Channel Equalisation For Software Defined Radio Computer Science Essay

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Abstract - We provide a brief overview over the development of software-defined radio system and channel equalization techniques. Software Defined Radio is an all new technology being developed in the 21st century. The primary goal of SDR is to replace as many analog components and hardwired digital VLSI devices of the transceiver as possible with programmable devices. One of the major practical problems in digital communication systems is channel distortion which causes errors due to intersymbol interference (ISI). In order to restore the transmitted sequence and given the observed sequence at the channel output which is accomplished by equalizers. In this paper Nonlinear equalizers, Adaptive equalizers, Fuzzy equalizers, Neural equalizers are underlined. We discuss that which equalizer is best out of these and the reasons for this also proposed.

Keywords- SDR, Nonlinear Equalizers, Adaptive Equalizers, Fuzzy Equalizers,Neural Equalizers. ISI.

Introduction to SDR :

Over the last decade as semiconductor technology has improved both in terms of performance capability and cost, new radio technologies have emerged from military and research, development labs and become mainstream technologies. One of these technologies is Software Defined Radio. As the name implies, a SDR is a radio that has the ability to be transformed through the use of software or re-definable logic and this is done with general purpose DSPs. SDR is an advanced radio technology in which the modulation and demodulation of radio signals is performed exclusively by software. A communication link consists of three components: the transmitter, the channel, and the receiver. The transmitter element process an information signal in order to produce a signal most likely to pass reliably and efficiently through the channel. [2]

Many definitions have appeared that might cover a definition for a software defined radio, SDR. The SDR Forum themselves have defined the two main types of radio containing software in the following fashion [3]:

Software Controlled Radio:   Radio in which some or all of the physical layer functions are Software Controlled. In other words this type of radio only uses software to provide control of the various functions that are fixed within the radio.

Software Defined Radio:   Radio in which some or all of the physical layer functions are Software Defined. In other words, the software is used to determine the specification of the radio and what it does. If the software within the radio is changed, its performance and function may change.

Another definition of the Software Defined radio is that SDR has a generic hardware platform on which software runs to provide functions including modulation and demodulation, filtering and other functions such as frequency selection and if required frequency hopping.

A software-defined radio is characterized by its flexibility that is simply modifying or replacing software programs can completely change its functionality. This allows easy upgrade to new modes and improved its performance without the need to replace hardware. An SDR can also be easily modified to fulfill the operating needs of individual applications.

SDR technology can be used to implement military, commercial and civilian radio applications. A wide range of radio applications like Bluetooth, WLAN, GPS, Radar, WCDMA, GPRS, etc. can be implemented using SDR technology. In today's world, radios exist in a multitude of items such as cell phones, computers, car door openers, vehicles, and televisions.

DISTINCT FEATURES OF SOFTWARE DEFINED RADIO:

There are a number of characteristics that an SDR possesses. While it is not required that an SDR have these entire characteristics[4]. It only has one or more of them. These below characteristics tell us about the different facets of the SDR that it can be of what type.

Multi-Band Radios

Most traditional radio architectures operate on a single band. There are many applications where multiple frequencies of operations are desired like cellular communications. This application requires multiple radios to operate in one specified band.

A multi-band radio has the ability to operate on two or more bands either sequentially or simultaneously.

Multi-Mode Radios

A multi-mode radio has the ability to process several different kinds of standards. These standards include AM, FM, GMSK, CDMA but are limited to none of these. An SDR has the ability to work with many different standards and be continuously reprogrammed.

Multi-Carrier Radios

A multi-carrier or multi-channel radio has the ability to simultaneously operate on more than one frequency at a time. This may be within the same band or in the case of a multiband radio, in two different bands at the same time.

IMPORTANT BENEFITS OF SDR:

SDRs have several advantages over today's hard-wired radios. It has the ability to receive and transmit using various modulation methods using common set of hardware. SDRs can recognize and avoid interference with other communication channels. Detailed benefits are mentioned below [3]:

Flexibility:

The most obvious advantage is flexibility. They can be easily and fastly upgraded with enhanced features. They can talk and listen to multiple channels simultaneously. We can build new kinds of radios that have never existed before. Smart radios or cognitive radios are looked at the utilization of the RF spectrum in their immediate neighborhood and configure themselves for best performance.

