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.

The information that one finds in the standards and specifications of a technology are frequently only part of the story. While standardization is critical to achieve industry-wide interoperability, it is also very important for individual manufacturers to be able to differentiate their products. For this reason, standardization bodies generally restrict themselves to describing those elements that are absolutely required to establish interoperability or common customer experience and usually remain silent about the rest of a design.

In the case of UWB, there are several points that are not described in the standards, but which one might wish to be aware of. For instance, there are many cases in which it will be necessary to place a UWB radio alongside one or more other radios as part of a general system. Some effort is required to get these devices to co-exist. There are also trade-offs that a designer will need to make on topics such as the level of integration that is desirable, the chip-packaging trade-offs and the antenna-selection options. Each of these issues is discussed in the sections which follow.

Co-location with other radios on the same platform

Because it is a wireless technology, UWB is subject to more scrutiny in terms of interference – including its effects on neighbouring devices as well as their effects on it. This is to reduce negative effects both to other UWB receivers and non-UWB receivers.

Douglas Adams once said, “Anything that was invented before you're born is normal and ordinary and is just part of the way the world works, anything that's invented between when you're 15 and 35 is technology, anything invented after you're 35 is against the natural order of things.”

This chapter aims to explain the ‘natural order’ of some of the protocols that enable wireless connections. Consumers rarely see the technology or read the specification for a wireless protocol. Instead, they are in contact with the application layer that determines the normal and ordinary behaviour of the product and its high-level features. If you imagine a layered radio product, the top layer is the application layer. Underneath that, a standardized protocol determines how the radio interacts with the rest of the network. Examples of protocols include CW USB (or wireless USB), WiMedia Layer2 Protocol (WLP), Bluetooth, Wireless 1394 and ZigBee. Below the protocol is a common radio platform, which, depending on its capabilities, could simultaneously support multiple protocols. A diagram of this structure is included in Figure 3.1.

To help explain the various available protocols, consider a common application – sharing photographs. In this sample application, a consumer uses a camera phone to take pictures, and then wants to print the pictures on a local printer and send them to a website. The initial step is to establish a wireless connection between the camera and the printer.

To start this discussion on standards, one should understand that UWB standards and specifications will not be generated by a single standards organization. There are a number of organizations involved in the effort and each of these is engaged for a specific purpose. The following overview gives a flavour of how the division of labour is structured between Ecma International (Ecma), the International Standards Organization (ISO) and the European Standards and Technical Institute (ETSI) in the development of UWB.

Ecma International

The standards organization leading the effort in the development of the UWB physical layer (PHY) and the media access control layer (MAC) is Ecma International. Ecma was initially focused on developing standards for the European computer markets when it was created in 1961, [1] but has since expanded its charter to cover standards in software, consumer electronics and communications on an international scale. Ecma is responsible for developing such well known standards as the DVD (digital video disc) and emerging standards such as near-field communications (NFC), which uses inductive coupling to establish a link between smart cards (credit cards) and their readers. As a side note, the NFC techniques are likely to emerge as a means of association in next-generation wireless LAN and PAN devices.

In UWB, the two principal standards that Ecma is developing are ECMA 368 [2] and ECMA 369.