Joseph Parfait Owoundi
email: owoundijoseph@yahoo.fr

The Weight of Health Expenditures on Household Income in Cameroon

Keywords: Health Expenditures, Household Income, Average Propensity of Total Medical consumption

African leaders pledged at the Abuja conference in 2001, to mobilize more financial resources to allocate at least 15% of their national budgets to the health sector to achieve the Millennium Development Goals (MDGs), seem to have difficulty meeting this commitment because of weakness and fragmentation of health systems. These commitments were renewed in Gaborone, Botswana in 2005 and in Ouagadougou, Burkina Faso in 2006. Indeed, donor funding is still a large part of public health spending on the continent. In some countries, 50% or more of their budgets come from foreign or private assistance. In about half the countries, the private health financing is equal to or exceeds largely public funding, up to 70% in some states like Sudan, Côte d'Ivoire, Cameroon, Chad, Liberia and Uganda. Only five countries (Rwanda, Malawi, Zambia, Burkina Faso, and Togo) have so far respected the promise made to the Abuja conference. In Cameroon, where 51% of the population lives on less than two dollars per day, the average propensity of the total medical consumption is very high. Indeed, 32% of households spend less than half of income on health, while 16% of households spend more than half of the income and 52% spend more than the total income. This corresponds to a weight of 68% in health care spending.

Introduction

Sufficient funds to health are a fundamental issue. In many countries, health expenditures remain below the defined threshold for the provision of a series of basic health services. For poor countries, the challenge is to increase the funding available to provide the necessary services and ensure adequate quality of care2 (treatment, prevention, rehabilitation and health promotion). For rich countries, the challenge is that health spending does not increase with the aging of the population (which has implications on revenue and expenditure) and increased costs as a result of progress technology (a challenge that also face some poorer countries).

The Bamako Initiative, adopted in 1988, has developed a strategy of community participation and cost recovery to promote primary health care. The actual membership of sub-Saharan Africa has been very progressive in the 90s. Cost recovery still the problem of affordability by the absence of a political risk-sharing coupled with geographic accessibility by lack of monitoring of health coverage. The price elasticity of demand excludes disadvantaged classes for which the indigent and support measures that accompany it are often remained at the project stage. The establishment of financing schemes guaranteeing the reduction of inequalities is limited by an economy based primarily on an informal basis (A. [START_REF] Richard | Financing of Health in Sub-Saharan Africa. Cost recovery, 4p[END_REF].

According to a study by Adam Leive and Xu Ke in 2008, the system of financing health expenditure in Africa is too weak to protect households against catastrophic expenses. The borrowing or selling assets to finance health care is a common practice. The proportion of households who have paid their health spending by borrowing or selling assets ranged from 23% in Zambia to 68% in Burkina Faso.

Socio-economic context

The economic crisis of the 1980 years drove Cameroon into recession. Accentuated by the 60% decrease in salaries of civil servants in 1993 and the devaluation of the CFA franc in 1994, this crisis has significantly changed the structure of consumption and access to care. Thus, the decline in households' financial capacity was accompanied by a decline in the state budget devoted to health. The share of health expenditure in the household budget increased from 4% in 1984 to 6% in 1996 and 7.2% in 2001. Between 1995 and 1996, total health expenditure was 250 billion CFA francs, 72% financed by households, 22% by the state and 6% partners [START_REF] Commeyras | Study of the determinants of accessibility and use of health care and medicine in Cameroon[END_REF]. Figure 1 below shows the evolution of the national expenditure on health as percentage of GDP.

Graphic 1 : Evolution of the National Health Expenditure as % of GDP

Source: The using of OMS database of Health expenditure, sort by world countries; Update of 2012.

In 1982, Cameroon has adopted and implemented the primary health care (health for all by the year 2000) to provide free care. In 1993, the country adopted the Bamako Initiative through policy reorientation of primary health care, through: the recovery of costs for care and medications, community participation, and the organization of health districts [START_REF] Sieleunou | Health care financing in Cameroon: Trends analysis and overview of main challenges?[END_REF]. In addition, Cameroon has benefited from innovative financing of health after numerous international initiatives aimed at achieving the Millennium Development Goals (MDGs) by 2015. The Health Sector Wide Approach (SWAP), adopted in 2010, emerges as a key instrument for mobilizing and optimizing the use of resources for the implementation of the Health Sector Strategy. Despite these measures, the "Average Propensity of Total Medical Consumption" of households is high in Cameroon, where 51% of the population lives on less than two dollars a day.

The objective of this study is to provide some answers about the relationship between "Population and Health" through the analysis of the weight of health expenditure on household income. Thus, this study aimed to answer some specific to the situation in Cameroon questions, including: What is the impact of health expenditures on household income? What is the share of household income devoted to health care spending? What is the "average propensity of the total medical consumption"? What is the profile of households who bear the brunt of health spending? To this end, we have structured our work in five party, namely: (i) introduction, (ii) general information on health expenditures in Africa, (ii) the inequality of income distribution in Cameroon (iii) methodology, (iv) data analysis, and (iv) the conclusion and recommendations. 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Health expendidures as % of GDP

Definition of concepts

Health Expenditures

As defined by the World Bank, the total health expenditure is the sum of expenditures on public and private health. It covers the provision of health services (preventive and curative), family planning activities, related to nutrition and reserved for emergency health assistance but excludes the provision of water services and hygiene.

Health expenditure, recorded in the accounts of health, covering different types of services: hospital charges, costs "outpatient" drug spending, the expenditure approach, prostheses and small equipment medical, medical transportation costs [START_REF] Irdes | [END_REF].

In this study, the health expenditure of households include expenses related to medical care and goods (hospital care, outpatient care, medical transportation and medical goods), as well as preventive medicine.

Health financing

According to WHO, health financing is the way financial resources are generated, allocated and used in health systems. The following issues relating to health financing, should be asked: how and from what sources raise sufficient funds for health? How to overcome financial barriers that exclude many poor accesses to health services? How to offer a range of services of fair and effective health?

Household income

In economics, the primary income (primary income) of household income that households derive from their contribution to economic activity either directly (income from employment or self-employment) or indirectly (income furniture placement or real estate). It does not include social security benefits -this is income before redistribution. In national accounts, the primary household income includes income directly or indirectly related to household participation in the production process (INSEE, 2013). In our study, the income mentioned here is imputed income to the main activity of household members.

Average propensity of the total medical consumption

The average propensity of medical consumption is the share of income spent on medical consumption. The total medical consumption includes the consumption of medical care and goods (CSBMs), as well as preventive medicine. It is hospital care, outpatient care, medical transportation and medical goods.

Weight health expenditure

At the individual level, the weight of health care costs can be considered as the relationship between health expenditure and income. But in order to define classes vulnerable and disadvantaged social groups to which measures should be taken to improve the quality of health care spending, we can define the weight of health expenditure as a proportion of households or persons that support health spending well over half of the household income.

General information on health financing in Africa

The key issue that arises in terms of health care spending is not how to reduce but rather how we choose to finance (issue of receipts) and optimized (question of the organization), with the goal of an inclusive health fair system. So, thinking about the financial aspects of expenditure and revenue of the health system cannot be achieved without addressing the problem of the organization of this system. Indeed, whether it is able to bring new resources to the system or to consider how spending facing the community versus individual, any direction that is acceptable only if the financial effort is fair and optimized what it will be used.

However, three fundamental developments appear to maintain "upstream" the dynamism of the expense. These are: (i) changes in health (epidemiology and aging) that shape the needs and demand for health care, (ii) the standard of living through the rise in demand for health care driven by a higher income, and (iii) the progress of medical knowledge (technical progress), allowing both to better diagnose diseases and better treatment [START_REF] Albouy | Health spending in France: Determinants and impact of aging in[END_REF]. Thus, knowledge of the sheer scale of health spending by households is an avenue that should be explored, given the importance of private health expenditure in development planning.

African leaders pledged at the Abuja conference in 2001, to mobilize more financial resources for the achievement of the Millennium Development Goals (MDGs) by allocating at least 15% of their national budgets to the sector health, seem to have difficulty meeting their commitments, because of weakness and fragmentation of health systems. These commitments were renewed in Gaborone, Botswana in 2005 and in Ouagadougou, Burkina Faso in 2006. Indeed, donor funding is still a large part of public health spending on the continent. Thus, in some countries, 50% or more of their budgets come from the private or foreign aid, according to the 2013 WHO report on global health statistics (see table below).

In nearly half of the country is 40%, the private health financing is equal to or exceeds largely public funding, up more than 70% (see Graph 1.2) in some states such as: Cameroon (70, 4%), the DRC (71.6%), Sao Tome and principles (64.7%), the Chad (75%), Ivory Coast (75.5%), the Guinea (67.5% ) to Bissau (66.9%), Guinea, Liberia (81%), Nigeria (68.5%), Sierra Leone (84.7%) in Burundi (65%), Uganda (76 , 9%), Sudan (72.4%), Tanzania (60.8%), Mauritius (62.9%). The expenditure of social security against health are very low or non-existent in most of these countries. Furthermore, only 10.63% (5 of 47 countries) fulfilled at the Abuja commitments to allocate at least 15% of their budgets to health. This is Rwanda, Malawi, Zambia and Burkina Faso and Togo. In addition, many efforts are expected in most countries. Thus, 38% (18 out of 47 countries) still spend less than 10% of their budgets to health, while 51% (24 out of 47 countries) currently allocate between 10 and 14% of their budgets to health. Thus, for the Abuja commitment is respected by African States, other sources of funding are essential. Map1 below gives the distribution of the percentage of the budget allocated to health by African countries. Data source 3 The frame used consists of enumeration areas (EAs) of the last census of population and housing Census (PHC 2005). In each borough or district, the urban EAs are numbered from 001 to 699 and those in rural areas from 700 to 999. A recovery rate of raw data on the population of 1.2 was used. First degree in each stratum, is derived enumeration areas (EAs) in proportion to their size to account for remaining differences between these sizes, the second stage, a sample of households is drawn in each selected first-degree ZD. Thus, in all 12 survey areas, 12 609 households were selected in 742 ZD, a staff of 12 households per EA Douala / Yaoundé and 18 in the rest of the country. Purchases and acquisitions were recorded daily for 15 days in each urban household and two passages in rural households covering 10 days should be supplemented by statements of historical costs. The investigation lasted three months, in order to take into account seasonal fluctuations. 3 The general methodology in the context of the third national survey on poverty reality by the National Institute of Statistics in 2007 is detailed in the report of the investigation and downloaded from the website of the INS Cameroon. This is the data from this survey will be used as the basis for our analysis.

