Beam Adaptive Algorithms For Smart Antennas Computer Science Essay

Beam Adaptive Algorithms For Smart Antennas Computer Science Essay Smart antennas employed in space division multiple access systems can cater the high demand in terms of capacity in mobile applications without further increase in radio frequency spectrum allocation. Furthermore, smart antennas provide better quality of service (QoS) and better coverage. Therefore, beam adaptive algorithms used in smart antenna systems are of great interests. 1.2 Project Aims The project aim is to study and understand the adaptive algorithms for beamforming for Smart Antenna Systems and to develop research skills, by reading research papers and journal papers related to the topic. Moreover, to produce the technical draft that contains the analysis, results and discussion of several adaptive algorithms employed in Smart Antenna Systems. The research work conducted and a simulation will provide the better understanding of the subject and can be a possible contribution to the existing work. 1.3 Project Objectives The project objective is to attempt systematic comparison of the performance of different Adaptive Algorithms for beamforming for Smart Antenna System. The algorithms that will be under investigation in this project will be training sequence algorithms like Recursive Least Squares (RLS) and Least Mean Squares (LMS), and Constant Modulus Algorithm (CMA). Simulations will be done to find out which algorithms are best for beamforming i.e. to form main lobes towards desired user and for convergence rate. The comparison of algorithms will be made on the basis of formation of main lobe and the convergence rate. The effect of jammers in algorithms will also be studied. 1.4 Project Outcomes The outcome of the project is come up with simulation software that will calculate the parameters of Smart Antenna and tell us which algorithm performs better in a particular situation. Moreover to prepare a research report that contains critical analysis different beam adaptive algorithms. 1.5 Structure of the report The first chapter of the final report is the Introduction which describes the motivation for research, project aims and objectives. The second chapter is the Literature review. The overview of recent work conducted in this domain and the brief summary of several research papers studied for conducting this research has been presented. Reading list is appended in the references section. Third chapter is the introduction to antennas and smart antennas. The fundamentals parameters of antennas and smart antennas are briefly described. Fourth chapter is on smart antenna system. Types of smart antennas and their comparison are done in detail. In fifth chapter description about the signal processing algorithms is presented. Lastly given are the results and conclusion. Chapter 2: Literature Review Smart antenna is the most efficient leading innovation for maximum capacity and improved quality and coverage [1]. They can adapt to varying traffic requirements dynamically. Smart antennas radiate narrow beam to serve different users and are normally employed at base stations. The complex weight computations that are based on different criteria are integrated in the signal processor in the form of software algorithms [3]. Due to the rapid growing demand in mobile applications not only for capacity but for high quality of service (QoS) and better coverage without increasing the radio frequency spectrum allocation the wireless systems that uses fixed antenna systems will no longer be in use [3] discuss the need for smart antennas in space division multiple access systems. The article focused on adaptive beam forming approach based on smart antennas the adaptive algorithms that are employed to compute the complex weight are discussed and LMS and RLS in particular. The fact that error c omputed from the filter at time n is used to provide the filter coefficient at time n+1 provides a nontraditional way to understand adaptive algorithms. [2] describe the approach of seeing classical adaptive algorithm like LMS, RLS, CMA, Decision directed) as recursive structures. [4] Explain the normalized least mean square (NLMS) algorithm for smart antenna system. The algorithm was implemented on the StarCore SC3400 DSP core and the performance was evaluated on MSC8144 DSP and the antenna coverage pattern was obtained and analyzed to find towards the desired signal source. With adaptive beamforming algorithm to name LMS for smart antenna the downlink multiple-input multiple-output (MIMO) multi-carrier code division multiple access system is proposed [5]. [1] Describes the sequential Studies of beamforming algorithms for smart antenna systems. 