Making devices adaptable:

Users want to send or receive information at minimal costs. So, a multi-standard radio could scan all frequency bands for available wireless infrastructure and use the one which meets the user demands at minimal costs. In addition, such a radio can adapt to the radio environment, thereby minimizing power.

EVOLUTION OF SOFTWARE DEFINED RADIO:

The term "Software Defined Radio" was coined in 1991 by Joseph Mitola, who published the first paper on the topic in 1992[2]. Though the concept was first proposed in 1991, software-defined radios have their origins in the defense sector. Joe Mitola of Mitre is generally credited with being the father of software defined radio[1,4]. Mitola's SDR can receive every band and channel concurrently that may be important for military, but not necessary for civilian uses.

One of the first public software radio initiatives was a U.S. military project named SpeakEasy. The primary goal of the SpeakEasy project was to use programmable processing to emulate more than 10 existing military radios, operating in frequency bands between 2 and 2000 MHz[4]. It has a programmable digital processing capacity on the order of one billion 16-bit integer operations per second and 200 million 32-bit operations per second. Another design goal was to be able to easily incorporate new coding and modulation standards in the future, so that military communications can move with advances in coding and modulation techniques. At the same time the US DoD initiated SPEAKeasy as the first publicly announced military software radio, then DARPA continued with the SPEAKeasy II program

In the last decade, there are many enabling technologies and these technologies was forcing on development of low cost Digital Signal Processors. From a market point of view, the rapid growth of the telecommunications, industry, particularly cellular communications, provided a demand for low cost equipment both from a user and infrastructure point of view.

Although first generation cellular was based on analog modulation, it became clear that due to the limit amount of spectrum and the relative inefficiency of those standards that more efficient means of spectral usage were required. Therefore second generation cellular systems such as GSM and IS-95 were developed that took advantage of the emerging DSP technologies. In these early systems, the DSP was responsible for taking the complex baseband data I and Q and determining what bit stream was being sent and correcting for errors introduced due to noise, interference and fading. These modem functions were based on programs running on a DSP and therefore could be changed simply by changing the program. These standards and variations of the standards were allowed better efficiency and higher data transmission rates. The transition from an analog to a digital radio can be viewed as a graduated evolution, as depicted in Figure 1[18]. It maps the SDR hardware along the vertical axis and increasing functional capability along the horizontal axis so that the ultimate in a SDR culminates at the upper right corner of the square. The farther a radio approaches the upper-right corner, the closer it comes to epitomizing an SDR.

Fig 1: Graduated evolution of SDR

The evolution of the modems used for GSM and CDMA is an aspect of SDR, other factors such as incompatibility of these two standards drives the second aspect of SDR.These standards are incompatible in terms of bandwidth, modulation format and data rate. Traditional radios, even those with DSP modems, operate with fixed bandwidths and therefore prevent cross functionality.

Since these radio devices are fixed, it is not possible to change the channel bandwidth characteristics. Therefore an SDR needs additional circuitry that provides incompatibility in terms of bandwidth, modulation format and data rate between different standards. In practice, the optimal way to do this is to digitize the signal and use digital techniques for manipulating the channel of interest.

Hence, the Modified SDRs are:

Good enough to receive one channel at a time, but from any band, with any channel bandwidth, and any modulation

Tunes channel of interest to zero IF

Wideband receiver (no RF preselect)

Table 1 provides information of potential services and applications, as they apply to each user community, which now become possible with the implementation SDR technology[6].

CHANNEL EQUALIZATION:

Channel equalization is the process of compensating for the effect of physical channel between a transmitter and receiver. Basically, it improves the quality of transmission. Also channel equalization can define as the process of reducing amplitude, frequency and phase distortion in the radio channel with the intent of improving transmission performance. In digital communications, an equalizer is a device that attempts to recover a signal transmitted through an ISI channel. It may be a simple linear filter or a complex algorithm.

