σ 2 λ 12 P2 )] , ( 5
)
where the exponential integral function is defined as, [START_REF] Abramowitz | Handbook of mathematical functions[END_REF],

E 1 (x) ∫ +∞ 1 exp (-x t) t dt, x ≥ 0. (6)

B. Constrained spectrum-sharing

Now we investigate a spectrum-sharing scenario where the secondary transmission is subject to some interference constraints in order to protect the primary user. In this case, estimating the secondary-toprimary link gain, g 12 , may be crucial. In general, depending on the type of constraint, primary protection should require different CSI to the secondary transmitter.

1) Primary mean-rate loss constraint: This constraint is useful when improving the primary mean rate is in concern. It consists in setting a maximum loss of the primary mean rate:

C 1,max -C 1 ≤ C 1,loss , (7)
where

C 1,max E [ log ( 1 + P1 g11 σ 2 )]
is the mean rate of the primary user without interfering signal.

C 1,loss denotes the maximum mean-rate loss allowed by the primary user. Maximizing the secondary mean rate, subject to [START_REF] Ghasemi | Capacity of fading channels under spectrum-sharing constraints[END_REF], may require primary link gain estimation ĝ11 , [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF], that might demand sophisticated techniques. In the sequel, we do not use this constraint.

2) Interference constraints: The primary transmission can be also protected by using the dimensions time and space of the spectrum to manage the secondary user interference to the primary receiver. More general spatial spectrum-sharing problem is considered in [START_REF] Gastpar | On Capacity Under Receive and Spatial Spectrum-Sharing Constraints[END_REF]: given two different networks (for instance two MAC), to enable coexistence, we can regulate their transmission power, such that a network may not create an interference that exceeds a prescribed level Q I outside of a predefined zone. For the two-user spectrum-sharing problem, peak and average interference constraints, stated by ( 8) and ( 9), are commonly used to protect the primary transmission, [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] to [START_REF] Ghasemi | Capacity of fading channels under spectrum-sharing constraints[END_REF], :

p 2 g 12 ≤ Q peak (8) E [p 2 g 12 ] ≤ Q avg , (9)
where Q peak denotes the instantaneous interference threshold and Q avg the average interference threshold.

Specially, performing a power control under the instantaneous interference constraint ( 8) requires the secondary-to-primary link gain estimation ĝ12 .

3) Lower bound: In order to protect the primary transmission, we assume that the secondary-to-primary link gain estimation ĝ12 is available for secondary power control. Therefore, to achieve a desired rate under interference constraints, the secondary user performs a power scheduling/control scheme such that the transmit power p 2 can be expressed as:

p 2 = ψ (2) (ĝ 22 , ĝ21 , ĝ12 ), (10)
thanks to appropriate techniques to estimate g 22 , g 21 and g 12 . ψ (2) is a {ĝ 22 , ĝ21 , ĝ12 }-dependent function or operator. It includes all power control schemes that depend either on ĝ22 only, or on ĝ21 only, or on ĝ12 only, or any combination of ĝ22 , ĝ21 , ĝ12 , and constant power control scheme. The primary mean rate verifies:

C 1 = E g11 [ E {g22, g21, g12}/g11 [ log ( 1 + P1 g 11 σ 2 + p 2 g 12 )]
]

≥ E g11 [ log ( 1 + P1 g 11 σ 2 + E [p 2 g 12 ] )] ≥ C (2) 1,min E [ log ( 1 + P1 g 11 σ 2 + Q avg )] ,
where the first inequality is due to Jensen. The second inequality is due to the mean interference power constraint [START_REF] Gastpar | On Capacity Under Receive and Spatial Spectrum-Sharing Constraints[END_REF]. The lower bound C

1,min can be expressed (appendix A) as 3 :

C (2) 1,min = exp ( λ 11 ( σ 2 + Q avg ) P1 ) E 1 ( λ 11 ( σ 2 + Q avg ) P1 ) . ( 11
)

IV. POWER CONTROL FOR SPECTRUM SECONDARY USE

In this section, we investigate secondary user power control and compare the achieved primary mean rate to its lower bounds found previously.

A. Power control with mean-transmit-power constraint only

We assume that there is only one constraint for secondary access to the spectrum: the mean transmit power constraint, stated by 3.