2.1 Scope of Smart Antennas Smart antennas assured to award the significant increment in the capacity of system and its performance in the wireless communication system [11]. Which will eventually lead into increase profits for the telecommunications companies and also a decline in blocked and dropped calls. Its been about 45 years Antenna was first used in applications related to radar communication in the form of fixed array. In later years many researches on antenna helped into smart antennas and tiled the way for their uses in commercial wireless systems [12]. These are the main reasons smart antennas got so much interest over the few years. At present the application of smart antennas are predominant at the mobile base stations due to compact area and processing power requirements [13]. Currently, a lot of research is going on the mobile terminal based smart antennas. In the coming future we can expect smart antenna skills to be present at the base station and mobile terminal too. Chapter 3: Introduction to Antennas and Smart Antenna 3.1 Antenna According to IEEE Standard Definitions of Terms for Antenna, it is defined as a means for radiating or receiving radio waves. 3.2 Fundamental parameters of Antenna The fundamental parameters of Antenna are: Radiation pattern Radiation intensity Directive gain and directivity Power gain and Radiation efficiency Front to Back ratio Antenna beamwidth Antenna beam efficiency 3.2.1Radiation Pattern Radiation pattern show the angular variation of field strength. They are drawn at some distance r proportional to field intensity in the direction ÃŽÂ ¸ and à Ã¢â‚¬  . Normalized Field Pattern: It is obtained by dividing the field component by its maximum value. It is dimensionless. EÃŽÂ ¸(ÃŽÂ ¸,à Ã¢â‚¬  ) = EÃŽÂ ¸(ÃŽÂ ¸,à Ã¢â‚¬  ) / EÃŽÂ ¸(ÃŽÂ ¸,à Ã¢â‚¬  ) max Power Radiation Pattern: Power density Pd is defined as power flow per unit area and is given by: Pd(ÃŽÂ ¸,à Ã¢â‚¬  ) =1/2*[E (ÃŽÂ ¸,à Ã¢â‚¬  )] 2/ÃŽÂ ·0 Pattern may also be expressed in terms of power per unit solid angle .The normalized power pattern can also be expressed in terms of this parameter as the ratio of radiation intensity U(ÃŽÂ ¸,à Ã¢â‚¬  ) as function of angle to its max value. Pn(ÃŽÂ ¸,à Ã¢â‚¬  ) = Pd(ÃŽÂ ¸,à Ã¢â‚¬  ) / Pd max(ÃŽÂ ¸,à Ã¢â‚¬  ) The co-ordinates are ÃŽÂ ¸ and dB are used to draw the pattern and calculate by following relation: dB= 10log10Pn(ÃŽÂ ¸,à Ã¢â‚¬  ) 3.2.2 Radiation Intensity The power radiated from an antenna per unit solid angle is called the radiation intensity U. 3.2.3 Directive Gain and Directivity For omnidirectional antenna: the power density at all the points on the surface of a sphere will be same. Pavg = Prad/4à Ã¢â€š ¬r2 The directive gain is defined as the ratio of the power density Pd(ÃŽÂ ¸,à Ã¢â‚¬  ) to the average power radiated. For isotropic antenna, the value of directive gain is unity. The directive gain can be defined as a measure of the concentration of the radiated power in a particular direction (ÃŽÂ ¸,à Ã¢â‚¬  ) . The ratio of the maximum power density to the average power radiated is called maximum directive gain or directivity of the antenna. GDmax = Pdmax/Pavg 3.2.4 Power Gain and Radiation efficiency The relation between input power and power radiated is given as: Prad=ÃŽÂ ·r Pin ÃŽÂ ·r=Prad/ Pin Pin=Prad +Ploss ÃŽÂ ·r=Prad/(Prad +Ploss) The power radiated and the ohmic power loss can be expressed in terms of r.m.s. current as: ÃŽÂ ·r=Rrad/(Rrad +Rloss) The ratio of the power radiated in a particular direction (ÃŽÂ ¸,à Ã¢â‚¬  ) to the actual power input to the antenna is called power gain of antenna. 3.2.5 Front to Back ratio (FBR) It is the ratio of the power radiated in the desired direction to the power radiated in the opposite direction. FBR = Power radiated in desired direction/Power radiated in opposite direction FBR depends on frequency of operation, spacing between antenna elements (inversely proportional) and electrical length of the parasitic elements of the antenna. FBR is an important consideration especially in receiving antennas 3.2.6 Antenna Beamwidth Antenna beamwidth is the measure of the directivity of the antenna. The antenna beamwidth is an angular width in degrees. It is measured on a radiation pattern on a major lobe. 3.2.7 Beam efficiency Total beam area ÃŽÂ ©A consists of the main beam area ÃŽÂ ©M plus the minor lobe area ÃŽÂ ©m. ÃŽÂ ©A= ÃŽÂ ©M+ ÃŽÂ ©m So Beam Efficiency is, M= ÃŽÂ ©M/ ÃŽÂ ©A Stray Factor: The ratio of minor lobe ÃŽÂ ©m area to the total beam area ÃŽÂ ©A is called stray factor. Em= ÃŽÂ ©m/ ÃŽÂ ©A EM + Em= 1 3.3 Smart Antenna The smart antenna is defined as an antenna array system that is supported by processing system that deals with the received and transmitted signal by the array using proper array algorithms to advance wireless system performance. [4] Numbers of distributive antenna elements are combined to make arrays of antennas called smart antenna. The different signals collected by individual antenna are calculated in such a manner that increases the signal strength of desired signal and reduces interference from other signals. [12] A smart antenna can be observed as a combination of antennas, whose transmitted or received signals are processed using smart algorithms. These smart algorithms make antenna work efficiently, and reliable in communication [13]. Main purpose of smart antenna is that the signal comes from the source hit the target in that way if the target is moving the antenna is such a smart and intelligent that it would change its direction according to the target movement and the main lobe of the antenna must be toward the target so that the maximum strength of the signal strikes the target. Mostly Smart antennas are being used in wireless communication systems to provide interference reduction and enhance user capacity and the data rates [6]. 