Lecture 4 Capacity of Wireless Channels
|
|
- Vivian Dixon
- 6 years ago
- Views:
Transcription
1 Lecture 4 Capacity of Wireless Channels I-Hsiang Wang ihwang@ntu.edu.tw 3/0, 014
2 What we have learned So far: looked at specific schemes and techniques Lecture : point-to-point wireless channel - Diversity: combat fading by exploiting inherent diversity - Coding: combat noise, and further exploits degrees of freedom Lecture 3: cellular system - Multiple access: TDMA, CDMA, OFDMA - Interference management: orthogonalization (partial frequency reuse), treat-interference-as-noise (interference averaging)
3 Information Theory Is there a framework to - Compare all schemes and techniques fairly? - Assert what is the fundamental limit on how much rate can be reliably delivered over a wireless channel? Information theory! - Provides a fundamental limit to (coded) performance - Identifies the impact of channel resources on performance - Suggest novel techniques to communicate over wireless channels Information theory provides the basis for the modern development of wireless communication 3
4 Historical Perspective G. Marconi C. Shannon First radio built 100+ years ago Great stride in technology But design was somewhat ad-hoc Information theory: every channel has a capacity Provides a systematic view of all communication problems Engineering meets science New points of view arise 4
5 Modern View on Multipath Fading Channel quality Time Classical view: fading channels are unreliable - Diversity techniques: average out the variation Modern view: exploit fading to gain spectral efficiency - Thanks to the study on fading channel through the lens of information theory! 5
6 Plot Use a heuristic argument (geometric) to introduce the capacity of the AWGN channel Discuss the two key resources in the AWGN channel: - Power - Bandwidth The AWGN channel capacity serves as a building block towards fading channel capacity: - Slow fading channel: outage capacity - Fast fading channel: ergodic capacity 6
7 AWGN Channel Capacity Outline Resources of the AWGN Channel Capacity of some LTI Gaussian Channels Capacity of Fading Channels 7
8 AWGN Channel Capacity 8
9 Channel Capacity Capacity := the highest data rate can be delivered reliably over a channel - Reliably Vanishing error probability Before Shannon, it was widely believed that: - to communicate with error probability 0 - data rate must also 0 Repetition coding (with M-level PAM) over N time slots on AWGN channel: r! 6N M 3 SNR - Error probability Q - Data rate = log M N - As long as M N ⅓, the error probability 0 as N - But, the data rate! =! log N! still 0 as N 3N 9
10 Channel Coding Theorem For every memoryless channel, there is a definite number C that is computable such that: - If the data rate R < C, then there exists a coding scheme that can deliver rate R data over the channel with error probability 0 as the coding block length N - Conversely, if the data rate R > C, then no matter what coding scheme is used, the error probability 1 as N We shall focus on the additive white Gaussian noise (AWGN) channel - Give a heuristic argument to derive the AWGN channel capacity 10
11 AWGN Channel Power constraint: NX x[n] apple NP n=1 x[n] z[n] N (0, ) y[n] =x[n]+z[n] We consider real-valued Gaussian channel As mentioned earlier, repetition coding yield zero rate if the error probability is required to vanish as N Because all codewords are spread on a single dimension in an N-dimensional space How to do better? 11
12 Sphere Packing Interpretation y = x + z R N By the law of large numbers, as N, most y will lie p N(P + ) inside the N-dimensional sphere of radius p N(P + ) p N Also by the LLN, as N, y will lie near the surface of the N-dimensional sphere centered at x with radius p N Vanishing error probability non-overlapping spheres How many non-overlapping spheres can be packed into the large sphere? 1
13 Why Repetition Coding is Bad R N p N(P + ) y = x + z It only uses one dimension out of N! 13
14 Capacity Upper Bound R N p N(P + ) p N y = x + z Maximum # of non-overlapping spheres = Maximum # of codewords that can be reliably delivered p N(P + NR ) N apple p N N =) R apple 1 N log p N(P + ) N p N N! = 1 log 1+ P This is hence an upper bound of the capacity C. How to achieve it? 14
