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Channel Equalization in Underwater Acoustic Communications using Multiple Antennas

A Thesis

Presented in Partial Fulfillment of the Requirements for the

Degree of Master of Engineering Science

with a

Major in Electrical Engineering

in the

College of Graduate Studies

University of Idaho

by

Leili Baghaei Rad

August 2006

Major Professor: Professor Richard Wells.

AUTHORIZATION TO SUBMIT

THESIS

This thesis of Leili Baghaei Rad, submitted for the degree of Master of Science with a major in Electrical Engineering and titled " Channel Equalization in Underwater Acoustic Communications using Multiple Antennas," has been reviewed in final form. Permission, as indicated by the signatures and dates given below, is now granted to submit final copies to the College of Graduate Studies for approval.

Major Professor _____________________________________Date______________

Richard B. Wells

Committee Members

_____________________________________Date______________

Egolf, David

_____________________________________Date_______________

Anderson, Michael

Department Administrator

_____________________________________Date______________

Brian Johnson

Discipline's College Dean

_____________________________________Date_______________

Aicha Elshabini

Final Approval and Acceptance by the College of Graduate Studies

_____________________________________Date_____________

Margrit von Braun

ACKNOWLEDGMENTS

Firstly, I would like to thank my supervisor Professor Richard B. Wells for proof reading this theses. Also for his guidance, patience and constant support throughout the time I have worked with him. I am very grateful for every thing I have learned from him both inside and outside the classroom.

Next, I would like to thank Professor David Egolf for his support and help throughout my degree at the University of Idaho and also for all of his encouragement in my studies. Thank you to Professor Michael Anderson for helpful discussions on problems related to the work in this thesis and also his support. Thank you to Isaac Spurgeon Kodavaty from the Center for Intelligent Systems Research at the University of Idaho for providing the data used to model the underwater channel.

I would like to thank my mother-in-law, Gaye Downes, for proof reading my thesis at very short notice. I am also grateful to Mr. and Mrs. Lynch for enabling me to come to the USA through the Rebecca Lynch Memorial Scholarship. Last but not least, I would like to greatly thank my family, especially my parents for their unwavering support and enthusiasm throughout my life.

This work is dedicated to my husband and best friend, Ian Downes. Without his love and support none of this work would have been possible.

Contents

1. Introduction 1

1.1 Motivation and problem description 1

1.2 Literature Review 3

1.3 Thesis outline 5

1.4 Contributions 7

2. Underwater Acoustic Communication and Channel Model 8

2.1 Channel Model 8

2.2 The Baseband channel 11

2.3 FSK Signal modulation and demodulations: 18

3. Receiver and Transmitter Diversity Techniques: 21

3.1 Antenna Diversity 22

3.2 Temporal Diversity 23

3.3 Branch Combining 24

3.4 Diversity gain and failure 25

3.5 Conventional multiple antenna systems 27

3.6 Parallel stream MIMO 28

4. Review of Adaptive Equalizers 32

4.1 Linear equalization 36

4.1.1 Peak distortion criterion 38

4.1.2 Mean square error criterion 40

4.1.3 Fractionally spaced equalizer 42

4.2 Decision feedback equalization 43

4.2.1 Mean square error optimization criterion 46

4.3 Filter coefficient update methods 48

4.3.1 LMS 48

4.3.2 RLS 51

4.4 Selected experimental results 53

5 Diversity Equalization 57

5.1 Baseband communication system model 57

5.2 Joint channel equalizer (JCE) 63

5.3 Diversity with separate optimization using an independent channel equalizer (ICE) 77

6 Conclusions and Topics for Future Research 83

6.2 Further research directions 84

References 86

ABSTRACT

Currently available commercial underwater acoustic modems have been optimized for long distance communications, on the order of tens to hundreds of kilometers, trading transmission rate for increased range, reliability and robustness. However, new under-water applications are arising which do not require such long distance links. One example is the use of small fleets of closely spaced Autonomous Underwater Vehicles (AUVs) for tasks such as mine detection where the vehicle spacing is more likely to be measured in hundreds of meters at most.

This thesis investigates ways to improve transmission rates over this much shorter link. In particular, it considers the use of an additional hydrophone, currently used for navigation, to achieve diversity at the receiver. We propose an adaptive receiver structure that is capable of reliable asynchronous communication with improved efficiency.

Existing underwater acoustic equalization studies are limited to optimizing the mini-mum Mean-Square Error (MSE) jointly among all spatial diversity channels, called the Joint Channel Equalizer (JCE). In this study we propose a new sub-optimal equalizer that separately optimizes the diversity channels. We have called this the Independent Channel Equalizer (ICE). It ultimately results in a higher MSE but the system is more robust to step changes. This is beneficial to allow rapid re-establishment of communications. Re-sults are presented both in terms of the MSE and the probability of Bit Error Rate (BER). The latter is important, as it is the ultimate measure for a digital communication system.