1) Optimal power control: the optimal power control maximizing the secondary mean rate C 2 , under the power constraint (3), is expressed by the well known water filling [START_REF] Cover | Elements of Information Theory[END_REF]:

p 2 =   ζ - σ 2 + P1 g 21 g 22   + , ( 12
)
where the constant ζ is obtained such that the mean power constraint is met. (.) + denotes max (., 0). Let

w g 22 σ 2 + P1 g 21
, the constant ζ is obtained as:

P2 = ∫ +∞ 1 ζ   ζ - 1 w   f W (w) dw, ( 13
)
where f W is the probability density function of the random variable W with sample w. The probability density function of W is given by (appendix A):

f W (w) =                      1 + b + b a w a   1 + 1 a w   2 exp   - b a w   if w ≥ 0 0 if w < 0 (14) with a = λ 21 P1 λ 22 and b = σ 2 λ 21 P1 .
2) A scheduling approximating the optimal power control: the difficulty of performing the optimal power allocation ( 12) is due to the uncertain knowledge of the information w = g22 σ 2 + P1 g21 . Using an adequate estimation technique, assume ŵ is the estimated value of w. We can reduce the impact of estimation errors on the power control (12) by using the following scheduling:

p 2 =            c if ŵ > 1 ζ 0 if ŵ ≤ 1 ζ (15)
where the constant c is obtained such that

E [p 2 ] = P2 = ∫ +∞ 1 ζ c f W (w) dw, September 9, 2010 DRAFT
thus, it can be expressed as:

c = P2 ∫ +∞ 1 ζ f W (w) dw .
Using expression [START_REF] Vu | The primary exclusive region in cognitive networks[END_REF] of f W , we obtain:

∫ +∞ 1 ζ f W (w) dw = λ 21 λ 22 P1 λ 21 λ 22 P1 + 1 ζ exp   - λ 22 σ 2 ζ   . ( 16
)
Therefore, constant c is expressed as:

c = P2   1 + λ 22 λ 21 1 ζ P1   exp   λ 22 σ 2 ζ   . ( 17
)
In the scheduling [START_REF] Bagayoko | Impact of shadowing on the primary exclusive region in cognitive networks[END_REF], constant c does not depend on the channel realizations. Moreover, the binary condition ŵ 1 ζ is less sensitive to the estimation errors. This relatively easy-done scheduling, for the secondary link, should achieve a primary mean rate close to the one achieved using the optimal water-filling.

3) Numerical examples: both the theoretical optimal allocation (12) and the scheduling (15) are functions of the channels gains ĝ22 and ĝ12 . So they have the form of (4). C

(1) 1,min is a lower bound of such kinds of power control/scheduling. Now, we give numerical examples to compare the primary mean rates achieved, using [START_REF] Sahai | Fundamental design tradeoffs in cognitive radio systems[END_REF] and [START_REF] Bagayoko | Impact of shadowing on the primary exclusive region in cognitive networks[END_REF], with the lower bound C As it can be noticed in Fig. 2 and3, the proposed scheduling [START_REF] Bagayoko | Impact of shadowing on the primary exclusive region in cognitive networks[END_REF] provides a performance that is very close to the optimal water-filling. Moreover, we can see the gap level between the lower bound C

(1) 1,min and the considered power controls. The optimal power control at the secondary side does not cause the most harmful interference to the primary transmission, as we should imagine. On the contrary, for same mean power, P1 = P2 = 1 for instance, the optimal water-filling provides nearly 1 nat/s/Hz protection, to the primary user, more than the constant power control (Cf. Fig. 2). These results do not take into account the primary protection since there is no interference constraint.

B. Power control with outage performance requirement and direct links CSI

In this part, we propose a novel power control under the requirements (1) and [START_REF] Goldsmith | Breaking spectrum gridlock with cognitive radios: an information theoretic perspective[END_REF]. We assume that the secondary user can estimate the secondary-to-secondary and the secondary-to-primary links gains only.

