CHAPTER 1: INTRODUCTION

Transcription

1 CHAPTER 1: INTRODUCTION 1.1 SCOPE AND CONTENT Counications and sensing systes are ubiquitous. They are found in ilitary, industrial, edical, consuer, and scientific applications eploying radio frequency, infrared, visible, and shorter wavelengths. Even acoustic systes operate under siilar principles. Counications eaples range fro optical fiber or satellite systes to wireless radio. Radio astronoy, radar, lidar, and sonar systes probe the environent and have counterparts in analytical instruents and eory systes used for a wide variety of purposes. HUMAN PROCESSOR TRANSDUCER A B C Radio Optical, Infrared Acoustic, other Electroagnetic Environent HUMAN PROCESSOR TRANSDUCER G F E D Figure 1.1: Architecture of counications and sensing systes. Figure 1.1 characterizes the ajor electroagnetic and signal processing eleents of such systes, not all which of are necessarily involved in any particular case. For eaple, in counication systes a huan (A) (or coputer counter-part) typically generates signals which are first processed (B) and then coupled to an electroagnetic environent (D) by a transducer or antenna (C). After propagating through the environent (D) the signals are intercepted by another transducer (E) which usually consists of an antenna followed by a detector which converts these electroagnetic signals into voltages and currents. The signals fro the transducer (E) are then generally anipulated in processor (F) before transission to the huan or coputer recipient (G). Counications and active systes that probe the environent generally involve all seven eleents (A)-(G). Passive systes generally involve only the last four, fro the environent (D) to the user (G). Eaples of the latter include environental or astronoical observations, edical systes seeking spectral or theral signatures of disease, and the readout of inforation fro eory systes such as copact disks. To copletely analyze such a broad range of systes would require several tetbooks. Here the fundaentals for each of the eleents in Figure 1.1 are presented in a generally coplete 1

2 way but, for efficiency, only few of their possible cobinations are presented in any detail. For eaple, the probability of sybol detection error is analyzed for counications systes, but this analysis is not repeated for other systes. It is hoped that readers of this book will acquire sufficient understanding of the eleents of Figure 1.1 to be able to conceive, design, and analyze a wide variety of electroagnetic signal-based systes by cobining these eleents appropriately. The chapters of this book can be divided into three groups. First Chapter 1 defines the basic notation and surveys briefly soe of the basic notation fundaental to signal processing and electroagnetic waves. The second group of chapters focuses on the fundaental eleents of counications and sensing systes. Chapter discusses basic noise processes and the devices used for detection of radio, infrared, and visible signals, including those first-stage signal processing operations that yield the desired signal, energy, or power spectral density estiates. Chapter 3 then discusses the transducers and antennas that link these detectors to the electroagnetic environent, including wire antennas, apertures, siple optics, coon propagation phenoena, and how the transitting and receiving properties of systes are related in a siple way. The third group of chapters deals with coplete systes applied to counications (Chapter 4) and both active and passive sensing (Chapter 5). Estiation techniques for both sensing and counication systes are then discussed separately in Chapter MATHEMATICAL NOTATION Because physical signals are generally analog, we rely in this tet ore heavily on continuous functions and operators than on discrete signals and the z transfor. Physical signals in tie or space are generally represented by lower case letters followed by their arguents in parentheses, whereas their transfors are generally represented by capital letters, again followed by their arguents in parentheses. Cople quantities are generally indicated by underbars. For eaple, the Fourier transfor relating a voltage pulse v(t) to its spectru V(f) is: jft Vf v t e dt volts Hzvoltsec (1..1) jft v t V f e df volts (1..) v(t) V(f ) (1..3) where frequency is generally represented by f(hz) or f (radians/ second). We abbreviate this Fourier relationship as: v(t) V(f ). These relations apply for pulses of finite energy, i.e.:

3 v(t) dt (1..4) Sf Vf and has units volts Hz for the case where v(t) has units [volts]. This energy density spectru is the Fourier transfor of the voltage autocorrelation function R(), where: The energy spectru Vf Sf R v t v t dt v sec or J, etc. (1..5) Parseval s theore, which says that the integral of power over tie equals the integral of energy spectral density S(f) over frequency, follows easily fro Equation (1..5) and the definition of a Fourier transfor (1..) for t = 0: R(0) v t dt = Sf df. (1..6) These relationships for analytic pulse signals can be represented copactly by the following notation: v R t V f V f S f (1..7) The single-headed arrows pointing downward indicate that the transforations fro v(t) to R(), and fro V to V f are irreversible. The units of these quantities depends on the units f associated with v(t). For eaple, if v(t) represents volts as a function of tie, then the units in clockwise order in (1..7) for these four quantities are: volts, volts/hz, (volts/hz), and volts seconds. If this voltage v(t) is across a 1-oh resistor, then we can associate the autocorrelation function R() with the units Joules, and the energy density spectru S(f) with the units Joules/Hz. Another iportant operator is convolution, represented by an asterisk, where: 3