Military

Commercial

Public Safety

• Secure, encrypted communicat-ions

• Mission flexibility

• Options to select communicat-ions channel by availability

• Real-time flexibility

• Portable command post

•Integrated radio, router, and computer

•International connectivity to prevailing networks

• International connectivity

• Location awareness

• Multimedia applications

• Virtual private networks- Closed user groups

• Media distribution

• Combined delivery of e-mail, voice mail, messages, and faxes

• Browser capability

• Nationwide portable station for response crisis management

• Improved emergency communications by the use of one device that can operate on multiple systems

• Closed user groups

• Database access

• International connectivity (especially for Federal use, search and rescue operations)

Table 1: Potential Services and Applications

In a communication system, the transmitter sends the information over an RF channel. High speed data transmissions over communication channels distort the transmitted signals in both amplitude and phase due to presence of Inter Symbol Interference (ISI) befores it reaches the receiver. The overlapping of the transmitted data produced due to the limited bandwidth of the transmission channel [10]. When the channel is band limited, symbols transmitted through will be dispersed. This causes previous symbols to interfere with the next symbols, yielding the ISI. The receiver task is to figure out what signal was transmitted and turn the received signal in understandable information. This approach to remove ISI is usually known as equalization.

NEED OF CHANNEL EQUALIZERS:

One of the major practical problems in digital communication systems is channel distortion which causes errors due to Inter Symbol interference. Since the source signal is passed through the channel, causing distortion in the received message. This distortion translates into errors in the received sequence. Our problem as communication engineers is to restore the transmitted sequence given the observed sequence at the channel output. This task is accomplished by channel equalizers.The purpose of an equalizer is to reduce the ISI as much as possible and to minimize the effect of the additive noise for better demodulation. The main advantage of this approach to remove ISI is that a digital filter is easy to build and is easy to alter for different equalization schemes, as well as to fit different channel conditions[10].

EQUALIZER NOISE ENHANCEMENT

The goal of equalization is to mitigate the effects of ISI. However, this goal must be balanced so that in the process of removing ISI, the noise power in the received signal is not enhanced. A simple analog equalizer, shown in Figure 2, illustrates the pitfalls of removing ISI without considering this effect on noise[19]. Consider a signal s(t) that is passed through a channel with frequency response H(f). At the receiver front end white Gaussian noise n(t) is added to the signal, so the signal input to the receiver is

Y (f) = S(f)H(f)+N(f) (1)

where N(f) has power spectral density . If the bandwidth of s(t) is B then the noise power within the signal bandwidth of interest is B. Suppose we wish to equalize the received signal so as to completely remove the ISI introduced by the channel. This is easily done by introducing an analog equalizer in the receiver defined by

= 1/H(f) (2)

The receiver signal Y (f) after passing through this equalizer becomes

[S(f)H(f) + N(f)] = S(f) +N'(f) (3)

where N'(f) is colored Gaussian noise with power spectral density /. Thus, all ISI has been removed from the transmitted signal S(f).

However, if H(f) has a spectral null i.e. H() = 0 for some at any frequency within the bandwidth of s(t), then the power of the noise N'(f) is infinite. Even without a spectral null, if some frequencies in H(f) are greatly attenuated, the equalizer = 1/H(f) will greatly enhance the noise power at those frequencies. In this case even though the ISI effects are removed, the equalized system will perform poorly due to its greatly reduced SNR.

n(t)

Channel

H(f)

+

Equalizer

Heq (f)

Demod

s(t) r(t) y(t) s(t)+n(t)

Fig 2: Analog Equalizer Illustrating Noise Enhancement

Thus, the true goal of equalization is to balance mitigation of the effects of ISI with maximizing the SNR of the post-equalization signal. Linear digital equalizers in general work by inverting the channel frequency response and therefore have the most noise enhancement. Nonlinear equalizers do not invert the channel frequency response, and thus tend to suffer much less from noise enhancement.

TYPES OF EQUALIZERS:

Two main classes of equalizers are known: linear and non-linear equalizers. Several equalizer types are listed below [12,13]:

Linear Equalizer: Linear equalizer processes the incoming signal with a linear filter. These are simplest to implement.

Zero Forcing Equalizer: approximates the inverse of the channel with a linear filter.

