Understanding Digital Signal Processing with MATLAB® and Solutions

Alexander D. Poularikas

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351623278
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The book discusses receiving signals that most electrical engineers detect and study. The vast majority of signals could never be detected due to random additive signals, known as noise, that distorts them or completely overshadows them. Such examples include an audio signal of the pilot communicating with the ground over the engine noise or a bioengineer listening for a fetus' heartbeat over the mother's. The text presents the methods for extracting the desired signals from the noise. Each new development includes examples and exercises that use MATLAB to provide the answer in graphic forms for the reader's comprehension and understanding.

1 Continuous and Discrete Signals

1.1 CONTINUOUS DETERMINISTIC SIGNALS

PERIODIC SIGNALS

The most fundamental periodic signal is the sinusoidal one. Any periodic signal is defined by

(1.1) $f(t\pm n{T}_p)=f(t)n=0,\pm 1,\pm 2,\dots T_p=\text{period}$

This means that the values of the function at t, t + T_p, t + 2T_p,…, t + nT_p,… are identical.

Two sinusoidal functions with different amplitudes are

(1.2) $f(t)=a\sin (\omega t+\theta )\text{or}f(t)=b\cos (\omega t+\theta ),T_p=\frac{2\pi }{\omega },\omega =2\pi f,f=\frac{1}{T_p}$

where a and b are the amplitudes, θ the phase, T_p the period in seconds, ω frequency in radians per second (rad/s), and f frequency in cycles per second, Hz.

If we use the Maclaurin series expansion exp(x) = 1 + (x/1!) + (x²/2!) + (x³/3!) + ⋯, where (!) indicates factorial, e.g., 4! = 1 × 2 × 3 × 4, substituting jω for x in the expansion, we obtain

(1.3) $\begin{array}{ccc}\hfill e^{j\omega t}& \hfill =\hfill & 1+j\omega t+\frac{{(j\omega t)}^2}{2!}+\cdots +\frac{{(j\omega t)}^n}{n!}+\cdots \hfill \\ & \hfill =\hfill & \underset{\cos \omega t}{\underbrace{[1-\frac{{(j\omega t)}^2}{2!}+\frac{{(j\omega t)}^4}{4!}-\cdots ]}}+j\underset{\sin \omega t}{\underbrace{[\omega t-\frac{{(j\omega t)}^3}{3!}+\frac{{(j\omega t)}^5}{5!}-\cdots ]}}=\cos \omega t+j\sin \mathrm{\omega t}\hfill \end{array}$

From the above equation we obtain the Euler’s equation:

(1.4) $\cos \omega t=\frac{e^{j\omega t}+e^{-j\omega t}}{2},\sin \omega t=\frac{e^{j\omega t}-e^{-j\omega t}}{2j},\mathrm{Re}\left\{e^{j\omega t}\right\}=\cos \omega t,\mathrm{Im}\left\{e^{j\omega t}\right\}=\sin \omega t$

Figure 1.1 shows four different deterministic types of periodic signals. Figure 1.1d includes the following four musical signals: (1) the dotted one (…) is the A note f(t) = sin(2π440t), (2) the dash-dot line ( _ . _ . ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Abbreviations
• Author
• Chapter 1 Continuous and Discrete Signals
• Chapter 2 Fourier Analysis of Continuous and Discrete Signals
• Chapter 3 The z-Transform, Difference Equations, and Discrete Systems
• Chapter 4 Finite Impulse Response (FIR) Digital Filter Design
• Chapter 5 Random Variables, Sequences, and Probability Functions
• Chapter 6 Linear Systems with Random Inputs, Filtering, and Power Spectral Density
• Chapter 7 Least Squares-Optimum Filtering
• Chapter 8 Nonparametric (Classical) Spectra Estimation
• Chapter 9 Parametric and Other Methods for Spectral Estimation
• Chapter 10 Newton’s and Steepest Descent Methods
• Chapter 11 The Least Mean Square (LMS) Algorithm
• Chapter 12 Variants of Least Mean Square Algorithm
• Chapter 13 Nonlinear Filtering
• Appendix 1: Suggestions and Explanations for MATLAB Use
• Appendix 2: Matrix Analysis
• Appendix 3: Mathematical Formulas
• Appendix 4: MATLAB Functions
• Bibliography
• Index