In the previous chapter we looked at some of the parameters that affect the choice of a wireless standard. This chapter explains how to get the best performance by tailoring it for the specific implementation. It explains the trade-offs that can be made for some of the common parameters in a wireless design. As before, many of the comments and techniques are valid across the range of standards.

9.1 Range and throughput

Invariably, the first question that is asked is, ‘What is the range?’ In Chapter 2, I looked at the fundamentals of range, which are essentially: transmit power, receive sensitivity and matching. In this chapter, I'll look at how to put them into practice and discuss the other key influence – the choice of antenna.

9.1.1 Power amplifiers and low noise amplifiers

The first thought of most designers coming to wireless is how to shout louder; in other words, how they can add additional amplification to boost the transmit power. A number of points should be borne in mind when doing this:As the radio link is symmetrical (i.e., each radio needs to receive as well as transmit), increasing the output power only gives a real benefit if it is done at both ends, otherwise the second radio will not be able to transmit at a level that allows the first unit to hear whether or not its transmissions have been received. It is back to the issue of asymmetric link budgets.

5.1 Introduction

The 802.11 and Wi-Fi standards have become immensely successful in providing Internet connectivity for laptops. In recent years they have also started to appear in mobile phones and other portable devices to provide a moderate speed connection to Internet hotspots. They are also finding new uses that take advantage of the widely deployed infrastructure, notably in the M2M space, and in some low-power incarnations for asset tracking. The most recent release of the standard – 802.11n is beginning to garner some degree of success for audio or video streaming applications in the home. Despite these uses, almost all current deployments are targeted solely at Internet access.

802.11 is the oldest of the wireless standards covered in this book. Its genesis grew out of a proprietary wireless LAN called WaveLAN that first appeared on the market in 1988, having been started back in 1986. In its early days, it was not intended for Internet access, but as a wireless replacement for Ethernet cables, with the potential markets of factory warehousing and connection to an office network. The concept was to replace the wired physical connection of the 802.11 standard with a wireless alternative that would slot into the same 802 protocol stack. In 1991, efforts were begun to evolve it into a wireless networking standard, which led to the release of the 802.11 specification in 1997.

Many designers rush into wireless without any knowledge or consideration of the practical issues they will face in manufacturing and selling a wireless product. Wireless introduces a number of requirements over and above those of normal electronics design. These need to be understood if manufacturers wish to place their products on the market and conform to legal requirements.

This chapter highlights these areas, so that a designer can assess the most practical route when embarking on a wireless design. If they are ignored, (as they frequently are), the resulting cost in putting things right after the event can be greater than the cost of the rest of the design effort. In the worst case, a national regulator can stop shipment of products within its country.

10.1 Regulatory approval

To the best of my knowledge, it is legal to sell a cable anywhere in the world. Plugging in a cable doesn't generate any significant amount of electromagnetic radiation that could interfere with other products. Replace that cable with a radio transmitter and everything changes.

Although we are talking about radios that work in the unlicensed ISM bands, that does not grant designers a right of laissez-faire. Products still need to adhere to strict rules and manufacturers must be able to prove that they meet them. These rules exist to try to ensure open access to anyone who wants to use that spectrum, minimising the possibility and severity of interference and to prevent any single product from monopolising too much of the spectrum.

Bluetooth low energy is the latest short-range wireless specification to appear on the market, having been ratified at the end of 2009. Although written by the Bluetooth Special Interest Group, it is a fundamentally different radio standard from the one covered in Chapter 5, both in terms of how it works and the applications it will enable. Hence it merits its own chapter.