Table1: Health Expenditure in Sub-Saharan Africa

WHO states members

Ratios of health spending in Sub-Saharan Africa

Macroeconomic approach to overall expenditure

The macroeconomic approach used here is that the overall expenditure of Keynes 4 , that "the level of real income (ignoring the price) is the main determinant of the overall expenditure cause." To this end, the overall expenditure (AE) is the sum of what households plan to buy (or consumption C), and that the companies plan to consume in the form of capital (or investment I): AE = C + I. Thus, the desire to use a proportion of income (Y) for consumption (C) is called the average propensity to consume (APC): APC = C / Y. As incomes rise, the average propensity to consume decreases. Indeed, one can observe this phenomenon among affluent individuals who spend a smaller share of their income on consumption than do the poorest (the latter may in fact be obliged to receive money from others). The marginal propensity to consume (MPC) is the proportion of additional consumption (dC) which will be added in an increase in income (dY): MPC = dC / dY.

The main assumption in this study is as follows: "The level of income determines the overall health care spending."

To verify the said assumption, we use the multivariate analysis methods, using SPSS (Statistical Package Social Sciences) software on Windows using statistical techniques of regression of variance (ANOVA one-factor), and the Multiple Correspondence Factor Analysis (MCA). This will allow us to highlight the change in weight of health expenditures on household income. The data to be used will be those of the third survey on the living conditions of the population and the poverty profile in Cameroon, conducted in 2007 (ECAM III).

Analysis of Weight Health Costs on Household Income

Income distribution in Cameroon

The average annual income of Cameroonian society has increased between 1996 and 2001, from 243,240.44 FCFA in 1996 to 294,403.41 in 2001, an increase of 21% (ECAM 1996 and2001), before moving to 659,376 FCFA in 2007. Despite this increase in income, however, there many inequalities between income distributions at the national level. The Gini coefficient rose from 0.406 in 1996 to 0.408 in 2001, a positive variation of 0.002, indicating the increasing inequality in Cameroon. To account for the distribution of income accurately, we calculate the distribution of Theil index, Gini and Hoover in 2007.

Theil index

The Theil index (T i ) is a measure of inequality based on the Shannon entropy 5 : 4 The main approach presented here is the use of Keynes to propose a solution for excessive periods of unemployment (ie d. During recession) aims. This solution is related to the idea that employment depends on what companies must produce while the production level depends on what households and businesses plan to buy. This is what Keynes called the overall expense. 5 The Shannon entropy, due to Claude Shannon (circa 1948) is a mathematical function that intuitively corresponds to the amount of information contained in or supplied by a source of information. This source can be written in a particular language text, an electrical signal or any computer file (collection of bytes). Is a fundamental discrete Ω={ } and P is a probability measure on (Ω, P (Ω)), Shannon entropy is defined by

I(P) = -∑
où p i = P({ }). In addition, entropy is combinatorial version of discrete probabilities or by continuous probability.

 An index of 0 represents absolute equality;  An index of 0.5 indicates inequality represented by a society where 74% of people have 26% resources and 26% of individuals 74% of resources;  An index of 1 indicates inequality represented by a society in which 82.4% of individuals were 17.6% and 17.6% of the resources individuals have 82.4% of the resources.

We calculated the Theil index from the following formula:

T=Ln ( ) -

∑  N: Number of quartiles;  Ei: Resources for the quartile i ;  Ai: effective in quartile i ;  Etotal: resources for all quartiles in a society (a nation, etc...);  Atotal: size of the company (the nation, etc..).

The Theil index obtained from this formula is Ti = 1. We thus conclude that the Cameroonian society is represented by the inequality where 82.4% of individuals have 17.6% and 17.6% of the resources individuals have 82.4% of the resources.

The Gini Index

The Gini coefficient (G) in turn is a measure of the degree of inequality of income distribution in a given society, developed by the Italian statistician Corrado Gini. The Gini coefficient is a number ranging from 0 to 1, where 0 means perfect equality (everyone has the same income) and 1 corresponds to perfect inequality (one person has all the income, others have nothing Extreme cases of master and slaves).

We calculated the coefficient from the following formula:

G = 1-∑ Where k is from 1 to n,

.)/m

We always have  p =  p = 1. For convenience we take  0 =  0 = 0.  k is the percentage of individuals having a value of less than or equal to x k character: these individuals have a fraction of the total mass equal to  k . We can also say that  k is the mass (in %) owned by the  first k individuals (in %), these are sorted by increasing value of the character. By applying the above formula, we obtain the value of G = 0.712 in 2007, an increase of 0.3 between 2001 and 2007. This result shows that income inequalities are quite high in Cameroon.

Index Hoover

Hoover index (H) is the simplest of all indices measuring income inequality. It is equal to the income portion of the population that should be redistributed. This is the portion that must be taken to the richest part of the population (above average income) and given to the poorest (below average income) part so that there is perfect equality (everyone has to income, average income). We used the following formula to calculate this index:

H = ∑ - │
 N: Number of quartiles;  Ei: Resources for the quartile i ;  Ai: effective in quartile i ;  Etotal: resources for all quartiles in a society (a nation, etc...)  Atotal: size of the company (the nation, etc...).

 Advantage: The index is between 0 (uniform distribution) and 1 (maximum concentration). Note that the value 1 could only be achieved if we had an infinite number of individuals possessing nothing and a person having any;  Disadvantages: The index does not provide information on the structure of inequalities (2 different income groups or one person rich and many poor people).

The Hoover index is generally used in applications related to socio-economic classes and health. But this index does not measure inequality within the richest or poorest parts. This is why we can to prefer the Gini coefficient, which is based on the Lorenz curve. By applying the above formula, we find H = 0.698. Once again, the result shows that income inequality is quite important in 2007 in Cameroon. These results can be illustrated by the Lorenz curve below: The above results are the fruit of a Cameroonian economy slowed during the period between 1997 and 2005, when real GDP growth slowed from 5.31% to 2.3% [START_REF] Ins | Data from the National Accounts of Cameroon[END_REF]. This slowdown is due to the deterioration of the trade balance, the sluggish international economic environment and worsening fiscal constraints due to the combined effects of the global economic and financial crisis, the food crisis and the energy deficit. Figure 4 below shows the trend in GDP growth in Cameroon.

Graphic 4 : Evolution of Cameroon's GDP between 1994 and 2010

Source : Exploitation des données d'ECAM III.

Despite reaching the completion point of the HIPC initiative in 2006 and the signing of the C2D with France, unemployment and underemployment remain very important in Cameroon, especially among young people. The result is a general discontent of youth that has little hope of social advancement. This led to riots in Cameroon in February 2008, reflecting the discontent of people facing hunger and rising fuel prices and that of commodities. Also, the government has taken emergency measures to boost the agricultural sector, including the production prioritized such as maize, rice, cassava, potato, oil palm and plantain.

Marginal and average propensity of the total medical consumption

Marginal propensity Medical Consumption

As we defined in the methodology section reserved for the marginal propensity to consume (MPC) is the proportion of additional consumption (dC) which will be added in an increase in income (dY): MPC = dC / dY. We calculated between the two surveys that took place

Evolution of GDP between 1994 and 2010

Thus, MPC = (12775-22000) / (54948-37000) = -9225/17948 = -0.5. This result shows that consumption decreased between 2001 and 2007, but at a slower pace compared to the increase in income. The value of the elasticity obtained by the English law is as follows:

ε = ΔC / C / ΔR / R = ((C 2 -C 1 ) /C 1 ) / ((R 2 -R 1 )/R 1 ε = ((12775 -22000) /12775) / ((54948 -37000)/54948 = -0,86
We get ε = -0.86 ˂ 0. We find ourselves in a situation of inferior goods, that is to say, the demand decreases when income increases.

Average propensity of Total Medical Consumption

As we noted earlier, APC = C / Y. We created this indicator using the analysis software SPSS data. To this end, we excluded households with zero health expenditure (20% of households surveyed) to avoid overstaffing for the first quartile of health expenditure because only individuals who spent feel the weight of health spending. Also, to avoid bias related to the measurement of health care spending, we selected only the expenses of the household head, since it supports most of the household expenses. In addition, we obtained the following results: among the 11,391 households surveyed, 9,018 households or 80% actually spent on health in 2007. Of these 9,018 households, 32.2% spent less than half of their income on health, 16.3% in turn spent more than half of their income on health, and finally, 51.5% of households spent most of their total income on health.

Spending overall health national level amounted to FCFA 711,343,126 while the total annual income of FCFA 565,795,940, representing an annual average propensity of the total medical consumption of 1.25. This result shows that the Cameroonian population spends more than all of the income to health. The tables below summarize the situation.

Source: Operation of ECAM III data

From Table 2, so we see that the weight of health care spending to income is (16.3 +51.5 = 68%). This result reflects the fact that Cameroon is a country where the problem of the rational use of resources. In addition, incentives are not sufficiently relevant to the optimization of the efficiency and cost moderation. To this end, the development of mechanisms that promote the best value for money for medical care is needed. In the same vein, the result is in line with the WHO data according to which the private health expenditures represent 70.4% of total government expenditure in Cameroon. These results are also shown in Map 2 below, where seven out of 10 regions (70%) bear the weight of health care spending:

Analysis of the correlation between overall health expenditure and

income, using the one-way ANOVA.

Analysis of variance is to determine the average, which differs most from the other or average torques which oppose the most between a dependent variable and one or more independent variables. This analysis will allow us to determine the different behavior in terms of consumption expenditure according to income groups (quartiles of income) households.

We start from the hypothesis H0 according to which "the average health expenditure observed in the various income groups are all equal" and wants to confront the alternative that "the average health expenditure observed in various groups are not all equal "(hypothesis H1). This calculates the statistical Fisher observed (Fobservé) as compared with the theoretical Fisher (Fthéorique). For this purpose, if:  If Fobservé> Fthéorique then H0 is rejected at the risk of error α. In other words, if the p-value <α then H1 is accepted and the averages are statistically different;  If Fobservé ≤ Fthéorique hypothesis H0 is accepted: p-value> α then H1 is rejected and the averages are statistically equal.

The results of the ANOVA model led us to the following conclusions:

Health expenditures vary from one income class to another and so this growth (Health expenditures increases with income level); The Fisher statistic for a 95% chance is Fobservé (3, 10293) = 146> Fthéorique (3, 10293) = 2.61 (P-value = 0.00 ˂ 5%).

We can therefore reject the hypothesis H0 (and accept H1). It concludes that the average income quartiles health expenditures are equal. The graph below shows us well that "more income increases on average, households spend on health." The diagram below illustrates the theory of Duesenberry (demonstration effect), that economic agents of a particular social group tend to imitate the consumption of high-income social group by demonstrating their social status.

Multiple correspondence analysis (MCA) and the profile of the population according to health spending

Factor analysis methods 6 include search subspaces of small dimensions that best fit the scatter of individuals and the scatterplot of variables. The proximities measured in these subspaces should best reflect the actual proximity. The representation space obtained is called factorial space.