15 Achieving Capacity (1/3) (random) Encoding: randomly generate NR codewords {x1, x,...} lying inside the x-sphere of radius Decoding: Performance analysis: WLOG let x 1 is sent - By the LLN, := P P + y! MMSE! y! Nearest Neighbor! bx y x 1 = w +( 1)x 1 N +( 1) NP = N P - As long as αy lies inside the uncertainty sphere centered at x1 r with radius!!!, decoding will be correct - Pairwise error probability (see next slide) = N P P + P + p NP P + N/ 15
16 Achieving Capacity (/3) x-sphere p NP When does an error occur? Ans: when another codeword falls inside the uncertainty sphere of x1 r N P P + x 1 What is that probability (pairwise error probability)? Ans: the ratio of the volume of the two spheres Pr {x 1! x } = = p NP /(P + p N NP P + N/ ) N Union bound: Total error probability apple NR P + N/ 16
17 Achieving Capacity (3/3) Total error probability (by union bound) Pr {E} apple NR P + N/ = N R+ 1 log 1 1+ P!! As long as the following holds, Pr {E}! 0 as N!1 R< 1 log 1+ P Hence, indeed the capacity is C = 1 1+ log P bits per symbol time 17
18 Resources of AWGN Channel 18
19 Continuous- Time AWGN Channel System parameters: - Power constraint: P watts; Bandwidth: W Hz - Spectral density of the white Gaussian noise: N0/ Equivalent discrete-time baseband channel (complex) - 1 complex symbol = real symbols Capacity: Power constraint: NX x[n] apple NP n=1 x[n] z[n] CN(0,N 0 W ) y[n] =x[n]+z[n] C AWGN (P, W) = 1 log 1+ P/ bits per symbol time N 0 W/ SNR := P/N = W log 1+ P 0 W SNR per complex symbol bits/s = log (1 + SNR) bits/s/hz N 0 W 19
20 Complex AWGN Channel Capacity C AWGN (P, W) =W log 1+ P N 0 W bits/s = log (1 + SNR) bits/s/hz Spectral Efficiency The capacity formula provides a high-level way of thinking about how the performance fundamentally depends on the basic resources available in the channel No need to go into details of specific coding and modulation schemes Basic resources: power P and bandwidth W 0
21 Power SNR = P N 0 W log (1 + SNR) Fix W: High SNR: - Logarithmic growth with power Low SNR: - Linear growth with power SNR C = log(1 + SNR) log SNR C = log(1 + SNR) SNR log e 1
22 Bandwidth Fix P C(W )=W log 1+ P N 0 W W P N 0 W log e = P N 0 log e P N 0 log e Power limited region 1 C(W ) (Mbps) Capacity Limit for W 0.4 Bandwidth limited region Bandwidth W (MHz) 5 30
23 Bandwidth- limited vs. Power- limited C AWGN (P, W) =W log SNR = When SNR 1: (Power-limited regime) C AWGN (P, W) W P N 0 W 1+ P N 0 W - Linear in power; Insensitive to bandwidth When SNR 1: (Bandwidth-limited regime) - Logarithmic in power; Approximately linear in bandwidth P N 0 W C AWGN (P, W) W log bits/s log e = P N 0 log e P N 0 W 3
24 Capacity of Some LTI Gaussian Channels 4
25 SIMO Channel x h y y = hx + w C L # h MRC, h # ey = h x + ew, ew CN 0, Power constraint: P w CN 0, I L MRC is a lossless operation: we can generate y from ey : y = ey (h/ h ) Hence the SIMO channel capacity is equal to the capacity of the equivalent AWGN channel, which is C SIMO = log 1+ h P Power gain due to Rx beamforming 5
26 MISO Channel x h h = h 1 y h y = h x + w C, # Tx Beamforming x = xh/ h # y = x h + w x, h C L Power constraint: NX x apple NP n=1 Goal: maximize the received power h* x - The answer is h P! (check. Hint: Cauchy Schwarz inequality) Achieved by Tx beamforming - Send a scalar symbol x on the direction of h - Power constraint on x : still P Capacity: C MISO = log 1+ h P 6
27 Frequency- Selective Channel y[m] = LX 1 h l x[m l]+w[m] l=0 Key idea 1: use OFDM to convert the channel with ISI into a bunch of parallel AWGN channels - But there is loss/overhead due to cyclic prefix Key idea : CP overhead 0 as N c First focus on finding the capacity of parallel AWGN channels of any finite Nc Then take N c to find the capacity of the frequencyselective channel 7
28 Recap: OFDM x [1] = d[n L + 1] y[1] x [L 1] = d[n 1] y[l 1] d 0 d[0] Cyclic prefix x [L] = d[0] Channel y[l] Remove prefix y[l] ỹ 0 IDFT DFT d N 1 d[n 1] x [N + L 1] = d[n 1] y[n + L 1] y [N + L 1] ỹ N 1 y := y[l : N c + L 1], w := w[l : N c + L 1], h := h 0 h 1 h L T ey n = e h n e dn + ew n, n =0, 1,...,N c 1 Nc parallel AWGN channels ey n := DFT (y) n, e dn := DFT (d) n, ew n := DFT (w) n, e hn := p N c DFT (h) n 8
29 Parallel AWGN Channels e h0 ew 0 [m] Parallel Channels ed 0 [m] ey 0 [m] ey n = e h n e dn + ew n, n [0 : 1 : N c 1] ed 1 [m] e h1 ew 1 [m] ey 1 [m] Equivalent Vector Channel ey = H e d e + ew ew CN 0, eh = diag eh0,..., e h Nc 1 I... ed Nc 1[m] e hnc 1 ew Nc 1[m] ey Nc m =1,,...,M (M channel uses) 1[m] Power Constraint MX d[n] e apple MN c P n=1 Due to Parseval theorem of DFT 9
