# ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process

Size: px
Start display at page:

Download "ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process"

Transcription

1 1 ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 3 Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process

2 2 Multipath-Fading Mechanism local scatterers mobile subscriber base station A typical macrocellular mobile radio environment.

3 3 Multipath-Fading Mechanism local scatterers local scatterers mobile station mobile station Typical mobile-to-mobile radio propagation environment.

4 4 local mean Ω (db) area mean µ (db) envelope fading Path loss, shadowing, envelope fading.

5 5 Doppler Shift y n th incoming wave mobile station v θ n x 2-D Model of a typical wave component incident on a mobile station (MS). Assuming2-Dpropagation,theDoppler shiftisf D,n = f m cosθ n,wheref m = v/λ c (λ c is the carrier wavelength, v is the mobile station velocity).

6 Multipath Propagation Consider the transmission of the band-pass signal s(t) = Re { s(t)e j2πf ct } At the receiver antenna, the nth plane wave arrives at angle θ n and experiences Doppler shift f D,n = f m cosθ n and propagation delay τ n. If there are N propagation paths, the received bandpass signal is r(t) = Re N n=1 C n e jφ n j2πcτ n /λ c +j2π(f c +f D,n )t s(t τ n ), where C n, φ n, f D,n and τ n are the amplitude, phase, Doppler shift and time delay, respectively, associated with the nth propagation path, and c is the speed of light. The delay τ n = d n /c is the propagation delay associated with the nth propagation path, where d n is the length of the path. The path lengths, d n, will depend on the physical scattering geometry which we have not specified at this point. 6

7 7 Multipath Propagation The received bandpass signal r(t) has the form where the received complex envelope is and r(t) = Re [ r(t)e j2πf ct ] r(t) = N n=1 C n e jφ n(t) s(t τ n ) φ n (t) = φ n 2πcτ n /λ c +2πf D,n t is the time-variant phase associated with the nth propagation path. Note that the nth component varies with the Doppler frequency f D,n. The term cτ n /λ c is the propagation distance cτ n normalized by the carrier wavelength λ c. For cellular frequencies (900 MHz), λ c is on the order of a foot. The phase φ n is a random introduced by the nth scatterer and can be assumed to be uniformly distributed on [ π, π). The received phase at any time t, φ n (t) is uniformly distributed on [ π,π).

8 8 Flat Fading - impulse response The received waveform is given by the convolution r(t) = t 0 g(t,τ) s(t τ)dτ It follows that the channel can be modeled by a linear time-variant filter having the time-variant impulse response g(t,τ) = N n=1 C n e jφ n(t) δ(τ τ n ) If the differential path delays τ i τ j are all very small compared to the modulation symbol period, T, then the τ n can be replaced by the mean delay µ τ inside the delta function. Note that this approximation is not applied to the channel phases φ n (t), since small changes in τ n result in large changes in φ n (t). The channel impulse response has the approximate form g(t,τ) = g(t)δ(τ µ τ ), The received complex envelope is g(t) = N n=1 C n e jφ n(t). r(t) = g(t) s(t µ τ ) (1) which experiences fading due to the time-varying complex channel gain g(t).

9 9 Flat Fading - frequency domain By taking Fourier transforms of both sides of (1), the received complex envelope in the frequency domain is R(f) = G(f) S(f)e j2πfµ τ Since the channel component g(t) changes with time, it follows that G(f) has a finite non-zero width in the frequency domain. Due to the convolution operation, the output spectrum R(f) will be wider than the input spectrum S(f). This broadening of the transmitted signal spectrum is caused by the channel time variations and is called frequency spreading or Doppler spreading. If the maximum Doppler frequency f m is much less than the signal bandwidth W c, then the Doppler spreading will not distort S(f). Fortunately, this is often the case.

10 10 Channel Transfer Function - Flat Fading The time-variant channel transfer function is obtained by taking the Fourier transform of the time-variant channel impulse response g(t, τ) with respect to the delay variable τ, i.e., T(t,f) = g(t)e j2πfµ τ. Since the magnitude response is T(t,f) = g(t), all frequency components in the received signal are subject to the same time-variant amplitude gain g(t) and phase response g(t) = 2πfµ τ. The received signal is said to exhibit flat fading, because the magnitude of the time-variant channel transfer function T(t, f) is constant (or flat) with respect to frequency variable f. The phase response g(t) = 2πfµ τ is linear in f meaning that the channel simply delays the input signal and attenuates it.