3.4 Uses of Smart Antenna Smart antennas are considered useful in the wireless communication systems. The area coverage and the capacity of a system are increased by Smart antennas. Maximum data rate is increased by using smart antennas in multipath and diminish fading due to the terminating the component of multipath. One of its most useful applications is direction finding with the applications including emergency services and traffic monitoring [11]. In areas with less population, extended coverage is predominant in those areas than increased capacity. I such areas the gain provided by the antennas can extend the range covered by a cell and hence more users can communicate with less system capacity than any other typical antenna. It is used for the interference reduction and rejection as well. Finding the location of user will be a useful application of smart antenna, geo-locations of user and to make downlink beam forming easy [14]. Smart antennas are currently used in Radars, Radio astronomy and mostly used in cellular systems to keep the system speed equal with the number of increasing subscribers. It is also used in defense for safe communication purpose. 3.5CHARACTERISTICS OF SMART ANTENNA There are four main characteristics of smart antenna, DOA (direction of arrival) evaluation Beam forming, antenna diversity beam-width 3.5.1 DIRECTION OF ARRIVAL / ANGLE OF ARRIVAL A measure of the direction of the propagation of electro magnetic radiation upon arrival at a receiver; it is the angle between the plan of the phase front and some plan of reference, usually the horizontal, at the receiver antenna. [9] Direction of arrival indicates the direction from which usually a signal arrives at a point [3]. We can guess the direction of arrival of the signal by using different techniques like multiple signal classification. In this technique an algorithm is used to for frequency estimation and the location of emitter, matrix pencil method or their derivatives. [9] Usage Geodesic location or Geo-location of the cell phones is the main application of direction of arrival nowadays [9].Multiple base stations calculate the angle of arrival of the cell phone signal and this information is combined to locate the cell phone anywhere on the earth. It is generally used to locate the pirate or military radio transmission point. In submarines acoustics, angle of arrival is the method to localize objects with active and passive ranging [9]. 3.5.2 BEAM-FORMING Beam forming is a technique of digital signal processing and its is used for directional signal transmission or reception form the preferred signal direction as compared to some undesired direction [3]. It mean that the techniques which we used have the capability to make the radiation pattern of the antenna by constructive and destructive interference for reception and transmission and to accept moving beams in the direction of preferred signal and put nulls in the direction of interfering signals. This mean due to this co-channel and interference reduce and antenna gain is increased in the direction of desired signal [13]. Figure 3.1 These rebounds from different surfaces can set up time delays, attenuations, phase shifts, and distortions that can interfere with each other at the receiver end of the antenna. It can be set by the digital signal processing techniques used in smart antenna system [15]. 3.5.3 ANTENNA DIVERSITY Antenna diversity uses more then two antennas to develop the quality and dependability of a wireless link [3]. Specially in the areas where LOS is not clear between transmitter and receiver. Before reaching at the receiver end signal is reflected along several paths. This result in introducing phase shift, time delays, attenuation, and distortion which results in the interference between signal arriving before and after the particular signal [9].As two or more antennas receive several observations of the same signal they will calculate the most strengthen signal give the output. 3.5.4 BEAM-WIDTH It means the half-power beam-width [7]. The maximum radiation strength is found, and then the points on both side of the maximum, represent half power of the maximum strength are located. The distance between the half power points is known as the beam-width [16].Half of the power expressed in decibels is -3dB, so half power beam-width is often referred to as the 3dB beam-width. We considered both horizontal and vertical beam-widths. [6] 3.6 TYPES OF SMART ANTENNAS There are two major types of smart antenna switched beam smart antenna and adaptive array smart antenna. 3.6.1 SWITCHED BEAM SMART ANTENNA There are different permanent beam patterns presented in switched beam system. A decision is made as to which beam to be taken, at any given point in time, depend upon the necessities of the system [3]. 