Detection and Estimation of Signals in Noise Dr. Robert Schober Department of Electrical and Computer Engineering University of British Columbia Vancouver, August 24, 2010
2 Contents 1 Basic Elements of a Digital Communication System 1 1.1 Transmitter...................................... 1 1.2 Receiver........................................ 3 1.3 Communication Channels.............................. 4 1.3.1 Physical Channel............................... 4 1.3.2 Mathematical Models for Communication Channels............ 5 2 Probability and Stochastic Processes 9 2.1 Probability...................................... 9 2.1.1 Basic Concepts................................ 9 2.1.2 Random Variables.............................. 14 2.1.3 Functions of Random Variables....................... 21 2.1.4 Statistical Averages of RVs.......................... 26 2.1.5 Gaussian Distribution............................ 31 2.1.6 Chernoff Upper Bound on the Tail Probability............... 39 2.1.7 Central Limit Theorem............................ 41 2.2 Stochastic Processes................................. 42 2.2.1 Statistical Averages.............................. 43 2.2.2 Power Density Spectrum........................... 50 2.2.3 Response of a Linear Time Invariant System to a Random Input Signal. 52 2.2.4 Sampling Theorem for Band Limited Stochastic Processes........ 56 2.2.5 Discrete Time Stochastic Signals and Systems............... 57 2.2.6 Cyclostationary Stochastic Processes.................... 60 3 Characterization of Communication Signals and Systems 63 3.1 Representation of Bandpass Signals and Systems.................. 63 3.1.1 Equivalent Complex Baseband Representation of Bandpass Signals... 64 3.1.2 Equivalent Complex Baseband Representation of Bandpass Systems... 74 3.1.3 Response of a Bandpass Systems to a Bandpass Signal.......... 76
3 3.1.4 Equivalent Baseband Representation of Bandpass Stationary Stochastic Processes................................... 77 3.2 Signal Space Representation of Signals....................... 85 3.2.1 Vector Space Concepts A Brief Review.................. 85 3.2.2 Signal Space Concepts............................ 88 3.2.3 Orthogonal Expansion of Signals...................... 90 3.3 Representation of Digitally Modulated Signals................... 103 3.3.1 Memoryless Modulation........................... 104 3.3.1.1 M-ary Pulse Amplitude Modulation (MPAM)......... 104 3.3.1.2 M-ary Phase Shift Keying (MPSK)............... 109 3.3.1.3 M-ary Quadrature Amplitude Modulation (MQAM)...... 112 3.3.1.4 Multi Dimensional Modulation.................. 115 3.3.1.5 M-ary Frequency Shift Keying (MFSK)............. 116 3.3.2 Linear Modulation With Memory...................... 122 3.3.2.1 M ary Differential Phase Shift Keying (MDPSK)....... 122 3.3.3 Nonlinear Modulation With Memory.................... 124 3.3.3.1 Continuous Phase FSK (CPFSK)................. 124 3.3.3.2 Continuous Phase Modulation (CPM).............. 128 3.4 Spectral Characteristics of Digitally Modulated Signals.............. 141 3.4.1 Linearly Modulated Signals......................... 141 3.4.2 CPFSK and CPM.............................. 150 4 Optimum Reception in Additive White Gaussian Noise (AWGN) 151 4.1 Optimum Receivers for Signals Corrupted by AWGN............... 151 4.1.1 Demodulation................................. 153 4.1.1.1 Correlation Demodulation..................... 153 4.1.1.2 Matched Filter Demodulation................... 158 4.1.2 Optimal Detection.............................. 164 4.2 Performance of Optimum Receivers......................... 174 4.2.1 Binary Modulation.............................. 174 4.2.2 M ary PAM.................................. 182 4.2.3 M ary PSK.................................. 186
4 4.2.4 M ary QAM................................. 187 4.2.5 Upper Bound for Arbitrary Linear Modulation Schemes.......... 188 4.2.6 Comparison of Different Linear Modulations................ 190 4.3 Receivers for Signals with Random Phase in AWGN................ 192 4.3.1 Channel Model................................ 192 4.3.2 Noncoherent Detectors............................ 194 4.3.2.1 A Simple Noncoherent Detector for PSK with Differential Encoding (DPSK)........................... 195 4.3.2.2 Optimum Noncoherent Detection................. 201 4.3.2.3 Optimum Noncoherent Detection of Binary Orthogonal Modulation................................ 205 4.3.2.4 Optimum Noncoherent Detection of On Off Keying....... 210 4.3.2.5 Multiple Symbol Differential Detection (MSDD) of DPSK... 212 4.4 Optimum Coherent Detection of Continuous Phase Modulation (CPM)..... 218 5 Signal Design for Bandlimited Channels 225 5.1 Characterization of Bandlimited Channels..................... 226 5.2 Signal Design for Bandlimited Channels...................... 228 5.3 Discrete Time Channel Model for ISI free Transmission............. 240 6 Equalization of Channels with ISI 244 6.1 Discrete Time Channel Model............................ 245 6.2 Maximum Likelihood Sequence Estimation (MLSE)................ 247 6.3 Linear Equalization (LE)............................... 257 6.3.1 Optimum Linear Zero Forcing (ZF) Equalization............. 258 6.3.2 ZF Equalization with FIR Filters...................... 263 6.3.3 Optimum Minimum Mean Squared Error (MMSE) Equalization..... 268 6.3.4 MMSE Equalization with FIR Filters.................... 278 6.4 Decision Feedback Equalization (DFE)....................... 283 6.4.1 Optimum ZF DFE.............................. 288 6.4.2 Optimum MMSE DFE............................ 295 6.5 MMSE DFE with FIR Filters............................ 301