11 11 Invoking the Central Limit Theorem Consider the transmission of an unmodulated carrier, s(t) = 1. For flat fading channels, the received band-pass signal has the quadrature representation r(t) = g I (t)cos2πf c t g Q (t)sin2πf c t where g I (t) = N n=1 g Q (t) = N n=1 and where φ n (t) = φ n 2πcτ n /λ c +2πf D,n t. C n cosφ n (t) C n sinφ n (t) The phases φ n are independent and uniform on the interval [ π,π), and the path delays τ n are all independent with f c τ n 1. Therefore, the phases φ n (t) at any time t can be treated as being independent and uniformly distributed on the interval [ π,π). In the limit N, the central limit theorem can be invoked and g I (t) and g Q (t) can be treated as Gaussian random processes, i.e., at any time t, g I (t) and g Q (t) are Gaussian random variables. The complex faded envelope is g(t) = g I (t)+jg Q (t)

12 12 Rayleigh Fading Forsometypesofscatteringenvironments,g I (t)andg Q (t)atanytimet 1 areindependent identically distributed Gaussian random variables with zero mean and identical variance b 0 = E[g 2 I (t 1)] = E[g 2 Q (t 1)]. This typically occurs in a rich scattering environment where there is no line-of-sight or strong specular component in the received signal (i.e., there is no dominant C n ) and isotropic antennas are used. Under such conditions, the channel exhibits Rayleigh fading. The probability density function the envelope α = g(t 1 ) = g 2 I(t 1 )+g 2 Q(t 1 ) can be obtained by using a bi-variate transformation of random variables (see Appendix in textbook). The envelope α = g(t 1 ) = gi(t 2 1 )+gq(t 2 1 ) is Rayleigh distributed at any time t 1, i.e., p α (x) = x exp b x2 0 2b = 2x exp 0 Ω x2 p Ω, x 0, p where Ω p = E[α 2 ] = E[gI 2(t 1)]+E[gQ 2(t 1)] = 2b 0 is the average envelope power. The squared-envelope α 2 at any time t 1 has the exponential distribution p α 2(x) = 1 exp Ω x p Ω, x 0. p

13 13 Ricean Fading θ 0 mobile direction of motion A line-of-sight (LoS) or specular (strong reflected) component arrives at angle θ 0.

14 14 For scattering environments that have a specular or LoS component, g I (t) and g Q (t) are Gaussian random processes with non-zero means m I (t) and m Q (t), respectively. If we again assume that g I (t 1 ) and g Q (t 1 ) at any time t 1 are independent random variables with variance b 0 = E[(g I (t 1 ) m I (t 1 )) 2 ] = E[(g Q (t 1 ) m Q (t 1 )) 2 ], then the magnitude of the envelope α = g(t 1 ) at any time t 1 has a Rice distribution. With Aulin s Ricean fading model m I (t) = E[g I (t)] = s cos(2πf m cos(θ 0 )t+φ 0 ) m Q (t) = E[g Q (t)] = s sin(2πf m cos(θ 0 )t+φ 0 ) where f m cos(θ 0 ) and φ 0 are the Doppler shift and random phase offset associated with the LoS or specular component, respectively. The envelope α(t) = g(t) = g 2 I(t)+g 2 Q(t) has the Rice distribution p α (x) = x exp { x2 +s 2 } xs b 2b Io 0, x 0 0 b 0 s 2 = m I (t) 2 +m 2 Q (t) is the specular power. 2b 0 is the scatter power. The Rice factor, K = s 2 /2b 0, is the ratio of the power in the specular and scatter components.

15 15 The average envelope power is E[α 2 ] = Ω p = s 2 +2b 0 and Hence, p α (x) = 2x(K +1) Ω p exp s 2 = KΩ p K +1, K (K +1)x2 Ω p 2b 0 = Ω p K +1 I o 2x K(K +1) Ω p, x 0 The squared-envelope α 2 (t) has non-central chi-square distribution with two degrees of freedom p α 2(x) = (K +1) Ω p exp K (K +1)x Ω p I o 2 K(K +1)x Ω p, x 0 The squared-envelope is important for the performance analysis of digital communication systems because it is proportional to the received signal power and, hence, the received signal-to-noise ratio.

