From the Ground Up!

0.1 Introduction

In our modern world, signals of all kinds emanate from different types of devices—radios and TVs, cell phones, global positioning systems (GPS), radars, and sonars. These systems allow us to communicate messages, to control processes, and to sense or measure signals. In the last 65 years, with the advent of the transistor, of the digital computer and of the theoretical fundamentals of digital signal processing the trend has been toward digital representation and processing of data, which in many applications is in analog form. Such a trend highlights the importance of learning how to represent signals in analog as well as in digital forms and how to model and design systems capable of dealing with different types of signals.

The year 1948 is considered the year when technologies and theories responsible for the spectacular advances in communications, control, and biomedical engineering since then were born. Indeed, in June of that year, the Bell Telephone Laboratories announced the invention of the transistor. Later that month, a prototype computer built at Manchester University in the United Kingdom became the first operational stored-program computer. Also in that year fundamental theoretical results were published: Claude Shannon’s mathematical theory of communications, Richard W. Hamming’s theory on error-correcting codes, and Norbert Wiener’s Cybernetics comparing biological systems to communication and control systems [53].

Digital signal processing advances have gone hand-in-hand with progress in electronics and computers. In 1965, Gordon Moore, one of the founders of Intel, envisioned that the number of transistors on a chip would double about every two years [35]—Intel, the largest chip manufacturer in the world, has kept that pace for over four decades. It is these advances in digital electronics and in computer engineering that have permitted the proliferation of digital technologies. Today, digital hardware and software process signals from cell phones, high definition television (HDTV) receivers, digital radio, radars, and sonars, just to name a few. The use of digital signal processors (DSP) and more recently of field-programmable gate arrays (FPGAs) have been replacing the use of application-specific integrated circuits (ASIC) in industrial, medical, and military applications.1

It is clear that digital technologies are here to stay. The abundance of algorithms for processing digital signals, and the pervasive presence of DSPs and FPGAs in thousands of applications make digital signal processing theory a necessary tool not only for engineers but for anybody who would be dealing with digital data—soon, that will be everybody! This book serves as an introduction to the theory of signals and systems—a necessary first step in the road toward understanding digital signal processing.

0.2 Examples of Signal Processing Applications

With the availability of digital technologies for processing signals, it is tempting to believe there is no need to understand their connection with analog technologies. That it is precisely the opposite is illustrated by considering the following three interesting applications: the compact-disk (CD) player, software-defined and cognitive radio, and computer-controlled systems.

0.2.1 Compact-Disk (CD) Player

Compact disks [10] were first produced in Germany in 1982. Recorded voltage variations over time due to an acoustic sound is called an analog signal given its similarity with the differences in air pressure generated by the sound waves over time. Audio CDs and CD-players illustrate best the conversion of a binary signal—unintelligible—into an intelligible analog signal. Moreover, the player is a very interesting control system.

To store an analog audio signal, e.g., voice or music, on a CD the signal must be first sampled and converted into a sequence of binary digits—a digital signal—by an analog–to–digital (A/D) converter and then specially encoded to compress the information and to avoid errors when playing the CD. In the manufacturing of a CD, pits and bumps—corresponding to the ones and zeros from the quantization and encoding processes—are impressed on the surface of the disk. Such pits and bumps will be detected by the CD-player and converted back into an analog signal that approximates the original signal when the CD is played. The transformation into an analog signal uses a digital-to-analog (D/A) converter.

As we will see in Chapter 8, an audio signal is sampled at a rate of about 44,000 samples/second (corresponding to a maximum frequency around 22 kHz for a typical audio signal) and each of these samples is represented by a certain number of bits (typically 8 bits/sample). The need for stereo sound requires that two channels be recorded. Overall, the number of bits representing the signal is very large and needs to be compressed and especially encoded. The resulting data, in the form of pits and bumps impressed on the CD surface, are put into a spiral track that goes from the inside to the outside of the disk.

Besides the binary-to-analog conversion, the CD-player exemplifies a very interesting control system (See Figure 0.1). Indeed, the player must: (i) rotate the disk at different speeds depending on the location of the track within the CD being read; (ii) focus a laser and a lens system to read the pits and bumps on the disk, and (iii) move the laser to follow the track being read. To understand the exactness required, consider that the width of the track is typically less than a micrometer (10−6 meters or 3.937 × 10−5 inches), and the height of the bumps is about a nanometer (10−9 meters or 3.937 × 10−8 inches).

0.2.2 Software-Defined Radio and Cognitive Radio

Software-defined and cognitive radio are important emerging technologies in wireless communications [44]. In software-defined radio (SDR), some of the radio functions typically implemented in hardware are converted into software [65]. By providing smart processing to SDRs, cognitive radio (CR) will provide the flexibility needed to more efficiently use the radio frequency spectrum and to make available new services to users. In the United States the Federal Communication Commission (FCC), and likewise in other parts of the world the corresponding agencies, allocates the bands for different users of the radio spectrum (commercial radio and TV, amateur radio, police, etc.). Although most bands have been allocated, implying a scarcity of spectrum for new users, it has been found that locally at certain times of the day the allocated spectrum is not being fully utilized. Cognitive radio takes advantage of this.

Conventional radio systems are composed mostly of hardware, and as such cannot be easily reconfigured. The basic premise in SDR as a wireless communication system is its ability to reconfigure by changing the software used to implement functions typically done by hardware in a conventional radio. In an SDR transmitter, software is used to implement different types of modulation procedures, while A/D and D/A coverters are used to change from one type of signal into another. Antennas, audio amplifiers, and conventional radio hardware are used to process analog signals. Typically, an SDR receiver uses an A/D converter to change the analog signals from the antenna into digital signals that are processed using software on a general purpose processor. See Figure 0.2.