Introduction

1.1 Motivation and problem description

As technology progresses our ability to utilize and operate within the underwater environment is advancing. Once confined to merely operating on the surface we are now rap-idly exploring the depths of the oceans for military, commercial and scientific purposes.

The use of Autonomous Underwater Vehicles (AUVs) is showing great potential among many new areas of underwater research. Possible applications for this technology include dangerous and/or routine tasks such as minesweeping and reconnaissance, neither of which is suitable for human operators. However, there are many challenges that need to be addressed. These include the autonomous control of a fleet of AUVs and consequently the underwater communications required for this.

In [1,2] the problem of control algorithms for the formation flying of AUVs is considered. As part of the solution a pair of hydrophones is used to provide relative headings to aid in maintaining the formation. The presence of the two hydrophones also present possible opportunities to improve communications between AUVs, using techniques such as receiver diversity, transmitter diversity or Multiple-Input-Multiple-Output (MIMO) communications.

In this application the hydrophones are located at the bow and stern of the AUV. Due to the physical dimensions of the AUV the separation is approximately one meter. This is much closer than typical diversity receiver separations [3]. However, other researchers have shown that receiver separations as low as 0.35 meters can provide additional sources of information in the underwater acoustic channel [4].

The fading and multi-path underwater acoustic channel has always been a great impediment to building reliable underwater communication systems. Many complex physical phenomena cause the propagating acoustic wave intensity and phase to vary temporally and spatially. Thus, one advantage of spatial diversity equalization is its ability to improve the limited signal-to-noise ratio at the receiver through coherent combining. The multi-path spread in these channels is largely caused by reflections from ocean boundaries and refraction due to sound speed variations as a function of depth. Since the degradation in the signal is caused by multi-path intersymbol interference (ISI), simply increasing the signal-to-noise-ratio will not alleviate the problem.

It is well known that coherent equalization techniques improve the bandwidth efficiency of the communication system, thus increasing the data rate [6]. Equalization techniques are well understood in radio communications [5]-[7]. A variety of equalization techniques are available, such as zero-forcing equalization, MMSE equalization, and block equalization. Decision Feedback Equalization (DFE) can be considered an effective technique because it can help to eliminate causal ISI in addition to compensating for the channel [5].

When the communication channel is unknown to the designer, adaptive equalization techniques can be used to first extract the channel response from a training sequence and then compensate the channel distortion in the incoming data symbols [7]. In this thesis the problems above are considered and a sub-optimal decision feedback adaptive equalizer with spatial diversity has been proposed and the results are compared to the optimal equalizer.

1.2 Literature Review

A general overview of the current state of underwater communications is presented in [8]. The underwater channel and its characteristics are discussed and examples of a number of communication systems of various ranges are given. It also outlines some of the current research topics, namely, receiver complexity reduction, interference cancellation and multi-user communication, system self-optimization, modulation and coding and mobile underwater communication.

Among current research a common approach to improving communication rate and reliability is to use multiple diversity channels and a jointly optimized receiver structure. Researchers have shown that for the underwater acoustic channels receiver separations as low as 0.35 meters can provide valid diversity channels [4]. In [3] a joint channel equalizer is formulated where the equalizer is optimized across all diversity channels with the receivers spaced apart in depth by between 9.4 and 55.2 meters. The separation between transmitter and receiver is 8 nautical miles. This separation is significantly greater than any expected separation within the AUV fleet. The receiver was found to have excellent performance. However, it was computationally complex and thus a sub-optimal design was presented with much reduced complexity. This sub-optimal equalizer consisted of a set of single channel equalizers followed by a MMSE combiner. Satisfactory performance was found with a DFE equalizer composed of 25 feed-forward taps and 15 feedback taps.

In [9] and [10] the authors argue that carrier phase is the most rapidly changing parameter in the underwater channel. The variation in phase can be higher than the convergence rate of the equalizer leading to tap rotation and poor performance. The authors suggest one possible solution is to jointly perform synchronization and equalization. They present a receiver that jointly performs MMSE multi-channel combining, carrier phase recovery and fractionally spaced decision feedback equalization. Their results indicate that superior performance for coherent reception can be obtained through joint diversity combining and equalization. The resulted presented are limited to coherent PSK and QAM constellations.

A further problem is the possibility of the equalizer entering a so-called degenerative state, where the output does not depend of the input. In [11], this phenomenon is discussed in the context of radio communications. The authors present an overview of current receiver structures that attempt to avoid this problem. The paper continues further to present a new algorithm for blind decision feedback equalization, which is based on constrained optimization and does not admit degenerative solutions. The possibility of carrying these ideas over to underwater acoustic communication is explored as part of the research presented in this thesis.

1.3 Thesis outline

This study will investigate several possible potential benefits of multiple hydrophones with the goal of improving the communication data rate, either by increasing the raw data rate or by improving efficiency.

This thesis is organized into four main chapters following the introductory chapter as below.

