As discussed in the last chapter, in a Gaussian or flat fading channel, multicode channels are orthogonal. In a realistic wireless channel, however, the orthogonality no longer maintains. Thus, MCI is caused. In this chapter, a novel detection method, called hybrid MCI cancellation and MMSE detection, is proposed and compared with pure MMSE detection. The system performance is analytically studied with imperfect channel estimation to show how it is affected by parameters such as the window size in the channel estimation, Doppler shift, the number of stages of the hybrid detection, the power ratio of pilot to data channels, spreading factor, and so on.

Introduction

Although MMSE detection can suppress MCI as discussed in the last chapter, so far MMSE detection has not been investigated for the OFCDM system with interference cancellation. Figure 10.1 illustrates the situation when MMSE detection is employed with MCI cancellation. It can be seen that the input to the MCI canceller is a combination of the useful signal, MCI and background noise. After MCI cancellation, the useful signal and background noise remain unchanged, but MCI is reduced. The combined signal is input to the MMSE detector with estimated signal plus interference power. Then the weighted output is generated after MMSE detection. Since the weight of MMSE is related to the input signal plus interference power, it should be updated stage by stage because the input power of MCI changes due to MCI cancellation.

In this chapter we illustrate the relevance of stochastic resonance to auditory neural coding. This relates to natural auditory coding and also to coding by cochlear implant devices. Cochlear implants restore partial hearing to profoundly deaf people by trying to mimic, using direct electrical stimulation of the auditory nerve, the effect of acoustic stimulation.

Introduction

It is not surprising that the study of auditory neural coding involves stochastic phenomena, due to one simple fact – signal transduction in the ear is naturally a very noisy process. The noise arises from a number of sources but principally it is the Brownian motion of the stereocilia (hairs) of the inner hair cells that has the largest effect (Hudspeth 1989). Although the Brownian motion of the stereocilia appears small – typically causing displacements at the tips of the stereocilia of 2–3 nm (Denk and Webb 1992) – this is not small compared with the deflection of the tips at the threshold of hearing. Evidence exists that suggests, at threshold, the tip displacement is of the order of 0.3 nm (Sellick et al. 1982, Crawford and Fettiplace 1986), thus yielding a signal-to-noise ratio (SNR) of about –20 dB at threshold. Of course, under normal operating conditions the SNR will be greater than this, but, at the neural level, is typically of the order of 0 dB (DeWeese and Bialek 1995).

Given the level of noise and the fact that neurons are highly nonlinear makes the auditory system a prime candidate for observing stochastic resonance (SR) type behaviour.

Quantization of a signal or data source refers to the division or classification of that source into a discrete number of categories or states. It occurs, for example, when analogue electronic signals are converted into digital signals, or when data are binned into histograms. By definition, quantization is a lossy process, which compresses data into a more compact representation, so that the number of states in a quantizer's output is usually far fewer than the number of possible input values.

Most existing theory on the performance and design of quantization schemes specifies only deterministic rules governing how data are quantized. By contrast, stochastic quantization is a term intended to pertain to quantization where the rules governing the assignment of input values to output states are stochastic rather than deterministic. One form of stochastic quantization that has already been widely studied is a signal processing technique called dithering. However, the stochastic aspect of dithering is usually restricted so that it is equivalent to adding random noise to a signal prior to quantization. The term stochastic quantization is intended to be far more general, and applies to the situation where the rules of the quantization process are stochastic.

The inspiration for this study comes from a phenomenon known as stochastic resonance, which is said to occur when the presence of noise in a system provides a better performance than the absence of noise. Specifically, this book discusses a particular form of stochastic resonance – discovered by Stocks – known as suprathreshold stochastic resonance, and demonstrates how and why this effect is a form of stochastic quantization.

The coherent MRC reception of PSAM MQAM systems with antenna diversity is studied in this chapter. A general fast time varying fading channel model is assumed. Pilot symbols are periodically inserted during the transmission of data symbols, which are used to track the time varying fading, and to provide channel estimation for data decisions at the receiver. Based on a digital implementation, a coherent demodulation scheme is presented. Channel estimation error due to fast fading and additive noise is studied. System performance is evaluated in terms of BER. The analysis shows that in perfect channel estimation cases, with the antenna diversity technique, the BER performance improves significantly, and higher-order QAM can be employed for higher throughput. It is also found that inaccurate channel estimation limits the benefit of diversity when the modulation order is large. By increasing the length of the channel estimator and the amplitude of the pilot symbol, more accurate channel estimation can be achieved, so that the BER performance is improved. Moreover, when the Doppler frequency is less than 1 / 2ST, where S is the number of symbols per time slot and T is the symbol duration, the performance is flat since the channel estimator is robust to fading rate.

Introduction

During the past several decades, MQAM has been considered for high rate data transmission over wireless links due to its high spectral efficiency [1–4].