The methods differ depending on the nature of the analyzed variables can be continuous variables, categorical variables or categories in the case of contingency tables. The lines can be individuals or groups.

To build our analytical model, we used seven variables:  Continuous-variables: household income, health expenditure;  Illustrative-nominal variables: the weight of health expenditure in household income;  Nominal variables: the age of the household head group, his place of residence, sociooccupational category, level of education and standard of living of the household.

Focused on the choice of these variables is justified by the fact that these are the main determinants of health expenditure and well-being of the population. Pathway following model has guided our choice (figure 1).

Three criteria allowed us to retain five (05) factorial axes: (i) the method of Cattle (method or the elbow), which helps to retain the number of axes that lie after breaking information, (ii ) Kaiser method, which holds the number of axes in the axis value of the last question is greater than the average of the eigenvalues (iii) the method of the cumulative percentage of inertia (which holds the number of axes as the last line reveals considered at least 60% of the inertia of the point cloud. Based on these three criteria, we finally selected two factorial axes (axis 1 and axis2) among the five because they reflected the maximum information. The data analysis allowed us to identify two major patterns of health expenditure of households, the following sociodemographic variables shown in Table 4 below:

Social Determinants of health and health equities

We obtain a classical result, characteristic of sub-Saharan Africa, where we have one hand, a poor aging population in rural areas facing demographic vulnerabilities, as opposed to the other a young population residing in urban, educated generally working in the public, private and non-agricultural informal. This result illustrates somehow "theory of urban bias 7 ."

The graphical representation of Factor1 and Factor2 obtained from the SPAD software in Multiple Correspondence Analysis (MCA) is given below: Graphic 5: Factorial consisting axe1 and axis2.

Source: Analysis of data from ECAM III

General Conclusion

The issue of the proper functioning of the health system is to improve the quality of life of individuals, families and communities. It is for this reason that the performance of the health system should be the top priorities for government action (Hamidou Baba et al., 2012). However, the health systems of the Member States of the WHO African Region States are struggling to effectively perform their duties because of their weakness and fragmentation. Weak health systems is explained by a number of factors, namely: deficiencies in planning and forecasting, the lack of human, financial and material resources, the lack of data in order to inform informed decision making; gaps in institutional processes and weaknesses of performance monitoring and impact of interventions.

In Cameroon, the expense of overall health at the national level is high, or 711 343 126 FCFA, corresponding to an overall annual income of FCFA 565,795,940, representing an annual average propensity of the total medical consumption of 1.25. Cameroon's population spends more than all of the income to health. The marginal propensity to consumption is -0.5, highlighting the reduction of medical consumption between 2001 and 2007, but at a slower pace compared to the increase in income. Despite the fact that demand decreases when income increases (elasticity coefficient ε = -0.86), Cameroon is found in a situation of inferior goods in health. "As income increases on average, households spend on health." The macroeconomic approach to the overall expense of Keynes, that "the level of real income (ignoring the price) is the main determinant of the overall expenditure" has been checked. So we finally two profiles of households facing health spending: first, households with poor aging rural population who spend more than the total income on health, as opposed to the other households consisting of a young population residing in urban areas, educated, usually working in the public, private and non-agricultural informal spending less than half of income on health. This highlights the "theory of urban bias."

Strengthening health systems is therefore a prerequisite for any significant progress in health in the African Region and even the world. In this sense, the health system must be approached holistically and approached as a whole and not item by item. Without such a vision, most initiatives focus only on resources related to specific diseases or specific target groups, usually letting the components to account for organizational processes. But it is in these processes plays the structuring of the system, that is to say, the articulation of human and material resources, the functions of the various organs and their methods of cooperation.

Recommendations

Major reforms are needed to allow for adaptation of the organization of health systems in Africa so that they guarantee sustainable way health solidarity in 21st century. To this end, we recommend:

To the World Health Organization (WHO) members States,  the exemption from social charges on salaries;  compliance with the Abuja commitments and launch studies traceability and impact to optimize the management of the budget allocated to health;  the acceleration of the operation of health insurance in the states where the need is still felt;  Awareness of how key contextual factors, including budgetary constraints, affect the ability of a country to achieve its policy objectives.

Bibliographic references:

1. Adam Leive, Ke Xu, 2008: How do they deal with household health expenditure in their care: empirical evidence from 15 African countries (WHO) Bulletin of the World Organization of the previous Volume Health, Volume 86, November, 817-908.

Map 1 :Graphic 2 :

 12 Budget allocated to health by African countries. Source: Analysis of data from the 2013 report of the WHO World Health Statistics using MapInfo software. Private Health Expenditure SSA countries between 60 and 70% of GDP. Source: Analysis of data from the 2013 report of the WHO World health statistics

  we calculate:  k = (n1 + n2 + ... + nk)/n and k =  (n1.x1+ n2.x2 + …..+ nk.xk ..

Graphic 3 :

 3 Lorenz curveSource : Operation of ECAM III data

  in 2001 and 2007. The average monthly income obtained from the frequency distribution amounted to 54,948 FCFA in 2007 against 37,000 in 2001, the average annual health expenditure to 12,775 FCFA in 2007, against 22,000 FCFA in 2001, according to reports General surveys in 2001 and 2007.

Graphic 4 :

 4 Average income by expenditure quartile Health.Source: Operation of ECAM III data

Figure 1 :

 1 Figure 1: The Pathway model of the Commission on Social Determinants (CDSS)

WHO states members Ratios of health spending in Sub-Saharan Africa

		Global health	Spending of	Private	Spending of	External		Social security
		expenditures	public heath	health	PHA (as % of	health		health
	Somalie	Global health expenditures (GHE) (as % of GDP) (GHE) (as % of GDP) 2000 2010 2,4 …	Spending of public heath administrations (PHA) (as % of GHE) administrations (PHA) (as % of GHE) 2000 2010 2000 2010 2000 Private health expenditures (as % of GHE) Spending of PHA (as % of global expenditure of Public Administration expenditures global (as % of expenditure of GHE) Public Administration 2010 44,8 … 55,2 … 4,2 …	External health financing (as % of GHE) financing (as % of GHE) 2000 2010 2000 2010 spending Social security (as % of health PHA) spending (as % of PHA) 40,1 … 0 …
	Soudan	2000 3,3	2010 7,2	2000 25,8	2010 2000 2010 2000 27,6 74,2 72,4 7,6	2010 10,3 4,5 2000 2010 2000 2010 2,9 … 11,1
	Tanzanie	3,4	7,2	43,4	Central Africa 39,2 56,6 60,8 10,2 11,1 27,8 39,6	0	4,5
	Cameroon Zambie	4,4 5,7	5,1 6,0	19,3 51,3	29,6 80,7 70,4 5,5 60,8 48,7 39,2 9,4	8,5 16,0 17,8 43,7 0 4,3 7,0 4,2	2,6 0
	Congo	2,1	2,3	57,5 60,5 42,5 39,5 4,8 Southern Africa	6,5	4,6	4,6	0	0
	Gabon Afrique du Sud 8,1 2,5	3,5 8,7	42,0 42,3	51,8 58,0 48,2 4,8 46,6 57,7 56,4 10,9 12,4 0,3 6,6 2,6	2,3 2,2	5,7 3,3	24,9 2,9
	Guinée-Angola	1,9 2,9	4,2 3,4	46,3 58,1	57,4 53,7 42,6 7,8 61,0 41,9 39,0 2,9	7,0 6,3	8,8 3,0	2,1 2,3	0 0	0 0
	Equatorial Comores	3,6	5,3	42,1	57,2 57,9 42,8 9,3	13,1 20,5 15,3 0	0
	République Lesotho	2,2 7,0	3,8 11,5	93,1 51,2 70,3 48,8 29,7 6,3 51,0 6,9 49,0 12,9	10,4 35,4 25,3 … 13,8 3,1 18,2 0	… 0
	Centrafricaine Madagascar	3,7	3,6	66,5	58,1 33,5 41,9 15,5 13,5 20,1 9,5	…	…
	RDC Malawi	4,7 3,0	7,5 8,4	4,2 91,2	28,4 95,8 71,6 1,8 74,2 8,8 25,8 9,0	10,0 2,8 18,5 53,4 58,1 0 33,0 …	… 0
	Sao Tomé et Mozambique Principes Tchad Maurice Mauritanie Bénin Namibie Burkina Faso Swaziland Botswana Zimbabwe	8,4 6,2 6,3 3,7 6,0 1,9 6,1 5,1 5,3 4,7 …	7,5 6,3 4,0 5,6 6,1 4,3 5,5 7,4 7,8 5,1 …	43,2 70,0 42,5 52,0 66,5 99,9 68,9 39,6 56,3 62,2 …	35,3 56,8 64,7 9,0 52,8 30,0 47,2 17,0 10,4 25,3 62,2 0,3 5,6 34,8 20,3 0 25,0 57,5 75,0 13,1 3,3 24,9 7,9 … 37,1 48,0 62,9 8,7 8,3 1,4 1,6 0 66,0 33,5 34,0 12,9 13,1 11,2 7,4 8,7 West Africa 51,2 0,1 48,8 10,0 10,5 38,5 32,8 0,5 58,0 31,1 42,0 6,9 6,5 3,8 25,5 1,8 55,4 60,4 44,6 8,8 15,7 13,9 36,1 0,8 71,1 43,7 28,9 10,5 13,7 5,8 14,4 0 64,5 37,8 35,5 7,3 8,7 0,5 8,0 … … … … … … … … …	0 20,3 0 … 8,5 2,7 0,4 0 0,4 … …
	Cap-Vert	4,8	2,3	73,3	49,1 26,7 50,9 9,9	2,5	13,0 19,5 34,9 87,3
	Côte d'Ivoire	1,4	6,2	94,9	24,5 5,1 75,5 7,2	6,8	19,4 8,3	…	…
	Gambie	3,6	4,4	34,2	56,1 65,8 43,9 10,4 11,3 19,1 39,2 0	0
	Ghana	4,7	5,2	48,7	58,2 51,3 41,8 8,3	12,1 14,5 16,5 0	25,2
	Guinée	5,7	6,2	18,7	32,5 81,3 67,5 6,4	6,8	13,4 26,7 1,1	4,5
	Guinée-Bissau 4,9	7,0	10,5	33,1 89,5 66,9 2,3	11,1 30,0 28,5 5,4	1,1
	Libéria	5,9	16,4	24,5	19,0 75,5 81,0 6,7	11,1 9,2	42,4 0	0
	Mali	6,0	6,5	29,2	43,5 70,8 56,5 7,5	12,3 8,2	22,2 1,8	0,7
	Niger	3,4	4,8	44,5	49,2 55,5 50,8 8,4	11,1 40,4 32,7 3,3	1,3
	Nigeria	4,6	5,4	33,5 31,5 66,5 68,5 4,2	5,7	16,2 8,7 0	0
	Sénégal	4,3	5,8	36,8	56,9 63,2 45,1 8,5	12,1 17,4 18,4 8,8 3,9
	Sierra Leone	17,5	20,8	22,6	15,3 77,4 84,7 14,2 11,7 5,5	26,8 0	0
	Togo	5,3	7,5	28,5	45,9 71,5 54,1 8,5	15,4 5,9	18,2 11,7 6,5
						East Africa
	Burundi	6,2	9,1	29,3	35,0 70,7 65,0 7,5	8,1	19,1 40,9 25,1 23,0
	Djibouti	4,0	7,9	99,2	68,5 0,8	31,5 12,0 14,1 47,7 22,1 11,3 9,6
	Erythrée	4,5	2,9	39,1	45,2 60,9 54,8 2,6	3,6	29,8 35,0 0	0
	Ethiopie	4,3	4,8	53,6	52,9 46,4 47,1 8,9	13,7 16,5 36,1 0	0
	Kenya	4,7	4,4	46,3	40,2 53,7 59,8 10,5 5,9	8,0	37,9 10,9 13,0
	Maurice	3,7	6,2	52,0
								9,3	4,3	4,2	5,0	1,1