MMSE equalizer: designs the filter to minimize mean square error i.e. E[|e|2], where e is the error signal, which is the filter output minus the transmitted signal.

Non-Linear Equalizer: These equalizers generally less suffered from noise enhancement than linear equalizers.

Decision Feedback Equalizer: augments a linear equalizer by adding a filtered version of previous symbol estimates to the original filter output.

Maximum Likelihood Sequence Estimation: tests all possible data sequences rather than decoding each received symbol by itself and chooses the data sequence with the maximum probability as the output.

Other Equalizers:

Adaptive Equalizer: It is typically a linear equalizer or a DFE. It updates the equalizer parameters such as the filter coefficients as it is processes the data. It assumes that it makes the correct symbol decisions, and uses its estimate of the symbols to compute 'e', which is defined the error signal.

Fuzzy Equalizers: The structure of fuzzy equalizer provides scope for co-channel interference (CCI) suppression in the presence of intersymbol interference (ISI) and additive white Gaussian noise (AWGN) with the help of a pre-processor. The advantages associated with the fuzzy equalizers can make them a candidate for mobile communication applications.

Neural Equalizer: Artificial Neural Networks (ANN) can be applied to non-invertible channels for achieving better performance than conventional methods. A model of neural equalizer using MLP (multi layer perceptron) reduces the mean square error to minimum and eliminates the effects of ISI.

NON-LINEAR EQUALIZERS:

Linear equalization techniques typically suffer from more noise enhancement than non-linear equalizers and are therefore not used in most wireless applications. A non-linear equalizer corrects for intersymbol interference in a digital data transmission system by introducing baud rate samples into an N-stage tapped delay line. Both the samples from the tapped delay line and the contents of the prior decision register are coupled to a weighting matrix processor which is updated to permit adaptation to different channels conditions. Moreover, not only can it remove intersymbol interference. But it has the ability to modify previously made decisions upon the receipt of

additional data from the channel.

Decision Feedback Equalizer:

It is a simple nonlinear equalizer which is particularly useful for channel with severe amplitude distortion. It uses decision feedback to cancel the interference from symbols which have already have been detected or we can say a decision-feedback equalizer (DFE) is a nonlinear equalizer that employs previous decisions as training sequences. The equalized signal is the sum of the outputs of the forward and feedback parts of the equalizer. The forward part is like the linear transversal equalizer. Decisions made on the equalized signal are fed back via a second transversal filter.

The basic idea is that if the values of the symbols already detected are known, then the ISI contributed by these symbols can be canceled exactly by subtracting past symbol values from the equalizer output[9,10]. Since the output of the feedback section of the DFE is a weighted sum of noise-free past decisions.

In a Decision Feedback Equalizer Architecture (DFE), shown in figure 3,[12] consists of a linear feed forward filter (FFF) and a feedback filter (FBF). The FFF

Fig 3: Decision Feedback Equalizer

suppresses the contribution of the pre-cursor ISI, namely the interference caused by the symbols transmitted after the symbol of interest. The FBF cancels the post-cursor ISI by subtracting a weighted linear combination of the previous symbol decisions, assumed to be correct. The result is then applied to a threshold device to determine the symbol of interest. The FFF enhances the noise, but the noise gain is not as severe as in the case of a linear equalizer. Both the forward and feedback filters may be adjusted simultaneously to minimize the Mean Square Error.

The coefficients of a linear transversal equalizer are selected to force the combined channel and equalizer impulse response to approximate a unit pulse. In a DFE, the ability of the feedback section is to cancel the ISI, because of a number of past symbols, allows more freedom in the choice of the coefficients of the forward section. The combined impulse response of the channel and the forward section may have nonzero samples following the main pulse. That is, the forward section of a DFE need not approximate the inverse of the channel characteristics, and so avoids excessive noise enhancement and sensitivity to sampler phase.

When a particular incorrect decision is fed back, the DFE output reflects this error during the next few symbols as the incorrect decision traverses the feedback delay line. Thus there is a greater likelihood of more incorrect decisions following the first one. While better in performance in general, a major drawback of DFE is its potential catastrophic behavior due to error propagation.

The detrimental potential of error propagation is the most serious drawback for decision feedback equalization.