Chapter 2 first outlines a selection of relevant aspects of underwater acoustic communications, in particular the modeling of the underwater channel. It also provides an overview of wireless channel multipath fading characteristics.

Chapter 3 provides background information for transmit and receive diversity. It introduces the main techniques and receiver structures that are used to gain benefit from channels containing diversity. It analyzes the trade offs between the complexity and performance for each technique.

Chapter 4 provides an overview of adaptive signal processing. It introduces the concept of an adaptive channel equalizer and formulates several equalizer structures. Simple linear equalizers are discussed before the more suitable but nonlinear decision feedback equalizers are introduced.

Chapter 5 presents the main work of this thesis. It investigates possible receiver structures to utilize the second hydrophone including a fully jointly optimized equalizer and a sub-optimal equalizer. Performance of the two equalizers is evaluated both in terms of MSE and robustness to step changes. The improved robustness of the sub-optimal equalizer is demonstrated.

Finally the conclusion and future research directions are outlined in chapter 6.

1.4 Contributions

The contributions of this thesis to the underwater acoustics communications community can be summarized in two parts. The first part is a comprehensive review of the current state of underwater acoustic communications. The progression from simple receiver structure to more complex receiver structures employing receiver diversity and joint optimization is discussed. Further discussion investigates the possible trade off in robustness against lower mean square error performance for alternative receiver structures.

The second and major part of the contribution includes the development and analysis of a new diversity equalizer structure. The equalizer is sub-optimal in the mean square sense. However it is shown to be more robust to sudden changes in the impulse response of the channel. The receiver structure consists of two or more separately optimized feed-forward filters, which are equally combined and then followed by a feedback filter.

2. Underwater Acoustic Communication and Channel Model

In general, the shallow underwater acoustic channel is a challenging environment for communication. The channel experiences a large degree of multipath interference due to reflections from the water surface and the sea floor as well as other medium changes. The multi-path signals can be of comparatively large amplitude and spread over a significant time period (many symbols), leading to severe intersymbol interference. Signal fading is present due to several complex physical phenomena, which cause the propagating acoustic wave phase and intensity to vary temporally and spatially. These phenomena include variation in sound speed and channel geometry. In addition, noise is present from ocean surface waves, shipping noise, bubbles, and aquatic wildlife [3], [4], and [16].

2.1 Channel Model

The method-of-images was used for modeling the acoustic pressure in an infinite-range underwater channel with a pressure-release surface and rigid bottom wave-guide. It has perfect reflection of the sound waves from the surface and bottom of the channel [13]. Acoustic pressure caused by a harmonic point source located in the r-z plane is given by

[pic] (2-1)

where k is the wave number given by k=(/c, c is the wave speed, and [pic]is the distance traveled by the corresponding image source. Index l is given as 0, 1, 2…[pic] , m = [1, 2, 3, 4] and i =[pic] .

Figure 2.1 shows how the reflected waves can be thought of as being transmitted from separate virtual sources located above and below the real source. The channel impulse response is formed by the summation of the received direct and reflected signals. The signal phase is such that two out of phase arrivals can add either destructively or constructively.

[pic]

Figure 2-1: Method of images.

It was assumed that this method-of-images model suffices for development of acoustic communication systems at short range because it provided a conservative estimate of multipath effects. The geometric positioning of a pair of AUVs is shown in Figure 2.2.

s[pic]

Figure 2-2: View of the AUVs.

D is the depth of the ocean, Zs is the distance of the transmitter from the ocean floor, Zr is the distance of the receiver from the ocean floor and R is the distance between the AUVs. Throughout this research the water depth is 100 meters. The AUVs are nominally located at a depth of 30 meters and are separated by approximately 30-50 meters. Figure 2.3 shows the view of a pair of AUVs from behind. It illustrates a possible change in the channel due to a relative motion of one of the AUVs.

[pic]

Figure 2-3: Diagram of the relative AUV positions.

A channel impulse response is initially obtained for the passband carrier frequency of 8 kHz. The research was performed using the equivalent baseband model to reduce computational requirement and hence the channel was converted to its baseband equivalent.

2.2 The Baseband channel

The analytic equivalent channel [pic]is given as

[pic] (2-2)

where [pic]is the Hilbert transform [pic]. The baseband equivalent channel at any carrier frequency (c is given by

[pic]. (2-3)

For a valid baseband representation the carrier frequency should be sufficiently large to ensure that [pic]has no significant energy at frequencies greater than the carrier frequency. The baseband equivalent channel is found in Matlab by using the Hilbert function to obtain the analytic equivalent channel. This channel is then multiplied by the exponential over the time period [0;M] with samples spaced at 1/fs. M is the desired length of the impulse response. Figure 2.4 and Figure 2.5 show examples of the passband and baseband equivalent channels for D=100, R=50, Zr=50 and Zs=50 and 55 respectively.