30 Independent Uses of Parallel Channels One way to code over such parallel channels (a special case of a vector channel): treat each channel separately - It turns out that coding across parallel channels does not help! Power allocation: - Each of the Nc channels get a portion of the total power NX - c 1 Channel n gets power Pn, which must satisfy P n apple N c P For a given power allocation {P n}, the following rate can be achieved: R = NX c 1 n=0 log 1+ e h n P n! n=0 30
31 Optimal Power Allocation Power allocation problem: max P 0,...,P N c 1 subject to NX c 1 n=0 NX c 1 n=0 log 1+ e h n P n It can be solved explicitly by Lagrangian methods!, P n = N c P, P n 0, n =0,...,N c 1 Final solution: let (x)+ := max(x, 0) P n = e h n! +, satisfies NX c 1 n=0 e h n! + = N c P 31
32 Water]illing e h n P* 1 = 0 * P * P 3 Note: e h n = H b nw N c Subcarrier n Baseband frequency response at f = nw/nc 3
33 Frequency- Selective Channel Capacity Final step: making N c - Replace all!!!!! by Hb(f), summation over [0 : Nc 1] becomes integration from 0 to W Power allocation problem becomes Optimal solution becomes P (f) = max P (f) Z W 0 subject to e hn = H b nw N c log H(f) 1+ H(f) P (f) Z W 0 +, df, P (f) =P, P(f) 0, f [0,W] satisfies Z W 0 H(f) + df = P 33
34 Water]illing over the Frequecy Spectrum H(f) P *( f ) W 0.W 0 0.W 0.4W Frequency ( f ) 34
Lecture 4 Capacity of Wireless Channels
Lecture 4 Capacity of Wireless Channels I-Hsiang Wang ihwang@ntu.edu.tw 3/0, 014 What we have learned So far: looked at specific schemes and techniques Lecture : point-to-point wireless channel - Diversity:
More informationLecture 2. Capacity of the Gaussian channel
Spring, 207 5237S, Wireless Communications II 2. Lecture 2 Capacity of the Gaussian channel Review on basic concepts in inf. theory ( Cover&Thomas: Elements of Inf. Theory, Tse&Viswanath: Appendix B) AWGN
More informationLecture 6 Channel Coding over Continuous Channels
Lecture 6 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 9, 015 1 / 59 I-Hsiang Wang IT Lecture 6 We have
More informationAppendix B Information theory from first principles
Appendix B Information theory from first principles This appendix discusses the information theory behind the capacity expressions used in the book. Section 8.3.4 is the only part of the book that supposes
More informationLecture 4. Capacity of Fading Channels
1 Lecture 4. Capacity of Fading Channels Capacity of AWGN Channels Capacity of Fading Channels Ergodic Capacity Outage Capacity Shannon and Information Theory Claude Elwood Shannon (April 3, 1916 February
More information16.36 Communication Systems Engineering
MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.36: Communication
More informationELEC546 Review of Information Theory
ELEC546 Review of Information Theory Vincent Lau 1/1/004 1 Review of Information Theory Entropy: Measure of uncertainty of a random variable X. The entropy of X, H(X), is given by: If X is a discrete random
More informationLecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH
MIMO : MIMO Theoretical Foundations of Wireless Communications 1 Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview MIMO
More informationLecture 8: Orthogonal Frequency Division Multiplexing (OFDM)
Communication Systems Lab Spring 2017 ational Taiwan University Lecture 8: Orthogonal Frequency Division Multiplexing (OFDM) Scribe: 謝秉昂 陳心如 Lecture Date: 4/26, 2017 Lecturer: I-Hsiang Wang 1 Outline 1
More informationSingle-User MIMO systems: Introduction, capacity results, and MIMO beamforming
Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming Master Universitario en Ingeniería de Telecomunicación I. Santamaría Universidad de Cantabria Contents Introduction Multiplexing,
More informationMultiple-Input Multiple-Output Systems
Multiple-Input Multiple-Output Systems What is the best way to use antenna arrays? MIMO! This is a totally new approach ( paradigm ) to wireless communications, which has been discovered in 95-96. Performance
More informationLecture 4 Channel Coding
Capacity and the Weak Converse Lecture 4 Coding I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 15, 2014 1 / 16 I-Hsiang Wang NIT Lecture 4 Capacity
More informationLecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH
: Antenna Diversity and Theoretical Foundations of Wireless Communications Wednesday, May 4, 206 9:00-2:00, Conference Room SIP Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication
More informationLecture 9: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH
: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1 Rayleigh Wednesday, June 1, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication
More informationLecture 4 Noisy Channel Coding
Lecture 4 Noisy Channel Coding I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 9, 2015 1 / 56 I-Hsiang Wang IT Lecture 4 The Channel Coding Problem
More informationLecture 9: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1
: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1 Rayleigh Friday, May 25, 2018 09:00-11:30, Kansliet 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless
More informationLecture 5: Channel Capacity. Copyright G. Caire (Sample Lectures) 122
Lecture 5: Channel Capacity Copyright G. Caire (Sample Lectures) 122 M Definitions and Problem Setup 2 X n Y n Encoder p(y x) Decoder ˆM Message Channel Estimate Definition 11. Discrete Memoryless Channel
More informationLecture 1: The Multiple Access Channel. Copyright G. Caire 12
Lecture 1: The Multiple Access Channel Copyright G. Caire 12 Outline Two-user MAC. The Gaussian case. The K-user case. Polymatroid structure and resource allocation problems. Copyright G. Caire 13 Two-user
More informationLecture 18: Gaussian Channel
Lecture 18: Gaussian Channel Gaussian channel Gaussian channel capacity Dr. Yao Xie, ECE587, Information Theory, Duke University Mona Lisa in AWGN Mona Lisa Noisy Mona Lisa 100 100 200 200 300 300 400
More informationLecture 6: Gaussian Channels. Copyright G. Caire (Sample Lectures) 157
Lecture 6: Gaussian Channels Copyright G. Caire (Sample Lectures) 157 Differential entropy (1) Definition 18. The (joint) differential entropy of a continuous random vector X n p X n(x) over R is: Z h(x
More informationLecture 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
More information18.2 Continuous Alphabet (discrete-time, memoryless) Channel
0-704: Information Processing and Learning Spring 0 Lecture 8: Gaussian channel, Parallel channels and Rate-distortion theory Lecturer: Aarti Singh Scribe: Danai Koutra Disclaimer: These notes have not
More informationMulti-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems
Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User
More informationChapter 9 Fundamental Limits in Information Theory
Chapter 9 Fundamental Limits in Information Theory Information Theory is the fundamental theory behind information manipulation, including data compression and data transmission. 9.1 Introduction o For
More informationLimited Feedback in Wireless Communication Systems
Limited Feedback in Wireless Communication Systems - Summary of An Overview of Limited Feedback in Wireless Communication Systems Gwanmo Ku May 14, 17, and 21, 2013 Outline Transmitter Ant. 1 Channel N
More informationDiversity Combining Techniques
Diversity Combining Techniques When the required signal is a combination of several plane waves (multipath), the total signal amplitude may experience deep fades (Rayleigh fading), over time or space.
More informationELEC546 MIMO Channel Capacity
ELEC546 MIMO Channel Capacity Vincent Lau Simplified Version.0 //2004 MIMO System Model Transmitter with t antennas & receiver with r antennas. X Transmitted Symbol, received symbol Channel Matrix (Flat
More informationLecture 7 MIMO Communica2ons
Wireless Communications Lecture 7 MIMO Communica2ons Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2014 1 Outline MIMO Communications (Chapter 10
More informationMultiple Antennas in Wireless Communications
Multiple Antennas in Wireless Communications Luca Sanguinetti Department of Information Engineering Pisa University luca.sanguinetti@iet.unipi.it April, 2009 Luca Sanguinetti (IET) MIMO April, 2009 1 /
More informationInterleave Division Multiple Access. Li Ping, Department of Electronic Engineering City University of Hong Kong
Interleave Division Multiple Access Li Ping, Department of Electronic Engineering City University of Hong Kong 1 Outline! Introduction! IDMA! Chip-by-chip multiuser detection! Analysis and optimization!