43,7 48,0 56,3 8,7 10,8 1,4 1,9 … … Ouganda 6,6 9,2 26,8 23,1 73,2 76,9 7,3 10,8 28,3 27,6 0 0 Rwanda 4,2 10,4 39,2 54,8 60,8 45,2 8,2 23,3 52,0 48,0 6,4 9,2 Seychelles 4,8 3,3 82,7 91,9 17,3 8,1 7,3 Source: Analysis of the 2013 Report of the WHO health statistics.

Table 2 : Average Propensity of total Medical Consumption by region Regions Average Propensity of medical consumption of households by region (%) Less than half of income More than half of income Over the entire income

 2

	Adamaoua	48	18,7	33,3
	Far-north	32,6	14,7	52,7
	North	33,2	16,9	49,9
	East	33,1	19	47,9
	Centre	33,6	15,8	50,7
	Littoral	31,6	16,4	52
	North-West	21,6	12,2	66,2
	West	20,4	16,8	62,7
	South	37,6	18,8	43,6
	South-West	27,9	16,7	55,4
	National	31,96	16,6	51,44

Table 3 : Weight of health expenditures by region Regions Health expenditures by income for any region (%) Average monthly income by region FCFA Average health expenditure by region Average propensity by region Adamaoua

 3

		72192,20	47272,20	0,65
	Far-north	52995,60	44064,93	0,83
	North	48961,20	53438,69	1,09
	East	34687,00	42778,16	1,23
	Centre	51229,28	66961,02	1,31
	Littoral	41345,03	55135,62	1,34
	North-West	31943,82	52571,87	1,65
	West	43827,30	68359,52	1,56
	South	76352,24	59523,65	0,78
	South-West	51387,20	64011,14	1,25
	National	50492,08	55411,68	1,097

Table 4 :

 4 Profile of health expenditures according to certain sociodemographic variables

1 st Profile: Heads of households who spend less than half of income on health 2 nd profile : : Heads of households who spend more than the entire income on health

	Political and
	socio-economic context	Social position
			Material
	governance policy macroeconomic social health Cultures, standards and societal	Education Work Income Ethnicity / race Sexe	conditions factors social cohesion factors psychosocial behavior biological	Distribution to Health and well-being
	values
				Health care
				system
	Age : under 30 years old		Age : 60 years old and over
	Level of education: Secondary and Higher	Level of education ; out of school
	Place of residence: urban	Milieu de résidence : rural
	Occupational category: non-agricultural	Occupational	Category :	agricultural
	informal, formal private and public sector	informal
	Income: 3rd and 4th quartile	Income : 1 st quartile

Source: Operation of ECAM III data

These treatments refer to what WHO calls Universal Health Coverage, which is to ensure that the entire population has access to preventive, curative, rehabilitative and health promotion it needs and that these services be of sufficient quality to be effective, their cost without causing financial difficulties for users. Universal coverage has its roots in the WHO Constitution, adopted in 1948, which made health one of the fundamental rights of every human being, and the Global Strategy for Health for All, launched in 1979 (www . who.int).

For more information, please refer to the user's work Lebart, Morineau and Piron (1995).

According to Lipton (1976) cited by Cris Beauchemin (1993), "Cities play a negative role in the development because they absorb all the resources at the expense of rural areas and the agricultural sector."


Ming Liu

Jean-Franc ¸ois Hélard
email: jean-francois.helard@insa-rennes.fr

Matthieu Crussière
email: matthieu.crussiere@insa-rennes.fr

Maryline Hélard
email: maryline.helard@insa-rennes.fr

Reduced-Complexity Maximum-Likelihood Decoding for 3D MIMO Code

The 3D MIMO code is a robust and efficient spacetime coding scheme for the distributed MIMO broadcasting. However, it suffers from the high computational complexity if the optimal maximum-likelihood (ML) decoding is used. In this paper we first investigate the unique properties of the 3D MIMO code and consequently propose a simplified decoding algorithm without sacrificing the ML optimality. Analysis shows that the decoding complexity is reduced from O(M 8 ) to O(M 4.5 ) in quasi-static channels when M -ary square QAM constellation is used. Moreover, we propose an efficient implementation of the simplified ML decoder which achieves a much lower decoding time delay compared to the classical sphere decoder with Schnorr-Euchner enumeration.

I. INTRODUCTION

Multiple-input multiple-output (MIMO) technology in combination with the space-time block code (STBC) offers increased spectral efficiency and improved reliability without requiring additional spectrum bandwidth [START_REF] Tarokh | Space-time codes for high data rate wireless communication: Performance criterion and code construction[END_REF]. Hence, it has been widely adopted by many state-of-the-art communication systems such as IEEE 802.11n, 3GPP Long Term Evolution (LTE) and WiMAX etc. It is also considered as a core technique for the future TV broadcasting system [START_REF]DVB-NGH: Next Generation Handheld[END_REF].

A so-called Space-Time-Space (3D) MIMO code has been proposed for the 4 × 2 distributed MIMO broadcasting in which MIMO modulated signal is sent from two cooperating transmission sites to the receivers in the coverage are. Each site has two transmit antennas and each receiver equips two receive antennas, as well [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF]. The 3D MIMO code combines the robustness of Alamouti scheme [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] with the efficiency of the Golden code [START_REF] Belfiore | The golden code: a 2×2 fullrate space-time code with nonvanishing determinants[END_REF]. Hence it offers reliable performance even in presence of strong received signal power imbalances. The latest study [START_REF] Liu | D2.3: Report on advanced concepts for DVB-NGH[END_REF] shows that the 3D MIMO code is the most efficient and robust in distributed MIMO broadcasting scenarios compared with other state-of-the-art STBCs such as DjABBA code [START_REF] Hotinen | Multi-antenna Transceiver Techniques for 3G and Beyond[END_REF], BHV code [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] and Srinath-Rajan code [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], which suggests it is a promising candidate for the future distributed MIMO broadcasting systems.

However, as eight M -QAM modulated information symbols are stacked within one 3D MIMO codeword, the computational complexity is as high as O(M 8 ) when maximumlikelihood (ML) decoding is adopted. No study on reducing the ML decoding complexity of 3D MIMO code has been conducted in the literature. [START_REF] Polonen | Reduced complexity space-time coding in single-frequency networks[END_REF] proposed another fast-decodable STBC for the distributed MIMO broadcasting. However, it does not achieve full-rate and its performance is not as good as 3D MIMO code.

In this work, we first reveal some unique properties of the 3D MIMO code which have not been presented in any previous work. Based on these properties, we propose a novel simplified ML detection method. The complexity reduction is achieved by three means: 1) embedded orthogonality coming from Alamouti-like block structure in the codeword enables a group-wise detection; 2) embedded orthogonality between real and imaginary parts of the symbol group inheriting from the Golden code enables independent detection of real and imaginary parts in parallel; 3) adaptive searching radius avoids the cumbersome exhaustive search.

The reminder of the paper is organized as follows. The 3D MIMO code and the MIMO system model is introduced in Section II. The novel simplified ML decoding algorithm is proposed in Section III. Complexity analysis and simulation results are presented in Section IV. Finally, conclusions are drawn in Section V.

In this paper, x R and x I represent the real and imaginary parts of the complex number x, respectively. The function x performs the complex-real conversion as:

x [[x R x I ] T [-x I x R ] T ].
For a complex vector x = [x 1 , x 2 , . . . , x n ] T , the function x separates the real and imaginary parts of the complex vector, i.e. x [x R 1 , x I 1 , . . . , x R n , x I n ] T . For a matrix X = [x 1 , x 2 , . . . , x n ] where x j is the jth column of X, the function vec(X) denotes stacking the columns of X one below another, i.e. vec(X) [x T 1 , x T 2 , . . . , x T n ] T . Consequently, vec(X) denotes vectorizing matrix X followed by the real/imaginary part separation. The inner product of two vectors x and y is denoted by x, y .

II. 3D MIMO CODE AND SYSTEM MODEL

The codeword matrix of the 3D MIMO code is given in [START_REF] Tarokh | Space-time codes for high data rate wireless communication: Performance criterion and code construction[END_REF] which is shown on next page, where θ

= 1+ √ 5 2 , θ = 1 -θ, α = 1 + i(1 -θ) and ᾱ = 1 + i(1 -θ) with i = √ -1.
The codeword is formed by arranging two Golden codewords X 1 and X 2 in an Alamouti manner. It achieves full-diversity. Since eight information symbols s = [s 1 , . . . , s 8 ] T are transmitted over four (T = 4) uses, it achieves a space-time (ST) coding rate of two which is full-rate for 4 × 2 MIMO transmission.

The 3D MIMO code is a linear STBC and its codeword

X 2 X * 1 = 1 √ 5   α(s1+θs2) α(s3+θs4) -α * (s * 5 +θs * 6 ) -α * (s * 7 +θs * 8 ) i ᾱ(s3+ θs4) ᾱ(s1+ θs2) i ᾱ * (s * 7 + θs * 8 ) -ᾱ * (s * 5 + θs * 6 ) α(s5+θs6) α(s7+θs8) α * (s * 1 +θs * 2 ) α * (s * 3 +θs * 4 ) i ᾱ(s7+ θs8) ᾱ(s5+ θs6) -i ᾱ * (s * 3 + θs * 4 ) ᾱ * (s * 1 + θs * 2 )   (1)
matrix can be constructed as [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF]:

vec(X) = G s, (2)
where G = [ vec(A 1 ), vec(B 1 ), . . . , vec(B 8 )] is the generator matrix with A j and B j being the the weight matrices representing the contribution of the real and imaginary parts of the jth information symbol s j in the final codeword matrix [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]. For the MIMO system with N t transmit and N r receive antennas, the signal transmission over quasi-static flat-fading channel is expressed as:

Y = HX + N, (3)
where Y and N are N r × T matrices representing received signal and complex-valued additive white Gaussian noise (AWGN) component, respectively; X is an N t × T matrix representing a codeword of the STBC; H is an N r × N t matrix in which the (j, k)th element h j,k is the gain of the channel link between the kth transmit antenna and jth receive antenna. Separating the real and imaginary parts and stacking the columns of the transmitted/received signal, it yields the signal expression in real-value form:

y = H eq s + n, (4)
where y = vec(Y), n = vec(N) and H eq is the equivalent channel matrix and is written as:

H eq = (I T ⊗ Ȟ)G, (5)
where I T is the T × T identity matrix, ⊗ is the Kronecker product.