Figure 2.4 and 2.5 illustrate the dramatic change in the channel that can occur even over small translations. In this case initially a pair of AUVs were both located at the mid-point between the ocean surface and the ocean floor. When one of the AUVs is moved away from this mid-point the multipath arrivals will not add at the same instant leading to a more complex channel.

Figure 2.6 shows the effect of the channel on a burst of data symbols. The intersymbol interference caused by the channel is evident. However, the equalizer effectively removes this ISI. Further details about the equalizer are discussed in chapter 4.

The existing system is based around modems supplied by the Woods Hole Institute. The modulation method for these modems is binary frequency-shift keying (FSK) with frequency hopping. The power spectrum for the BFSK is shown in Figure 2.7.

[pic]

Figure 2-6: Effect of channel on data symbols.

2.3 FSK Signal modulation and demodulations:

Binary FSK is a form of frequency modulation in which the modulating signal shifts the output frequency between predetermined values. Generally, the instantaneous frequency is shifted between two discrete values termed the “mark” frequency and the “space” frequency. The mark and space correspond to binary one and zero respectively.

FSK can be transmitted coherently or non-coherently. Coherency implies that the phase of each mark or space tone has a fixed phase relationship with respect to a reference. This is similar to generating an FSK signal by switching between two fixed-frequency oscillators to produce the mark and space frequencies. While this method is sometimes used, the constraint that transitions from mark to space and vice versa must be phase continuous (“glitch” free) requires that the shift and keying rate be interrelated. A synchronous FSK signal which has a shift in Hertz equal to an exact integral multiple (n= 1, 2,…) of the keying rate in bauds, is the most common form of coherent FSK. Coherent FSK is capable of superior error performance but non-coherent FSK is simpler to generate and is used for the majority of FSK transmissions and is the modulation method used in the Woods Hole Oceanographic Institute (WHOI) modems. Non-coherent FSK has no special phase relationship between consecutive elements, and, in general, the phase varies randomly.

Many different coding schemes are used to transmit data with FSK. They can be classified into two major groups: synchronous and asynchronous. Synchronous transmissions have mark-to-space and space-to-mark transitions synchronized with a reference clock. Asynchronous signals do not require a reference clock but instead rely on special bit patterns to control timing during decoding

For the WHOI modems [2] the symbol rates are chosen to be integer multiples of each other and allow for an even number of A/D samples using common clock rates. Symbol rates of 80 and 160 symbols per second (sps) are used for all three-frequency bands. The symbol durations corresponding to the two rates are 12.5 and 6.25 milliseconds respectively. At the 80 sps rate the separation between frequencies is fixed at 160 Hz. At the 160 sps rate the separation is 320 Hz.

The number of hops in each hopping table and the frequency bin separation defines the bandwidth occupied by the signals. The number of hops is 7 or 13, depending upon the symbol rate. The upper band edges are calculated based on this and listed in Table 2.1.

There are three frequency bands defined in the modem specification. The bandwidth is approximately 4 kHz for all bands. Band A is intended for compatibility with underwater telephone transducers. The bands are summarized in Table 2.1 where Fl denotes the carrier of the lowest frequency bin and Fh is the highest carrier frequency. The total bandwidth is slightly larger than Fh – Fl and is also different for each of the two symbol rates.

[pic]

Table 2.1: WHOI modem parameters.

The channel clearing time is a function of the number of hops and the symbol duration. The clearing time is defined as the period from the end of a symbol in a particular bin to the start of the next symbol in that same bin. The values for Tclear for each of the bands and rates are also listed in the table. Commercial transducers identified as fitting these bands are listed in Table 2.2 as a convenience.

[pic]

Table 2-2: Commercial transducers fitting the frequency bands of WHOI modems.

3. Receiver and Transmitter Diversity Techniques:

The use of multiple antennas in wireless links combined with appropriate space-time (ST) coding/modulation and decoding/demodulation is rapidly becoming the new frontier of wireless communication. A growing awareness of the performance gains possible with ST techniques has spurred efforts to integrate this technology into practical systems. The gradual evolution of systems follows the quest for high data rate, measured in bits/sec (bps), and high spectral efficiency. The capacity of a channel defines the amount of information that can be reliably transmitted over a channel. Shannon has shown that for a single transmitter single receiver system operating over a channel with bandwidth B , the capacity is

[pic] bits/s. (4.1)

Until recently multipath fading was the scourge of wireless communication. In a fading channel, signals experience fades (i.e., they fluctuate in their strength). Fading results from the superposition of multipath signals, which have experienced differences in attenuation, delay and phase shift while traveling from the source to the receiver. The best way to overcome the effects of multipath fading is to actually exploit the multiple copies of the transmitted signal that are available from different channels (diversity branches). The multiple copies are called diversity branches and can be combined to improve the performance, i.e. Signal to Noise Ratio. This is generally termed as diversity and is acquired through multiple antennas.