More informationPhysical Layer and Coding
Physical Layer and Coding Muriel Médard Professor EECS Overview A variety of physical media: copper, free space, optical fiber Unified way of addressing signals at the input and the output of these media:
More informationLECTURE 18. Lecture outline Gaussian channels: parallel colored noise inter-symbol interference general case: multiple inputs and outputs
LECTURE 18 Last time: White Gaussian noise Bandlimited WGN Additive White Gaussian Noise (AWGN) channel Capacity of AWGN channel Application: DS-CDMA systems Spreading Coding theorem Lecture outline Gaussian
More informationChapter 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
More informationMulti-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems
Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User
More informationApproximately achieving the feedback interference channel capacity with point-to-point codes
Approximately achieving the feedback interference channel capacity with point-to-point codes Joyson Sebastian*, Can Karakus*, Suhas Diggavi* Abstract Superposition codes with rate-splitting have been used
More informationECE Information theory Final (Fall 2008)
ECE 776 - Information theory Final (Fall 2008) Q.1. (1 point) Consider the following bursty transmission scheme for a Gaussian channel with noise power N and average power constraint P (i.e., 1/n X n i=1
More information12.4 Known Channel (Water-Filling Solution)
ECEn 665: Antennas and Propagation for Wireless Communications 54 2.4 Known Channel (Water-Filling Solution) The channel scenarios we have looed at above represent special cases for which the capacity
More informationEE 5407 Part II: Spatial Based Wireless Communications
EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang E-mail: rzhang@i2r.a-star.edu.sg Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture II: Receive Beamforming
More informationLecture 5 Channel Coding over Continuous Channels
Lecture 5 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 14, 2014 1 / 34 I-Hsiang Wang NIT Lecture 5 From
More informationWireless Communications Lecture 10
Wireless Communications Lecture 1 [SNR per symbol and SNR per bit] SNR = P R N B = E s N BT s = E b N BT b For BPSK: T b = T s, E b = E s, and T s = 1/B. Raised cosine pulse shaper for other pulses. T
More informationHierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks
Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks Ayfer Özgür, Olivier Lévêque Faculté Informatique et Communications Ecole Polytechnique Fédérale de Lausanne 05 Lausanne, Switzerland
More informationL interférence dans les réseaux non filaires
L interférence dans les réseaux non filaires Du contrôle de puissance au codage et alignement Jean-Claude Belfiore Télécom ParisTech 7 mars 2013 Séminaire Comelec Parts Part 1 Part 2 Part 3 Part 4 Part
More informationOne Lesson of Information Theory
Institut für One Lesson of Information Theory Prof. Dr.-Ing. Volker Kühn Institute of Communications Engineering University of Rostock, Germany Email: volker.kuehn@uni-rostock.de http://www.int.uni-rostock.de/
More informationHarnessing Interaction in Bursty Interference Networks
215 IEEE Hong Kong-Taiwan Joint Workshop on Information Theory and Communications Harnessing Interaction in Bursty Interference Networks I-Hsiang Wang NIC Lab, NTU GICE 1/19, 215 Modern Wireless: Grand
More informationSecrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View
Secrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View P. Mohapatra 9 th March 2013 Outline Motivation Problem statement Achievable scheme 1 Weak interference
More informationOn the Secrecy Capacity of Fading Channels
On the Secrecy Capacity of Fading Channels arxiv:cs/63v [cs.it] 7 Oct 26 Praveen Kumar Gopala, Lifeng Lai and Hesham El Gamal Department of Electrical and Computer Engineering The Ohio State University
More informationWhen does vectored Multiple Access Channels (MAC) optimal power allocation converge to an FDMA solution?
When does vectored Multiple Access Channels MAC optimal power allocation converge to an FDMA solution? Vincent Le Nir, Marc Moonen, Jan Verlinden, Mamoun Guenach Abstract Vectored Multiple Access Channels
More informationPerformance Analysis of Physical Layer Network Coding
Performance Analysis of Physical Layer Network Coding by Jinho Kim A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Electrical Engineering: Systems)
More informationPOWER ALLOCATION AND OPTIMAL TX/RX STRUCTURES FOR MIMO SYSTEMS
POWER ALLOCATION AND OPTIMAL TX/RX STRUCTURES FOR MIMO SYSTEMS R. Cendrillon, O. Rousseaux and M. Moonen SCD/ESAT, Katholiee Universiteit Leuven, Belgium {raphael.cendrillon, olivier.rousseaux, marc.moonen}@esat.uleuven.ac.be
More informationSecure Degrees of Freedom of the MIMO Multiple Access Wiretap Channel
Secure Degrees of Freedom of the MIMO Multiple Access Wiretap Channel Pritam Mukherjee Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 074 pritamm@umd.edu
More information2. Wireless Fading Channels
Fundamentals of Wireless Comm. Andreas Biri, D-ITET.07.7. Introduction Modulation (frequency shift to carrier frequency): - enables multiple (slightly shifted) simultaneous channels - better channel characteristics