That is, only ĝ22 and ĝ12 are available for the secondary user power control. 1) Outage performance constraints: the primary and secondary outage constraints are modeled by [START_REF] Tortelier | Spectrum Policy Task Force Report[END_REF] and [START_REF] Goldsmith | Breaking spectrum gridlock with cognitive radios: an information theoretic perspective[END_REF]. By replacing C 1 and C 2 by theirs formulas, events "C 1 ≤ C 0 " and "C 2 ≤ C ′ 0 " can be expressed, respectively, as:

C 1 ≤ C 0 ⇒ g 11 ≤ α 0 ( σ 2 + p 2 ĝ12 ) P1 , (18)
C 2 ≤ C ′ 0 ⇒ g 21 ≥ 1 P1   p 2 ĝ22 α ′ 0 -σ 2   , ( 19
)
with

α 0 = exp (C 0 ) -1 and α ′ 0 = exp (C ′ 0 ) -1.
The outage probabilities become:

Prob g11, g21 (C 1 ≤ C 0 ) = ∫ γ 0 λ 11 exp (-λ 11 x) dx = 1 -exp (-λ 11 γ) , ( 20
)
where γ = α0 (σ 2 +p2 ĝ12)

P1

, and

Prob g11, g21 ( C 2 ≤ C ′ 0 ) = ∫ +∞ γ ′ λ 21 exp (-λ 21 x) dx = exp ( -λ 21 γ ′ ) , ( 21
) with γ ′ = 1 P1 ( p2 ĝ22 α ′ 0 -σ 2 )
. Then, outage constraints ( 1) and ( 2) can be expressed, respectively, as:

1 -exp   -λ 11 α 0 ( σ 2 + p 2 ĝ12 ) P1   ≤ ϵ, ( 22
) exp   - λ 21 P1   p 2 ĝ22 α ′ 0 -σ 2     ≤ ϵ ′ . ( 23
)
After some manipulations, expressions ( 22) and (23) become

p 2 ĝ12 ≤ Q peak , ( 24
)
p 2 ĝ22 ≥ K. ( 25
)
Where the peak interference threshold is defined as:

Q peak = P1 λ 11 α 0 log   1 1 -ϵ   -σ 2 , ( 26
)
and the minimum received power K as:

K = α ′ 0   σ 2 - P1 λ 21 log ( ϵ ′ )   . ( 27
)
Therefore, the primary outage constraint (1) consists in forcing the instantaneous interference p 2 ĝ12 , from the secondary user, to be lower than a threshold Q peak , while secondary outage constraint (2) consists September 9, 2010 DRAFT in forcing the secondary instantaneous received power p 2 ĝ22 to be greater than a threshold K. For a given network and system, the peak interference threshold Q peak is determined by the primary minimum required rate C 0 , the outage probability ϵ and the mean transmit power P1 . Specially, Q peak is proportional to P1 and log-increasing in ϵ. Otherwise, when the outage probability ϵ ′ increases, the secondary service quality is low, and thus, the threshold K decreases.

2) Power control: previously, we found the constraints ( 24) and (25) to ensure given outage performance to both the primary and the secondary users. In this respect, transmit power p 2 of the secondary user must fulfill the set of inequalities

   p 2 ĝ12 ≤ Q peak p 2 ĝ22 ≥ K (28)
We verify the compatibility of both the equations in (28):

• if

( ĝ22 ĝ12 ≥ K Q peak )
, then 4 power p 2 can be greater than the minimum required p 2,min K ĝ22 . But to meet the interference constraint, power p 2 must always fulfill p 2 ĝ12 ≤ Q peak . So, the cognitive user can opportunistically communicate with

p 2 = Q peak ĝ12 ; • if ( ĝ22 ĝ12 < K Q peak )
, then the minimum power p 2,min can not meet the interference constraint. Consequently, we set p 2 = 0 (CR transmission is off).

