Theory of Telecommunications Networks

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1 Theory of Telecommunications Networks Anton Čižmár Ján Papaj Department of electronics and multimedia telecommunications

2 CONTENTS Preface Introduction Mathematical models for communication channels Channel capacity for digital communication Shannon Capacity and Interpretation Hartley Channel Capacity Solved Problems Noise in digital communication system White Noise Thermal Noise Solved Problems Summary Exercises Signal and Spectra Deterministic and random signals Periodic and nonperiodic signals Analog and discrete Signals Energy and power Signals Spectral Density Energy Spectral Density Power Spectral Density Solved Problems Autocorrelation Autocorrelation of an Energy Signal Autocorrelation of a Periodic Signal Baseband versus Bandpass Summary Exercises Probability and stochastic processes Probability Joint Events and Joint Probabilities Conditional Probabilities Statistical Independence Solved Problems Random Variables, Probability Distributions, and probability Densities Statistically Independent Random Variables

3 3.2.2 Statistical Averages of Random Variables Some Useful Probability Distributions Stochastic processes Stationary Stochastic Processes Statistical Averages Power Density Spectrum Response of a Linear Time-Invariant System (channel) to a Random Input Signal Sampling Theorem for Band-Limited Stochastic Processes Discrete-Time Stochastic Signals and Systems Cyclostationary Processes Solved Problems Summary Exercises Signal space concept Representation Of Band-Pass Signals And Systems Representation of Band-Pass Signals Representation of Band-Pass Stationary Stochastic Processes Introduction of the Hilbert transform Different look at the Hilbert transform Hilbert Transform, Analytic Signal and the Complex Envelope Hilbert Transform in Frequency Domain Hilbert Transform in Time Domain Analytic Signal Solved Problems Signal Space Representation Vector Space Concepts Signal Space Concepts Orthogonal Expansions of Signals Gram-Schmidt procedure ) Solved Problems Summary Exercises Digital modulation schemes Signal Space Representation Memoryless Modulation Methods Pulse-amplitude-modulated (PAM) signals (ASK)

4 5.2.2 Phase-modulated signal (PSK) Quadrature Amplitude Modulation (QAM) Multidimensional Signals Orthogonal multidimensional signals Linear Modulation with Memory Non-Linear Modulation Methods with Memory Spectral Characteristic Of Digitally Modulated Signals Power Spectra of Linearly Modulated Signals Power Spectra of CPFSK and CPM Signals Solved Problems Summary Exercises Optimum Receivers for the AWGN Channel Optimum Receivers For Signals Corrupted By Awgn Correlation demodulator Matched-Filter demodulator The Optimum detector The Maximum-Likelihood Sequence Detector Performance Of The Optimum Receiver For Memoryless Modulation Probability of Error for Binary Modulation Probability of Error for M-ary Orthogonal Signals Probability of Error for M-ary Biorthogonal Signals Probability of Error for Simplex Signals Probability of Error for M-ary Binary-Coded Signals Probability of Error for M-ary PAM Probability of Error for M-ary PSK Probability of Error for QAM Solved Problems Summary Exercises Performance analysis of digital modulations Goals Of The Communications System Designer Error Probability Plane Nyquist Minimum Bandwidth Shannon-Hartley Capacity Theorem Shannon Limit Bandwidth-Efficiency Plane

5 7.5.1 Bandwidth Efficiency of MPSK and MFSK Modulation Analogies Between Bandwidth-Efficiency and Error-Probability Planes Modulation And Coding Trade-Offs Defining, Designing, And Evaluating Digital Communication Systems M-ary Signaling Bandwidth-Limited Systems Power-Limited Systems Requirements for MPSK and MFSK Signaling Bandwidth-Limited Uncoded System Example Power-Limited Uncoded System Example Solved Problems Summary Exercise Why use error-correction coding Trade-Off 1: Error Performance versus Bandwidth Trade-Off 2: Power versus Bandwidth Coding Gain Trade-Off 3: Data Rate versus Bandwidth Trade-Off 4: Capacity versus Bandwidth Code Performance at Low Values of E b /N Solved problem Exercise Appendix A The Q-function The Error Function Appendix B Comparison of M-ary signaling techniques Error performance of M-ary signaling techniques References