16 K = 0 K = 3 K = 7 K = 15 p α (x) x The Rice distribution for several values of K with Ω p = 1.

17 17 Nakagami Fading Nakagami fading describes the magnitude of the received complex envelope by the distribution p α (x) = 2mm x 2m 1 exp Γ(m)Ω m mx2 p Ω m 1 p 2 When m = 1, the Nakagami distribution becomes the Rayleigh distribution, when m = 1/2 it becomes a one-sided Gaussian distribution, and when m the distribution approaches an impulse (no fading). The Rice distribution can be closely approximated with a Nakagami distribution by using the following relation between the Rice factor K and the Nakagami shape factor m K m 2 m+m 1 (K +1)2 m (2K +1). The squared-envelope has the Gamma distribution p α 2(x) = m 2Ω p m x m 1 Γ(m) exp mx 2Ω p.

18 m = 1 m = 2 m = 4 m = 8 m = p α (x) x The Nakagami pdf for several values of m with Ω p = 1.

19 F α 2 (x) m=2 ~ (K=3.8 db) m=4 ~ (K=8.1 db) m=8 ~ (K=11.6 db) m=16 ~ (k=14.8 db) x Comparison of the cdf of the squared-envelope with Ricean and Nakagami fading.

20 20 X 1 ( t) sample space S s s 1 2 s ξ X ( t ) 2 t t X ( ) ξ t t Ensemble of Sample Functions for a Random Process.

21 21 Autocorrelation Functions Let X(t) denote a random process. At any time t, X(t) is a random variable with probability density function f X(t) (x). The ensemble mean of X(t) is µ X (t) = E[X(t)] = xf X(t) (x)dx The autocorrelation of X(t) is obtained from the random variables X(t 1 ) and X(t 2 ) as φ XX (t 1,t 2 ) = E[X(t 1 )X(t 2 )] = x 1 x 2 f X(t1 ),X(t 2 )(x 1,x 2 )dx 1 dx 2 Often the same random variable will appear in both X(t 1 ) and X(t 2 ), meaning that X(t 1 ) and X(t 2 ) are correlated. For a wide sense stationary random process where τ = t 2 t 1. µ X (t) µ X φ XX (t 1,t 2 ) = φ XX (t 2 t 1 ) φ XX (τ)

22 22 Properties of φ XX (τ) The autocorrelation function, φ XX (τ), of a stationary random process satisfies the following properties. φ XX (0) = E[X 2 (t)]. This is the total power in the random process. φ XX (τ) = φ XX ( τ). The autocorrelation function must be an even function. φ XX (τ) φ XX (0). This is a variant of the Cauchy-Schwartz inequality. φ XX ( ) = E 2 [X(t)] = µ 2 X. This holds if X(t) contains no periodic components and is equal to the d.c. power.

23 Complex-valued Random Process A complex-valued random process is given by Z(t) = X(t) + jy(t) where X(t) and Y(t) are real-valued random processes. The autocorrelation function of a wide sense stationary complexvalued random process is φ ZZ (τ) = 1 2 E[Z(t 1)Z (t 2 )] = 1 2 (φ XX(τ)+φ YY (τ)+j(φ YX (τ) φ XY (τ))). The factor of 1/2 in included for convenience, when Z(t) is a complexvalued Gaussian random process. The crosscorrelation function between two complex-valued wide sense stationary random process satisfies the following properties φ ZW(τ) = φ WZ ( τ) φ ZZ (τ) = φ ZZ( τ) 23

24 24 Power Spectral Density The power spectral density (psd) of a wide-sense stationary random process X(t) is the Fourier transform of the autocorrelation function, i.e., S XX (f) = = φ XX(τ)e j2πfτ dτ φ XX (τ) = S XX(f)e j2πfτ df. In our case, we are interested in the bandpass process where r(t) = g I (t)cos2πf c t g Q (t)sin2πf c t g(t) = g Q (t)+jg Q (t) is the complex fading envelope of the channel. For g(t), we are interested in its autocorrelation function and Doppler or power spectrum φ gg (τ) = φ gi g I (τ)+jφ gi g Q (τ) S gg (f) = S gi g I (f)+js gi g Q (f)

### ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 4. Envelope Correlation Space-time Correlation

ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 4 Envelope Correlation Space-time Correlation 1 Autocorrelation of a Bandpass Random Process Consider again the received band-pass random process r(t) = g

### EE6604 Personal & Mobile Communications. Week 5. Level Crossing Rate and Average Fade Duration. Statistical Channel Modeling.