However, the maximum transmit power Q peak ĝ12 can be infinitely high (when ĝ12 is very low), while in real system instantaneous transmit power is limited. To alleviate this problem, we set the practical constraint p 2 ≤ p 2,peak . Finally, we propose the following original power control policy:

p 2 =                        p 2,peak if ĝ22 ĝ12 ≥ K Q peak and p 2,peak ≤ Q peak ĝ12 Q peak ĝ12 if ĝ22 ĝ12 ≥ K Q peak and p 2,peak > Q peak ĝ12 0 if ĝ22 ĝ12 < K Q peak ( 29
)
Where p 2,peak is the secondary-user maximum transmit power. Contrary to the optimal power control, derived in [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF] and [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF], and the non-cooperative games in [START_REF] Etkin | Spectrum Sharing in Unlicensed Bands[END_REF], the goal of the allocation strategy (29) is neither to achieve, in any case, maximum possible rate, nor to maximize selfish utilities. But the particularity of (29) is to ensure, at some occurrence predefined by the outage probabilities ϵ and ϵ ′ , at least given minimum instantaneous rates to the two users, while using only the direct links gains 4 When p2 = p 2,min

K ĝ22 , then p2 ĝ12 ≤ Qpeak ⇔ ĝ22 ĝ12 ≥ K Q peak
September 9, 2010 DRAFT estimations ĝ22 and ĝ12 (that is not considered in the previous works such as [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF], [START_REF] Zhang | Optimal power control over fading cognitive radio channel by exploiting primary user CSI[END_REF] and [START_REF] Etkin | Spectrum Sharing in Unlicensed Bands[END_REF]). It is then more appropriate for spectrum-sharing systems that carry out real-time delay-sensitive applications, e.g. voice and video.

Now, we will study some typical parameters of this power control.

3) Mean transmit and mean interference power: now, we study the evolution of the mean transmit power and the mean received interference power, according to the parameters K, p 2,peak and Q peak , which are imposed by the desired performance of the network, and according to the parameters λ 11 , λ 22 , λ 12

and λ 21 , which are imposed by the channel fades.

Let x = ĝ12 and y = ĝ22 . The mean transmit power can be expressed as:

E [p 2 ] = ∫ Q peak p 2,peak 0 ∫ +∞ K Q peak x λ 22 λ 12 p 2,peak exp (-λ 22 y) exp (-λ 12 x) dx dy + ∫ +∞ Q peak p 2,peak ∫ +∞ K Q peak x λ 22 λ 12 Q peak x exp (-λ 22 y) exp (-λ 12 x) dx dy.
After some manipulations (Cf. appendix B), we obtain:

E [p 2 ] = p 2,peak 1 + λ22 λ12 K Q peak   1 -exp   - λ 22 K + λ 12 Q peak p 2,peak     + λ 12 Q peak E 1   λ 22 K + λ 12 Q peak p 2,peak   (30)
The mean received interference power is obtained similarly as follows:

E [p 2 ĝ12 ] = ∫ Q peak p 2,peak 0 ∫ +∞ K Q peak
x λ 22 λ 12 x p 2,peak exp (-λ 22 y) exp (-λ 12 x) dx dy

+ ∫ +∞ Q peak p 2,peak ∫ +∞ K Q peak x λ 22 λ 12 Q peak exp (-λ 22 y) exp (-λ 12 x) dx dy. ( 31
)
After some manipulations (Cf. appendix B), it can be expressed as:

E [p 2 ĝ12 ] = p 2,peak /λ 12 ( 1 + λ22 λ12 K Q peak ) 2   1 -exp   - λ 22 K + λ 12 Q peak p 2,peak     . ( 32
)
Therefore, the mean transmit power E [p 2 ] and the mean interference power E [p 2 ĝ12 ] are connected via the following equation:

E [p 2 ] = λ 12 Q peak     1 + λ 22 λ 12 K Q peak   E [p 2 ĝ12 ] Q peak + E 1   λ 22 K + λ 12 Q peak p 2,peak     . ( 33
)
In practical situations, we assume λ 12 ≥ 1. Therefore, from (33), the mean transmit power is greater than the mean interference power, especially when E 1

( λ22 K+λ12 Q peak p 2,peak
) is high or equivalently when

λ22 K+λ12 Q peak p 2,peak
is low. As we can see below with numerical examples, this situation is profitable because the challenge in spectrum-sharing and cognitive networks is to achieve better services to the secondary user while minimizing the interference towards the licensee-primary user.