6 PREFACE Providing the theory of digital communication systems, this textbook prepares senior undergraduate and graduate students for the engineering practices required in the real word. With this textbook, students can understand how digital communication systems operate in practice, learn how to design subsystems, and evaluate end-to-end performance. The book contains many examples to help students achieve an understanding of the subject. The problems are at the end of the each chapter follow closely the order of the sections. The entire book is suitable for one semester course in digital communication. All materials for teaching texts were drawn from sources listed in References. 5

7 2 SIGNAL AND SPECTRA 2.1 DETERMINISTIC AND RANDOM SIGNALS A signal can be classified as deterministic, meaning that there is no uncertainty with respect to its value at any time, or as random, meaning that there is some degree of uncertainty before the signal actually occurs. Deterministic signals or waveforms are modeled by explicit mathematical expressions, such as () =510. For a random waveform it is not possible to write such an explicit expression. However, when examined over a long period, a random waveform, also referred to as a random process, may exhibit certain regularities that can be described in terms of probabilities and statistical averages. Such a model, in the form of a probabilistic description of the random process, is particularly useful for characterizing signals and noise in communication systems. 2.2 PERIODIC AND NONPERIODIC SIGNALS A signal (t) is called periodic in time if there exists a constant T 0 > 0 such that () =( ) (2.1) where t denotes time. The smallest value of T 0 that satisfies this condition is called the period of x(t). The period T 0 defines the duration of one complete cycle of x(t). A signal for which there is no value of T 0 that satisfies Equation 2.1 is called a nonperiodic signal. 2.3 ANALOG AND DISCRETE SIGNALS An analog signal x(t) is a continuous function of time; that is, x(t) is uniquely defined for all t. An electrical analog signal arises when a physical waveform (e.g., speech) is converted into an electrical signal by means of a transducer. By comparison, a discrete signal () is one that exists only at discrete times; it is characterized by a sequence of numbers defined for each time, where k is an integer and T is a fixed time interval. 2.4 ENERGY AND POWER SIGNALS An electrical signal can be represented as a voltage v(t) or a current i(t) with instantaneous power p(t) across a resistor R defined by Or () = () (2.2) () = () ( 2.3) In communication systems, power is often normalized by assuming R to be 1Ω, although R may be another value in the actual circuit. If the actual value of the power is needed, it is obtained by denormalization of the normalized value. For the normalized case, Equations 2.2 and 2.3 have the same form. Therefore, regardless of whether the signal is a voltage or current waveform, the normalization convention allows us to express the instantaneous power as 23

8 () = () (2.4) where x(t) is either a voltage or a current signal. The energy dissipated during the time interval ( T/2, T/2) by a real signal with instantaneous power expressed by Equation 2.4 can then be written as = () (2.5) and the average power dissipated by the signal during the interval is = = () The performance of a communication system depends on the received signal energy; higher energy signals are detected more reliably (with fewer errors) than are lower energy signals - the received energy does the work. On the other hand, power is the rate at which energy is delivered. It is important for different reasons. The power determines the voltages that must be applied to a transmitter and the intensities of the electromagnetic fields that one must contend with in radio systems (i.e., fields in waveguides that connect the transmitter to the antenna, and fields around the radiating elements of the antenna). In analyzing communication signals, it is often desirable to deal with the waveform energy. We classify x(t) as an energy signal if, and only if, it has nonzero but finite energy (0 ) for all time, where = lim = () () In the real world, we always transmit signals having finite energy (0 ). However, in order to describe periodic signals, which by definition (Equation 2.1) exist for all time and thus have infinite energy, and in order to deal with random signals that have infinite energy, it is convenient to define a class of signals called power signals. A signal is defined as a power signal if, and only if, it has finite but nonzero power (0 ) for all time, where =lim () The energy and power classifications are mutually exclusive. An energy signal has finite energy but zero average power, whereas a power signal has finite average power but infinite energy. A waveform in a system may be constrained in either its power or energy values. As a general rule, periodic signals and random signals are classified as power signals, while signals that are both deterministic and nonperiodic are classified as energy signals. Signal energy and power are both important parameters in specifying a communication system. The classification of a signal as either an energy signal or a power signal is a convenient model to facilitate the mathematical treatment of various signals and noise. (2.6) (2.7) (2.8) 24

