Math Notes for ECE 278 G. C. Papen September 6, 2017 c 2017 by George C. Papen All rights reserved. No part of this manuscript is to be reproduced without written consent of the author. Contents 1 Background 1 1.1 Linear Systems.......................................1 1.1.1 Bandwidth and Timewidth............................7 1.1.2 Passband and Complex-Baseband Signals.................... 12 1.1.3 Signal Space.................................... 15 1.2 Random Signals....................................... 23 1.2.1 Probability Distribution Functions........................ 23 1.2.2 Random Processes................................. 35 1.3 Electromagnetics...................................... 46 1.3.1 Material Properties................................ 47 1.3.2 The Wave Equation................................ 50 1.3.3 Random Electromagnetic field Fields....................... 55 1.4 References.......................................... 56 1.5 Problems.......................................... 56 2 Examples 65 2.1 Filter Estimation...................................... 65 2.2 Constant-Modulus Objective Function.......................... 68 2.3 Adaptive Estimation.................................... 69 Bibliography 71 iii iv 1 Background The study of communication systems is rich and rewarding, bringing together a broad range of topics in engineering and physics. Our development of this subject draws on the understanding of basic material in the subjects of linear systems, random signals, electromagnetics. The emphasis in this chapter is on the concepts that are relavent to the understanding of modern digital communication systems. This background chapter also introduces and reinforces various and alternative sets of notation that are used throughout the book. Much of the understanding of the various topics in this book depends on the choice of clear and appropriate notation and terminology. 1.1 Linear Systems A communication system conveys information by embedding that information into temporal and perhaps spatial variations of a propagating signal. We begin with a discussion of the properties of signals and systems. A signal is a real-valued or complex-valued function of a continuous or discrete variable called time. A system responds to a signal s(t) at its input producing one or more signals r(t) at its output. The most amenable systems are linear systems because a linear mathematical model can support Figure 1.1: A block diagram of a linear system characterized by an impulse response function h(t). Using the properties of homogeneity and additivity, an input ax1(t) + bx2(t) produces an output ay1(t) + by2(t). powerful methods of analysis and design. We are interested in both discrete systems and continuous systems expressed in a variety of mathematical forms such as by using continuous integral equations, continuous differential equations, or discrete difference equations. A communication signal may be a real function of time or a complex function of time. The rectangular form of a complex function is a(t) = aR(t) + iaI (t) where aR(t) is the real part and aI (t) 2 iφ(t) p 2 2 is the imaginary part and i = 1. The polar form is a(t) = A(t)e where A(t) = aR(t) + aI (t) −−1 is the amplitude and φ = tan (aI =aR) is the phase. Systems can be classified by the properties that relate the input s(t) to the output r(t). Linearity A system, either real or complex, is linear if it is homogeneous and additive: 1 1. Homogeneous systems If input s(t) has output r(t), then for every scalar a, real or complex, input as(t) has output ar(t). 2. Additive systems If input x1(t) has output y1(t) and input x2(t) has output y2(t), then input x1(t) + x2(t) has output y1(t) + y2(t). The output r(t) of a linear continuous-time system can be written as a superposition integral of the input s(t) and a function h(t; τ) Z 1 r(t) = h(t; τ)s(τ)dτ (1.1.1) −∞ where h(t; τ), called the time-varying impulse response, is defined as the output of the system at time t in response to a Dirac impulse δ(t τ). − The Dirac impulse δ(t) is a not a proper functiona. It is defined by the formal integral relationship Z 1 s(t) = δ(t τ)s(τ)dτ; (1.1.2) −∞ − for any function s(t). This integral is referred to as the sifting property of a Dirac impulse. For the treatment of discrete-time signals, a Kronecker impulse δmn is useful, defined by δmn equal to one if m is equal to n, and δmn equal to zero otherwise. Shift Invariance Under appropriate conditions, a system described