Maximum Likelihood Sequence Estimation:

Maximum-likelihood sequence estimation (MLSE) avoids the problem of noise enhancement since it doesn't use an equalizing filter instead it estimates the sequence of transmitted symbols. The MSE-based linear equalizers are optimum with respect to the criterion of minimum probability of symbol error when the channel does not introduce any amplitude distortion. Yet this is precisely the condition in which an equalizer is needed for a mobile communications link. This limitation on MSE-based equalizers led researchers to investigate optimum or nearly optimum nonlinear structures. These equalizers use various forms of the classical maximum likelihood receiver structure. Using a channel impulse response simulator within the algorithm, the MLSE tests all possible data sequences rather than decoding each received symbol by itself and chooses the data sequence with the maximum probability as the output. An MLSE usually has a large computational requirement, especially when the delay spread of the channel is large.

Using the MLSE as an equalizer was first proposed by Forney in which he set up a basic MLSE estimator structure and implemented it with the Viterbi algorithm[12]. This algorithm was recognized to be a maximum likelihood sequence estimator (MLSE) of the state sequences of a finite state Markov process observed in memoryless noise. It has recently been implemented successfully for equalizers in mobile radio channels.

Estimated data

Matched Filter

MLSE

Delay

Channel Estimator

Channel Estimator

Delay

MLSE

Matched Filter

Channel Estimator

Delay

MLSE

Channel Estimator

Delay

MLSE

MLSE

Matched Filtery(t) z(t) sequence

Delay

Channel Estimator + e

_

Fig 4: The structure of a MLSE with an adaptive matched filter

The block diagram of a MLSE receiver based on the DFE is shown in Figure 4[12]. The MLSE is optimal in the sense that it minimizes the probability of a sequence error. The MLSE requires knowledge of the channel characteristics in order to compute the metrics for making decisions. The MLSE also requires knowledge of the statistical distribution of the noise corrupting the signal. Thus, the probability distribution of the noise determines the form of the metric for optimum demodulation of the received signal. Notice that the matched filter operates on the continuous time signal, whereas the MLSE and channel estimator rely on discretized (nonlinear) samples.

ADAPTIVE EQUALIZERS:

An adaptive equalizer is an equalizer that automatically adapts to time-varying properties of the communication channel. Typically, adaptive equalizers used in digital communications require an initial training period, during which a known data sequence is transmitted. A replica of this sequence is made available at the receiver in proper synchronism with the transmitter, thereby making it possible for adjustments to be made to the equalizer coefficients in accordance with the adaptive filtering algorithm employed in the equalizer design. When the training is completed, the equalizer is switched to its decision directed mode. They are more powerful than linear equalizers especially for severe inter-symbol interference (ISI) channels without as much noise enhancement as the linear equalizers.

Here, the training sequence or pilot sequence is inserted periodically during data transmission. This particular set of symbols is therefore known to the receiver and thus can be used to train the equalizer i.e.

Fig 5: Adaptive Equalizer

update its coefficients. The Least Mean-Square (LMS) algorithm carries out the MSE by recursively updating the coefficients.

There are two modes that adaptive equalizers work:

Training Mode: To make equalizer suitable in the initial acquisition duration, a training signal is needed. In this mode of operation, the transmitter generates a data symbol sequence known to the receiver. Once an agreed time has elapsed, the slicer output is used as a training signal and the actual data transmission begins.

Decision Directed Mode: The receiver decisions are used to generate the error signal. Decision directed equalizer adjustment is effective in tracking slow variations in the channel response.

The table 2 below briefly describes the nature of the reference signal for each of the two operation modes.

Operation Mode of Equalizer

Reference Signal

Training mode

Preset known transmitted sequence

Decision-directed mode

Detected version of the output signal,

Table 2: Nature of reference signal

In typical applications, the equalizer begins in training mode to gather information about the channel and later switches to decision-directed mode.