EE6604 Personal & Mobile Communications Week 5 Level Crossing Rate and Average Fade Duration Statistical Channel Modeling Fading Simulators 1 Level Crossing Rate and Average Fade Duration The level crossing

### Lecture 2. Fading Channel

1 Lecture 2. Fading Channel Characteristics of Fading Channels Modeling of Fading Channels Discrete-time Input/Output Model 2 Radio Propagation in Free Space Speed: c = 299,792,458 m/s Isotropic Received

### Chapter 2 Propagation Modeling

Chapter 2 Propagation Modeling The design of spectrally efficient wireless communication systems requires a thorough understanding of the radio propagation channel. The characteristics of the radio channel

### Digital Band-pass Modulation PROF. MICHAEL TSAI 2011/11/10

Digital Band-pass Modulation PROF. MICHAEL TSAI 211/11/1 Band-pass Signal Representation a t g t General form: 2πf c t + φ t g t = a t cos 2πf c t + φ t Envelope Phase Envelope is always non-negative,

### Lecture 7: Wireless Channels and Diversity Advanced Digital Communications (EQ2410) 1

Wireless : Wireless Advanced Digital Communications (EQ2410) 1 Thursday, Feb. 11, 2016 10:00-12:00, B24 1 Textbook: U. Madhow, Fundamentals of Digital Communications, 2008 1 / 15 Wireless Lecture 1-6 Equalization

### Mobile Radio Communications

Course 3: Radio wave propagation Session 3, page 1 Propagation mechanisms free space propagation reflection diffraction scattering LARGE SCALE: average attenuation SMALL SCALE: short-term variations in

### Point-to-Point versus Mobile Wireless Communication Channels

Chapter 1 Point-to-Point versus Mobile Wireless Communication Channels PSfrag replacements 1.1 BANDPASS MODEL FOR POINT-TO-POINT COMMUNICATIONS In this section we provide a brief review of the standard

### Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University

Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise

### EAS 305 Random Processes Viewgraph 1 of 10. Random Processes

EAS 305 Random Processes Viewgraph 1 of 10 Definitions: Random Processes A random process is a family of random variables indexed by a parameter t T, where T is called the index set λ i Experiment outcome

### EE4601 Communication Systems

EE4601 Communication Systems Week 4 Ergodic Random Processes, Power Spectrum Linear Systems 0 c 2011, Georgia Institute of Technology (lect4 1) Ergodic Random Processes An ergodic random process is one

### EE303: Communication Systems

EE303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE303: Introductory Concepts

### EE401: Advanced Communication Theory

EE401: Advanced Communication Theory Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE.401: Introductory

### UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011

UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process,

### SRI VIDYA COLLEGE OF ENGINEERING AND TECHNOLOGY UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS

UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS 1. Define random process? The sample space composed of functions of time is called a random process. 2. Define Stationary process? If a random process is divided

### ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function

### UCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei)

UCSD ECE 53 Handout #46 Prof. Young-Han Kim Thursday, June 5, 04 Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei). Discrete-time Wiener process. Let Z n, n 0 be a discrete time white

### 2016 Spring: The Final Exam of Digital Communications

2016 Spring: The Final Exam of Digital Communications The total number of points is 131. 1. Image of Transmitter Transmitter L 1 θ v 1 As shown in the figure above, a car is receiving a signal from a remote

### 2A1H Time-Frequency Analysis II

2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period

### s o (t) = S(f)H(f; t)e j2πft df,

Sample Problems for Midterm. The sample problems for the fourth and fifth quizzes as well as Example on Slide 8-37 and Example on Slides 8-39 4) will also be a key part of the second midterm.. For a causal)

### ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process

Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random

### CS6956: Wireless and Mobile Networks Lecture Notes: 2/4/2015

CS6956: Wireless and Mobile Networks Lecture Notes: 2/4/2015 [Most of the material for this lecture has been taken from the Wireless Communications & Networks book by Stallings (2 nd edition).] Effective

### Signals and Systems: Part 2

Signals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms Some important Fourier transform theorems Convolution and Modulation Ideal filters Fourier transform definitions

### Chapter 5 Random Variables and Processes

Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability

### Fading Statistical description of the wireless channel

Channel Modelling ETIM10 Lecture no: 3 Fading Statistical description of the wireless channel Fredrik Tufvesson Department of Electrical and Information Technology Lund University, Sweden Fredrik.Tufvesson@eit.lth.se

### Introduction to Probability and Stochastic Processes I

Introduction to Probability and Stochastic Processes I Lecture 3 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark Slides

Chapter 3 Direct-Sequence Spread-Spectrum In this chapter we consider direct-sequence spread-spectrum systems. Unlike frequency-hopping, a direct-sequence signal occupies the entire bandwidth continuously.

### CHAPTER 14. Based on the info about the scattering function we know that the multipath spread is T m =1ms, and the Doppler spread is B d =0.2 Hz.

CHAPTER 4 Problem 4. : Based on the info about the scattering function we know that the multipath spread is T m =ms, and the Doppler spread is B d =. Hz. (a) (i) T m = 3 sec (ii) B d =. Hz (iii) ( t) c

### EEM 409. Random Signals. Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Problem 2:

EEM 409 Random Signals Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Consider a random process of the form = + Problem 2: X(t) = b cos(2π t + ), where b is a constant,

### This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 08 December 2009 This examination consists of

### UCSD ECE250 Handout #27 Prof. Young-Han Kim Friday, June 8, Practice Final Examination (Winter 2017)

UCSD ECE250 Handout #27 Prof. Young-Han Kim Friday, June 8, 208 Practice Final Examination (Winter 207) There are 6 problems, each problem with multiple parts. Your answer should be as clear and readable

### 3F1 Random Processes Examples Paper (for all 6 lectures)

3F Random Processes Examples Paper (for all 6 lectures). Three factories make the same electrical component. Factory A supplies half of the total number of components to the central depot, while factories

### ON THE SIMULATIONS OF CORRELATED NAKAGAMI-M FADING CHANNELS USING SUM OF-SINUSOIDS METHOD

ON THE SIMULATIONS OF CORRELATED NAKAGAMI-M FADING CHANNELS USING SUM OF-SINUSOIDS METHOD A Thesis Presented to the Faculty of Graduate School of University of Missouri-Columbia In Partial Fulfillment

### Communications and Signal Processing Spring 2017 MSE Exam

Communications and Signal Processing Spring 2017 MSE Exam Please obtain your Test ID from the following table. You must write your Test ID and name on each of the pages of this exam. A page with missing

### 7 The Waveform Channel

7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel

### EE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet

EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five

### EE6604 Personal & Mobile Communications. Week 12a. Power Spectrum of Digitally Modulated Signals

EE6604 Personal & Mobile Communications Week 12a Power Spectrum of Digitally Modulated Signals 1 POWER SPECTRUM OF BANDPASS SIGNALS A bandpass modulated signal can be written in the form s(t) = R { s(t)e

### Outline. 1 Path-Loss, Large-Scale-Fading, Small-Scale Fading. 2 Simplified Models. 3 Multipath Fading Channel

Outline Digital Communiations Leture 11 Wireless Channels Pierluigi SALVO OSSI Department of Industrial and Information Engineering Seond University of Naples Via oma 29, 81031 Aversa CE), Italy homepage:

### Lecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process

Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes

### On Coding for Orthogonal Frequency Division Multiplexing Systems

On Coding for Orthogonal Frequency Division Multiplexing Systems Alan Clark Department of Electrical and Computer Engineering A thesis presented for the degree of Doctor of Philosophy University of Canterbury

### EE6604 Personal & Mobile Communications. Week 13. Multi-antenna Techniques

EE6604 Personal & Mobile Communications Week 13 Multi-antenna Techniques 1 Diversity Methods Diversity combats fading by providing the receiver with multiple uncorrelated replicas of the same information

