1.6. Notes to the Reader
1.6.1. Types of Signals Considered
In describing the algorithms we will assume that the physical signal is a real, lowpass signal that has been sampled at an appropriate rate (at least twice the highest frequency, i.e., at least the Nyquist rate). For applications such as radar and sonar that process bandpass signals, it is customary to use complex demodulation, followed by sampling of the in phase and quadrature signals. This leads to a complex signal representation, which is slightly more complicated. The required extensions are described in Chapter 12. Also, because the signals are discrete-time in nature, having been sampled by some device and stored in a digital computer, we will always assume the received data has the form x[n], which is a sequence of real numbers indexed by the integer n. Typically, we will use the index set n = 0, 1,...,N –1 if the data record consists of N successive time samples. Note that we have referred to the samples as being indexed by time. However, the n index could equally well represent spatial samples such as would be obtained from N sensors equally spaced along a line with the spatial samples having been obtained by sampling all the sensor outputs at a given and fixed time.
1.6.2. Book Features and Notation
The MATLAB version used throughout the textbook is 7.8 (R2009A). Toolboxes are not required to run the MATLAB code, and where subprograms are called for, they are provided. A brief introduction to MATLAB is contained in Appendix B. A description of the code for all programs contained on the CD is given in Appendix C. In addition, a readme.txt file contained on the CD describes its contents. The “typewriter” font, such as used in run_simulation.m, indicates MATLAB program names and code.
Throughout the book there are exercises to provide the reader some practice in simple analytical manipulations and MATLAB algorithmic implementations. The solutions to the analytical exercises are contained in the corresponding chapter appendix. The MATLAB exercise solutions are only summarized, with more complete solutions found on the CD. Note the solutions were obtained using MATLAB version R2009A, and so future versions of MATLAB may produce slightly different results.
The reader is strongly encouraged to try the exercises as they form an important pathway to understanding the material.
At the end of each chapter there is a section entitled “Lessons Learned”. These are important results, many of which have become “rules of thumb”, and thus, should be committed to memory. For some applications they may form the basis on which an algorithm is either explored for its efficacy or otherwise rejected as not being suitable for the problem at hand.
The mathematical notation for all common symbols is summarized in Appendix A. The distinction between a continuous-time waveform and a discrete-time waveform or sequence is made through the symbolism x(t) for continuous-time and x[n] for discrete-time. Plots of discrete-time data such as x[n], however, may appear continuous in time, the points having been connected by straight lines for easier viewing. An example of this is given in Figure 1.1. All vectors and matrices are boldface, with all vectors being column vectors. When a random variable needs to be contrasted with its value, we will use a capital letter, say X, to denote the random variable, and a lower case letter, say x, to denote its value. All other symbolism is defined within the context of the discussion. Also, the reader will frequently be warned of potential “pitfalls”. Common misconceptions leading to design errors will be noted and described. The pitfall or caution symbol shown below should be heeded!!