I EECE 564 Signal Detection & Estimation Instructor: Dr. Robert Schober, Kaiser Building, Room 4103, Phone: 604-822-3515, Email: rschober@ece.ubc.ca. Office hours: (Almost) any time or after lecture. Text book: John G. Proakis, Digital Communications, 4th Edition, McGraw Hill. Course notes and homework assignments: Download from http://www.ece.ubc.ca/ elec564/ Additional reading: 1. Oppenheim and Willsky, Signals & Systems, 2nd Edition, Prentice Hall. 2. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw Hill. 3. Moon and Stirling, Mathematical Methods and Algorithms for Signal Processing, Prentice Hall. 4. Haykin, Communications Systems, 4th Edition, Wiley. 5. Wozencraft and Jacobs (1965), Principles of Communication Engineering, Wiley. Grading: Homework Assignments 10% Midterm 30% Final Exam 60%
II Course Outline 1. Basic Elements of a Digital Communication System 2. Probability and Stochastic Processes a Brief Review 3. Characterization of Communication Signals and Systems 4. Detection of Signals in Additive White Gaussian Noise 5. Bandlimited Channels 6. Equalization
1 1 Basic Elements of a Digital Communication System Info Source Source Encod. Chan. Encod. Digital Mod. Channel Info Sink Source Decod. Chan. Decod. Digital Demod. 1.1 Transmitter a) Information Source analog signal: e.g. audio or video signal digital signal: e.g. data, text b) Source Encoder Objective: Represent the source signal as efficiently as possible, i.e., with as few bits as possible minimize the redundancy in the source encoder output bits
2 c) Channel Encoder Objective: Increase reliability of received data d) Digital Modulator add redundancy in a controlled manner to information bits Objective: Transmit most efficiently over the (physical) transmission channel map the input bit sequence to a signal waveform which is suitable for the transmission channel Examples: Binary modulation: bit 0 s 0 (t) bit 1 s 1 (t) 1 bit per channel use M ary modulation: we map b bits to one waveform we need M = 2 b different waveforms to represent all possible b bit combinations b bit/(channel use)
3 1.2 Receiver a) Digital Demodulator Objective: Reconstruct transmitted data symbols (binary or M ary from channel corrupted received signal b) Channel Decoder Objective: Exploit redundancy introduced by channel encoder to increase reliability of information bits Note: c) Source Decoder In modern receivers demodulation and decoding is sometimes performed in an iterative fashion. Objective: Reconstruct original information signal from output of channel decoder
4 1.3 Communication Channels 1.3.1 Physical Channel a) Types wireline optical fiber optical wireless channel wireless radio frequency (RF) channel underwater acoustic channel storage channel (CD, disc, etc.) b) Impairments noise from electronic components in transmitter and receiver amplifier nonlinearities other users transmitting in same frequency band at the same time (co channel or multiuser interference) linear distortions because of bandlimited channel time variance in wireless channels For the design of the transmitter and the receiver we need a simple mathematical model of the physical communication channel that captures its most important properties. This model will vary from one application to another.
5 1.3.2 Mathematical Models for Communication Channels a) Additive White Gaussian Noise (AWGN) Channel s(t) α r(t) n(t) r(t) = α s(t) + n(t) The transmitted signal is only attenuated (α 1) and impaired by an additive white Gaussian noise (AWGN) process n(t). b) AWGN Channel with Unknown Phase e jϕ s(t) α r(t) n(t) r(t) = α e jϕ s(t) + n(t) In this case, the transmitted signal also experiences an unknown phase shift ϕ. ϕ is often modeled as a random variable, which is uniformly distributed in the interval [ π, π).
6 c) Linearly Dispersive Channel (Linear Filter Channel) s(t) c(t) r(t) n(t) c(t): channel impulse response; : linear convolution r(t) = c(t) s(t) + n(t) = c(τ) s(t τ)dτ + n(t) Transmit signal is linearly distorted by c(t) and impaired by AWGN. d) Multiuser Channel Two users: s 1 (t) r(t) s 2 (t) n(t) K user channel: r(t) = K s k (t) + n(t) k=1
7 e) Other Channels time variant channels stochastic (random) channels fading channels multiple input multiple output (MIMO) channels...
8 Some questions that we want to answer in this course: Which waveforms are used for digital communications? How are these waveforms demodulate/detect? What performance (= bit or symbol error rate) can be achieved?