4) Overall outage probability: previously, the strategy for the power control ( 29) is stated by firstly set-

ting Prob g11, g21 (C 2 ≤ C ′ 0 ) = ϵ ′ or equivalently p 2 = p 2,min . Then, to transmit if Prob g11, g21 (C 1 ≤ C 0 ) ≤ ϵ.
The overall outage probability of (29) can be expressed as:

P out = Prob ( Prob g11, g21 (C 1 ≤ C 0 ) > ϵ / Prob g11, g21 ( C 2 ≤ C ′ 0 ) = ϵ ′ ) . ( 34
)
Let x = ĝ12 , y = ĝ22 , z = y/x and z 0 = K/Q peak . From (29), the outage probability P out is obtained as follows:

P out = Prob (z < z 0 ) = ∫ z0 0 f Z (z) dz,
where f Z is the probability density function of the ratio ĝ22 /ĝ 12 . The ratio of two independent exponential random variables ĝ22 and ĝ12 , with parameters λ 22 and λ 12 , is a random variable Z with the following probability density function:

f Z (z) = ∫ +∞ 0 x f Y (z x) f X (x) dx = λ 22 λ 12 ∫ +∞ 0 x exp (-(λ 22 z + λ 12 ) x) dx = (λ 12 /λ 22 )   z + λ 12 λ 22   2 (35)
The outage probability is then expressed as:

P out = ∫ z0 0 (λ 12 /λ 22 )   z + λ 12 λ 22   2 dz = 1 - (λ 12 /λ 22 ) λ 12 λ 22 + z 0 .
Finally, we obtain:

P out = K K + λ12 λ22 Q peak . ( 36
)
The outage occurrence depends on the thresholds K and Q peak that model the quality requirements of the services for the two users. The cut-off value z 0 of the ratio ĝ22 /ĝ 12 is function of the outage probability and of the channel parameters λ 22 and λ 12 : z 0 = λ 12 λ 22 P out 1 -P out .

5) Connection with TIFR transmission policy: now, we investigate a special case where the primary-tosecondary link is sufficiently attenuated to neglect the primary interference P1 g 21 to the secondary user.

Such a situation occurs for instance when the secondary receiver is located outside an exclusive region around the primary transmitter, [START_REF] Sahai | Fundamental design tradeoffs in cognitive radio systems[END_REF], [START_REF] Vu | The primary exclusive region in cognitive networks[END_REF], [START_REF] Bagayoko | Impact of shadowing on the primary exclusive region in cognitive networks[END_REF]. In this case, we can define a delay-limited capacity (also referred to as zero-outage capacity) which represents the constant-rate that is achievable in all fading states [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF]. Assuming the secondary user transmits with the minimum required power p 2,min in non-outage states, to fulfill the set of constraints (28) we propose:

p 2 =        K ĝ22 if z ≥ z 0 0 if z < z 0 . ( 37
)
The adaptive transmission technique (37) is called truncated channel inversion with fixed rate (TIFR), [START_REF] Musavian | Ergodic and outage capacities of spectrum-sharing systems in fading channels[END_REF], [START_REF] Goldsmith | Capacity of fading channels with channel side information[END_REF]. Since the secondary user transmits p 2,min in non-outage events, then, power transmission policy (37) is a variant of (29) in which primary user receives always the weakest instantaneous interference.

This case is interesting because it protects, the best, primary user. We derive the mean transmit power of (37) as follows: (

E [p 2 ] = ∫ +∞ 0 ∫ y z 0 0 λ 12 λ 22 K y exp (-λ 12
) Since ∫ y z 0 0 λ 12 exp (-λ 12 x) dx = 1 -exp   -λ 12 y z 0   , 38
we have

E [p 2 ] = ∫ +∞ 0 λ 22 K y exp (-λ 22 y) dy - ∫ +∞ 0 λ 22 K y exp   -   λ 22 + λ 12 z 0   y   dy. ( 39
)
The first integral can be calculated as:

∫ +∞ 0 λ 22 K y exp (-λ 22 y) dy = λ 22 K [ lim y-→0 E 1 (λ 22 y) -lim y-→+∞ E 1 (λ 22 y) ] . ( 40
)
The exponential integral function verifies, [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]:

lim y-→+∞ E 1 (λ 22 y) = 0.
September 9, 2010 DRAFT So, we obtain the following expression for the first integral in (39):

∫ +∞ 0 λ 22 K y exp (-λ 22 y) dy = λ 22 K lim y-→0 E 1 (λ 22 y) .
The second integral has the same form as the first one. Then,

