For many years, rumors have been circulating in the realm of digital signal processing about quantization noise:

1. (a) the noise is additive and white and uncorrelated with the signal being quantized, and

2. (b) the noise is uniformly distributed between plus and minus half a quanta, giving it zero mean and a mean square of one–twelfth the square of a quanta.

Many successful systems incorporating uniform quantization have been built and placed into service worldwide whose designs are based on these rumors, thereby reinforcing their veracity. Yet simple reasoning leads one to conclude that:

1. (a) quantization noise is deterministically related to the signal being quantized and is certainly not independent of it,

2. (b) the probability density of the noise certainly depends on the probability density of the signal being quantized, and

3. (c) if the signal being quantized is correlated over time, the noise will certainly have some correlation over time.

In spite of the “simple reasoning,” the rumors are true under most circumstances, or at least true to a very good approximation. When the rumors are true, wonderful things happen:

1. (a) digital signal processing systems are easy to design, and

2. (b) systems with quantization that are truly nonlinear behave like linear systems.

In order for the rumors to be true, it is necessary that the signal being quantized obeys a quantizing condition. There actually are several quantizing conditions, all pertaining to the probability density function (PDF) and the characteristic function (CF) of the signal being quantized.

We have seen when discussing the sampling theorem in Section 2.2 that the conditions of the theorem are exactly met only if the signal being sampled is perfectly bandlimited. This is rarely the case, since perfect bandlimitedness implies that the signal cannot be time–limited. Such a signal can be easily defined mathematically, but measured signals are always time–limited, so the condition of the sampling theorem can be met only approximately. While the sinc function wave is theoretically bandlimited, its time truncated versions are not, so the sampling theorem can be applied only approximately. However, sampling theory proved to be very powerful despite its approximate applicability.

The situation is similar with the quantizing theorems. Bandlimitedness of the CF would imply that the PDF is not amplitude–limited. Since measured signals are always amplitude–limited, the quantization theorems can be applied only approximately. Similarly to the sampling theorem, this does not prevent the quantizing theorems from being very powerful in many applications.

Most distributions that are applied in practice, like the Gaussian, exponential or chi–squared are not bandlimited. This fact does not prevent the application of the quantizing theorems if the quantum step size is significantly smaller than the standard deviation. Nevertheless, it is of interest to investigate the question whether there are distributions whose characteristic functions are perfectly bandlimited, similarly to the sinc function. In the following paragraphs we will discuss some examples of distributions whose CFs are perfectly bandlimited.

In the previous chapter, certain relations between the quantization noise and the quantizer input and output were established. These relations are in the form of moments, particularly crosscorrelations and covariances. These moments are extremely useful for the analysis of quantization and for the analysis of control and signal processing systems containing quantizers. The similarities between quantization and the addition of independent uniformly distributed noise when certain quantizing theorems are satisfied are very useful for analysis of quantization.

The properties of these moments are not totally descriptive of the behavior of the quantizer however, because the quantization noise is deterministically related to the quantizer input, and the moments do not show this. To gain a complete and deeper understanding, we dedicate this chapter to the derivation of joint PDFs and CFs of quantization noise and the quantizer input and output.

JOINT PDF AND CF OF THE QUANTIZER INPUT AND OUTPUT

In this section, we will derive the joint PDF and CF of x and x', i.e. fx,x' (x, x'), and Φx,x' (ux, ux'). The statistical relationship between x and x' is like a statistical version of an input–output “transfer function” for the quantizer.

The derivation will proceed in the following manner. Refer to Fig. 7.1. This figure compares quantization (the addition of actual quantization noise ν to x) with the PQN model (the addition of independent noise n to x).

Scientific computation is done these days with floating–point arithmetic. Generally double precision is used. Of concern are quantization effects in the solutions of equations caused by limited precision in the representation of dependent variables (equation solutions) and limited precision in the representation of coefficients (equation parameters).

The full scope of the issues raised here is so great that it would be impossible to encompass it all in a single chapter. There are too many different kinds of equations to be dealt with. Some are linear, some nonlinear. Some have a unique solution, some have solutions of many values such as sampled time functions. Some have driving functions, some are homogeneous with given initial conditions. Some have feedback from the output to decrease the error, some are open–loop. Some have a single input and a single output, some have many inputs and many outputs, etc.

In this brief chapter, we will employ what we have learned about floating–point quantization to analyze roundoff errors in a nonlinear system. The ideas will be developed by studying a few simple cases. It is hoped that the reader will be able to use these ideas in solving new and unusual problems.

DEFINITION OF THE QUANTIZER

Quantization or roundoff occurs whenever physical quantities are represented numerically. The time displayed by a digital watch, the temperature indicated by a digital thermometer, the distances given on a map etc. are all examples of analog values represented by discrete numbers.

The values of measurements may be designated by integers corresponding to their nearest numbers of units. Roundoff errors have values between plus and minus one half unit, and can be made small by choice of the basic unit. It is apparent, however, that the smaller the size of the unit, the larger will be the numbers required to represent the same physical quantities and the greater will be the difficulty and expense in storing and processing these numbers. Often, a balance has to be struck between accuracy and economy. In order to establish such a balance, it is necessary to have a means of evaluating quantitatively the distortion resulting from rough quantization. The analytical difficulty arises from the inherent nonlinearities of the quantization process.