9 2.5 SPECTRAL DENSITY The spectral density of a signal characterizes the distribution of the signal s energy or power in the frequency domain. This concept is particularly important when considering filtering in communication systems. We need to be able to evaluate the signal and noise at the filter output. The energy spectral density (ESD) or the power spectral density (PSD) is used in the evaluation Energy Spectral Density The total energy of a real-valued energy signal x(t), defined over the interval, (, ) is described by Equation 2.7. Using Parseval s theorem [1], we can relate the energy of such a signal expressed in the time domain to the energy expressed in the frequency domain, as = () = () where X(f) is the Fourier transform of the nonperiodic signal x(t). Let () denote the squared magnitude spectrum, defined as (2.9) () = () (2.10) The quantify () is the waveform energy spectral density (ESD) of the signal x(t). Therefore, from Equation 2.9, we can express the total energy of x(t) by integrating the spectral density with respect to frequency: = () (2.11) This equation states that the energy of a signal is equal to the area under the () versus frequency curve. Energy spectral density describes the signal energy per unit bandwidth measured in joules/hertz. There are equal energy contributions from both positive and negative frequency components, since for a real signal, (), () is an even function of frequency. Therefore, the energy spectral density is symmetrical in frequency about the origin, and thus the total energy of the signal () can be expressed as Power Spectral Density =2 () (2.12) The average power of a real-valued power signal () is defined in Equation 2.8. If () is a periodic signal with period T 0, it is classified as a power signal. The expression for the average power of a periodic signal takes the form of Equation 2.6, where the time average is taken over the signal period T 0, as follows: = () (2.13) Parseval s theorem for a real-valued periodic signal [1] takes the form = () = (2.14) 25

10 where the terms are the complex Fourier series coefficients of the periodic signal. To apply Equation 2.14, we need only know the magnitude of the coefficients,. The power spectral density (PSD) function G (f) of the periodic signal () is a real, even, and nonnegative function of frequency that gives the distribution of the power of () in the frequency domain, defined as G () = ( ) (2.15) Equation 2.15 defines the power spectral density of a periodic signal () as a succession of the weighted delta functions. Therefore, the PSD of a periodic signal is a discrete function of frequency. Using the PSD defined in Equation 2.15, we can now write the average normalized power of a realvalued signal as = () = 2 () (2.16) Equation 2.15 describes the PSD of periodic (power) signals only. If () is a nonperiodic signal it cannot be expressed by a Fourier series, and if it is a nonperiodic power signal (having infinite energy) it may not have a Fourier transform. However, we may still express the power spectral density of such signals in the limiting sense. If we form a truncated version () of the nonperiodic power signal () by observing it only in the interval ( /2, /2), then () has finite energy and has a proper Fourier transform (). It can be shown that the power spectral density of the nonperiodic () can then be defined in the limit as Solved Problems Problem 1 Solution G () = lim () (2.17) a) Find the average normalized power in the waveform, () = 2, using time averaging. b) Repeat part (a) using the summation of spectral coefficients. a) Using Equation (2.13), we have = 1 = 2 b) Using Equation 2.16 and 2.17 give us 2 (1 4 ) = ( 2 ) = 2 26

11 G () = ( ) = = 2 = 0 = 0, 2, 3, G () = 2 ( ) 2 ( ) 2.6 AUTOCORRELATION = Autocorrelation of an Energy Signal () = 2 Correlation is a matching process; autocorrelation refers to the matching of a signal with a delayed version of itself. The autocorrelation function of a real-valued energy signal () is defined as Φ () = ()( ) (2.18) The autocorrelation function Φ () provides a measure of how closely the signal matches a copy of itself as the copy is shifted units in time. The variable plays the role of a scanning or searching parameter. Φ () is not a function of time; it is only a function of the time difference between the waveform and its shifted copy. The autocorrelative function of a real-valued energy signal has the following properties: 1. Φ () =Φ () symmetrical in τ about zero 2. () (0) for all maximum value occurs at the origin 3. () () autocorrelation and PSD form a Fourier transform pair 4. (0) = () value at the origin is equal to the energy of the signal If items 1 through 3 are satisfied, Φ () satisfies the properties of an autocorrelation function. Property 4 can be derived from property 3 and thus need not be included as a basic test Autocorrelation of a Periodic Signal The autocorrelation function of a real-valued power signal () is defined as Φ () = lim ()( ) (2.19) When the power signal () is periodic with period T 0, the time average in Equation 2.19 may be taken over a single period T 0, and the autocorrelation function can be expressed as 27