III. SIMPLIFIED ML DETECTION FOR 3D MIMO CODE

A. ML decoding of STBC Since the received signal y in (4) can be viewed as lattice points perturbed by the noise, the maximum-likelihood (ML) solution of the transmitted signal is the combination of the information symbol s which has minimal Euclidean distance to the received signal y, namely:

ŝML = arg min s∈Θ 8 y -H eq s 2 , ( 6
)
where Θ is the set of the constellation of complex-valued information symbols, and Θ 8 indicates that the symbol vector s consists of eight independently selected constellation points. It means that the optimal solution is found by jointly determining eight information symbols. Specifically, when the M -QAM modulation is adopted by the information symbols, a bruteforce searching of ŝML requires testing all M 8 possibilities of the signal vector, which is computationally intensive. Fast decoding of the STBC based on orthogonal-triangular (QR) decomposition has been discussed in literatures [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF]. More precisely, by performing Gram-Schmidt procedure to the columns of the equivalent channel matrix H eq , it yields an unitary matrix Q and an upper triangular matrix R, i.e. H eq = QR where Q [q 1 , . . . , q 16 ] and

R      r 1 2 q 1 , h 2 • • • q 1 , h 16 0 r 2 2 • • • q 2 , h 16 . . . . . . . . . . . . 0 0 • • • r 16 2      , (7)
where

r 1 = h 1 , r j = h j - j-1 k=1 q k , h j q k , q j = r j / r j , j = 1, . . . , 16.
Instead of solving ( 6), the ML solution can be alternatively obtained by:

ŝML = arg min s∈Θ 8 z -R s 2 , (8)
where z = Q H y is the real-valued received signal after a linear operation Q H . For a well-designed STBC, some elements of R are equal to zero, which permits some information symbols to be determined independently from others.

In other words, a joint detection in high dimension is turned to several independent detections in low dimension, leading to a significant reduction of decoding complexity [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF].

B. Important properties of 3D MIMO code

With the definitions in (1), ( 2) and ( 5), we can derive the real-valued 16 × 16 equivalent channel matrix H eq . Rewrite the upper triangular matrix R:

R =     R 11 R 12 R 13 R 14 0 R 22 R 23 R 24 0 0 R 33 R 34 0 0 0 R 44     , (9)
where R jk 's are 4 × 4 matrices containing q m , h n 's with m = 4(j -1) + 1, . . . , 4j and n = 4(k -1) + 1, . . . , 4k. More importantly, R matrix has the following interesting properties. Theorem 1: R 11 is an upper triangular matrix with

q 1 , h 2 = q 1 , h 4 = q 2 , h 3 = q 3 , h 4 = 0.
Proof: According to the definition of QR decomposition, R 11 is an upper triangular matrix.

With some straightforward computations based on H eq , it yields

h 1 , h 2 = h 1 , h 4 = h 2 , h 3 = h 3 , h 4 = 0. From the definition of QR decomposition, q 1 = h 1 / h 1 . Hence, q 1 , h 2 = q 1 , h 4 = 0.
In addition,

r 2 = h 2 -q 1 , h 2 q 1 = h 2 , q 2 = r 2 / r 2 = h 2 / h 2 . Therefore, using h 2 , h 3 = 0, it yields q 2 , h 3 = 0.
Finally, r 3 = h 3 -2 j=1 q j , h 3 q j = h 3q 1 , h 3 q 1 , q 3 = (h 3q 1 , h 3 q 1 )/ r 3 . Therefore, q 3 , h 4 = ( h 3 , h 4q 1 , h 3 q 1 , h 4 )/ r 3 = 0.

Remark: Theorem 1 suggests that the real and imaginary parts of information symbol s 1 and s 2 can be decoded independently. Using similar idea and procedure, the same property can be derived for R jj , j = 2, 3, 4.

Theorem 2: R 13 is a null matrix when the channel is quasi-static, i.e. q j , h k = 0 with j = 1, 2, 3, 4 and k = 9, 10, 11, 12.

Proof: Using the same method as in the proof of Theorem 1, it yields q j , h k = 0, ∀j = 1, 2, 3, k = 9, 10, 11, 12.

Taking into account that q 1 , h 4 = q 3 , h 4 = 0, it yields r 4 = h 4 -3 j=1 q j , h 4 q j = h 4q 2 , h 4 q 2 . Hence, q 4 , h k = ( h 4 , h kq 2 , h 4 q 2 , h k )/ r 4 = 0, ∀k = 9, 10, 11, 12.

Remark: Theorem 2 suggests that z 1 and z 2 do not contain contribution from s 5 and s 6 . It enables separating decoding into groups. The orthogonalities between columns partially come from the Alamouti structure embedded in the codeword which requires the quasi-staticity of the channel. Proof: Due to the length limitation, we omit the details of basic manipulations and only present some sketches of the proof.

Denote the jth column of R 23 as p j , i.e.

R 23 = [p 1 , p 2 , p 3 , p 4 ].
From the definition of H eq and using previous two theorems, it is easy to prove that: q j , h k = q j+1 , h k+1 , ∀j = 5, 7 and k = 9, 11, (10) q j+1 , h k =q j , h k+1 , ∀j = 5, 7 and k = 9, 11, (11)

h 5 , h 9 h 6 , h 11 -h 5 , h 11 h 6 , h 9 = h 6 , h 9 h 7 , h 9 -h 5 , h 9 h 8 , h 9 = h 6 , h 11 h 7 , h 9 -h 5 , h 11 h 8 , h 9 . (12)
(10) and [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF] suggest that the first and the second columns of matrix R 23 are orthogonal, i.e. p 1 , p 2 = 0. Hence, it is sufficient to assert that the (1, 2)th element of matrix F is zero.

In addition, using the properties in ( 10) and ( 11), the inner product of matrix R 23 's first and fourth columns writes: p 1 , p 4 = q 6 , h 9 q 5 , h 11q 5 , h 9 q 6 , h 11 + q 8 , h 9 q 7 , h 11q 7 , h 9 q 8 , h 11 With the definition of H eq , it can be shown that:

= 1 r 5
r 7 2 = r 5 2 1+ 1 h 5 2 h 5 , h 7 - 1 h 5 4 h 5 , h 7 2 . ( 14
)
Taking into account the equalities in ( 12) and ( 14), ( 13) is turned to p 1 , p 4 = 0, which is sufficient to assert that the (1, 4)th element of matrix F is also zero.

Following similar procedure, we can prove that the (2, 3)th and (3, 4)th elements of F are equal to zero, as well.

The aforementioned properties are illustrated in Fig. 1 where zero and nonzero entries of matrix R are easily seen. These properties can be exploited to achieve low-complexity ML decoding, which will be demonstrated in the following parts.

C. Simplified ML decoding

Based on the theorems provided in the previous part, a simplified ML detection for the 3D MIMO code is proposed in this subsection. The basic idea is that, using Theorems 2 and 3, the joint detection of eight information symbols is converted into several detections in lower searching dimension in parallel, which results in a lower global detection complexity. Moreover, using Theorem 1 and its inferences, the detection of complex information symbols is turned to independent detections of real and imaginary parts in parallel, which further reduces the complexity.

More precisely, with the knowledge of matrix R in ( 9) and taking into account Theorem 2, the ML detection metric in (8) can be expressed as: 15) to (18), it can be seen that symbol groups a and c can be determined independently from each other for given b and d. For instance, a is obtained by using Illustration of the modified ML decoding metric in quasi-static channel.

z -R s 2 = z 78 -R 44 d 2 (15) + z 56 -R 33 c -R 34 d 2 (16) + z 34 -R 22 b -R 23 c -R 24 d 2 (17) + z 12 -R 11 a -R 12 b -R 14 d 2 , ( 18
)
where a = [s 1 , s 2 ] T , b = [s 3 , s 4 ] T , c = [s 5 , s 6 ] T , d = [s 7 , s 8 ] T , z 12 = [z 1 , z 2 ] T , z 34 = [z 3 , z 4 ] T , z 56 = [z 5 , z 6 ] T and z 78 = [z 7 , z 8 ] T . From (
only (18), if d is already known. This motivates us to perform conditional detection [START_REF] Sirianunpiboon | Fast optimal decoding of multiplexed orthogonal designs by conditional optimization[END_REF] to realize group-wise decoding.

Specifically, the ML detection ( 8) can be rewritten in an equivalent form:

ŝML = arg min [b,d]∈Θ 4 z 78 -R 44 d 2 (19) + arg min a∈Θ 2 v 12 -R 11 a 2 (20)
+ arg min

c∈Θ 2 v 56 -R 33 c 2 + v 34 -R 23 c 2 , ( 21
)
where

v 56 = z 56 -R 34 d, v 34 = z 34 -R 22 b -R 24 d and v 12 = z 12 -R 12 b -R 14 d.
It suggests that the joint searching of eight information symbols is turned into two independent searching of two information symbols (shown by the two minimum operations inside the parentheses) conditioned on the other four information symbols (the first minimum operation outside the parentheses). Therefore, the ML decoding complexity is reduced from O(M 8 ) to O(M 6 ). Note that this complexity reduction is achieved without any constraint on the constellation of information symbols. In addition, Theorem 1 suggests that the real and imaginary parts of a can be determined independently, which results in a further reduction in complexity. Interestingly, according to Theorem 3, the real and imaginary parts of c can be obtained independently, as well. Specifically, the minimum operation for searching c in ( 21) is turned equivalently to:

arg min c∈Θ 2 v 56 -R 33 c 2 + u 34 -Fc 2 , ( 22
)
where u 34 = E T v 34 . The resulting ML detection metric is illustrated in Fig. 2, where the independency of the real and imaginary parts of a and c is clearly shown. Provided that the real and imaginary parts of the information symbols are independently modulated (such as in the square QAM case), the ML decoding complexity is then reduced to O(M 5 ). Moreover, as far as the square QAM constellation is concerned, the decoding of real (imaginary) parts of information symbols can be further simplified. Take the detection of a as an example. The square M -QAM complex symbols a are [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF] O(M 6 ) O(M 4.5 ) DjABBA [START_REF] Hotinen | Multi-antenna Transceiver Techniques for 3G and Beyond[END_REF] O(M 7) O(M 6 ) BHV [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] O(M 6 ) O(M 4.5 ) Srinath-Rajan [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF] O(M 5 ) O(M 4.5 )

separated into √ M -PAM real symbols on both real and imaginary axes, denoted as

a R = [s R 1 , s R 2 ]
T and a I = [s I 1 , s I 2 ] T , respectively. The searching for a in (20) is converted into [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF]:

arg min a∈Θ 2 v 12 -R 11 a 2 = arg min a R ∈Ψ 2 v R 12 -R R 11 a R 2 +arg min a I ∈Ψ 2 v I 12 -R I 11 a I 2 , ( 23
)
where we slightly abuse the notation by denoting v R 12 (v I 12 ) as the first and third (second and fourth) elements of v 12 , R R 11 (R I 11 ) is tailored accordingly, Ψ is the set of √ M -PAM constellation points. Furthermore, the conditional detection is applied again here. For a given s R 2 , the metric of the real part writes:

v R 12 -R R 11 a R 2 = w 12 (1)-R 11 (1, 1)s R 1 2 +w 12 (3) 2 , (24)
where

w 12 (1) = v 12 (1) -R 11 (1, 3)s R 2 and w 12 (3) = v 12 (3) -R 11 (3, 3)s R 2 .
The metric is a quadratic function of s R 1 . Therefore, the best PAM symbol that minimizes the metric is easily found by:

ŝR 1 = Q v 12 (1) -R 11 (1, 3)s R 2 R 11 (1, 1) , (25)
where Q(•) is the slicing operation providing the PAM symbol that is closest to the given value. Following the same procedure, for a given s R 6 we have:

ŝR 5 = Q w 56 (1)R 33 (1, 1) + w 34 (1)F(1, 1) R 33 (1, 1) 2 + F(1, 1) 2 , (26)
where w 56 (1) = v 56 (1) -R 33 (1, 3)s R 6 and w 34 (1) = u 34 (1) -F(1, 3)s R 6 . Similar expressions can be derived for ŝI 1 and ŝI 5 , as well. Obviously, decoding of real symbol groups such as a R is turned into two searchings over √ M -PAM constellation points requiring a complexity of O( √ M ). Therefore, the decoding complexity is reduced to O(M 4.5 ). Eventually, fully exploiting all the aforementioned properties, the 3D MIMO code turns out to be a fast decodable STBC.

The ML decoding complexities of state-of-the-art 4×2 rate-2 STBCs are compared in Table I. Note that all these STBCs require the quasi-staticity of the channel to achieve the claimed low complexities. It can be seen that the 3D MIMO code requires equivalent complexity as other fast-decodable STBCs.

D. Efficient implementation of the simplified ML decoder

The pseudocode of an implementation of the simplified ML decoder is illustrated in Algorithm 1 which is presented at the end of this paper. Its major part follows the derivation in the previous subsection. The two outermost 'for' loops performs The detection of a R , a I , c R and c I is implemented by realvalued sphere decoder [START_REF] Agrell | Closest point search in lattices[END_REF], as shown e.g. from line 13 to 25 (or from line 27 to 39). Moreover, the Schnorr-Euchner (S-E) enumeration arranges the searching sequence according to the distances between the constellation points and the received signal to speed up the searching convergence [START_REF] Agrell | Closest point search in lattices[END_REF]. It can be simply implemented by look-up table S-E(x) where x is the zero-forcing (ZF) result of the received signal (lines 13 and 27 ) [START_REF] Wiesel | Efficient implementation of sphere demodulation[END_REF]. Note that the sort function is actually implemented using the same technique to reduce the complexity.

Once a combination of information symbols having a smaller distance than the minimum distance in the previous search is found, the current solution x and the minimum distance d min are updated (line 42). In other words, the searching radius is adaptively adjusted in the decoding progress, helping the fast convergence of the searching (lines 7, 17 and 31).

Note that Algorithm 1 is a straightforward implementation of the proposed simplified decoder without sacrificing ML optimality. Other techniques such as statistical tree pruning [START_REF] Ghaderipoor | A statistical pruning strategy for schnorr-euchner sphere decoding[END_REF] and sorted QR decomposition [START_REF] Wubben | Efficient algorithm for decoding layered space-time codes[END_REF] can also be incorporated in the implementation providing various performance-complexity trade-offs.

IV. SIMULATION

We evaluate the proposed low-complexity ML decoder by simulation in this section. The proposed decoder is implemented according to the pseudocode in Algorithm 1. The sphere decoder with S-E enumeration is realized based on Guo-Nilsson's algorithm [START_REF] Guo | Reduced complexity Schnorr-Euchner decoding algorithms for MIMO systems[END_REF] which is an improved algorithm of the classical implementation [START_REF] Agrell | Closest point search in lattices[END_REF] achieving much lower complexity than the original version. As the proposed decoder contains four parallel searching branches, the processing time delay is determined by the maximum visited nodes among all searching branches. In contrast, since the classical sphere decoder does not exploit the embedded properties of the code and follows a serial implementation, the delay is the time spent by the whole decoding process. The comparison takes the common assumption that the processing time for checking each possible solution (referred to as 'node') is approximately the same for both methods. The channel is modeled as quasistatic i.i.d. Rayleigh fading channel. Symbol constellation is 4-QAM.

Fig. 3 presents the bit error rate (BER) performance of the proposed simplified decoder, Guo-Nilsson's sphere decoder and the optimal ML decoder without taking into account the channel coding. This is to show the 'pure' decoding performance of the STBC decoders. The three decoders achieve almost the same performance. Especially, the curve of proposed decoder overlaps with that of ML decoder, which suggests that the proposed simplified decoder provides the optimal decoding performance.

Fig. 4 presents the decoding complexity in terms of processing time delay. As can be seen from the figure, both decoders spend much less complexity than the ML decoder which needs to check 4 8 = 65536 times. Moreover, the proposed decoder achieves a lower complexity than the classical sphere decoder within the whole signal-to-noise ratio (SNR) range. Especially, the improvement is more significant in low SNR region. For instance, the average visited nodes is reduced from 2738.9 to 550.7 at SNR of 0 dB, namely about 80% reduction in processing time. The time reduction is over 53% at SNR of 10 dB. The improvement decreases in higher SNR region i.e. 15∼30 dB. It is due to the fact that the ZF solution is more accurate at higher SNR and hence the S-E enumeration helps greatly improving the sphere decoding speed. Nevertheless, in high SNR region, the nodes visited by the proposed method approaches to 29 which is less than 37.6, the amount required by the classical sphere decoder. This still leads to 23% reduction of processing time.

V. CONCLUSION

In this work, we first explore some interesting properties of the 3D MIMO code. With this knowledge, we propose a simplified ML decoder which reduces the decoding complexity from O(M 8 ) to O(M 4.5 ) in quasi-static channel. Consequently we propose an implementation of the simplified ML decoder. Simulation results show that the proposed simplified decoder needs less processing time, especially in the low SNR region, than the classical sphere decoder with S-E enumeration without sacrificing the ML decoding optimality.

Theorem 3 :

 3 Performing QR decomposition R 23 = EF, the yielding upper triangular matrix F has similar structure as R 11 , namely its (1, 2),[START_REF] Tarokh | Space-time codes for high data rate wireless communication: Performance criterion and code construction[END_REF][START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF],[START_REF]DVB-NGH: Next Generation Handheld[END_REF][START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF] and[START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF][START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] elements equal to zero.

Fig. 1 .

 1 Fig. 1. Illustration of the ML decoding metric in quasi-static channel.

  Fig. 2.Illustration of the modified ML decoding metric in quasi-static channel.

Fig. 3 .

 3 Fig. 3. BER comparison of sphere decoder with S-E, proposed simplified ML decoder and the ML decoder in quasi-static Rayleigh channel with 4-QAM.

Fig. 4 .

 4 Fig. 4. Computational complexity required by sphere decoder with S-E and proposed simplified ML decoder, in quasi-static Rayleigh channel with 4-QAM constellation.

  2 h 5 , h 11 h 6 , h 9h 5 , h 9 h 6 , h 11 + 1 r 7 2 h 6 , h 9 h 7 , h 9h 5 , h 9 h 8 , h 9

+ 1 h 5 2 h 5 , h 7 h 6 , h 11 h 7 , h 9h 5 , h 11 h 8 , h 9 -1 h 5 4 h 5 , h 7 2 h 5 , h 9 h 6 , h 11h 5 , h 11 h 6 , h 9 .

ACKNOWLEDGMENT

The authors would like to thank the support of French ANR project "Mobile Multi-Media (M3)" and "Pôle Images & Réseaux".

Algorithm 1: Implementation of the proposed simplified ML decoder for 3D MIMO code.


Ming Liu

Maryline Hélard

Jean-Franc ¸ois Hélard
email: jean-francois.helard@insa-rennes.fr

Matthieu Crussière
email: matthieu.crussiere@insa-rennes.fr

A Fast Decodable Full-Rate STBC with High Coding Gain for 4 × 2 MIMO Systems

In this work, a new fast-decodable space-time block code (STBC) is proposed. The code is full-rate and full-diversity for 4 × 2 multiple-input multiple-output (MIMO) transmission. Due to the unique structure of the codeword, the proposed code requires a much lower computational complexity to provide maximum-likelihood (ML) decoding performance. It is shown that the ML decoding complexity is only O(M 4.5 ) when Mary square QAM constellation is used. Finally, the proposed code has highest minimum determinant among the fast-decodable STBCs known in the literature. Simulation results prove that the proposed code provides the best bit error rate (BER) performance among the state-of-the-art STBCs.

I. INTRODUCTION

Quasi-orthogonal space-time block codes (STBCs) [START_REF] Jafarkhani | A quasi-orthogonal space-time block code[END_REF] support full coding rate for the multiple-input multiple-output (MIMO) transmissions with more than two transmit antennas. However, they suffer from high maximum-likelihood (ML) decoding complexity when the high order constellation is used.

