When we hear the word signal we may first think of a phenomenon that occurs continuously over time that carries some information. Most phenomena observed in nature are continuous in time, as for instance, our speech or heart beating, the temperature of the city where we live, the car speed during a given trip, the altitude of the airplane we are traveling in – these are typical continuous-time signals. Engineers are always devising ways to design systems, which are in principle continuous time, for measuring and interfering with these and other phenomena.

One should note that, although continuous-time signals pervade our daily life, there are also several signals which are originally discrete time, as, for example, the stock-market weekly financial indicators, the maximum and minimum daily temperatures in our cities, and the average lap speed of a racing car.

If an electrical or computer engineer has the task of designing systems to interact with natural phenomena, his/her first impulse is to convert some quantities from nature into electric signals through a transducer. Electric signals, which are represented by voltages or currents, have a continuous-time representation. Since digital technology constitutes an extremely powerful tool for information processing, it is natural to think of processing the electric signals generated using it. However, continuous-time signals cannot be processed using computer technology (digital machines), which are especially suited to deal with sequential computation involving numbers.

Introduction

This chapter deals with the design methods in which a desired frequency response is approximated by a transfer function consisting of a ratio of polynomials. In general, this type of transfer function yields an impulse response of infinite duration. Therefore, the systems approximated in this chapter are commonly referred to as infinite-duration impulse-response (IIR) filters.

In general, IIR filters are able to approximate a prescribed frequency response with fewer multiplications than FIR filters. For that matter, IIR filters can be more suitable for some practical applications, especially those ones involving real-time signal processing.

In Section 6.2, we study the classical methods of analog filter approximation, namely the Butterworth, Chebyshev, and elliptic approximations. These methods are the most widely used for approximations meeting prescribed magnitude specifications. They originated in the continuous-time domain and their use in the discrete-time domain requires an appropriate transformation.

We then address, in Section 6.3, two approaches that transform a continuous-time transfer function into a discrete-time transfer function, the impulse-invariance and bilinear transformation methods.

Section 6.4 deals with frequency transformation methods in the discrete-time domain. These methods allow the mapping of a given filter type to another, for example the transformation of a given lowpass into a desired bandpass filter.

In applications where magnitude and phase specifications are imposed, we can approximate the desired magnitude specifications by one of the classical transfer functions and design a phase equalizer to meet the phase specifications.

Introduction

In many applications of digital signal processing, it is necessary for different sampling rates to coexist within a given system. One common example is when two subsystems working at different sampling rates have to communicate and the sampling rates must be made compatible. Another case is when a wideband digital signal is decomposed into several non-overlapping narrowband channels in order to be transmitted. In such a case, each narrowband channel may have its sampling rate decreased until its Nyquist limit is reached, thereby saving transmission bandwidth.

Here, we describe such systems which are generally referred to as multirate systems. Multirate systems are used in several applications, ranging from digital filter design to signal coding and compression, and have been increasingly present in modern digital systems.

First, we study the basic operations of decimation and interpolation, and show how arbitrary rational sampling-rate changes can be implemented with them. The design of decimation and interpolation filters is also addressed. Then, we deal with filter design techniques which use decimation and interpolation in order to achieve a prescribed set of filter specifications. Finally, MATLAB functions which aid in the design and implementation of multirate systems are briefly described.

Basic principles

Intuitively, any sampling-rate change can be effected by recovering the band-limited analog signal xa(t) from its samples x(m), and then resampling it with a different sampling rate, thus generating a different discrete version of the signal, x′(n).

Introduction

In the previous chapter, we dealt with multirate systems in general, that is, systems in which more than one sampling rate coexist. Operations of decimation, interpolation, and sampling-rate changes were studied, as well as some filter design techniques using multirate concepts.

In a number of applications, it is necessary to split a digital signal into several frequency bands. After such decomposition, the signal is represented by more samples than in the original stage. However, we can attempt to decimate each band, ending up with a digital signal decomposed into several frequency bands without increasing the overall number of samples. The question is whether it is possible to exactly recover the original signal from the decimated bands. Systems which achieve this are generally called filter banks.

In this chapter, we first deal with filter banks, showing several ways in which a signal can be decomposed into critically decimated frequency bands, and recovered from them with minimum error. Later, wavelet transforms are considered. They are a relatively recent development of functional analysis that is generating great interest in the signal processing community, because of their ability to represent and analyze signals with varying time and frequency resolutions. Their digital implementation can be regarded as a special case of critically decimated filter banks. Finally, we provide a brief description of functions from the MATLAB Wavelet toolbox which are useful for wavelets and filter banks implementation.