The initial research into suprathreshold stochastic resonance described in Chapter 4 considers the viewpoint of information transmission. As discussed briefly in Chapter 4, the suprathreshold stochastic resonance effect can also be modelled as stochastic quantization, and therefore results in nondeterministic lossy compression of a signal. The reason for this is that the effect of independently adding noise to a common signal before thresholding the result a number of times, with the same static threshold value, is equivalent to quantizing a signal with random thresholds. This observation leads naturally to measuring and describing the performance of suprathreshold stochastic resonance with standard quantization theory. In a context where a signal is to be reconstructed from its quantized version, this requires a reproduction value or reproduction point to be assigned to each possible state of a quantized signal. The quantizing operation is often known as the encoding of a signal, and the assignment of reproduction values as the decoding of a signal. This chapter examines various methods for decoding the output of the suprathreshold stochastic resonance model, and evaluates the performance of each technique as the input noise intensity and array size change. As it is the performance criterion most often used in conventional quantization theory, the measure used is the mean square error distortion between the original input signal and the decoded output signal.

Introduction

We begin this chapter by very briefly reviewing the SSR model introduced in Chapter 4. We then introduce the concept of decoding the output of a quantizer's encoding to reconstruct the input signal, and consider measuring the performance of such a reconstruction.

Interference suppression is important to allow UWB devices to operate over the spectrum occupied by narrowband systems. In this chapter, the use of a notch filter in TH-IR for UWB communication is considered, where a Gaussian monopulse is employed with pulse position modulation (PPM). A lognormal fading channel is assumed and a complete analytical framework is provided for performance evaluation using a transversal-type notch filter to reject narrowband interference. A closed-form expression of BER is derived, and the numerical results show that the use of a notch filter can significantly improve system performance. Furthermore, a performance comparison between TH-IR and multicarrier CDMA UWB systems is also made for the same transmit power, the same data rate and the same bandwidth. It is shown that, in the presence of narrowband interference, the TH-IR system with a notch filter can achieve a performance similar to multicarrier CDMA.

Introduction

Over the last decade, there has been a great interest in ultra-wideband (UWB) time-hopping (TH) impulse radio (IR) communication systems [1–8]. These systems make use of ultra-short duration pulses (monocycles), which yield ultra-wide bandwidth signals characterized by extremely low power densities. UWB systems are particularly promising for short-range high-speed wireless communications as they potentially combine reduced complexity with low power consumption, low probability of intercept, high accuracy positioning and immunity to multipath fading due to discontinuous transmission.

This chapter studies MQAM for downlink multicode CDMA systems with interference cancellation to support high data rate services. In the current 3G WCDMA systems, in addition to multicode transmission, MQAM is employed for HSDPA due to its high spectral efficiency. In frequency selective fading channels, multipath interference seriously degrades the system performance. In this chapter, theoretical analysis is presented to show that with the help of interference cancellation technique, MQAM may be employed in high SNR cases to increase system throughput. Moreover, it is found that when using the interference cancellation technique, extra pilot power should be invested for more accurate channel estimation, and consequently better BER performance can be achieved.

Introduction

MQAM modulated multicode CDMA is proposed for HSDPA in the 3G standards, by which the throughput can be increased without extra bandwidth investment. As mentioned in Chapter 1, the introduction of multicode transmission causes multipath interference in frequency selective fading channels due to multipath propagation delays. In this chapter, a coherent Rake receiver with interference cancellation is studied.

Moreover, in WCDMA systems, a common pilot channel is used for channel estimation at the receiver. However, channel estimation error occurs since the received pilot channel signal suffers from the multipath interference and AWGN noise, which affects the coherent data decision and the regeneration of multipath interference, and thus degrades the system performance. The effects of imperfect channel estimation and additive multipath interference on system performance are investigated.

To conclude this book, we summarize our main results and conclusions, before briefly speculating on the most promising areas for future research.

Putting it all together

Stochastic resonance

Chapter 2 presents a historical review and elucidation of the major epochs in the history of stochastic resonance (SR) research, and discussion of the evolution of the term ‘stochastic resonance’. A list of the main controversies and debates associated with the field is given.

Chapter 2 also demonstrates qualitatively that SR can actually occur in a single threshold device, where the threshold is set to the signal mean. Although SR cannot occur in the conventional signal-to-noise ratio (SNR) measure in this situation, if ensemble averaging is allowed, then the presence of an optimal noise level can decrease distortion.

Furthermore, Chapter 2 contains a discussion and critique of the use of SNR measures to quantify SR, the debate about SNR gains due to SR, and the relationship between SNRs and information theory.

Suprathreshold stochastic resonance

Chapter 4 provides an up-to-date literature review of previous work on suprathreshold stochastic resonance (SSR). It also gives numerical results, showing SSR occurring for a number of matched and mixed signal and noise distributions not previously considered. A generic change of variable in the equations used to determine the mutual information through the SSR model is introduced. This change of variable results in a probability density function (PDF) that describes the average transfer function of the SSR model.