3.6.2 ADAPTIVE ARRAY SMART ANTENNA It allows the antenna to focus the beam to any direction of direction of desired signal continuously ignoring interfering signals. Beam direction can be estimated using the direction of arrival (DOA) estimation methods [3]. 3.7 ADVANTAGES For the most part smart antennas are employed at the base station in the mobile network to improve system capacity. Capacity means the number of the users that can be handled in a system. Using of Omni-directional antennas originate co-channel interference when two users use the same band of frequency that finally limits the user capacity in the system [8]. In case of smart antennas beams are focused towards the desired user minimize interference to other users using the same frequency band. 0: Figure 3.2 It helps against multipath fading noise which improves dependability of received signal. Reduced power consumption for cell phones, low probability of interception and detection improved location estimation and improved range of reception [1]. 3.8 DISADVANTAGE Most major disadvantages of smart antenna is in their design and completion in hardware. Multi RF chains can boost the price and make the transceiver bulkier [4]. Most of the devices in the making of a typical antenna used non-linear devices and using smart antenna also increases the components used. If not checked properly, this affects the performance of the antenna [15]. As the data bandwidth required for the digital signal processing increases with the number of antenna elements used. This can limit the data rate for different applications Chapter 4 SMART ANTENNA SYSTEM 4.1 INTRODUCTION Smart Antenna System is combination of multiple antennas which Transmit or Receive Signals using an Adaptation Algorithm [4]. A smart antenna system is combination of many antennas elements with a signal receiving and transmitting ability to optimize its radiation and reception pattern robotically in reaction to the signal environment [10]. 4.2 TYPES OF SMART ANTENNA SYSTEMS There are mainly two ways to implement antennas that dynamically change their antenna radiation pattern to minimize interference or multipath affects by increasing coverage area and range. à ¢Ã¢â€š ¬Ã‚ ¢ Switched beam: There are finite numbers of fixed patterns which are defined by the system (sectors) à ¢Ã¢â€š ¬Ã‚ ¢ Adaptive arrays: There are an infinite number of patterns (scenario-based) which are adjusted in real time The Switched beam approach is easy and simple then the adaptive approach. It increases the network capacity as compared to the usual Omni-directional antenna systems. In this technique, an antenna array produces over lapping beams that cover the neighboring area as in the figure [17]. When a signal is coming and detected, the base station determine the beam that is best associated in the signal-of-interest direction and then switch that beam toward the user for communication [6]. 0: Figure 4.1 [17] The Adaptive array system is efficient then the switched array technique [2]. A mobile user is tracked by this system constantly by routing the main beam towards the destination and at the same time sending no signal in the direction of interfering signals like switched beam. 0: Figure 4.2 [17] 4.2.1 SWITCHED BEAM SYSTEMS Switched beam antenna systems make many fixed beams with finely tuned sensitivity in particular directions [17]. These antenna systems detect strength of signal, choose one from numerous already determined, fixed beams, and switch his beam from one beam to another as the user moves during the sector [11]. The output of many antennas combine through Switched beam systems in such a way that it form finely directional beams with more selectivity than can be achieve with usual, single-element antenna technique. 4.2.1.1 WORKING In this type of adaptive approach in reality did not steer or scan the beam in the direction of the desired signal [17]. Switched beam use an antenna array which radiates a number of overlapping permanent beams covering a elected angular area. The directional beam leads to increase the factor of a frequency reuse in channel by decreasing possible interference and it also increases the range [14]. These antennas system not have a uniform gain in all directions but when they are compared to simple antenna system they have more gain or increased gain in the desired directions. The Switched beam antenna has a switching method that enables the system to select and then switch the desired beam which gives the best response for a mobile user. The selection is generally based on maximum received power for user [9]. 0: Figure 4.3 [3] A usual switched beam system for a base station consists of many antennas with each array covering a certain sector in cell. Take an example of switched beam-forming system below [7]. It consists of a phase shifting network, which forms many beams look in certain directions. The RF switch targets the correct beam in the direction of interest. The measurement for selecting of the correct beam is done by the control logic unit. The control logic unit is controlled by an algorithm which scans all the beams and then selects that beam which is strongest signal based on a measurement calculated by the algorithm. 