### MATHEMATICAL TOOLS FOR DIGITAL TRANSMISSION ANALYSIS

ch03.qxd 1/9/03 09:14 AM Page 35 CHAPTER 3 MATHEMATICAL TOOLS FOR DIGITAL TRANSMISSION ANALYSIS 3.1 INTRODUCTION The study of digital wireless transmission is in large measure the study of (a) the conversion

### 16.584: Random (Stochastic) Processes

1 16.584: Random (Stochastic) Processes X(t): X : RV : Continuous function of the independent variable t (time, space etc.) Random process : Collection of X(t, ζ) : Indexed on another independent variable

### Signal Processing Signal and System Classifications. Chapter 13

Chapter 3 Signal Processing 3.. Signal and System Classifications In general, electrical signals can represent either current or voltage, and may be classified into two main categories: energy signals

### LOPE3202: Communication Systems 10/18/2017 2

By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.

### 5 Analog carrier modulation with noise

5 Analog carrier modulation with noise 5. Noisy receiver model Assume that the modulated signal x(t) is passed through an additive White Gaussian noise channel. A noisy receiver model is illustrated in

### ECE-340, Spring 2015 Review Questions

ECE-340, Spring 2015 Review Questions 1. Suppose that there are two categories of eggs: large eggs and small eggs, occurring with probabilities 0.7 and 0.3, respectively. For a large egg, the probabilities

### Fundamentals of Noise

Fundamentals of Noise V.Vasudevan, Department of Electrical Engineering, Indian Institute of Technology Madras Noise in resistors Random voltage fluctuations across a resistor Mean square value in a frequency

### Stochastic Processes

Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic

### Multiple Antennas. Mats Bengtsson, Björn Ottersten. Channel characterization and modeling 1 September 8, Signal KTH Research Focus

Multiple Antennas Channel Characterization and Modeling Mats Bengtsson, Björn Ottersten Channel characterization and modeling 1 September 8, 2005 Signal Processing @ KTH Research Focus Channel modeling

### Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno

Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.

### Lecture 6: Modeling of MIMO Channels Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH

: Theoretical Foundations of Wireless Communications 1 Wednesday, May 11, 2016 9:00-12:00, Conference Room SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 1 Overview

### Perfect Modeling and Simulation of Measured Spatio-Temporal Wireless Channels

Perfect Modeling and Simulation of Measured Spatio-Temporal Wireless Channels Matthias Pätzold Agder University College Faculty of Engineering and Science N-4876 Grimstad, Norway matthiaspaetzold@hiano

### Principles of Communications

Principles of Communications Weiyao Lin, PhD Shanghai Jiao Tong University Chapter 4: Analog-to-Digital Conversion Textbook: 7.1 7.4 2010/2011 Meixia Tao @ SJTU 1 Outline Analog signal Sampling Quantization

### ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen. D. van Alphen 1

ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen D. van Alphen 1 Lecture 10 Overview Part 1 Review of Lecture 9 Continuing: Systems with Random Inputs More about Poisson RV s Intro. to Poisson Processes

### Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes

Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of

### Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

Module Signal Representation and Baseband Processing Version ECE II, Kharagpur Lesson 8 Response of Linear System to Random Processes Version ECE II, Kharagpur After reading this lesson, you will learn

### 2.1 Basic Concepts Basic operations on signals Classication of signals

Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz 2017-18 Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1

### Square Root Raised Cosine Filter

Wireless Information Transmission System Lab. Square Root Raised Cosine Filter Institute of Communications Engineering National Sun Yat-sen University Introduction We consider the problem of signal design

### Analog Electronics 2 ICS905

Analog Electronics 2 ICS905 G. Rodriguez-Guisantes Dépt. COMELEC http://perso.telecom-paristech.fr/ rodrigez/ens/cycle_master/ November 2016 2/ 67 Schedule Radio channel characteristics ; Analysis and

### This examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 6 December 2006 This examination consists of

### 2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf

Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic

### Problem Sheet 1 Examples of Random Processes

RANDOM'PROCESSES'AND'TIME'SERIES'ANALYSIS.'PART'II:'RANDOM'PROCESSES' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Problem'Sheets' Problem Sheet 1 Examples of Random Processes 1. Give