3.1 Antenna Diversity

The goal of antenna diversity is to overcome spatial selectivity caused by small signal fading. Antenna diversity is a technique in which the information-carrying signal is transmitted along different propagation paths. This can be achieved by using multiple receiver antennas and/or by using multiple transmitting antennas. The multipath signals can add constructively or destructively as shown in Figure 3.1. Variations can occur over distances of a fraction of a wavelength.

There are several ways to achieve antenna diversity:

▪ Space diversity: use multiple similar antennas spatially separated to receive/transmit multiple versions of the signal. Each signal has followed a different propagation path.

▪ Polarization: use differently polarized antennas to receive multiple versions with different polarization.

▪ Pattern: use antennas with different patterns to receive multiple version of a wireless transmitted signal.

Transmit: introduce controlled redundancies at the transmitter. This can be exploited by appropriate signal processing techniques at the receiver to achieve diversity gain [14].

3.2 Temporal Diversity

The goal of temporal diversity is to overcome multipath fading due to time varying signals and frequency effects. In this case replicas of the transmitted signal are provided across time by a combination of channel coding and time interleaving strategies. The key requirement for this form of diversity to be effective is that the channel must provide sufficient variations in time.

There are several ways to achieve temporal diversity:

▪ Frequency diversity: exploit a large frequency band and send multiple copies of the signal at different frequencies.

▪ Code: receive time-delayed copies of the signal and use a Rake equalizer [3] to combine them.

▪ Time: use error correcting coding and interleaving

3.3 Branch Combining

To obtain performance benefits the receiver must combine the multiple diversity branches. Various diversity combining techniques exists. They mainly fall into two categories:

1. Gain combining: the receiver sums all of the weighted diversity branches.

1. Maximum Ratio Combining (MRC): The received signals are weighted so that the output SNR is maximized. This results in the best performance but the algorithm is complicated and expensive to implement.

2. Equal Gain Combining (EGC): All the received signals are summed coherently. This algorithm is simpler but it has problems with unequal branch power.

2. Switch Combining: the receiver only demodulates one branch at a time.

2.1 Pure Selection: the receiver monitors all of the SNR’s and selects the strongest signal is selected.

2.2 Threshold I: the receiver retains a branch until the SNR falls below a certain threshold. It then keeps switching between the branches until it finds a branch with adequate SNR.

2.3 Threshold II: This can avoid the problem of continuously switching between the channels. It is the same as above but the receiver stays with a new branch until it is above then below the certain threshold.

3.4 Diversity gain and failure

The required BER can be achieved for a lower SNR when diversity is present. Diversity failure occurs when SNR drops below the minimum required value despite having diversity branches. Diversity failure can occur due to:

1. Branch correlation

2. Unequal branch powers.

To overcome this each diversity branch must be virtually uncorrelated, and even power distribution on the branches is desired.

Figure 3.2 shows the decreasing effect of added diversity [14]. For a given SNR as the number of diversity branches increase the BER gets smaller with a diminishing return.

[pic]

3.5 Conventional multiple antenna systems

Figure 3.3 illustrates different antenna configurations used in defining space-time communication systems. Single-input single-output (SISO) is the well-known wireless configuration. Single-input multiple-output (SIMO) uses a single transmitting antenna and multiple (MR) receive antennas. Multiple-input single-output (MISO) has multiple (MT) transmitting antennas and one receive antenna. Multiple-input multiple-output (MIMO) has multiple (MT) transmitting antennas and multiple (MR) receive antennas. Capacity equations for various configurations are given as [15]

• SISO – Single Input Single Output

; (4.2)

• SIMO – Single Input Multiple Output

; (4.3)

• MISO – Multiple Input Single Output

; (4.4)

• MIMO – Multiple Input Single Output

. (4.5)

Figure 3.3 illustrates different antenna configurations for ST wireless links.

[pic]

3.6 Parallel stream MIMO

The previous results examined MIMO channel for increasing the SNR for a single signal. Different signals can also be sent from each transmit antenna. Each signal will experience a different channel as shown in Figure 3.4. The component hi,j is the fading coefficient from the jth transmit antenna to the ith receive antenna.

[pic]

We assume that the received power for each of the receive antennas is equal to the total transmitted power Es. This implies we can ignore signal attenuation, antenna gains, and so on. The narrow band system with MT transmit antennas and MR receive antennas the received signal is given by

[pic] (4.6)

or simply as y = Hx + n. Here x represents the Mt dimensional transmitted symbol, n is the Mr dimensional noise vector, and H is the Mr × Mt matrix of channel gains hij representing the gain from transmit antenna j to receive antenna i.

When both the transmitter and receiver have multiple antennas the performance gain (multiplexing gain) is found by decomposing the MIMO channel into R parallel independent channels. By multiplexing independent data onto these independent channels, we get an R-fold increase in data rate in comparison to a SISO system. The increased data rate is called the multiplexing gain. The steps for obtaining the independent channels from a MIMO system are described below.

Consider a MIMO channel as shown in Figure 3.5 with Mr × Mt channel gain matrix H known to both the transmitter and the receiver.