E [p 2 ] = lim y-→0   E 1 (λ 22 y) -E 1   y   λ 22 + λ 12 z 0       λ 22 K.
The exponential integral function E 1 (.) can be approximated around zero, [START_REF] Abramowitz | Handbook of mathematical functions[END_REF], as

E 1 (y) ≈ -γ -log(y), (41)
where γ is the Euler-Mascheroni constant: γ = 0.57721.... Using this closed-form approximation, we obtain a closed-form expression of E [p 2 ] as follows:

E [p 2 ] ≈ λ 22 K log   1 + λ 12 λ 22 1 z 0   . ( 42
)
Therefore, for given mean transmit power E [p 2 ], we can determine the constant received power K as follows:

K = E [p 2 ] λ 22 log   1 + λ 12 λ 22 1 z 0   . ( 43
)
The mean interference power for (37) is derived as:

E [p 2 ĝ12 ] = ∫ +∞ z0 K z f z (z) dz = ∫ +∞ z0 K z (λ 12 /λ 22 )   z + λ 12 λ 22   2 dz = K       λ 22 λ 12 log   1 + λ 12 λ 22 1 z 0   - 1 z 0 + λ 12 λ 22       . ( 44
)
We can express E [p 2 ĝ12 ] in terms of P out as:

E [p 2 ĝ12 ] = λ 22 λ 12 (P out -1 -log (P out )) K. ( 45
)
The zero-outage capacity C 2,out is expressed as:

C 2,out = (1 -P out ) log   1 + K σ 2   . ( 46
)
This capacity is obviously increasing with the mean interference power and the increasing speed is function of P out .

6)

Numerical examples: now we give some numerical examples in order to evaluate some achievable performances of (29). We set P1 = 1 and σ 2 = 0.01. The channel is set as: λ 11 = λ 22 = 1, λ 21 = 5 (in the part IV-B6d, we will neglect the primary-to-secondary link, so λ 21 is not used there) and λ 12 = 10.

That is, we choose to attenuate the secondary-to-primary link in order to avoid very strong interference.

Some authors, e.g. [START_REF] Sahai | Fundamental design tradeoffs in cognitive radio systems[END_REF], [START_REF] Vu | The primary exclusive region in cognitive networks[END_REF], [START_REF] Bagayoko | Impact of shadowing on the primary exclusive region in cognitive networks[END_REF], advocate to set an exclusive region around the primary receiver. No secondary operation is possible inside this range. So we can consider that the choice of λ 12 = 10 ( the value of the channel gain ĝ12 is then 1 λ12 = 0.1) is due to the fact that the secondary transmitter is located outside the primary exclusive region. a) Mean rates: in figures 4 and 5, we plot respectively the primary mean rate and the secondary mean rate, versus the peak interference threshold Q peak for different values of the outage probability P out . We set p 2,peak = 1. As the peak interference threshold increases, the secondary mean rate increases too, and consequently the primary mean rate decreases. For higher Q peak , the cut-off value z 0 is weak and p 2,peak is more likely to be lower than Q peak g12 . Consequently, p 2 = p 2,peak in most of the channel fades. Therefore, primary mean rate is tending to

E [ log ( 1 + P1g11 σ 2 +p 2,peak g12 )]
and secondary mean rate is tending to E

[ log

( 1 + p 2,peak g22 σ 2 + P1 g21 )]
. For given Q peak , secondary mean rate C 2 decreases with P out while primary mean rate C 1 increases.

In figure 6, we compare the primary mean rate C 1 with the lower bound C

1,min . For given P out , when Q avg increases, Q peak increases to 5 . Therefore, we have high occurrence of events p 2,peak ≤ Q peak g12 and p 2 = p 2,peak . As a consequence, primary mean rate is more and more greater than the lower bound C and the mean interference power E [p 2 ĝ12 ] in order to evaluate the ratio between the achievable service for the secondary user and the protection level of the primary user. increases more speedily than E [p 2 ĝ12 ]. Then, we note that the secondary user can achieve important information without causing important interference to the primary user. c) Outage probability: in figure 8, we plot the outage probability, P out , versus the peak interference power Q peak for different values of the minimum received power K. As predicted, when the primary user is less demanding (Q peak is increasing), the outage probability is decreasing. Otherwise, for given Q peak , the more the secondary user is less demanding (K is decreasing), the more it can transmit frequently over the common spectrum (P out is decreasing). In particular, we note that for greater values of Q peak , is neglected. Moreover, figure 10 shows that little mean power is required to achieve C 2,out .