By itself, Bluetooth low energy is incompatible with a standard Bluetooth chip – it is a completely new radio and protocol stack. Some of the applications it enables, such as allowing sports equipment to talk to watches, will use Bluetooth low energy chips for both ends of the link, neither of which will be able to talk to existing Bluetooth chips. In these end-to-end applications, it is not dissimilar to other low-power proprietary standards, such as ANT. However, where it differs, and what gives it its power, is that the standard allows dual-mode chips to be designed, which support multiplexed Bluetooth and Bluetooth low energy connections. These will replace the Bluetooth chips in today's mobile phones and PCs, providing an infrastructure of billions of devices that can communicate with existing Bluetooth peripherals, as well as the new generation of dedicated Bluetooth low energy products. It gives Bluetooth low energy the ‘free ride’ that will lead to economies of scale for chip vendors and a vibrant ecosystem of devices for products to connect to.

Now that we have a large collection of algorithms for convolutions and for the discrete Fourier transform, it is time to turn to how these algorithms are used in applications of signal processing. Our major purpose in this chapter is to discuss the role of algorithms in constructing digital filters. We shall also study other tasks such as interpolation and decimation. By using the methods of nesting and concatenation, we will build large signal-processing structures out of small pieces. The fast algorithms for short convolutions that were studied in Chapter 5 will be used to construct small filter segments.

The most important device in signal processing is the finite-impulse-response filter. An incoming stream of discrete data samples enters the filter, and a stream of discrete samples leaves. The streams of samples at the input and output are very long; in some instances millions of samples per second pass through the filter. Fast algorithms for filter sections always break the input stream into batches of perhaps a few hundred samples. One batch at a time is processed. The input samples are clocked into an input buffer, then processed one block at a time after that input block has been completed. The resulting block is placed into an output buffer, and the samples are clocked out of the output buffer at the desired rate.

Convolution by sections

Many algorithms for the discrete Fourier transform were studied in Chapter 3.

A quarter of a century has passed since the previous version of this book was published, and signal processing continues to be a very important part of electrical engineering. It forms an essential part of systems for telecommunications, radar and sonar, image formation systems such as medical imaging, and other large computational problems, such as in electromagnetics or fluid dynamics, geophysical exploration, and so on. Fast computational algorithms are necessary in large problems of signal processing, and the study of such algorithms is the subject of this book. Over those several decades, however, the nature of the need for fast algorithms has shifted both to much larger systems on the one hand and to embedded power-limited applications on the other.

Because many processors and many problems are much larger now than they were when the original version of this book was written, and the relative cost of addition and multiplication now may appear to be less dramatic, some of the topics of twenty years ago may be seen by some to be of less importance today. I take exactly the opposite point of view for several reasons. Very large three-dimensional or four-dimensional problems now under consideration require massive amounts of computation and this computation can be reduced by orders of magnitude in many cases by the choice of algorithm. Indeed, these very large problems can be especially suitable for the benefits of fast algorithms.

Good algorithms are elegant algebraic identities. To construct these algorithms, we must be familiar with the powerful structures of number theory and of modern algebra. The structures of the set of integers, of polynomial rings, and of Galois fields will play an important role in the design of signal-processing algorithms. This chapter will introduce those mathematical topics of algebra that will be important for later developments but that are not always known to students of signal processing. We will first study the mathematical structures of groups, rings, and fields. We shall see that a discrete Fourier transform can be defined in many fields, though it is most familiar in the complex field. Next, we will discuss the familiar topics of matrix algebra and vector spaces. We shall see that these can be defined satisfactorily in any field. Finally, we will study the integer ring and polynomial rings, with particular attention to the euclidean algorithm and the Chinese remainder theorem in each ring.

Groups

A group is a mathematical abstraction of an algebraic structure that appears frequently in many concrete forms. The abstract idea is introduced because it is easier to study all mathematical systems with a common structure at once, rather than to study them one by one.

Definition 2.1.1A group G is a set together with an operation (denoted by ✻) satisfying four properties.