by a superposition integral can be reduced to a simpler form known as a shift-invariant system, or when appropriate, as a time-invariant or a space-invariant system. If input s(t) has output r(t), then for every τ, input s(t τ) has output − r(t τ). In this case, the form of the impulse response for a linear and shift-invariant system − depends only on the time difference so that h(t; τ) = h(t τ; 0) and (1.1.1) reduces to − Z 1 r(t) = h(τ)s(t τ)dτ: (1.1.3) −∞ − The output is then a convolution of the input s(t) and the shift-invariant impulse response h(t) and is denoted by r(t) = s(t) ~ h(t). The shift-invariant impulse response is also called, simply, the impulse response. Every linear shift-invariant system can described as a linear shift-invariant filter. Convolution has the following properties: 1. Commutative property h(t) ~ s(t) = s(t) ~ h(t): 2. Distributive property h(t) ~ (x1(t) + x2(t)) = h(t) ~ x1(t) + h(t) ~ x2(t). aA Dirac impulse is an example of a generalized function or a generalized signal. For the formal theory see Strichartz (2003). 2 3. Associative property h1(t) ~ (h2(t) ~ s(t)) = (h1(t) ~ h2(t)) ~ s(t): Using the distributive property of convolution, we can write for complex functions, a(t) = b(t) ~ c(t) aR(t) + iaI (t) = (bR(t) + ibI (t)) ~ (cR(t) + icI (t)) = b (t) c (t) b (t) c (t) + ib (t) c (t) + b (t) c (t): (1.1.4) R ~ R − I ~ I R ~ I I ~ R The class of shift-invariant systems includes all those described by constant-coefficient, linear differential equations. An example of a spatially-invariant system is free space because it has no boundaries and thus the choice of the spatial origin is arbitrary. Systems with spatial boundaries are spatially-varying in at least one direction, but may be spatially-invariant in the other directions. However, many spatial systems with boundaries can be approximated as spatially-invariant over a limited range of spatial inputs. Causality A causal filter h(t) is a linear filter whose impulse response has a value equal to zero for all times t less than zero. A causal impulse response cannot have an output before it has an input. A right- sided signal s(t) has a value equal to zero for all times less than zero. A linear time-invariant system is causal if and only if it has a right-sided impulse response. A causal h(t) can be defined using the unit-step function, which is 8 1 for t > 0 : < 1 u(t) = 2 for t = 0 (1.1.5) : 0 for t < 0: A linear shift-invariant system is causal if its impulse response h(t) satisfies h(t) = h(t)u(t) except at t = 0. For this case, the lower limit of the integral for the output signal given in (1.1.3) is equal to zero. A function related to the unit-step function is the signum function defined as : sgn(t) = 2u(t) 1 − 8 1 for t > 0 : < = 0 for t = 0 (1.1.6) : 1 for t < 0: − A system for which the output r(t) depends on only the current value of s(t) is called memoryless. The corresponding property in space is called local. 3 The Fourier Transform The Fourier transformb (or spectrum) S(f) of the temporal signal s(t) is defined, provided the integral exists, as Z 1 S(f) = s(t)e−i2πftdt: (1.1.7) −∞ The Fourier transform formally exists for any signal whose energyc E, given by Z 1 E = s(t) 2; (1.1.8) −∞ j j is finite. Such signals are called finite energy or square-integrable signals. The Fourier transform can be extended to include a large number of signals and generalized i2πfct signals with infinite energy, but finite power, such as cos(2πfct) and e by means of a limiting process that often can be expressed using the Dirac impulse δ(t). The signal s(t) can be recovered as an inverse Fourier transform Z 1 s(t) = S(f)ei2πftdf; (1.1.9) −∞ with s(t) S(f) denoting the transform pair. To this purpose, two signals whose difference has zero energy ! are regarded as the same signal. Another way to say this is that the two signals are equal almost everywhere. A Fourier transform can also be defined for spatial signals. For a one-dimensional spatial signal f(x), we haved Z 1 F (k) = f(x)eikxdx; (1.1.10) −∞ where k is the spatial frequency, which is the spatial equivalent of the temporal frequency ! = 2πf. Properties of the Fourier Transform Several properties of the Fourier transform used to analyze communication systems are listed below. 1. Scaling 1 f s(at) S (1.1.11) ! a a j j for any nonzero real value a. This scaling property states that the width of a function in one domain scales inversely with the width of the function in the other domain.