0: Figure 4.4 [17] This technique is not good when interference is high this technique is simple in operation. Let us consider an example where User 1 is at the side-edge of the beam receiving low power which is entertained by this beam [6]. If there is a second user were at the direction of the null then there will be no interference but if the second user also moves into the direction where user 1 is located then there will be interference occur therefore the switched beam system is suitable where there is no interference [17] 4.2.2 ADAPTIVE ARRAY ANTENNAS Switched beam systems only give a limited performance improvement when compared to common antenna systems in wireless communication [3]. Greater improved performance can be achieved by using superior signal processing technique to practice the information achieved by the antenna arrays. The adaptive array system is Opposite to switched beam systems, they are smarter because they are able to react to the changing RF environment. They have a huge amount of radiation patterns as compared to fixed finite patterns in switched beam systems to adjust in the changing radio frequency environment [9]. An Adaptive array is just like a switched beam system which use number of antennas but they are controlled by signal processing [17]. This signal processing moves the radiation beam towards a desired user as he changes his direction and ii limit the interference happen from other users by launch nulls in their directions. This is shown in figure below [17]. http://i.cmpnet.com/embedded/gifs/2005/0503/0503feat1fig2.jpg 0: Figure 4.5 [17] 4.3 COMPARISON Here are the differences between switched bean array and adaptive beam array 4.3.1 SWITCHED BEAM SYSTEM It uses many preset directional beams with slim beam-widths. The necessary phase shifts be provide by normal preset phase shifting networks e.g. the butler matrix [11]. They do not need difficult algorithms, easy algorithms are used for selection of beam It need only reasonable relations between mobile and base station as compared to adaptive array system [15]. Because low technology is used it has minor price and complication. Mixing is easy and cheap into existing cellular system. It give important increase in coverage and capacity compare normal antenna based systems [6]. Since many slim beams are used, normal intra-cell hand-offs occur between beam when a mobile moves from one beam to another [15]. It does not make a distinction between direct signal and interfering signal this leading to undesired improvement of the interfering signal more than the desired signal [6]. à ¢Ã¢â€š ¬Ã‚ ¢ Switched beam systems offers limited co-channel interference control as compared to the adaptive array system. 4.3.2 ADAPTIVE ARRAY SYSTEM A whole adaptive system; moves the beam towards desired signal and put nulls toward interfering signal directions [17]. It is implemented on DSP technology [17]. To steer the beam and the nulls complicated adaptive algorithm is required [6]. Rejection capably of interference is much better then the switching beams system [15]. It is difficult to impose on existing systems, i.e. up gradation is hard and costly. Since nonstop moving of the beam is required as the user moves, high contact between the mobile and base station is required [17]. It provides improved coverage and improved capacity because of improved interference elimination as compare to the Switched beam. It can either decline multipath components or add them by correcting the delays to improve the signal quality[6]. 4.4 ANTENNAS SMARTNESS Smart antenna systems using these adaptive systems are quite smart in the true sense and that is why they are referred as smart antennas [17]. The smartness of these systems is due to the intelligent processor using digital processing techniques which are integrated in these systems. These signal processing techniques use complex algorithms which are our main concern in our project [11]. As modern world is using advanced technology in different fields, smart antenna also comes with such type of digital formation algorithms that operate the signals in accurate and flexible manner. On the way to the smart antenna, the signal is received, converted and modulated for transmission into digital signal and at the other end it is reconverted in analog information [5]. 