FUZZY EQUALIZERS:

Fuzzy systems or fuzzy logic1 systems is the name for systems which have a direct relationship with fuzzy concepts (like fuzzy sets, linguistic variables) and fuzzy logic [13]. The basic building block of a fuzzy logic system is presented in Fig. 6. Here the fuzzifier converts the real-world crisp input sample to a fuzzy output described by the membership function . This provides the degree to which the input scalar belongs to the fuzzy set . The inference engine provides the relationship between the fuzzy input in terms of membership functions and the fuzzy output of the controller using a set of IF 2 THEN 2 rules derived from the rule base. The rule l in the fuzzy rule base can be defined as

IF is and …. And

is THEN y is .

The defuzzifier converts the inferences to provide the crisp output y(k). Generally, in a fuzzy system the rule base is generated in advance with expert knowledge of the system under consideration. Also on-line learning properties have been introduced which provide scope for training. This feature in fuzzy systems is achieved with the adaptation and learning block that uses the available information in the system The available linguistic rules can also be applied in the adaptation algorithm. These

Fuzzifier

Interference engine

Defuzzifier

Fuzzy rule

base

Adaptation

Learning

x(k) y(k)

X(k)

y(k) Teacher

Fig 6: A typical fuzzy logic system

type of systems are also called adaptive neuro fuzzy filters (ANFF) possessing the ability to incorporate training like neural networks[15] and can also use rule bases from human experts as in fuzzy systems. These adaptive fuzzy systems have been appliedto a variety of engineering applications such as medical diagnostics, image processing, pattern classification, clustering, control applications, time-series forecasting, etc.

ARTIFICIAL NEURAL NETWORK (ANN):

Artificial neural network (ANN) takes their name from the network of nerve cells in the brain. Recently, ANN has been found to be an important technique for classification and optimization problem. McCulloch and Pitts have developed the neural networks for different computing machines. There are extensive applications of ANN in the field of channel equalization, estimation of parameters of nonlinear systems, pattern recognition, etc. ANN is capable of performing nonlinear mapping between the input and output space due to its large parallel interconnection between different layers and the nonlinear processing characteristics.

An artificial neuron basically consists of a computing element that performs the weighted sum of the input signal and the connecting weight. The sum is added with the bias or threshold and the resultant signal is then passed through a non-linear function of sigmoid or hyperbolic tangent type. Each neuron is associated with three parameters whose learning can be adjusted; these are the connecting weights, the bias and the slope of the nonlinear function. For the structural point of view a NN may be single layer or it may be multilayer. In multilayer structure, there is one or many artificial neurons in each layer and for a practical case there may be a number of layers. Each neuron of the one layer is connected to each and every neuron of the next layer. The Functional link ANN is another type of single layer NN. In this type of network the input data is allowed to pass through a functional expansion block where the input data are nonlinearly mapped to more number of points. This is achieved by using trigonometric functions, tensor products or power terms of the input. The output of the functional expansion is then passed through a single neuron.

Rumelhart developed the Back propagation algorithm, which is central to much work on supervised learning in multilayer NN. A feed forward structure with input, output, hidden layers and nonlinear sigmoid functions are used in this type of network. Different types of NNs are discussed below.

Single Neuron Structure:

The basic structure of an artificial neuron is presented in figure 7. The operation in a neuron involves the computation of the weighted sum of inputs and threshold. The resultant signal is then passed through a nonlinear activation function.

This is also called as a preceptor, which is built around a nonlinear neuron; whereas the LMS algorithm described

Fig 7: Structure of a Single Neuron

in the next sections is built around a linear neuron.

Multilayer Perceptron (MLP)

In the multilayer neural network or multilayer perceptron (MLP), the input signal propagates through the network in a forward direction, on a layer-by-layer basis.

Fig 8: Structure of Multilayer perceptron (MLP)

This network has been applied successfully to solve some difficult and diverse problems by training in a supervised manner with a highly popular algorithm known as the error back-propagation algorithm [41, 5].

The scheme of MLP using four layers is shown in figure 8. (n) represents the input to the network, and represent the output of the two hidden layers and (n) represents the output of the final layer of the neural network. The connecting weights between the input to the first hidden layer, first to second hidden layer and the second hidden layer to the output layers are represented , and by respectively.