### A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model

A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model Matthias Pätzold and Bjørn Olav Hogstad Faculty of Engineering and Science Agder University College 4898 Grimstad, Norway

### Lecture 6: Modeling of MIMO Channels Theoretical Foundations of Wireless Communications 1

Fading : Theoretical Foundations of Wireless Communications 1 Thursday, May 3, 2018 9:30-12:00, Conference Room SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 23 Overview

### PROBABILITY AND RANDOM PROCESSESS

PROBABILITY AND RANDOM PROCESSESS SOLUTIONS TO UNIVERSITY QUESTION PAPER YEAR : JUNE 2014 CODE NO : 6074 /M PREPARED BY: D.B.V.RAVISANKAR ASSOCIATE PROFESSOR IT DEPARTMENT MVSR ENGINEERING COLLEGE, NADERGUL

### Name of the Student: Problems on Discrete & Continuous R.Vs

Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan

### EE6604 Personal & Mobile Communications. Week 15. OFDM on AWGN and ISI Channels

EE6604 Personal & Mobile Communications Week 15 OFDM on AWGN and ISI Channels 1 { x k } x 0 x 1 x x x N- 2 N- 1 IDFT X X X X 0 1 N- 2 N- 1 { X n } insert guard { g X n } g X I n { } D/A ~ si ( t) X g X

### 5.9 Power Spectral Density Gaussian Process 5.10 Noise 5.11 Narrowband Noise

Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability

### P 1.5 X 4.5 / X 2 and (iii) The smallest value of n for

DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X

### 2.1 Introduction 2.22 The Fourier Transform 2.3 Properties of The Fourier Transform 2.4 The Inverse Relationship between Time and Frequency 2.

Chapter2 Fourier Theory and Communication Signals Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Contents 2.1 Introduction 2.22

### Review of Doppler Spread The response to exp[2πift] is ĥ(f, t) exp[2πift]. ĥ(f, t) = β j exp[ 2πifτ j (t)] = exp[2πid j t 2πifτ o j ]

Review of Doppler Spread The response to exp[2πift] is ĥ(f, t) exp[2πift]. ĥ(f, t) = β exp[ 2πifτ (t)] = exp[2πid t 2πifτ o ] Define D = max D min D ; The fading at f is ĥ(f, t) = 1 T coh = 2D exp[2πi(d

### Copyright license. Exchanging Information with the Stars. The goal. Some challenges

Copyright license Exchanging Information with the Stars David G Messerschmitt Department of Electrical Engineering and Computer Sciences University of California at Berkeley messer@eecs.berkeley.edu Talk

### Estimation of the Capacity of Multipath Infrared Channels

Estimation of the Capacity of Multipath Infrared Channels Jeffrey B. Carruthers Department of Electrical and Computer Engineering Boston University jbc@bu.edu Sachin Padma Department of Electrical and

### Chapter 4: Continuous channel and its capacity

meghdadi@ensil.unilim.fr Reference : Elements of Information Theory by Cover and Thomas Continuous random variable Gaussian multivariate random variable AWGN Band limited channel Parallel channels Flat

### Residual carrier frequency offset and symbol timing offset in narrow-band OFDM modulation

Residual carrier frequency offset and symbol timing offset in narrow-band OFDM modulation Master s thesis iko Ohukainen 2264750 Department of Mathematical Sciences University of Oulu Spring 2017 Contents

### ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the

he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus

### ECE 541 Stochastic Signals and Systems Problem Set 11 Solution

ECE 54 Stochastic Signals and Systems Problem Set Solution Problem Solutions : Yates and Goodman,..4..7.3.3.4.3.8.3 and.8.0 Problem..4 Solution Since E[Y (t] R Y (0, we use Theorem.(a to evaluate R Y (τ

### SIGNAL PROCESSING FOR COMMUNICATIONS

SIGNAL PROCESSING FOR COMMUNICATIONS ii Delft University of Technology Faculty of Electrical Engineering, Mathematics, and Computer Science Circuits and Systems SIGNAL PROCESSING FOR COMMUNICATIONS ET4