The so-called fast decodable STBCs [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], [START_REF] Ismail | A new low-complexity decodable rate-1 full-diversity 4 × 4 STBC with nonvanishing determinants[END_REF], [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF]- [START_REF] Srinath | Generalized silver codes[END_REF] exploit the orthogonality embedded in the codewords and achieve ML decoding performance at a low decoding complexity. For instance, Biglieri-Hong-Viterbo (BHV) constructed a rate-2 STBC [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] based on the classical Jafarkhani code [START_REF] Jafarkhani | A quasi-orthogonal space-time block code[END_REF]. Exploiting the quasi-orthogonal structure embedded in the Jafarkhani codeword, the BHV code requires much less complexity than the real ML decoding without losing performance. Ismail-Fiorina-Sari (IFS) proposed a rate-1 STBC for four-transmitantenna systems [START_REF] Ismail | A new low-complexity decodable rate-1 full-diversity 4 × 4 STBC with nonvanishing determinants[END_REF] based on the rate-3/4 complex orthogonal design (COD) [START_REF] Tirkkonen | Square-matrix embeddable space-time block codes for complex signal constellations[END_REF]. Thanks to the underlying orthogonal structure, the ML decoding is achieved by using a low complexity conditional detector followed by hard decisions. This construction idea was then generalized to rate-2 STBC case [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF]. Srinath and Rajan (Srinath-Rajan) proposed a fastdecodable rate-2 STBC [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], [START_REF] Srinath | Generalized silver codes[END_REF] based on co-ordinate interleaved orthogonal design (CIOD) [START_REF] Khan | Single-symbol maximum likelihood decodable linear STBCs[END_REF]. The Srinath-Rajan code possesses high coding gain but needs low decoding complexity due to its orthogonal structure. Similarly, some other STBCs such as DjABBA code [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF], Golden code [START_REF] Belfiore | The Golden code: a 2 × 2 full-rate space-time code with nonvanishing determinants[END_REF] and 3D MIMO code [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF] were shown to be also fast-decodable because of the underlying orthogonality in the codewords.

In this paper, we investigate the full-rate STBC for fourtransmit-two-receive-antenna (4 × 2) MIMO transmission which is a typical asymmetric MIMO scenario considered in LTE, DVB-NGH [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF] etc. We propose a new fast-decodable STBC that requires low decoding complexity and possesses high coding gain. The main contributions of the paper are:

• A new STBC codeword that enables low-complexity MIMO decoding is proposed; • Theoretical analysis proves that the new code is one of the least complex STBCs with full-rate for 4 × 2 MIMO systems; • Corresponding low-complexity ML decoding method for the new code is presented; • The optimal rotation angle is proposed for the new codeword, which leads to the highest coding gain; • Simulation results prove that the new code provides the best BER performance. The reminder of the paper is organized as follows. Section II introduces the MIMO system model. The novel STBC is proposed in Section III. Low-complexity decoding method is also presented in this part. Simulation results are given in Section IV. Conclusions are drawn in section V.

Notations: X T and X H denote the transpose and conjugate transpose of a matrix X. I n and O n denote the identity matrix and null matrix of size n. x R and x I represent the real and imaginary parts of a complex variable x. The operator (•) realizes the complex to real conversion:

x x R -x I x I x R . (1)
The operation x

[x R 1 , x I 1 , . . . , x R n , x I n ]
T separates the real and imaginary parts of the complex vector x. The operation vec(X)

[x T 1 , x T 2 , . . . , x T n ]
T stacks the columns of X, i.e. x j 's, one below another. The combined operation vec(X)

[x T 1 , x T 2 , . . . , x T n ]
T first converts matrix X into stacked vector and then separates the real and imaginary parts. x, y

x T y denotes the inner product of two vectors x and y. ⊗ represents the Kronecker product.

II. SYSTEM MODEL

A. MIMO system model

The codeword matrix of a linear dispersion STBC that contains κ information symbols can be represented by a linear combination [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]:

X = κ j=1 s R j A 2j-1 + s I j A 2j , (2)
where A 2j-1 ∈ C Nt×T (A 2j ∈ C Nt×T ), j = 1, 2, . . . , κ, are the complex weight matrices representing the contribution of the real (imaginary) part of the jth information symbol s j in the final codeword matrix. One STBC codeword X ∈ C Nt×T is transmitted by N t transmit antennas over T channel uses.

Assuming that the receiver has N r receive antennas, the received signal can be expressed as:

Y = HX + W, (3)
where H ∈ C Nr×Nt is the MIMO channel matrix in which the (j,k)th element h j,k represents the gain of the channel link between the kth transmit antenna and jth receive antenna; Y, W ∈ C Nr×T represent the received signal and noise, respectively. The channel is assumed to be quasi-static, which is a common assumption that is guaranteed by the system design. The MIMO transmission in (3) can be rewritten in a real-valued equivalent form:

y = H eq s + w, (4)
where y = vec(Y), w = vec(W) and H eq ∈ R 2NrT ×2κ is the equivalent channel matrix and is computed by [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF]:

H eq = (I T ⊗ Ȟ)G, (5)
in which the generator matrix G ∈ R 2NtT ×2κ is obtained by:

G [ vec(A 1 ), vec(A 2 ), . . . , vec(A 2κ )]. (6)

B. ML detection using sphere decoder

We represent the equivalent channel matrix in column vectors, i.e. H eq [h 1 , h 2 , . . . , h 2κ ]. After QR decomposition, the equivalent channel matrix can be decomposed into an orthogonal matrix Q and an upper triangular matrix R, i.e. H eq = QR where Q [q 1 , q 2 , . . . , q 2κ ] and

R      r 1 2 q 1 , h 2 • • • q 1 , h 2κ 0 r 2 2 • • • q 2 , h 2κ . . . . . . . . . . . . 0 0 • • • r 2κ 2      , (7)
where

r 1 = h 1 , r j = h j - j-1
k=1 q k , h j q k , q j = r j / r j , j = 1, . . . , 2κ. From (4) and taking advantage of QR decomposition of H eq , the ML solution of the transmitted symbols can be acquired by:

ŝ = arg min s∈Θ κ z -R s 2 , (8)
where z = Q T y is a linear transform of received signal, and Θ is the set of constellation symbols. It is actually a joint search of κ information symbols. The resulting complexity is O(M κ ) when the constellation is M -QAM.

III. PROPOSED FAST-DECODABLE STBC

A. Codeword matrix of DjABBA code Let's recall the codeword matrix of the DjABBA code [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF]: where ρ is the a rotation angle dedicated to optimizing the pairwise error of the code; X A , X B , X C and X D are four Alamouti codewords [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] associated with four information symbol pairs

X = cos ρ X A + sin ρ X C cos ρ X B + sin ρ X D i(sin ρ X B -cos ρ X D ) sin ρ X A -cos ρ X C , (9)
[s 1 s 2 ], [s 3 s 4 ], [s 5 s 6 ]
and [s 7 s 8 ], respectively. For instance,

X A = s 1 s 2 -s * 2 s * 1 . (10)
Since eight information symbols are transmitted over four channel uses, leading to a space-time coding rate of 2 which means full-rate for two-receive-antenna systems.

After performing QR decomposition to the channel matrix, the resulting upper triangular matrix R is illustrated in Fig. 1. The contributions of information symbols s R 1 , s I 1 , s R 2 and s I 2 are not correlated in the received signal which could be exploited to achieve a ML decoding complexity of O(M 6 ), instead of O(M 8 ) [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]. More interestingly, we also notice that this uncorrelation also exists in other information symbol pairs. This motivates us to propose a new codeword that fully exploits the embedded orthogonality to achieve lower decoding complexity.

B. Proposed code

As mentioned previously, low decoding complexity is commonly achieved by exploiting the orthogonality between information symbols. In the literature, many fast-decodable STBCs [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], [START_REF] Ismail | A new low-complexity decodable rate-1 full-diversity 4 × 4 STBC with nonvanishing determinants[END_REF], [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF] were designed so that some information symbols are uncorrelated with others. This symbol independency enables group-wise detections in parallel which can significantly reduce the decoding complexity.

As far as the DjABBA code is concerned, the orthogonality embedded in the Alamouti structure can be exploited to achieve lower decoding complexity. For instance, as shown in Fig. 1, the 4 × 4 submatrices located in the main diagonal positions of the R matrix are identity matrices. This means that the two information symbols in each Alamouti codeword (e.g. X A , X B , X C and X D ) are uncorrelated because of the orthogonality in the Alamouti structure. The ML decoding complexity is reduced from O(M 8 ) to O(M 6 ) thanks to this orthogonality within each Alamouti codeword. Intuitively, even lower complexity can be achieved if we can build orthogonality among more information symbols. Before proposing the new codeword, let's recall a simple fact related to the Alamouti structure. The lemma can be proved via some simple manipulations. Its proof is omitted here. This lemma actually means that the linear combination of two Alamouti codewords preserves the Alamouti structure as well as the orthogonality between different parts. It gives a way to build orthogonality among more information symbols.

Moreover, it is known that the decoding complexity is mainly determined by the orthogonality among the first several information symbols when the conditional detection is used [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]. For example, in the design of BHV code, the low decoding complexity is achieved by the Jafarkhani-like quasiorthogonal structure embedded in the first four information symbols. This motivates us to build orthogonality among {s 1 , s 2 , s 3 , s 4 } in the DjABBA codeword in order to reduce the decoding complexity.

With the knowledge presented above, we propose a new codeword as:

X new = cos ρ X A + sin ρ X B cos ρ X C + sin ρ X D i(sin ρ X C -cos ρ X D ) sin ρ X A -cos ρ X B , (11)
where the rotation angle ρ is chosen to be ρ = tan -1 ( 1+ √ 5

2 ) in order to maximize the coding gain. Some discussion on the selection of ρ will be given later.

The new codeword is very similar to the original one presented in [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF] except that it is formed by the linear combinations of X A and X B , as well as those of X C and X D . These modifications, as will be shown later, yield orthogonality among more information symbols, and therefore lead to a decoding complexity reduction. A detailed expression of the codeword matrix is given in [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF] shown on next page. After QR decomposition, the R matrix has a very good structure, as depicted in Fig. 2. More precisely, the property of R matrix can be expressed in the following theorem.

Theorem 1: For j, k ∈ {1, 2, . . . , 8}, in the upper triangular matrix R we have q j , h k = 0, ∀k = j or ∀k = j + 4.

Proof : From the codeword matrix [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF], it can be easily verified that, for j, k ∈ {1, 2, . . . , 8}:

A j A H k + A k A H j = O 4 , ∀k = j or ∀k = j + 4. (13)
Using the Theorem 2 given in [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], it yields that the corresponding jth and kth columns of equivalent channel matrix H eq are orthogonal, i.e. for j, k ∈ {1, 2, . . . , 8}:

h j , h k , ∀k = j or ∀k = j + 4. (14)
Consequently, according to the definition of QR decomposition, it can be obtained that, for j ∈ {1, 2, 3, 4}:

r j = h j , q j = h j / h j . (15)
Using ( 14) and (15), it is obviously that, for j ∈ {1, 2, 3, 4}, k ∈ {1, 2, . . . , 8}:

q j , h k = 0, ∀k = j or ∀k = j + 4. (16)
Moreover, using (16), we have for j ∈ {5, 6, 7, 8}:

r j = h j -q j-4 , h j q j-4 . (17)
Therefore, using ( 14) and ( 16), it yields that, for j ∈ {5, 6, 7, 8} and k ∈ {1, 2, . . . , 8}:

q j , h k = ( h j , h k -q j-4 , h j q j-4 , h k )/ r j = 0, ∀ k = j or ∀k = j + 4. (18)
Combining ( 16) and ( 18), it completes the proof.