This book originated from a training course for engineers at the research and development center of TELEBRAS, the former Brazilian telecommunications holding. That course was taught by the first author back in 1987, and its main goal was to present efficient digital filter design methods suitable for solving some of their engineering problems. Later on, this original text was used by the first author as the basic reference for the digital filters and digital signal processing courses of the Electrical Engineering Program at COPPE/Federal University of Rio de Janeiro.

For many years, former students asked why the original text was not transformed into a book, as it presented a very distinct view that they considered worth publishing. Among the numerous reasons not to attempt such task, we could mention that there were already a good number of well-written texts on the subject; also, after many years of teaching and researching on this topic, it seemed more interesting to follow other paths than the painful one of writing a book; finally, the original text was written in Portuguese and a mere translation of it into English would be a very tedious task.

In recent years, the second and third authors, who had attended the signal processing courses using the original material, were continuously giving new ideas on how to proceed. That was when we decided to go through the task of completing and updating the original text, turning it into a modern textbook.

Introduction

In practice, a digital signal processing system is implemented by software on a digital computer, either using a general-purpose digital signal processor, or using dedicated hardware for the given application. In either case, quantization errors are inherent due to the finite-precision arithmetic. These errors are of the following types:

• Errors due to the quantization of the input signals into a set of discrete levels, such as the ones introduced by the analog-to-digital converter.

• Errors in the frequency response of filters, or in transform coefficients, due to the finite-wordlength representation of multiplier constants.

• Errors made when internal data, like outputs of multipliers, are quantized before or after subsequent additions.

All these error forms depend on the type of arithmetic utilized in the implementation. If a digital signal processing routine is implemented on a general-purpose computer, since floating-point arithmetic is in general available, this type of arithmetic becomes the most natural choice. On the other hand, if the building block is implemented on special-purpose hardware, or a fixed-point digital signal processor, fixed-point arithmetic may be the best choice, because it is less costly in terms of hardware and simpler to design. A fixed-point implementation usually implies a lot of savings in terms of chip area as well.

For a given application, the quantization effects are key factors to be considered when assessing the performance of a digital signal processing algorithm.

In the past ten years, the dramatic growth of the Internet has had a profound and lasting impact on the way in which organizations and individuals communicate and conduct their public and private affairs. Tax forms are available online, students may submit their exams electronically to a (possibly remote) campus network, and companies may use the Internet as a public channel for linking up internal computing facilities or processes. For example, an employee may dial into a company's intranet from a hotel room or her home via a public Internet service provider. Since the Internet protocol does not provide sufficient mechanisms for ensuring the privacy, authenticity, integrity, and (if desired) anonymity of data that are processed through a usually dynamically determined chain of computers, there is a need for tools that guarantee the confidentiality and authenticity of data and of their communication sources and targets. Cautious consumers of mobile or foreign code prefer to verify that downloaded programs (e.g., Java applets) abide by a formal set of safety rules, possibly defined by the individual consumer. These needs appear to be even more pressing in the recent evolution of electronic commerce, where the act of selecting and purchasing a product occurs online. Although online companies are still waiting to reap their first real profits, it is evident that companies in general need to offer this mode of business in order to survive in a new economy that is global and at the same time strengthens regional identity.

ELECTRONIC COMMERCE: THE MANTRA OF Y2K+

We are presently witnessing mergers and takeovers of unprecedented speed and extent between companies once thought to have national identities, or at least clearly identifiable lines of products or services. On the day this paragraph was written, the British Vodaphone Air Touch announced an Internet alliance with the French conglomerate Vivendi. The deal was conditional on Vodaphone's hostile takeover of Germany's Mannesmann and, in the end, did establish a branded multi-access portal in Europe. About a week later, the takeover of Mannesmann was official – the biggest ever, and friendly. MCI's attempted takeover of Sprint is another example of a strategically advantageous combination of different information technologies. January 2000 saw CNN, NTV, and the Deutsche Handelsblatt (a direct competitor to the Financial Times) launch a multimedia product for stock market news that is accessible via television, printed newspapers, and the World Wide Web. And so it goes. Although many differing views are held regarding the causes and consequences of these phenomena, we would probably all agree that they reflect a certain shift of emphasis from production-based economics to one grounded in the processing, marketing, and access of information. Whether the products themselves are merely “information” or systems for managing and processing vast amounts of data, information systems are seen as a crucial strategic means for organizing, improving, and maintaining more traditional production cycles.