V and UH are unitary matrices (their geometrical length is unchanged), H is the Hermitian operator defined as the conjugate transpose. In this system processing will be performed at both the transmitter and the receiver. The system has no control over H but it can control V and UH. V and UH are taken such that D = UHHV is diagonalized (i.e. the Singular Value Decomposition)

. (4.7)

Hence, mathematically the MIMO channel can be viewed as a set of Mt separate channels. Each diagonal in D is the square root of an eigenvalue of HHH [15] and specifies the channel gain given by

.

. (4.8)

Each signal in x(t) has a capacity of

. (4.9)

For Mt signals the total capacity is

. (4.10)

The basis of this thesis research is to explore how the diversity provided by the two hydrophones can be exploited. The hydrophones provide receiver diversity and thus form a SIMO system. Relevant areas discussed in this chapter include branch combining and spatial diversity. At the present the second hydrophone is being used as a receiver. However, in future research it can also be used as a transmitter and thus form a MIMO system.

4. Review of Adaptive Equalizers

Adaptive equalizers can be classified as either linear or non-linear. For each type of equalizer, there can be a structure, and corresponding to the structure there are algorithms. The main structures and the fundamental algorithms are illustrated in Figure 4.1. The review material presented in this chapter is taken principally from the Proakis textbook [5].

The general modes of operation of an equalizer involve training and tracking. In the training mode, a fixed length, pre-determined sequence is sent from the transmitter to the receiver to enable the equalizer to minimize a cost function. When an equalizer is properly trained, it is said to have converged. Once the equalizer is trained, the user data is transmitted and the equalizer employs an algorithm to track the changing channel and update the equalizer coefficients.

Furthermore, either of the operating modes can be divided into two stage processes. The first stage is filtering while the second is updating the filter coefficients as shown in Figure 4.2. The choice of adaptive algorithm, therefore, governs the performance of an equalizer and its ability to track channels. Other factors that should be considered when choosing an algorithm include numerical stability, implementation complexity and robustness [17].

[pic]

Figure 4-1: Equalizer types, structure and algorithms.

[pic]

Figure 4-2: The equalization process- filtering and updating equalizer coefficients.

In [5] the optimum receiver for digital transmission is derived based on the maximum-likelihood sequence detection criterion. It is shown that the receiver can be comprised of an optimum demodulator matched to the channel followed by a sampler operating at the symbol rate [pic] and some form of processing algorithm for estimating the information sequence from the sequence of samples. If it is assumed that the ISI affects only a finite number of symbols then the output of the demodulator can be viewed as the output of a finite state machine. Such an output may be represented as a trellis diagram with the maximum-likelihood estimate of the sequence corresponding to the most probable path through the trellis. One widely used algorithm to perform the trellis search is the Viterbi algorithm [18].

Although the MLSE (Maximum Likelihood Sequence Estimation) is optimum from a probability of error perspective the computational complexity of the equalizer grows exponentially with the length of the channel time dispersion [5]. For an alphabet size of M and with L symbols contributing to the ISI the Viterbi algorithm computes ML+1 metrics for each received symbol. Typically this complexity is regarded as not feasible for a radio channel. For an Underwater Acoustic (UWA) channel, which has significantly more ISI contributing symbols, it is regarded as prohibitive. Although there is some reason to think this opinion may be misplaced [26] it is nonetheless the wide consensus that acceptable practice today should avoid MLSE approaches.

A number of approaches have been developed to achieve sub-optimum equalization with achievable computational complexity [19]-[21]. The following discussion addresses two of the most widely used approaches and their implementations. The standard linear equalizer will be discussed first before the much improved decision feedback equalizer is considered.

4.1 Linear equalization

The most common implementation of the linear equalizer is as a transversal filter as shown in Figure 4.3. The input to the filter is the sequence vk of sampled values and the output is the estimate of the current data symbol. Each of the 2K+1 samples are weighted by [pic] and summed to form the estimate. Formally, the estimate for the kth data symbol is

[pic]. (4.1)

Typically the symbol estimate [pic]will not exactly represent a data symbol and a decision must be made as to which is the nearest data symbol. If the decision results in [pic] that is not identical to the transmitted data symbol [pic] then a symbol error has occurred. For a simple (non-adaptive) linear equalizer this error will not affect further data symbols. To optimize the filter coefficients some criterion must be used to measure the equalizer performance. As discussed previously the MLSE uses Euclidean distance or an equivalent metric [26] as the criterion but is regarded as too computationally complex. In a similar manner the optimization of the filter coefficients is also complex. Substantially more tractable criteria include the peak distortion criterion and the mean square error criterion, described in the next section.

[pic]

Figure 4-3: Linear equalizer structure.

