Analog and Digital Signals and Systems

The primary goal of this book is to introduce the reader on the basic principles of signals and to provide tools thereby to deal with the analysis of analog and digital signals, either obtained naturally or by sampling analog signals, study the concepts of various transforming techniques, filtering analog and digital signals, and finally introduce the concepts of communicating analog signals using simple modulation techniques. The basic material in this book can be found in several books. See references at the end of the book.

In this chapter we will consider two signal analysis concepts, namely convolution and correlation. Signals under consideration are assumed to be real unless otherwise mentioned. Convolution operation is basic to linear systems analysis and in determining the probability density function of a sum of two independent random variables. Impulse functions were defined in terms of an integral (see (1.4.4a)) using a test function \(\phi (t)\).

In this chapter we will consider approximating a function by a linear combination of basis functions, which are simple functions that can be generated in a laboratory. Joseph Fourier (1768–1830) developed the mathematical theory of heat conduction using a set of trigonometric (sine and cosine) series of the form we now call Fourier series (Fourier, J.B.J., 1955 (A. Freeman, translation)). He established that an arbitrary mathematical function can be represented by its Fourier series. This idea was new and startling and met with vigorous opposition from some of the leading mathematicians at the time, see Hawking (2005). Fourier series and the Fourier transform are basics to mathematics and science, especially to the theory of communications. For example, a phoneme in a speech signal is smooth and wavy. A linear combination of a few sinusoidal functions would approximate a segment of speech within some error tolerance. Suppose we like to build a structure that allows us to climb from the first floor to the second floor of a building. We can have a staircase approximating a ramp function using a linear combination of pulse functions. The amplitudes and the width of the pulses can be determined based on the error between the ramp and the staircase. Apart from the staircase problem, this type of analysis is important in electrical engineering, for example, when converting an analog signal to a discrete signal.

In Chapter 3 we have discussed the frequency representation of a periodic signal. Fourier series expansions of periodic signals give us a basic understanding how to deal with signals in general. Since most signals we deal with are aperiodic energy signals, we will study these in terms of their Fourier transforms in this chapter. Fourier transforms can be derived from the Fourier series by considering the period of the periodic function going to infinity. Fourier transform theory is basic in the study of signal analysis, communication theory, and, in general, the design of systems. Fourier transforms are more general than Fourier series in some sense. Even periodic signals can be described using Fourier transforms. Most of the material in this chapter is standard (see Carlson, 1992, Lathi, 1983, Papoulis, 1962, Morrison, 1994, Ziemer and Tranter, 2002, Haykin and Van Veen, 1999, Simpson and Houts, 1971, Baher, 1990, Poularikis and Seely, 1991, Hsu, 1967, 1993, Roberts, 2004, and others).

There are various ways of introducing a student to different forms of transforms. We chose the approximation of a function by using Fourier series first and then came up with the Fourier transforms. Fourier cosine and sine series were considered using the Fourier series. The next step is to study some of the other transforms that are related to the Fourier transforms. These include cosine, sine, Laplace, discrete, and fast Fourier transforms. Discrete and fast Fourier transforms will be included in Chapters 8 and 9. In many of the undergraduate engineering curricula, Laplace transforms are introduced first and then the Fourier transforms. Fourier transforms are considered more theoretical. The development of the sine and cosine transforms parallel to the Fourier cosine and sine series discussed in Chapter 3. For a good review on many of these topics, see the handbook by Poularikas (2000). We can consider the Laplace transform as an independent transform or a modified version of the Fourier transform. One problem with Fourier transforms is that the signal under consideration must be absolutely integrable. (the periodic functions are exceptions). Therefore, the transformation to the Fourier domain is limited to energy signals or to finite power signals that are convergent in the limit. Fourier and Laplace transforms have been widely used in engineering; Fourier transforms in the signal and communications area and the Laplace transform in the circuits, systems, and control area. Neither one is a generalization of the other. Both transforms have their own merits.

In this chapter we will consider systems in general, and in particular linear systems. Most systems are inherently nonlinear and time varying. A human being is a good example. He can run fast for a while and then speed comes down. If you plot speed versus time, the plot is not going to be a straight line, i.e., the function speed versus time is not linear. Humans are nonlinear and also time-varying systems. For example, if you want to ask your dad for a new car, you do not ask him when he is not happy. Moods change with time. These considerations are important in, for example, speaker identification. Human beings are not only nonlinear but also time-varying complicated systems. Nonlinear time-varying systems are very hard to deal with. Even though many of the systems may have nonlinear behavior characteristics, they can be approximated to be linear systems and they allow for transform analysis.

In the first part of this chapter we will consider a graphical representation of the transfer function in terms of its frequency response \(H(j\omega ) = \left| {H(j\omega )} \right|e^{\angle H(j\omega )}\). Bode diagrams or plots consist of two separate plots, the amplitude \(\left| {H(j\omega )} \right|\) and the phase angle \(\angle H(j\omega )\), with respect to the frequency \(\omega\) on a logarithmic scale. These plots are named after Bode, in recognition of his pioneering work Bode (1945). Bode’s basic work was based upon approximate representation of amplitude and phase response plots of a communication system. Wide range of frequencies of interest in a communication system dictated the use of the logarithmic frequency scale. Bode plots use the asymptotic behavior of the amplitude and the phase responses of simple functions by straight-line segments and are then approximated by smooth plots with ease and accuracy. Bode plots can be created by using computer software, such as MATLAB. The topic is mature and can be found in most circuits, systems, and control books. For example, see Melsa and Schultz (1969), Lathi (1998), Close (1966), Nilsson and Riedel (1966), and many others.

So far in this book we have concentrated on the continuous-time signals. These included continuous periodic and aperiodic time signals and the corresponding Fourier series and transform representations. We divided the continuous signals based on their energy and power properties. In the case of periodic signals, the Fourier series coefficients are discrete. In this chapter, we will start with continuous signals and their sampled versions. The advances in computers and the ease in implementing discrete algorithms using personal computers (PCs) made this as an essential area every electrical engineer should be interested in. Most discrete-time signals come from sampling continuous signals, such as speech, seismic, sonar, images, biological, and other signals. These days, telephone along with a computer forms an integral part of most communication systems. The advances in telemetry allow us to monitor remotely located patients. The ease of processing discrete-time signals made discrete-time implementations of analog operations, such as filtering, made it very popular. The analog signals are first converted to digital signals by making use of a device referred to as an analog-to-digital (A/D) converter.The reverse process of reconstructing an analog signal from a digital signal is achieved by a device referred to as a digital-to-analog (D/A) converter.Obviously if the source is an analog device and the end user requires an analog signal, the use of a digital processor requires the A/D and the D/A converters. Although user signals are usually analog, there are many situations wherein the discrete-time signals are source signals. For example, the stock prices, the temperatures at a particular time in a city, grades of students and many others are digital in nature. The transform study of the discrete-time signals is basic to our study.

In this chapter we will consider some of the fundamental concepts associated with analog modulation. Communication of analog signals from one location to another is accomplished by using either a wire channel or a radio channel. The source signals, such as voice, pictures, and in general baseband signals are not always suitable for direct transmission over a given channel. These signals are first converted by an input transducer into an electrical waveform referred to as the baseband signal or the message signal. The spectral contents of the baseband signals are located in the low-frequency region. The wire channels have a low-pass transfer function and can be used for transmitting signals that have a bandwidth less than the channel bandwidth. The radio channels have a band-pass characteristic. Low-pass signals can be transmitted through radio channels by using a modulator.