12 Φ () = ()( ) (2.20) The autocorrelation function of a real-valued periodic signal has properties similar to those of an energy signal: 1. Φ () =Φ () symmetrical in τ about zero 2. Φ (τ) Φ (0) for all τ maximum value occurs at the origin 3. Φ (τ) G (f) autocorrelation and ESD form a Fourier transform pair, as designated by the double-headed arrows 4. Φ (0) = x value at the origin is equal to the energy of the signal (t)dt 2.7 BASEBAND VERSUS BANDPASS An easy way to translate the spectrum of a low-pass or baseband signal () to a higher frequency is to multiply or heterodyne the baseband signal with a carrier wave 2, as shown in Figure 2.1. The resulting waveform, (), is called a double-sideband (DSB) modulated signal and is expressed as () =()2 (2.21) From the frequency shifting theorem, the spectrum of the DSB signal () is given by () = ( ) ( ) (2.22) The magnitude spectrum () of the baseband signal () having a bandwidth and the magnitude spectrum () of the DSB signal () having a bandwidth W DSB are shown in Figure 2.1 b) and c), respectively. In the plot of (), spectral components corresponding to positive baseband frequencies appear in the range to ( ). This part of the DSB spectrum is called the upper sideband (USB). Spectral components corresponding to negative baseband frequencies appear in the range ( ) to. This part of the DSB spectrum is called the lower sideband (LSB). Mirror images of the USB and LSB spectra appear in the negative frequency half of the plot. The carrier wave is sometimes referred to as a local oscillator (LO) signal, a mixing signal, or a heterodyne signal. Generally, the carrier wave frequency is much higher than the bandwidth of the baseband signal; that is, (2.23) From Figure 2.1, we can readily compare the bandwidth fm required to transmit the baseband signal with the bandwidth WDSB required to transmit the DSB signal; we see that =2 (2.24) That is, we need twice as much transmission bandwidth to transmit a DSB version of the signal than we do to transmit its baseband counterpart. 28

13 x(t) x c (t)=x(t)cos2πf c t cos2πf c t (local oscillator) (a) X(f) -f m 0 f m f Figure 2.1 Comparison of baseband and double -sideband spectra. (a) Heterodyning. (b) Baseband spectrum. (c) Doublesideband spectrum. 2.8 SUMMARY Two general classes of signals are deterministic and random. The former can be written as a completely known function of time, whereas the amplitudes of random signals must be described probabilistically. A periodic signal of period is one for which () =( ) for all t. The unit impulse function, (), can be thought of as a zero-width, infinite height pulse with unity area. The sifting property, ()( ) = ( ), where () is continuous at =, is a generalization of the defining relation for a unit impulse. The unit step function () is the integral of a unit impulse. A signal () for which () is finite is called an energy signal. If () is such that Baseband bandwidth (b) X c (f) USB LSB LSB USB f c f m f c f c +f m 0 f c f m f c f c +f m W DSB Double-sideband bandwidth =lim () is finite, the signal is known as a power signal. Example signals may be either or neither. The convolution of two signals, () and (), is () = = () ( ) = () (). The convolution theorem of Fourier transforms states that () = () (), where (), () () are the Fourier transforms of (), () (). f 29

14 2.9 EXERCISES 1. Classify the following signals as energy signals or power signals. Find the normalized energy or normalized power of each. a) () =(2 ) for b) () = (2 ) for, = 0 c) () = () for 0, 0 0 d) () =cos+5cos(2t) for 2. Determine the energy spectral density of a square pulse x(t) = rect (t/t), where (/) equals 1, for, and equals 0, elsewhere. Calculate the normalized energy in the pulse. 3. Find the autocorrelation function of () =(2 ) in terms of its period, =. Find the average normalized power of (), using =(0). 4. How does the plot of a signal s autocorrelation function reveal its bandwidth occupancy? 5. Find the power spectral densities and average powers of the following signals: a) () = 2cos (20 ) b) () = 3cos (30) c) () = 5 (10 ) d) () =3sin(30) 5cos (10 ) 6. Find the autocorrelation functions corresponding to the following signals: a) () = 2cos (10 ) b) () = 2sin (10 ) c) () = () () 30

15 TT S T KE M Department of electronics and multimedia telecommunications

Theory of Telecommunications Networks

Theory of Telecommunications Networks Anton Čižmár Ján Papaj Department of electronics and multimedia telecommunications CONTENTS Preface... 5 Introduction... 6. Mathematical models for communication channels...