4.1.1 Peak distortion criterion

The peak distortion criterion seeks to minimize the worst-case intersymbol interference present at the output. To determine the peak distortion it is beneficial to reformulate the expression for the data symbol estimate. This is achieved by combining the effect of the channel and the equalizer into a single linear filter [pic]. This can be shown to be simply the convolution of the channel impulse response [pic] and the impulse response of the linear filter in the equalizer [pic]. In formal analysis the equalizer is assumed to have an infinite number of taps (i.e. K =[pic]) and thus the output for the kth data symbol from the equalizer is

[pic]. (4.2)

The first term represents a scaled version of the desired symbol. The second term is the combined intersymbol interference from all past and present symbols (dependent on [pic]). The final term is the filtered equivalent discrete time noise sequence. The worst case for the intersymbol interference occurs when all of the ISI terms add constructively. The peak distortion is thus defined as

[pic] . (4.3)

From Nyquist's Criterion [22] the ISI can be completely removed if the following condition is met:

[pic]. (4.4)

By taking the z transform of the above equation it is seen that the equalizer has a transfer function given by

[pic]. (4.5)

Thus, the equalizer is simply the inverse of the channel response. Such an equalizer is known as a zero-forcing equalizer as it attempts to force the ISI to zero. The zero-forcing equalizer can be combined with a noise-whitening filter to give an equivalent zero-forcing equalizer

[pic]. (4.6)

Although simple in concept the zero forcing equalizer suffers from noise enhancement. In [5] it is shown the SNR depends on an integral containing the reciprocal of the channel response. If the channel contains any spectral nulls then the integrand becomes infinite and the SNR becomes zero. This occurs as the equalizer attempts to compensate for the nulls by producing an infinite gain at the null frequencies. The side effect is that the gain is also applied to the additive noise resulting in poor SNR for typical channels.

Furthermore, up to this point the equalizer has been assumed to contain an infinite number of taps. In any application this is not possible and a finite number of taps must be used. If cj = 0 for all [pic]> K then the convolution with the channel response, qn will be zero outside of the range [pic], where L is the length of the channel impulse response. The peak distortion will now be the sum over these non-zero terms. With a finite number of terms it will not generally be possible to completely remove all of the ISI even with the optimum set of filter weights and there will always be some residual ISI present. For the special case in which the peak distortion is less than unity it can be shown that the 2K+1 filter weights correspond to the coefficients from the infinite length zero-forcing equalizer in the range [pic][5].

4.1.2 Mean square error criterion

The error sequence,[pic] is defined as the difference between the transmitted symbols and the estimated symbol

[pic] . (4.7)

The mean square error criterion optimizes the filter coefficients to minimize the expected value of the squared error over all received symbols

[pic] (4.8)

where J is a quadratic function of the equalizer coefficients. The minimization of J can be formulated as an infinite set of linear equations either directly or, more elegantly, by using the orthogonality principle. By imposing the condition that the error sequence is orthogonal to the signal sequence the equalizer transfer function is found as

[pic] (4.9)

where N0 is the power spectral density function of the noise.

If a noise-whitening filter is also incorporated into the equalizer the transfer function becomes

[pic]. (4.10)

When compared to the equalizer transfer function for the linear equalizer using the peak distortion criterion the only difference is the inclusion of the noise spectral density factor [pic] in the numerator. If [pic] is very small or zero then this term has little impact and the filter coefficients will be similar to those obtained under the peak distortion criterion and the ISI will be completely eliminated. If [pic] is significant then the ISI will not be eliminated but the equalizer will not suffer from noise enhancement. The extra [pic] term serves to create an upper bound on the transfer function such that spectral nulls in the channel response will not lead to infinite gains.

As with the peak distortion criteria this result was obtained using an infinite length equalizer. For a finite length equalizer the minimization is over the 2K+1 equalizer coefficients and leads to a set of 2K+1 linear equations. This set of equations can be solved to find the optimum equalizer coefficients.

4.1.3 Fractionally spaced equalizer

In the preceding discussion the equalizer filter taps have been symbol spaced. This spacing is optimum if a matched filter precedes the equalizer. However, the matched filter requires knowledge of the channel, and often this is not available. For an unknown channel the receiver filter can be matched to the transmitted signal pulse shape with an associated optimum sampling time. The drawback of this approach is that the equalizer becomes sensitive to the choice and variation in the sampling times.

Using a fractionally spaced equalizer can mitigate the sensitivity on sampling phase

[pic] (4.12)

where [pic] is the fractional tap spacing. When aliasing is not a significant factor, it is shown in [5] that the equalized spectrum can be expressed as

[pic] (4.13)

where [pic]is the spectrum of the signal at the input to the equalizer [5].

The fractionally spaced equalizer is thus able to compensate for an arbitrary phase in the sampling time. It has in effect become a matched filter followed by a symbol rate equalizer and offers superior performance under some circumstances when compared to a symbol spaced equalizer.

4.2 Decision feedback equalization

The structure of a decision feedback equalizer is shown in Figure 4.4. The equalizer consists of a feed-forward section, which is identical to the linear equalizer discussed in the previous section. The feed-forward filter can either be symbol spaced or fractionally spaced. The feedback section contains a symbol-by- symbol detector and a symbol spaced feedback filter. The feedback section removes ISI due to previously detected symbols.