More informationELG7177: MIMO Comunications. Lecture 8
ELG7177: MIMO Comunications Lecture 8 Dr. Sergey Loyka EECS, University of Ottawa S. Loyka Lecture 8, ELG7177: MIMO Comunications 1 / 32 Multi-User Systems Can multiple antennas offer advantages for multi-user
More informationMulticarrier transmission DMT/OFDM
W. Henkel, International University Bremen 1 Multicarrier transmission DMT/OFDM DMT: Discrete Multitone (wireline, baseband) OFDM: Orthogonal Frequency Division Multiplex (wireless, with carrier, passband)
More informationSolutions to Homework Set #4 Differential Entropy and Gaussian Channel
Solutions to Homework Set #4 Differential Entropy and Gaussian Channel 1. Differential entropy. Evaluate the differential entropy h(x = f lnf for the following: (a Find the entropy of the exponential density
More informationExploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless
Exploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless SPAWC 2003 Rome, Italy June 18, 2003 E. Yoon, M. Vu and Arogyaswami Paulraj Stanford University Page 1 Outline Introduction
More informationMMSE estimation and lattice encoding/decoding for linear Gaussian channels. Todd P. Coleman /22/02
MMSE estimation and lattice encoding/decoding for linear Gaussian channels Todd P. Coleman 6.454 9/22/02 Background: the AWGN Channel Y = X + N where N N ( 0, σ 2 N ), 1 n ni=1 X 2 i P X. Shannon: capacity
More informationShannon s noisy-channel theorem
Shannon s noisy-channel theorem Information theory Amon Elders Korteweg de Vries Institute for Mathematics University of Amsterdam. Tuesday, 26th of Januari Amon Elders (Korteweg de Vries Institute for
More informationMaximum Achievable Diversity for MIMO-OFDM Systems with Arbitrary. Spatial Correlation
Maximum Achievable Diversity for MIMO-OFDM Systems with Arbitrary Spatial Correlation Ahmed K Sadek, Weifeng Su, and K J Ray Liu Department of Electrical and Computer Engineering, and Institute for Systems
More informationError Exponent Region for Gaussian Broadcast Channels
Error Exponent Region for Gaussian Broadcast Channels Lihua Weng, S. Sandeep Pradhan, and Achilleas Anastasopoulos Electrical Engineering and Computer Science Dept. University of Michigan, Ann Arbor, MI
More informationPrinciples of Communications
Principles of Communications Weiyao Lin Shanghai Jiao Tong University Chapter 10: Information Theory Textbook: Chapter 12 Communication Systems Engineering: Ch 6.1, Ch 9.1~ 9. 92 2009/2010 Meixia Tao @
More informationCapacity of multiple-input multiple-output (MIMO) systems in wireless communications
15/11/02 Capacity of multiple-input multiple-output (MIMO) systems in wireless communications Bengt Holter Department of Telecommunications Norwegian University of Science and Technology 1 Outline 15/11/02
More informationSimultaneous SDR Optimality via a Joint Matrix Decomp.
Simultaneous SDR Optimality via a Joint Matrix Decomposition Joint work with: Yuval Kochman, MIT Uri Erez, Tel Aviv Uni. May 26, 2011 Model: Source Multicasting over MIMO Channels z 1 H 1 y 1 Rx1 ŝ 1 s
More informationRandom Matrices and Wireless Communications
Random Matrices and Wireless Communications Jamie Evans Centre for Ultra-Broadband Information Networks (CUBIN) Department of Electrical and Electronic Engineering University of Melbourne 3.5 1 3 0.8 2.5
More informationAdvanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung
Advanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung Dr.-Ing. Carsten Bockelmann Institute for Telecommunications and High-Frequency Techniques Department of Communications
More informationEE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15
EE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15 1. Cascade of Binary Symmetric Channels The conditional probability distribution py x for each of the BSCs may be expressed by the transition probability
More informationEnergy State Amplification in an Energy Harvesting Communication System
Energy State Amplification in an Energy Harvesting Communication System Omur Ozel Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland College Park, MD 20742 omur@umd.edu
More informationDiversity-Multiplexing Tradeoff of Asynchronous Cooperative Diversity in Wireless Networks
Diversity-Multiplexing Tradeoff of Asynchronous Cooperative Diversity in Wireless Networks Shuangqing Wei Abstract Synchronization of relay nodes is an important and critical issue in exploiting cooperative
More information19. Channel coding: energy-per-bit, continuous-time channels
9. Channel coding: energy-per-bit, continuous-time channels 9. Energy per bit Consider the additive Gaussian noise channel: Y i = X i + Z i, Z i N ( 0, ). (9.) In the last lecture, we analyzed the maximum
More informationLecture 15: Thu Feb 28, 2019
Lecture 15: Thu Feb 28, 2019 Announce: HW5 posted Lecture: The AWGN waveform channel Projecting temporally AWGN leads to spatially AWGN sufficiency of projection: irrelevancy theorem in waveform AWGN:
More informationEE 4TM4: Digital Communications II. Channel Capacity
EE 4TM4: Digital Communications II 1 Channel Capacity I. CHANNEL CODING THEOREM Definition 1: A rater is said to be achievable if there exists a sequence of(2 nr,n) codes such thatlim n P (n) e (C) = 0.