4 . (1..8) a(t) b(t) a b t- d = c t Note that a unit ipulse convolved with any function yields the original function, where we define the unit ipulse (t) as a function which is zero for t > 0, and has an integral of value unity. Periodic signals with finite energy in each period T can be reversibly characterized by their Fourier series and irreversibly characterized by their autocorrelation function R() and its Fourier transfor, the energy density spectru. These are related as suggested in Equation 1..9 vt V(volts) -1 R watts Hz or S(f ) (Joules) (1..9) The Fourier series V can be siply coputed fro the original wavefor v(t) as: T 1 jf t o V T v(t)e dt (1..10) T where T equals f o -1 and: v(t) V e jf t o (1..11) T jfo (1..1) R( ) v(t)v (t )dt V e T T 1 jf T o V T R( )e d (1..13) Rando signals (t) can often be characterized by their autocorrelation function: Ett. (1..14) Such signals are called wide-sense stationary stochastic signals. For the special case where the signal (t) is the voltage across a 1-oh resistor, the autocorrelation function () for = 0 ay 4

5 be regarded as the average power dissipated in the resistor, and the Fourier transfor of the autocorrelation function can be regarded as the power spectral density f (watts/hz) where: v t v? f (1..15) In any cases we shall encounter Gaussian noise n(t), where the probability distribution of n is: 1 n P n e (1..16) E n pnn dn. (1..17) Band-liited Gaussian white noise, which is defined as having a unifor power spectral density (f) over a band of width B (Hz), can be characterized by the noise power spectral density N o, where: En NB (1..18) o Signal or wave powers are often characterized in ters of decibels, where if a signal increases its power of P 1 to P we say there has been a gain of: db gain 10 log P P 10 1 (1..19) Thus an aplifier having power output equal to the power input ehibits 0 db gain, where power gains of a factor of 10 or 100 correspond to 10 db and 0 db, respectively. 1.3 ELECTROMAGNETIC NOTATION We characterize electroagnetic phenoena in ters of the electric field E (volts/eter) and agnetic field H (aperes/eter), where these fields have both a agnitude and direction at each point in space and tie. We represent the electric displaceent by D (coulobs / eter ), where for siple edia D E and the perittivity for vacuu is o farads / eter ; F/. 4 B Tesla Weber 10 Gauss We represent the agnetic flu density by, where for siple edia B H and the pereability for 5

6 vacuu is o henries/eter; H/. The electric current density is represented by J(aperes / eter ; A / ) and the electric charge density by (coloubs/eter 3 ; C/ 3 ). This tet uses SI (ks) units throughout, in which case Mawell s equations becoe: B E t (1.3.1) D H J (1.3.) t D (1.3.3) B 0 (1.3.4) ˆ ŷ y ẑ z (1.3.5) where, ˆ y, ˆ and zˆ are unit vectors in Cartesian coordinates. In general these field quantities are functions of both space and tie and can be represented in different ways. For eaple, the coponent of a onochroatic electric field at frequency and position r can be represented as: E r,t Re E r e Re E r e j( t (r)) jt (1.3.6) where the operator Re{ } etracts the real part of its arguent, and the phasor can be represented as: j r E r E r e (1.3.7) In general, we ay cobine all three vector coponents of Er,t in a phasor representation to yield: E r,t Re E r e jt (1.3.8) E r has si nubers associated with it (three vectors, each with agnitude where we note that and phase). The other variables are also epressible as phasors when the signals are onochroatic. 6

7 Mawell s equations can be therefore rewritten in ters of phasors as: E j B (1.3.9) H jd J (1.3.10) D (1.3.11) B 0 (1.3.1) Many waves in counications or sensing systes travel on wires or transission lines where they ay be characterized in ters of voltages V(z, t) and currents I(z,t) as a function of position z and tie t. Most coonly such signals travel on transverse-electroagnetic-field (TEM) transission lines for which the voltage is easured between the two conductors at a particular position z and the currents I(z,t) in the two wires are equal and opposite at any position z. Such transission lines can be characterized by their inductance L per unit length (H/) and their capacitance C per unit length (C/). In general, LC =, where is a function of the ediu between and around the conductors. In general such transission lines satisfy the wave equation: v v LC z t (1.3.13) The general wave equation solution is the linear superposition of an arbitrarily shaped forward oving wave v z t LC and a backward oving wave where: which leads to: vz,t v zt LC v z t LC (1.3.14) z,t C L z t LC z t LC i v v (1.3.15) The instantaneous power at any position z on the transission line is siply the product of the voltage v and current i at that point in space and tie. The phasor equivalents of equations (1.3.14) and (1.3.15) are: and jkz jkz V z V e V e (1.3.16) 7

8 jkz jkz Iz Yo Ve Ve (1.3.17) where the wave nuber or propagation constant k LC, and the characteristic 1 adittance of the transission line Yo Zo C L, and where Z o (ohs) is called the characteristic ipedance of the TEM transission line. Waves are also characterized by their wavelength, where = c/f and c is the phase velocity of the electroagnetic waves in that ediu. 8