IMPORTANT EQUALIZATION ALGORITHMS:

Many adaptation strategies exist. But now days the most widely used algorithms are :

The LMS Algorithm:

The least mean squares algorithm searches for the optimum or near-optimum filter weights by performing the following iterative operation:

New weights = Previous weights + (constant) (Previous error) (current input vector)

where

Previous error = Previous desired output - Previous actual output

and constant may be adjusted by the algorithm to control the variation between filter weights on successive iterations. This process is repeated rapidly in a programming loop while equalizer attempt to converge. Upon reaching convergence, the adaptive algorithm freezes the filter weight until the error signal exceeds an acceptable level or until a new training sequence is sent.

In case of LMS algorithm the cost function (N) is given by

(4)

This cost function can be thought of as an instantaneous estimate of the MSE cost function, as JMSE (n) = EJLMS (n). Although it might not appear to be useful, the resulting algorithm obtained when JLMS (N) is used for J(N) given by (5)

(5)

is extremely useful for practical applications. Taking derivatives of with respect to the elements of W (n) and substituting the result into (3.2), we obtain the LMS adaptive algorithm given by

(n)X(n) (6)

Note that this algorithm requires only multiplications and additions to implement. In fact, the number and type of operations needed for the LMS algorithm is nearly the same as that of the FIR filter structure with fixed coefficient values, which is one of the reasons for the algorithm's popularity.

The behavior of the LMS algorithm has been widely studied, and numerous results concerning its adaptation characteristics under different situations have been developed. For now, we indicate its useful behavior by noting that the solution obtained by the LMS algorithm near its convergent point is related to the Wiener solution. In fact, analyses of the LMS algorithm under certain statistical assumptions about the input and desired response signals show that when the Wiener solution WMSE (n) is a fixed vector. Moreover, the average behavior of the LMS algorithm is quite similar to that of the steepest descent algorithm that depends that depends explicitly on the statistics of the input and desired response signals.

(7)

The problem with these approaches is that the resulting minimum transformed error used to update the adaptive filter can be biased from the true minimum output error and the algorithm may not be able to converge to the desired minimum error condition. These algorithms also tend to be complex, slow to converge, and may not be guaranteed to emerge from a local minimum.

Back-Propagation Algorithm:

Most neural network applications do not have a priori knowledge of the network correct weights that allow to perform a desired mapping. Thus, a learning procedure is necessary to obtain these weights. For that purpose, the BP algorithm [19] is used. A set of input-output pairs {x(n), d(n)} trains the network to implement the desired mapping. The BP algorithm is a supervised learning algorithm. It adjusts the multi layer neural network (MLNN) weights so as to minimize any differentiable cost function, e.g. the squared error energy function (the error power between the network output and the desired output), E(n)=, where is the MLNN output vector at time n and d(n) is the desired output.

The BP algorithm performs a gradient descent on the energy function in order to reach a minimum:

(8)

(9)

These equations can be explicitly expressed as

(10)

(11)

The so-called error term of the output layer is given by:

(12)

where denotes the derivative of the activation function (x)=df (x)/dx. The calculation of the error term of the hidden unit (i, k), can be easily expressed as a function of the next layer error terms:

(13)

Thus, the weight update is performed by propagating the error terms from the output layer to the input layer.

CONCLUSION

If the channel suffers from significant nonlinear distortions, the non-linear equalization techniques exhibit poor performance, and equalization techniques that better combat nonlinear channels are desirable. The adaptive equalization problem is typically viewed as an inverse filter problem. Traditional equalizers are based on finding the inverse of the channel and compensating the channel's influence using inverse filter technique. Hence there exists no equalizer for these non-invertible channels. This is the major disadvantage with normal adaptive channel equalizers.

The output of the equalizer is fed into a decision device, which attempts to estimate the transmitted symbol. This configuration of an inverse filter equalizer followed by a decision device results in the partitioning of the output signal space by linear decision boundaries between different symbols. When significant noise is added to the transmitted signal, linear boundaries are not optimal.

Artificial Neural Networks (ANN) can be applied to this field for achieving better performance than classical methods in some aspects. With this viewpoint to equalization, complete channel inversion is unnecessary, and the problem is tackled using classification techniques A few noticeable things of this equalizer are its computational simplicity, due to the small size of MLPs that can achieve good performance and efficient extraction of information from a small number of training samples.

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