More informationApproximate Capacity of Fast Fading Interference Channels with no CSIT
Approximate Capacity of Fast Fading Interference Channels with no CSIT Joyson Sebastian, Can Karakus, Suhas Diggavi Abstract We develop a characterization of fading models, which assigns a number called
More informationAdvanced 3 G and 4 G Wireless Communication Prof. Aditya K Jagannathan Department of Electrical Engineering Indian Institute of Technology, Kanpur
Advanced 3 G and 4 G Wireless Communication Prof. Aditya K Jagannathan Department of Electrical Engineering Indian Institute of Technology, Kanpur Lecture - 19 Multi-User CDMA Uplink and Asynchronous CDMA
More informationECS455: Chapter 5 OFDM. ECS455: Chapter 5 OFDM. OFDM: Overview. OFDM Applications. Dr.Prapun Suksompong prapun.com/ecs455
ECS455: Chapter 5 OFDM OFDM: Overview Let S = (S 1, S 2,, S ) contains the information symbols. S IFFT FFT Inverse fast Fourier transform Fast Fourier transform 1 Dr.Prapun Suksompong prapun.com/ecs455
More informationDiversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT. ECE 559 Presentation Hoa Pham Dec 3, 2007
Diversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT ECE 559 Presentation Hoa Pham Dec 3, 2007 Introduction MIMO systems provide two types of gains Diversity Gain: each path from a transmitter
More informationInteractive Interference Alignment
Interactive Interference Alignment Quan Geng, Sreeram annan, and Pramod Viswanath Coordinated Science Laboratory and Dept. of ECE University of Illinois, Urbana-Champaign, IL 61801 Email: {geng5, kannan1,
More informationUpper Bounds on the Capacity of Binary Intermittent Communication
Upper Bounds on the Capacity of Binary Intermittent Communication Mostafa Khoshnevisan and J. Nicholas Laneman Department of Electrical Engineering University of Notre Dame Notre Dame, Indiana 46556 Email:{mhoshne,
More informationMulti User Detection I
January 12, 2005 Outline Overview Multiple Access Communication Motivation: What is MU Detection? Overview of DS/CDMA systems Concept and Codes used in CDMA CDMA Channels Models Synchronous and Asynchronous
More informationMITOCW ocw-6-451_ feb k_512kb-mp4
MITOCW ocw-6-451_4-261-09feb2005-220k_512kb-mp4 And what we saw was the performance was quite different in the two regimes. In power-limited regime, typically our SNR is much smaller than one, whereas
More informationOptimization of Modulation Constrained Digital Transmission Systems
University of Ottawa Optimization of Modulation Constrained Digital Transmission Systems by Yu Han A thesis submitted in fulfillment for the degree of Master of Applied Science in the Faculty of Engineering
More informationDistributed Lossless Compression. Distributed lossless compression system
Lecture #3 Distributed Lossless Compression (Reading: NIT 10.1 10.5, 4.4) Distributed lossless source coding Lossless source coding via random binning Time Sharing Achievability proof of the Slepian Wolf
More informationAn introduction to basic information theory. Hampus Wessman
An introduction to basic information theory Hampus Wessman Abstract We give a short and simple introduction to basic information theory, by stripping away all the non-essentials. Theoretical bounds on
More informationELEC E7210: Communication Theory. Lecture 10: MIMO systems
ELEC E7210: Communication Theory Lecture 10: MIMO systems Matrix Definitions, Operations, and Properties (1) NxM matrix a rectangular array of elements a A. an 11 1....... a a 1M. NM B D C E ermitian transpose
More informationEntropies & Information Theory
Entropies & Information Theory LECTURE I Nilanjana Datta University of Cambridge,U.K. See lecture notes on: http://www.qi.damtp.cam.ac.uk/node/223 Quantum Information Theory Born out of Classical Information
More informationShannon s Noisy-Channel Coding Theorem
Shannon s Noisy-Channel Coding Theorem Lucas Slot Sebastian Zur February 13, 2015 Lucas Slot, Sebastian Zur Shannon s Noisy-Channel Coding Theorem February 13, 2015 1 / 29 Outline 1 Definitions and Terminology
More informationInformation Theory for Wireless Communications. Lecture 10 Discrete Memoryless Multiple Access Channel (DM-MAC): The Converse Theorem
Information Theory for Wireless Communications. Lecture 0 Discrete Memoryless Multiple Access Channel (DM-MAC: The Converse Theorem Instructor: Dr. Saif Khan Mohammed Scribe: Antonios Pitarokoilis I. THE