V. CONCLUSIONS

In future wireless communication systems, there will be a need of smart and adequate spectral usage due to the increasing demand in user data rates and to the penury of available spectrum resources. Spectrum sharing and cognitive radio, proposed as promising solutions for improving the spectrum efficiency, will continue to receive a lot of attention. In this paper, we considered the problem of spectrum secondaryuser power control in single-antenna flat-fading channels. The secondary user shares the spectrum with an existing spectrum-licensee or primary user. We found two lower bounds, for the primary mean rate, depending on the secondary user power control scheme. Several power control policies are investigated and the achievable primary mean rates are compared with the lower bounds. In particular, ensuring for each user given outage performance and assuming that only direct links gains estimations (secondaryto-secondary link and secondary-to-primary link) are available at the secondary transmitter, we have proposed an original secondary power control that is useful for real-time delay-sensitive applications.

APPENDIX A LOWER BOUNDS OF THE PRIMARY MEAN RATE

In this section we calculate the following integrals:

C (1) 1,min = E   log   1 + X σ 2 + Y     (47) C (2) 1,min = E   log   1 + X σ 2 + Q avg     , ( 48
)
where X = P1 g 11 and Y = P2 g 12 are exponentially distributed with parameters λ 11 P1 and λ 12 P2 .

A. Lower bounds C (1) 1,min

To calculate the integral (47), first, we derive the probability density function of the random variable Ω defined as

Ω = X σ 2 + Y . ( 49
)
Let T = σ 2 + Y . Since Y is exponentially distributed, T has a shifted-exponential distribution with the following probability density function:

f T (t) =          λ 12 P2 exp   λ 12 P2 σ 2   exp   - λ 12 P2 t   if t ≥ σ 2 0 if σ 2 < t (50)
The probability density function of the random variable Ω, for ω ≥ 0 , can be expressed as

f Ω (ω) = ∫ +∞ σ 2 t f X (ω t) f T (t) dt = λ 11 P1 λ 12 P2 exp   λ 12 P2 σ 2   ∫ +∞ σ 2 t exp   -   λ 11 P1 ω + λ 12 P2   t   dt,
thanks to the independence of X and T . After an integration by parts, we obtain The following equality holds:

f Ω (ω) =                      1 + b + b a ω a   1 + 1 a ω   2 exp   - b a ω   if ω ≥ 0 0 if ω < 0 (51) with 6 a = P1 λ 11 λ 12 P2 (52) b = σ 2 λ 12 P2 . ( 53
1 + b + b a ω a   1 + 1 a ω   2 exp   - b a ω   =   a (ω + a) 2 + b ω + a   exp   - b a ω   , (54) therefore,
C

,min = E [log (Ω + 1)] = ∫ +∞ 0   a (ω + a) 2 + b ω + a   exp   - b a ω   log (ω + 1) dω = a ∫ +∞ 0 log (ω + 1) (ω + a) 2 exp   - b a ω   dω + b ∫ +∞ 0 log (ω + 1) ω + a exp   - b a ω   dω. (55) Now, let I 1 = ∫ +∞ 0 log (ω + 1) ω + a exp   - b a ω   dω (56) I 2 = ∫ +∞ 0 log (ω + 1) (ω + a) 2 exp   - b a ω   dω. ( (1) 1
) 57
After an integration of I 1 by parts, we obtain:

I 1 = a b ∫ +∞ 0 1 (ω + 1) (ω + a) exp   - b a ω   dω - a b I 2 . ( 58
)
Then, we can express I 1 + a b I 2 as:

I 1 + a b I 2 = a b 1 a -1   ∫ +∞ 0 1 ω + 1 exp   - b a ω   dω - ∫ +∞ 0 1 ω + a exp   - b a ω   dω   , ( 59
)
thanks to the equality 1