### Name of the Student: Problems on Discrete & Continuous R.Vs

Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the

### Es e j4φ +4N n. 16 KE s /N 0. σ 2ˆφ4 1 γ s. p(φ e )= exp 1 ( 2πσ φ b cos N 2 φ e 0

Problem 6.15 : he received signal-plus-noise vector at the output of the matched filter may be represented as (see (5-2-63) for example) : r n = E s e j(θn φ) + N n where θ n =0,π/2,π,3π/2 for QPSK, and

### LECTURE 16 AND 17. Digital signaling on frequency selective fading channels. Notes Prepared by: Abhishek Sood

ECE559:WIRELESS COMMUNICATION TECHNOLOGIES LECTURE 16 AND 17 Digital signaling on frequency selective fading channels 1 OUTLINE Notes Prepared by: Abhishek Sood In section 2 we discuss the receiver design

### Summary II: Modulation and Demodulation

Summary II: Modulation and Demodulation Instructor : Jun Chen Department of Electrical and Computer Engineering, McMaster University Room: ITB A1, ext. 0163 Email: junchen@mail.ece.mcmaster.ca Website:

### Parameter Estimation

1 / 44 Parameter Estimation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay October 25, 2012 Motivation System Model used to Derive

### Digital Communications

Digital Communications Chapter 5 Carrier and Symbol Synchronization Po-Ning Chen, Professor Institute of Communications Engineering National Chiao-Tung University, Taiwan Digital Communications Ver 218.7.26

### Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this

### ECE353: Probability and Random Processes. Lecture 18 - Stochastic Processes

ECE353: Probability and Random Processes Lecture 18 - Stochastic Processes Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu From RV

### Ranging detection algorithm for indoor UWB channels

Ranging detection algorithm for indoor UWB channels Choi Look LAW and Chi XU Positioning and Wireless Technology Centre Nanyang Technological University 1. Measurement Campaign Objectives Obtain a database

### 3. ESTIMATION OF SIGNALS USING A LEAST SQUARES TECHNIQUE

3. ESTIMATION OF SIGNALS USING A LEAST SQUARES TECHNIQUE 3.0 INTRODUCTION The purpose of this chapter is to introduce estimators shortly. More elaborated courses on System Identification, which are given

### Signals and Spectra - Review

Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs

### Signal Design for Band-Limited Channels

Wireless Information Transmission System Lab. Signal Design for Band-Limited Channels Institute of Communications Engineering National Sun Yat-sen University Introduction We consider the problem of signal

### Massachusetts Institute of Technology

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ, April 1, 010 QUESTION BOOKLET Your

### Chapter 2 Random Processes

Chapter 2 Random Processes 21 Introduction We saw in Section 111 on page 10 that many systems are best studied using the concept of random variables where the outcome of a random experiment was associated

### ECE 636: Systems identification

ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental

### Fourier Methods in Digital Signal Processing Final Exam ME 579, Spring 2015 NAME

Fourier Methods in Digital Signal Processing Final Exam ME 579, Instructions for this CLOSED BOOK EXAM 2 hours long. Monday, May 8th, 8-10am in ME1051 Answer FIVE Questions, at LEAST ONE from each section.

### Kevin Buckley a i. communication. information source. modulator. encoder. channel. encoder. information. demodulator decoder. C k.

Kevin Buckley - -4 ECE877 Information Theory & Coding for Digital Communications Villanova University ECE Department Prof. Kevin M. Buckley Lecture Set Review of Digital Communications, Introduction to

### Fourier Analysis Linear transformations and lters. 3. Fourier Analysis. Alex Sheremet. April 11, 2007

Stochastic processes review 3. Data Analysis Techniques in Oceanography OCP668 April, 27 Stochastic processes review Denition Fixed ζ = ζ : Function X (t) = X (t, ζ). Fixed t = t: Random Variable X (ζ)

### A GENERALISED (M, N R ) MIMO RAYLEIGH CHANNEL MODEL FOR NON- ISOTROPIC SCATTERER DISTRIBUTIONS

A GENERALISED (M, N R MIMO RAYLEIGH CHANNEL MODEL FOR NON- ISOTROPIC SCATTERER DISTRIBUTIONS David B. Smith (1, Thushara D. Abhayapala (2, Tim Aubrey (3 (1 Faculty of Engineering (ICT Group, University