More informationChannel State Information in Multiple Antenna Systems. Jingnong Yang
Channel State Information in Multiple Antenna Systems A Thesis Presented to The Academic Faculty by Jingnong Yang In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of
More informationSingle-Carrier Block Transmission With Frequency-Domain Equalisation
ELEC6014 RCNSs: Additional Topic Notes Single-Carrier Block Transmission With Frequency-Domain Equalisation Professor Sheng Chen School of Electronics and Computer Science University of Southampton Southampton
More informationDiversity Multiplexing Tradeoff in ISI Channels Leonard H. Grokop, Member, IEEE, and David N. C. Tse, Senior Member, IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 55, NO 1, JANUARY 2009 109 Diversity Multiplexing Tradeoff in ISI Channels Leonard H Grokop, Member, IEEE, and David N C Tse, Senior Member, IEEE Abstract The
More informationEE 5407 Part II: Spatial Based Wireless Communications
EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang E-mail: rzhang@i2r.a-star.edu.sg Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture IV: MIMO Systems March 21,
More informationOn the Use of Division Algebras for Wireless Communication
On the Use of Division Algebras for Wireless Communication frederique@systems.caltech.edu California Institute of Technology AMS meeting, Davidson, March 3rd 2007 Outline A few wireless coding problems
More informationCognitive Radio: An Information-Theoretic Perspective
Cognitive Radio: An Information-Theoretic Perspective Aleksandar Jovičić and Pramod Viswanath May 11, 2009 Abstract We consider a communication scenario in which the primary and the cognitive radios wish
More informationA Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems
A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems Wei Zhang, Xiang-Gen Xia and P. C. Ching xxia@ee.udel.edu EE Dept., The Chinese University of Hong Kong ECE Dept., University of Delaware
More informationTitle. Author(s)Tsai, Shang-Ho. Issue Date Doc URL. Type. Note. File Information. Equal Gain Beamforming in Rayleigh Fading Channels
Title Equal Gain Beamforming in Rayleigh Fading Channels Author(s)Tsai, Shang-Ho Proceedings : APSIPA ASC 29 : Asia-Pacific Signal Citationand Conference: 688-691 Issue Date 29-1-4 Doc URL http://hdl.handle.net/2115/39789
More informationLecture 6 I. CHANNEL CODING. X n (m) P Y X
6- Introduction to Information Theory Lecture 6 Lecturer: Haim Permuter Scribe: Yoav Eisenberg and Yakov Miron I. CHANNEL CODING We consider the following channel coding problem: m = {,2,..,2 nr} Encoder
More informationCapacity Bounds for Power- and Band-Limited Wireless Infrared Channels Corrupted by Gaussian Noise
40th Annual Allerton Conference on Communications, Control, and Computing, U.Illinois at Urbana-Champaign, Oct. 2-4, 2002, Allerton House, Monticello, USA. Capacity Bounds for Power- and Band-Limited Wireless
More informationA Summary of Multiple Access Channels
A Summary of Multiple Access Channels Wenyi Zhang February 24, 2003 Abstract In this summary we attempt to present a brief overview of the classical results on the information-theoretic aspects of multiple
More informationAnalysis of Receiver Quantization in Wireless Communication Systems
Analysis of Receiver Quantization in Wireless Communication Systems Theory and Implementation Gareth B. Middleton Committee: Dr. Behnaam Aazhang Dr. Ashutosh Sabharwal Dr. Joseph Cavallaro 18 April 2007
More informationCooperative HARQ with Poisson Interference and Opportunistic Routing
Cooperative HARQ with Poisson Interference and Opportunistic Routing Amogh Rajanna & Mostafa Kaveh Department of Electrical and Computer Engineering University of Minnesota, Minneapolis, MN USA. Outline
More informationMinimum BER Linear Transceivers for Block. Communication Systems. Lecturer: Tom Luo
Minimum BER Linear Transceivers for Block Communication Systems Lecturer: Tom Luo Outline Block-by-block communication Abstract model Applications Current design techniques Minimum BER precoders for zero-forcing
More information(Classical) Information Theory III: Noisy channel coding
(Classical) Information Theory III: Noisy channel coding Sibasish Ghosh The Institute of Mathematical Sciences CIT Campus, Taramani, Chennai 600 113, India. p. 1 Abstract What is the best possible way
More information