For purposes of analysis, it has been found convenient to define the quantizer as a nonlinear operator having the input–output staircase relation shown in Fig. 1.1(a). The quantizer output x' is a single–valued function of the input x, and the quantizer has an “average gain” of unity. The basic unit of quantization is designated by q.

The decision to implement a UWB radio into a product is not simply a technical selection. There are a substantial number of business issues surrounding the selection of which one should also be aware. Intellectual-property obligations, price expectations and market development directions are all matters that intimately affect the potential for a successful outcome. This chapter is intended to highlight some of those issues and provide a distilled assessment.

Expected changes to the technology over time

As is the case with all technologies, UWB will evolve over time. What it is today is not what it will be in the three-year life expectancy of most computer products. In making product decisions, it is never enough simply to look at the way things are now. It is also necessary to look at trends that will occur over the expected life of a product. As an example, if one were to compare a UWB radio with an 802.11n radio today, the result will be far different from a comparison that will occur in the next three years. This section discusses some of the trends now visible, which will change the functionality of UWB.

Planned development in UWB

In addition to the trends that are happening as a result of general economic and market conditions, there will be focused efforts within UWB standards organizations and SIGs to evolve UWB as well. There are two primary directions along which UWB will develop.

The high-data-rate UWB products in development and shipping today are based on what is known as the WiMedia common radio platform (Figure 3.1). This chapter will provide a somewhat abbreviated technical overview of the physical layer, which makes up the lowest portion of the radio design. The intention here is to hit the highlights. For the reader who needs to understand the nuts and bolts of the radio's design, the ECMA-368 standards is recommended.[1] The content will focus on some of the more interesting facets of the design, which may be needed by individuals desiring a system view of the radio.

At the base of the common radio platform (Figure 3.2) lays the physical layer (PHY). Originally coined as a networking term, PHY refers to the combination of software and hardware programming that defines the electrical, mechanical and functional specifications to activate, maintain, and deactivate the transmission interfaces (or links) between communicating systems. The PHY may or may not include electromechanical devices, but in essence, it is the brains of the radio. Basically, the PHY's job is to transmit bits of data over a communication medium in either digital or analogue form. It makes no difference as to what those bits represent; the PHY operates in the same way regardless of the type of data. Physical-layer specifications typically define characteristics such as voltage levels, the timing of voltage changes, data rates, maximum transmission distances and physical connectors.

Any successful new technology can be described as a combination of features that allow the technology to perform a given application better than those technologies that precede it or that enable new applications to be performed. The consumer has the final word in the success of a technology. If the product that manufacturers are trying to sell to the consumer does not convey a strong sense of benefit or sex appeal, the product becomes a wallflower on the back of store shelves.

In this chapter, the discussion will centre on the features of UWB, a comparison of these features with competing technologies and the emerging applications that demand the improved performance that UWB provides. With UWB, the principal features of interest include speed, cost, location resolution and power consumption. Each of these characteristics will be covered separately.

Speed – specifying UWB

The exciting new applications that are emerging now or will emerge over the next few years will demand, more than any other single attribute, extremely fast speed. Speed is required for one of two reasons. Either the application involves a large file transfer, such as the download of a Blueray DVD (50 GB), [1] or high-resolution video streaming (Displayport up to 11 Gbps). [2] In the case of large file transfers, speed translates into consumer wait time.

If you are interested in a deep theoretical treatise on ultra-wideband, there are several excellent texts, which are listed at the end of this chapter, that we recommend [1, 2]. Essentials of UWB will definitely not fill that need. It is far too concise and practical and it fails to take up the requisite three inches of shelf space that are required to fill that niche in the literature.

If you are an engineer, business professional, regulator or marketing person who needs enough technical information to build, sell or regulate products that include a UWB radio, but don't aspire to become a radio frequency (RF) deity in your own right, this is the text that you are looking for. Our objective in writing this book is to provide a dependable overview of the data that you need to know to understand the technology and the industry. This includes technical overviews, industry organization, intellectual property overview, standardization and regulatory discussions. We will also attempt to provide pointers to source documents for deeper investigation for those who are so inclined. We know where the good data are buried because in many cases we had a hand in putting it there. Dr Aiello founded two UWB start-ups, contributed actively to the US regulatory processes, participated in the IEEE standardization wars and performed much of the early development of UWB modulation schemes and radio designs. He has also been a board member in the WiMedia Alliance for a number of years.

As with the PHY in the previous chapter, this discussion of the MAC is intended to be somewhat cursory. The full detail can be found in the ECMA 368 standard.[1] As demonstrated in Figure 4.1, the MAC layer sits immediately above the PHY.

The media access control (MAC) layer of the radio connects to the service access point (SAP) on top of the physical layer. The PHY SAP is nothing more than the logical gateway through which data flow in a specified format from the MAC to the PHY and back again. When data need to be communicated from one device to another, they must begin at the top of one of the protocol stacks shown above (WUSB, Bluetooth, etc.) and flow down through each layer, out over the connecting media (RF or wire) and up through each of the layers of the equivalent stack in the radio to whom the communication is being sent. Each layer of the stack has a specific task to perform in making sure that the data are successfully transferred.

Where the PHY is responsible for going through the physical steps of placing bits onto the air during a transmission effort and taking them off again during a receive operation, the MAC is responsible for the first level of processing that takes place on data coming out of the PHY.

For instance, a radio channel, by its nature, is relatively unreliable.