4.4.1 BASIC MECHANISM There are a lot of function a smart antenna can perform on the bases of techniques used for desired scenario. Some of them major functions smart antenna can perform are listed below First: Smart antenna can locate the direction as well as the angle of arrival of all the incoming signals [16]. Whether they are interfering in the signal of our interest or they are same signals arriving at the antenna from different directions after reflection from different surfaces. Their direction is estimated by the processing techniques used in smart antennas. Secondly: The signal of our interest can separated from the cluster of all the incoming signals using smart antenna systems. Lastly a beam is moving in the direction of the wanted signal and the user is traced when he moves in the area and placing noting at interfering signal directions by continually updating the weights [17]. It is pretty obvious that the direction of radiation beam of the main lobe in an array depends upon the phase difference between the elements of the array. Thus it is possible to constantly move the main beam in any direction by adjusting the phase difference between the elements. The same concept is used in adaptive array systems in which the phase is tuned to attain highest radiation in the most wanted direction. 4.5 ADAPTIVE ALGORITHM CLASSIFICATIONS The adaptive algorithms can be classify into category base on different approach given below Continuous adaptation: The algorithms based on this technique fine-tune the weights as the arriving data is sampled and continue updating it in such a way that it converges to an finest solution. This approach is suit able when the signal information is varying with time. Examples: The Least Mean Square (LMS) algorithm and the Recursive Least square (RLS) algorithm [17]. Block adaptation: The algorithms based on this technique calculate the weights depend on the approximation achieve from a temporary chunk of data [3]. This method can be used in a non-stationary situation provided the weights are calculated from time to time. Example: The Sample Matrix Inversion (SMI) algorithm [17]. Reference signal based algorithms: These types of algorithms worked on the principal of minimization of the mean square error between the received signal and the original signal. Hence it is necessary that an original signal is available which has high relationship with the desired signal [17]. Examples: The Least Mean Square (LMS) algorithm, The Recursive Least square (RLS) algorithm [17]. Blind adaptive algorithms: These algorithms do not want any original signal information. They are generating the necessary ordinal signal from the received signal to get the desired signal. Examples: The Constant Modulus Algorithm (CMA). CHAPTER 5ALGORITHMS It is obvious that themselves antennas or not much intelligent to perform such.It is the importance of antenna system which is such clever in aspect of advanced signal processing algorithms. In order to use the smart antenna to its full smart techniques superior and computationally smart algorithms must be used. There are number of algorithms which are optimized and specialized for different smart antenna system and for different scenarios. For this purpose here are the few algorithms we have studied till the day to get a brief idea about the smart antenna systems are given below. LMS (Least Mean Squares) RLS (Recursive Least Squares) CMA (Constant Modulus Algorithm) We will try to give a logical comparison on the performance of different Adaptive Algorithms for beam forming for our Smart Antenna. In this study we have exposed that these algorithms RLS and LMS are best for making beam forming like main lobe towards desired user but they have boundaries towards reject interference. In case of CMA has improved response towards beam forming and it gives improved results for interference rejection, but there is a problem the Bit Error Rate (BER) is greatest and high in case of single antenna element in CMA. It is confirmed that convergence rate of RLS is faster than LMS so RLS is proved the best choice for us. The effect of changing step size for LMS algorithm has also been studied. 5.1 CMA (CONSTANT MODULUS ALGORITHM) CMA is based on those methods which are old but many of them are modified or completely changed techniques. This is popular algorithm and we are using this algorithm for blind adaptive array processing for beam pattern. But there is a problem with this algorithm it has low convergence rate because it is a type of instant gradient searching method depend on performance surface. Later some enhanced algorithms such as orthogonal constant modulus (OCMA) and least square constant modulus algorithm (LSCMA) are proposed [4].OCMA use the Newton algorithm and LSCMA is a type of block-update interactive algorithm. Both use estimation of revision of covariance matrix. This means high calculation complexity and convergence rate is improved. But they will not converge when covariance matrix is positive and singular. Faster convergence rate, constant convergence steps, lower computational complexity and better numerical stability will be achieved with instant gradient searching and conjugate gradi ent algorithm [2]. Conclusion A constant modulus algorithm based on modified conjugate gradient for adaptive array processing. CMA algorithm has the similar performance as high convergence rate algorithm such as LSCMA; there is an advantage that they do not need to perform matrix inversion and its performance is improved then other type of conjugate gradient algorithms 5.2 LMS (Least Mean Squares) and RLS (Recursive Least Squares) An adaptive algorithm has a natural recursive formation even in the case of a limited impulse response modelization. As the error