A finite-state machine that puts out n elements from the field F at each time instant will generate a sequence of elements from the field F. The set of all possible output sequences from the finite-state machine can be represented on a kind of graph known as a trellis or, if the number of states is very large, on a kind of graph known as a tree. There are many applications in which such an output sequence from a finite-state machine is observed with errors or in noise, and one must estimate either the output sequence itself or the history of the finite-state machine that generated that sequence. This estimation task is a problem in searching a trellis or a tree for the particular path that best fits a given data sequence. Fast path-finding algorithms are available for such problems. This part of the subject of signal processing is quite different from other parts of signal processing. The trellis searching algorithms, which we introduce in this chapter, are quite different in structure from the other algorithms that we have studied.

Among the applications of trellis-searching and tree-searching algorithms are: the decoding of convolutional codes, demodulation of communication signals in the presence of intersymbol interference, demodulation of partial response waveforms or differential phase-shift-keyed waveforms, text character recognition, and voice recognition.

Trellis and tree searching

A finite-state machine consists of a set of states, a set of transitions between the states, and a set of output symbols from a field F assigned to each transition.

Signal-processing computations may arise naturally in a finite field, so it is appropriate to construct fast algorithms in the finite field GF(q). Computations in a finite field might also arise as a surrogate for a computation that originally arises in the real field or the complex field. In this situation, a computational task in one field is embedded into a different field, where that computational task is executed and the answer is passed back to the original field. There are several reasons why one might do this. It may be that the computation is easier to perform in the new field, so one saves work or can use a simpler implementation. It may be that devices that do arithmetic in one field may be readily available and can be used to do computations for a second field, if those computations are suitably reformulated. In other situations, one may want to devise a standard computational module that performs bulk computations and to use that module for a diversity of signal-processing tasks. In seeking standardization, one may want to fit one kind of computational task into a different kind of structure.

Another reason for using a surrogate field is to improve computational precision. Computations in a finite field are exact; there is no roundoff error. If a problem involving real or complex numbers can be embedded into a finite field to perform a calculation, it may be possible to reduce the computational noise in the answer.

A standard method of solving a system of n linear equations in n unknowns is to write the system of equations as a matrix equation A f = g, and to solve it either by computing the matrix inverse and writing f = A−1g or, alternatively, by using the method known as gaussian elimination. The standard methods of computing a matrix inverse have complexity proportional to n3. Sometimes, the matrix has a special structure that can be exploited to obtain a faster algorithm.

A Toeplitz system of equations is a system of linear equations described by a Toeplitz matrix A. The problem of solving a Toeplitz system of equations arises in a great many applications, including spectral estimation, linear prediction, autoregressive filter design, and error-control codes. Because the Toeplitz system is highly structured, methods of solution are available that are far superior to the general methods of solving systems of linear equations. These methods are the subject of this chapter, and are valid in any algebraic field.

The algorithms of this chapter are somewhat distant from those we have studied for convolution and for Fourier transforms. Convolutions and transforms are essentially problems of matrix multiplication, whereas this chapter deals with the solution of a system of linear equations. The solution of a system of linear equations is closer to the task of matrix inversion. It should be no surprise that we do not build on earlier algorithms directly, though techniques such as doubling may prove useful.

Number theory has already been seen in earlier chapters of this book. It was used in the design of fast Fourier transform algorithms. We did make use of some ideas that only now will be proved. This chapter, which is a mathematical interlude, will develop the basic facts of number theory – some that were used earlier in the book and some that we may need later.

We also return to the study of fields to develop the topic of an extension field more fully. The structure of algebraic fields will be important to the construction of number theory transforms in Chapter 10 and also to the construction of some multidimensional convolution algorithms in Chapter 11 and for some multidimensional Fourier transform algorithms in Chapter 12.

Elementary number theory

Within the integer quotient ring Zq, some of the elements may be coprime to q, and, unless q is a prime, others will divide q. It is important to us to know how many elements there are of each type.

Definition 9.1.1 (Euler)The totient function, denoted ϕ(q), where q is an integer larger than one, is the number of nonzero elements in Zq that are coprime to q. For q equal to one, ϕ(q) = 1.

When q is a prime p, then all the nonzero elements of Zq are coprime to p, and so ϕ(p) = p – 1 whenever p is a prime.