The equalizer output can be expressed as

[pic] (4.14)

where[pic]is the detected symbol. The equalizer is composed of (K1+1) feed-forward taps and K2 feedback taps. In contrast to the linear equalizer the decision feedback equalizer is non-linear due to the decision device in the feedback path.

[pic]

Figure 4-4: Decision Feedback Equalizer structure.

4.2.1 Mean square error optimization criterion

The optimization of the filter coefficients (both feed-forward and feedback) can be done with either the peak distortion criterion or the mean square error criterion. In practice the mean square error criterion is almost universally used and thus only the mean square error criterion is discussed here.

The minimization leads to the following set of equations for the feed-forward coefficients

[pic], [pic] (4.15)

where

[pic], [pic] . (4.16)

The coefficients of the feedback filter can be found from the feed-forward coefficients as follows

[pic] [pic]. (4.17)

Provided that the previous decisions are correct and that[pic] (i.e. the feedback filter length is greater than or equal to the channel impulse response length) then the ISI from previously detected symbols can be completely removed. In [5] the superiority of the decision feedback equalizer over the linear equalizer is demonstrated for a number of different channels.

As before a symbol spaced feed-forward section in a decision feedback equalizer leads to sensitivity in the sample timing. This can be overcome by using a fractionally spaced equalizer in the feed-forward section with the feedback section necessarily remaining symbol spaced.

4.3 Filter coefficient update methods

In the preceding sections the optimization of equalizer coefficients was derived for sub-optimal receivers. During the derivations it was implicitly assumed that the channel characteristics were known. In practice this is rarely the case and often the channel may be time varying. In such situations the equalizer must be able to be initially adjustable for the channel and, if necessary, adapt to time variations. Two common algorithms to achieve this are introduced.

4.3.1 LMS

During the derivation of the optimum coefficients in section 4.1.2 it was stated [5] that the coefficients can be found as the solution of a set of 2K+1 linear equations. This can be expressed in matrix form as follows

[pic] Q[pic] (4.18)

where Q is the (2K+1) x (2K+1) covariance matrix of the signal samples, C is the vector of equalizer coefficients and [pic]is a vector of channel filter coefficients. The solution of this set of equations forms the optimum equalizer coefficients and can be found by inverting the covariance matrix Q.

To avoid directly inverting the matrix an iterative algorithm can be used. One suitable algorithm is the method of steepest descent where an initial guess is refined. The initial guess lies on the 2k+1 dimensional quadratic MSE surface. The gradient at this point is computed and each coefficient is changed in the direction opposite to its corresponding gradient component. The magnitude of the change for each coefficient is proportional to the magnitude of the corresponding gradient. Thus, at each iteration the equalizer coefficient are updated using the following equation

[pic], [pic] (4.19)

where the gradient vector [pic] is written as

[pic]Q[pic]. (4.20)

[pic]represents the set of coefficients at the kth iteration, [pic]is the error signal at the kth iteration. Vk is the vector of received signal samples that make up the estimate [pic], and [pic] is a positive number selected small enough to guarantee the convergence of the iterative procedure.

When the minimum MSE is reached the gradient vector will be the zero-vector. For a finite number of iterations the minimum cannot generally be reached. However, it can be made arbitrary close.

The difficulty with this method lies in the dependence of the gradient vector on both the covariance matrix and the cross-correlation vector. Both of these in turn depend on the coefficients of the discrete time channel model. This returns us to the original problem whereby typically the channel was unknown.

This can be overcome by estimating the channel coefficients using estimates of the gradient vector as expressed below

[pic] (4.21)

where [pic] denotes an estimate of the gradient vector Gk and [pic] denotes the estimate of the vector of coefficients.

Finally, the update for the equalizer coefficients is given by the following equation

[pic] (4.22)

This forms the original basic LMS algorithm [23]. From the basic LMS a number of variations have been developed including the signed-LMS, normalized-LMS and the filtered gradient-LMS algorithms [24]-[25].

The convergence of the LMS algorithm is dependent on a single parameter, namely the step size parameter, Δ. A large value of Δ promotes fast convergence but large fluctuations from the minimum MSE in the steady state. On the other hand, a small value of Δ assures small fluctuations from the minimum MSE in the steady state at the expense of slower convergence.

4.3.2 RLS

To obtain an algorithm with faster convergence (while still maintaining an acceptable level of fluctuations) a more complex algorithm is required. One suitable criterion is the Least Squares (LS) approach that minimizes the quadratic performance index. In contrast to the LMS algorithm this is a time average rather than a statistical average.

The Recursive Least Squares (RLS) estimation of the data symbols may be formulated as follows. The algorithm seeks to minimize the time average weighted squared error given by

[pic] (4.23)

where the error is defined as

[pic]YN(n). (4.24)

w represents a weighting factor 0 ................
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