(ω + 1) (ω + a) = 1 a -1   1 ω + 1 - 1 ω + a   . ( 60
)
We can rewrite (59) in terms of integral exponential function E 1 , [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]:

I 1 + a b I 2 = a b 1 a -1   exp   b a   E 1   b a   -exp (b) E 1 (b)   , (61)
finally, we express the lower bounds C In this section, we calculate the mean transmit power and the mean interference power of (29). Let x = g 12 and y = g 22 , the mean transmit power of (29) can be expressed as: Finally, the mean interference power is expressed as:

E [p 2 ] = ∫ Q peak
E [p 2 ĝ12 ] = p 2,peak /λ 12 ( 1 + λ22 λ12 K Q peak ) 2   1 -exp   - λ 12 Q peak + λ 22 K p 2,peak     . ( 74)

Figure 1 .

 1 Figure 1. Spectrum sharing between a PR and a CR communication links

  . With the settings P1 = 1, σ 2 = 0.01 and λ 11 = λ 12 = λ 22 = λ 21 = 1, we obtain the figures 2 and 3.

Figure 2 . 4 P2Figure 3 .

 243 Figure 2. Primary mean rate versus secondary mean power for different power control schemes from the secondary user:(a) optimal power control water-filling; (b) proposed scheduling approximating the optimal power control; (c) constant power control that provides the lower bound of the primary mean rate. P1 = 1, σ 2 = 0.01 and λ11 = λ12 = λ22 = λ21 = 1.

  x) exp (-λ 22 y) dx dy =

  . b) Mean transmit and interference powers: in figure 7, we compare the mean transmit power E [p 2 ]

Figure 4 .Figure 5 .

 45 Figure 4. Primary mean rate, C1, versus peak interference power, Qpeak, for different values of outage probability Pout.

Figure 6 .

 6 Figure 6. Primary mean rate, C1, versus mean interference power, E [p2 ĝ12], for different values of outage probability Pout.

Figure 7 .Figure 8 .

 78 Figure 7. Mean transmit power, E [p2], and mean interference power, E [p2 ĝ12], versus peak interference power Qpeak. p2,peak = 1 and Pout = 0.1.

2 Figure 9 .

 29 Figure 9. Primary mean rate, C1, and secondary zero-outage capacity, C2,out, versus mean interference power, E [p2 ĝ12], for Pout = 0.1.

Figure 10 .

 10 Figure 10. Mean transmit power, E [p2], versus mean interference power, E [p2 ĝ12], for Pout = 0.1.

) 6

 6 In part IV-A1, we set w = g22 σ 2 + P1 g21 . The probability density function fW has the same expression as fΩ but with a = λ21 P1 λ22 and b = σ 2 λ21 P1 .

  a and b by theirs expressions in (52) allows us to write:

λλλ 2 λλ 22 λ 12 Q

 212 22 λ 12 p 2,peak exp (-λ 22 y) exp (-λ 12 x) dx dy + λ 22 y) exp (-λ 12 x) dx dy. 22 λ 12 p 2,peak exp (-λ 22 y) exp (-λ 12 x) dx dy, 22 λ 12 Q peak x exp (-λ 22 y) exp (-λ 12 x) dx dy. λ 22 K + λ 12 Q peak p 2,peak     + λ 12 Q peak E 1   λ 22 K + λ 12 Q peak p 22 λ 12 x p 2,peak exp (-λ 22 y) exp (-λ 12 x) dx dy, peak exp (-λ 22 y) exp (-λ 12 x) dx dy.

  The mean transmit power E [p 2 ] is very high (ratio>9) compared to the mean received interference power E [p 2 ĝ12 ]. Moreover, E [p 2 ]

	5 From (32), it follows that Qpeak = -p 2,peak λ 22 z 0 +λ 12 log ( 1 -	E[p 2 ĝ12 ] p 2,peak /λ 12 ( 1+ λ 22 λ 12	z 0	) 2

The systems are selfish in the sense that they only try to maximize their own utility[START_REF] Etkin | Spectrum Sharing in Unlicensed Bands[END_REF].September 9,

DRAFT

September 9, 2010 DRAFT

This case includes obviously the unconstrained spectrum-sharing case, consequently C (1) 1,min ≥ C (2) 1,min . September 9, 2010 DRAFT