Remark: Theorem 1 indicates that six real-valued information symbols are uncorrelated, as we designed. In fact, dividing the four complex symbols {s 1 , s 2 , s 3 , s 4 } into four groups {s R 1 , s R 3 }, {s I 1 , s I 3 }, {s R 2 , s R 4 }, and {s I 2 , s I 4 }, the two symbols within each group are correlated in the received signal, while symbols of different groups are uncorrelated, as illustrated in Fig. 2. This independency permits us to reduce the searching space in the decoding process, which will be presented in the following section.

C. Low complexity ML decoding

The independency between information symbols shown in Theorem 1 can be exploited to reduce the decoding complexity. For instance, the received signal z 1 does not contain any contribution from information symbols s 2 and s 4 . It means that the ML solutions of six information symbols {s 1 , s 3 , s 5 , s 6 , s 7 , s 8 } can be jointly determined regardless the choices of {s 2 , s 4 }. Similarly, z 2 does not have contribution from s 1 and s 3 . The six information symbols {s 2 , s 4 , s 5 , s 6 , s 7 , s 8 } can be decided together without considering the solutions of {s 1 , s 3 }. Hence, the overall detection of eight information can be carried out by a joint detection of four symbols {s 5 , s 6 , s 7 , s 8 } followed by two independent detections of {s 1 , s 3 } and {s 2 , s 4 } in parallel. In general, the ML decoding is realized by joint searches of six information symbols which results a complexity of O(M 6 ). Note that the parallel detections do not rely on the characteristic of the constellation. In other words, this complexity reduction is applicable for arbitrary constellation scheme. Furthermore, when the square M -QAM is adopted, the detections of real parts and imaginary parts of information symbols can be decoupled. Rewrite the 16 × 16 real-valued R matrix by:

X new =     cos ρ s 1 +
R = R 1 R 2 O R 4 , (19)
where R 1 R 2 and R 4 are 8 × 8 submatrices. Separating the symbol vectors s and z in two groups, i.e.

s (1) = [s 1 , s 2 , s 3 , s 4 ] T , s (2) = [s 5 , s 6 , s 7 , s 8 ] T , z (1) = [z 1 , z 2 , z 3 , z 4 ] T and z (2) = [z 5 , z 6 , z 7 , z 8 ]
T , the ML decoding in ( 8) is converted into a conditional detection:

arg min s∈Θ 8 ( z (1) -R 1 s (1) -R 2 s (2) 2 + z (2) -R 4 s (2) 2 ) = arg min s (2) ∈Θ 4 ( z (2) -R 4 s (2) 2 +arg min s (1) ∈Θ 4 v (1) -R 1 s (1) 2 ), (20)
where v (1) = z (1) -R 2 s (2) . In addition, if we take into account the property of R 1 given in Theorem 1, the inner search of four complex symbols s (1) is simplified to be four independent searches for {s

R 1 , s R 3 }, {s I 1 , s I 3 }, {s R 2 , s R 4 }
, and {s I 2 , s I 4 }, respectively. More precisely, we have: arg min

s (1) ∈Θ 4 v (1) -R 1 s (1) 2 = arg min s R 3 ∈Ψ ((v R 1 -R 1,1 sR 1 -R 1,5 s R 3 ) 2 + (v R 3 -R 5,5 s R 3 ) 2 )
+ arg min

s I 3 ∈Ψ ((v I 1 -R 2,2 sI 1 -R 2,6 s I 3 ) 2 + (v I 3 -R 6,6 s I 3 ) 2 ) + arg min s R 4 ∈Ψ ((v R 2 -R 3,3 sR 2 -R 3,7 s R 4 ) 2 + (v R 4 -R 7,7 s R 4 ) 2 ) + arg min s I 4 ∈Ψ ((v I 2 -R 4,4 sI 2 -R 4,8 s I 4 ) 2 + (v I 4 -R 8,8 s I 4 ) 2 ), (21)
where Ψ is the set of √ M -PAM constellation symbols; R j,k represents the (j, k)th element of the R matrix; sR 1 , sI 1 , sR 2 and sI 2 are the √ M -PAM constellation symbols that minimize the ML decoding metrics given s R 3 , s I 3 , s R 4 and s I 4 , respectively, and can be obtained via simple hard decisions:

s R 1 = Q v R 1 -R 1,5 s R 3 R 1,1 , s I 1 = Q v I 1 -R 2,6 s I 3 R 2,2 , (22)
s R 2 = Q v R 2 -R 3,7 s R 4 R 3,3 , s I 2 = Q v I 2 -R 4,8 s I 4 R 4,4 , (23)
where Q(x) is the hard decision function that returns the √ M -PAM symbol which is closest to the given value x. It can be seen that the parallel detections in ) O(M 4.5 ) DjABBA [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF] 0.8304 † O(M 7) O(M 6 ) Srinath-Rajan [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF] 10.24 O(M 5 ) O(M 4.5 ) 3D MIMO [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF] 0.0318 O(M 6 ) O(M 4.5 ) IFS rate 2 [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF] 0.0076 O(M 5 ) O(M 4.5 ) BHV [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] 0 O(M 6 ) O(M 4.5 ) † Using the best rotation ρ = cos -1 (0.8881) [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF] known in the literature.

QAM constellations are used. The decoding complexities of some state-of-the-art fast-decodable STBCs are given in Table I. It can be seen that the proposed code is among the least complex ones.

D. Optimization of minimum determinants

It is well known that in the STBC design the minimum determinant of the codeword difference matrix should be maximized to achieve higher coding gain which consequently leads to better pairwise error probability (PEP) performance [START_REF] Tarokh | Space-time codes for high data rate wireless communication: performance criterion and code construction[END_REF]. In the literature, constellation rotation is a common way to maximize the coding gain [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF], [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]. The rotated constellation brings additional diversities between in-phase and quadrature components of the signal. As far as the proposed codeword is concerned, the rotation angle ρ actually performs the constellation rotation and should be optimized in order to maximize the minimum determinant of the codeword difference matrix. It is worth noting that the choice of ρ does not affect the fast decodability of the codeword.

We propose to use the rotation angle ρ = tan -1 ( 1+ √

2 ), a value originated from the Golden number, in the new codeword [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in single-frequency networks[END_REF]. It has been shown in theory that a constellation rotation with Golden number actually gives best performance [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF], [START_REF] Belfiore | The Golden code: a 2 × 2 full-rate space-time code with nonvanishing determinants[END_REF]. In this work, it is proved through exhaustive search that the minimum determinant of the proposed codeword with the "Golden rotation" is 10.24 for unnormalized QAM constellations. This is the highest value for the 4 × 2 full-rate STBCs reported in the literature [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF], [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF]. A comparison of the minimum determinant of different 4 × 2 full-rate STBCs is given in Table I. It can be seen that the proposed code has the same determinant as the Srinath-Rajan code and has higher determinant than other STBCs. Finally, it is worth noting that the non-zero minimum determinant also indicates that the proposed new code achieves full-diversity for the fourtransmit-antenna MIMO transmissions. IV. SIMULATION RESULTS Fig. 3 and Fig. 4 present the bit error rate (BER) performance of different STBCs with QPSK and 16QAM, respectively. The channel model used in the simulation is the 4 × 2 MIMO channel with i.i.d. Rayleigh flat fading coefficients for all channel links. The ML decoder is used to decode the received MIMO signal. No channel coding scheme is used in the simulation to give the comparison of "pure" performance of the STBCs The rotation angle for the DjABBA code is chosen as ρ = cos -1 (0.8881) which is the best value known in the literature [START_REF] Hottinen | Precoder designs for high rate spacetime block codes[END_REF].

From the figures, it can be seen that the proposed new code provides the best BER performance among all STBCs considered in the comparison with both QPSK and 16QAM. This can be explained by the fact that the proposed code has the highest minimum determinant. In particular, the proposed new code achieves the same performance as the Srinath-Rajan code which has the same coding gain, while it outperforms other state-of-the-art STBCs such as 3D MIMO code, BHV code and IFS rate-2 code. Moreover, the new code performs better than the DjABBA code. It means that the proposed Golden rotation angle provides a better performance than the best proposal existing in the literature. In general, the simulation results prove the advantage of the proposed code in terms of the superior BER performance.

V. CONCLUSION

In this work, we propose a new fast-decodable full-rate fulldiversity STBC for 4 × 2 MIMO systems. The new code requires a ML decoding complexity of O(M 4.5 ) which is the least among all full-rate 4 × 2 STBCs in the literature. Moreover, with the proposed Golden rotation angle, the new code the possesses highest coding gain which provides a better PEP performance compared with other state-of-the-art STBCs. This is proved by the simulation results which show that the proposed code has superior BER performance.

Fig. 1 .

 1 Fig. 1. R matrix of DjABBA code in quasi-static channel.

Fig. 2 .

 2 Fig. 2. R matrix of the new code in quasi-static channel.

Lemma 1 :

 1 For two Alamouti codewords A and B, its linear combination C = aA + bB satisfies CC H = C H C = cI 2 , where a, b and c are scalar numbers.

  (21) are realized by searches over √ M -PAM symbols, resulting a complexity of O( √ M ). Combining (20) and (21), the ML decoding requires an overall detection complexity of O(M 4.5 ) when the square

Fig. 3 .

 3 Fig. 3. BER performance comparison of different 4 × 2 STBCs, QPSK, i.i.d. Rayleigh channel.

Fig. 4 .

 4 Fig. 4. BER performance comparison of different 4 × 2 STBCs, 16QAM, i.i.d. Rayleigh channel.

  sin ρ s 3 cos ρ s 2 + sin ρ s 4 cos ρ s 5 + sin ρ s 7 cos ρ s 6 + sin ρ s 8 -cos ρ s * 2 -sin ρ s * -cos ρ s 7 ) i(sin ρ s 6 -cos ρ s 8 ) sin ρ s 1 -cos ρ s 3 sin ρ s 2 -cos ρ s 4 -i(sin ρ s * 6 -cos ρ s * 8 ) i(sin ρ s * 5 -cos ρ s * 7 ) -sin ρ s * 2 + cos ρ s *

			
	4 i(sin ρ s 5 4 cos ρ s * 1 + sin ρ s * 3 -cos ρ s * 6 -sin ρ s * 8	cos ρ s * 5 + sin ρ s * 7 sin ρ s * 1 -cos ρ s * 3	  	(12)

ACKNOWLEDGMENT

The authors would like to thank the support of French ANR project "Mobile Multi-Media (M3)" and "Pôle Images & Réseaux".