As described and illustrated in Chapters 4–7, a form of stochastic resonance called suprathreshold stochastic resonance can occur in a model system where more than one identical threshold device receives the same signal, but is subject to independent additive noise. In this chapter, we relax the constraint in this model that each threshold must have the same value, and aim to find the set of threshold values that either maximizes the mutual information, or minimizes the mean square error distortion, for a range of noise intensities. Such a task is a stochastic optimal quantization problem. For sufficiently large noise, we find that the optimal quantization is achieved when all thresholds have the same value. In other words, the suprathreshold stochastic resonance model provides an optimal quantization for small input signal-to-noise ratios.

Introduction

The previous four chapters consider a form of stochastic resonance, known as suprathreshold stochastic resonance (SSR), which occurs in an array of identical noisy threshold devices. The noise at the input to each threshold device is independent and additive, and this causes a randomization of effective threshold values, so that all thresholds have unique, but random, effective values. Chapter 4 discusses and extends Stocks' result (Stocks 2000c) that the mutual information between the SSR model's input and output signals is maximized for some nonzero value of noise intensity. Chapter 6 considers how to reconstruct an approximation of the input signal by decoding the SSR model's output signal.

Stochastic resonance (SR), being an interdisciplinary and evolving subject, has seen many debates. Indeed, the term SR itself has been difficult to comprehensively define to everyone's satisfaction. In this chapter we look at the problem of defining stochastic resonance, as well as exploring its history. Given that the bulk of this book is focused on suprathreshold stochastic resonance (SSR), we give particular emphasis to forms of stochastic resonance where thresholding of random signals occurs. An important example where thresholding occurs is in the generation of action potentials by spiking neurons. In addition, we outline and comment on some of the confusions and controversies surrounding stochastic resonance and what can be achieved by exploiting the effect. This chapter is intentionally qualitative. Illustrative examples of stochastic resonance in threshold systems are given, but fuller mathematical and numerical details are left for subsequent chapters.

Introducing stochastic resonance

Stochastic resonance, although a term originally used in a very specific context, is now broadly applied to describe any phenomenon where the presence of internal noise or external input noise in a nonlinear system provides a better system response to a certain input signal than in the absence of noise. The key term here is nonlinear. Stochastic resonance cannot occur in a linear system – linear in this sense means that the output of the system is a linear transformation of the input of the system. A wide variety of performance measures have been used – we shall discuss some of these later.

By definition, signal or data quantization schemes are noisy in that some information about a measurement or variable is lost in the process of quantization. Other systems are subject to stochastic forms of noise that interfere with the accurate recovery of a signal, or cause inaccuracies in measurements. However stochastic noise and quantization can both be incredibly useful in natural processes or engineered systems. As we saw in Chapter 2, one way in which noisy behaviour can be useful is through a phenomenon known as stochastic resonance (SR). In order to relate SR and signal quantization, this chapter provides a brief history of standard quantization theory. Such results and research have come mainly from the electronic engineering community, where quantization needs to be understood for the very important process of analogue-to-digital conversion – a fundamental requirement for the plethora of digital systems in the modern world.

Information and quantization theory

Analogue-to-digital conversion (ADC) is a fundamental stage in the electronic storage and transmission of information. This process involves obtaining samples of a signal, and their quantization to one of a finite number of levels.

According to the Australian Macquarie Dictionary, the definition of the word ‘quantize’ is

1. Physics: a. to restrict (a variable) to a discrete value rather than a set of continuous values. b. to assign (a discrete value), as a quantum, to the energy content or level of a system. 2. Electronics: to convert a continuous signal waveform into a waveform which can have only a finite number (usually two) of values.

This chapter discusses the behaviour of the mutual information and channel capacity in the suprathreshold stochastic resonance model as the number of threshold elements becomes large or approaches infinity. The results in Chapter 4 indicate that the mutual information and channel capacity might converge to simple expressions of N in the case of large N. The current chapter finds that accurate approximations do indeed exist in the large N limit. Using a relationship between mutual information and Fisher information, it is shown that capacity is achieved either (i) when the signal distribution is Jeffrey's prior, a distribution which is entirely dependent on the noise distribution, or (ii) when the noise distribution depends on the signal distribution via a cosine relationship. These results provide theoretical verification and justification for previous work in both computational neuroscience and electronics.

Introduction

Section 4.4 of Chapter 4 presents results for the mutual information and channel capacity through the suprathreshold stochastic resonance (SSR) model shown in Fig. 4.1. Recall that σ is the ratio of the noise standard deviation to the signal standard deviation. For the case of matched signal and noise distributions and a large number of threshold devices, N, the optimal value of σ – that is, the value of σ that maximizes the mutual information and achieves channel capacity – appears to asymptotically approach a constant value with increasing N. This indicates that analytical expressions might exist in the case of large N for the optimal noise intensity and channel capacity.