TSKS01 Digital Communication Lecture 1

Transcription

1 TSKS01 Digital Communication Lecture 1 Introduction, Repetition, and Noise Modeling Emil Björnson Department of Electrical Engineering (ISY) Division of Communication Systems

2 Emil Björnson Course Director TSKS01 Digital Communication TSRT04 Introduction in Matlab TSKS12 Multiple Antenna Communication Associate Professor (Universitetslektor) Docent in Communication Systems, PhD in Telecommunications Coordinator of and Master programme in Communication Systems Master profile in Communication Systems Researcher on 5G communications (e.g., multiple antenna communications). Theoretical research and inventor TSKS01 Digital Communication - Lecture 1 2

3 TSKS01 Digital Communication - Formalities Information: Lecturer & examiner: Emil Björnson, emil.bjornson@liu.se Tutorials: Kamil Senel, kamil.senel@liu.se Teaching activities: 12 lectures, 12 tutorials, lab exercises Examination: Laboratory exercises (1hp): More details in HT2 Sign-up on the web (later) Written exam (5hp): 1 simple task (basics) Min: 50% 2 questions (5 points each) Min: 3p 4 problems (5 points each) Min: 6p Pass: 14 points from questions & problems TSKS01 Digital Communication - Lecture 1 3

4 Course Aims After passing the course, the student should be able to reliably perform standard calculations regarding digital modulation and binary (linear) codes for error control coding. (basics) should be able, to some extent, to perform calculations for solutions to practical engineering problems that arise in communication (primarily questions) be able to, with some precision, analyze and compare various choices of digital modulation methods and coding methods in terms of error probabilities, minimum distances, throughput, and related concepts. (problems) be able, to some extent, to implement and evaluate communication systems of the kinds treated in the course. (laboratory exercises) TSKS01 Digital Communication - Lecture 1 4

5 Course Book Introduction to Digital Communication New edition for 2017, A1671, SEK 214 Available for purchase in Building A, LiU Service Center, entrance 19C 207 pages dedicated to this course There is also a book with problems to be solved at tutorials (A1672, SEK 81) Older editions Version from 2016 can be used, along with the errata and the new tutorial problems found on last year s course webpage TSKS01 Digital Communication - Lecture 1 5

6 What is Communication? Information transfer n n n Classic technology Newspapers, books Telegraph Radio and TV broadcast Modern digital technology n n n CD, DVD, Blu-ray Optical fibres, network cables Wireless (4G, WiFi) TSKS01 Digital Communication - Lecture 1 6

7 Why Digital Communication? 1. Efficient Source Description Information has a built-in redundancy (e.g., text, images, sound) Compression: Represent information with minimal number of bits Sequence of bits: 0 and bits represent 2 "## # different messages 2. Efficient Transmission Nyquist-Shannon Sampling Theorem: B Hz signal 2B sample/s Sequence of bits is used to select these samples 3. Guaranteed Quality Protect signals against errors: All or nothing is received TSKS01 Digital Communication - Lecture 1 7

8 One-way Digital Communication System Error control Digital to analog Medium Source Channel encoder Modulator Channel coding Digital modulation Analog channel Destination Channel decoder Demodulator Error correction Analog to digital TSKS01 Digital Communication - Lecture 1 8

9 Digital Communication: An Exponential Story Overall IP traffic Source: Cisco VNI Global IP Traffic Forecast, Digital Communication Devices are Everywhere PCs, tablets, phones, machine-to-machine, TVs, etc. Total internet traffic: 23% growth/year Number of devices: 12% growth/year How can we sustain this exponential growth? TSKS01 Digital Communication - Lecture 1 9

10 Information-Bearing Signals Example: Mobile communication Analog electromagnetic signaling Bandwidth B Hz Sample principle in copper cables, optical fibers, etc. Nyquist Shannon Sampling Theorem Signal is determined by 2B real samples/second (or B complex samples) Information-bearing signals Select the samples in a particular way à Represent digital information Spectral efficiency: Number of bits transferred per sample [bit/s/hz] TSKS01 Digital Communication - Lecture 1 10

11 Designing Efficient Communication Systems Performance metric: Bit rate [bit/s] Formula: Bit rate [bit/s] = Bandwidth Hz < Spectral efficiency [bit/s/hz] How do we increase the bit rate? 1. Use more bandwidth 2. Increase spectral efficiency Wired: Use cable that handles wider bandwidths Wireless: Buy more spectrum Each sample represents more bits More sensitive to disturbances Improve signal quality Protect against bit errors This course primary deals with spectral efficiency! TSKS01 Digital Communication - Lecture 1 11

12 Preliminary Lecture Plan HT1: Introduction, repetition of key results (Lecture 1) Basic digital modulation (Lectures 2-3) Detection in AWGN channels (Lectures 3-5) Detection in dispersive channels (Lectures 6-7) HT2: Error control coding (Lectures 8-10) Practical aspects (e.g., synchronization) (Lecture 11) Link adaptation (Lecture 12) TSKS01 Digital Communication - Lecture 1 12

13 Outline Introduction Repetition: Signals and systems Repetition: Stochastic variables Repetition: Stochastic processes Noise modeling and detection example At the end: Examples related to repetition Read yourself! TSKS01 Digital Communication - Lecture 1 13

14 Repetition: Signals and Systems Signals: Systems: Voltages, currents, or other measurements Manipulate/filter signals x(t) System y(t) Complex exponential: e JKLMN = cos 2πft + j sin(2πft) 0, t < 0 Unit step: u(t) = V 1, t > 0 \ Unit impulse: δ(t): x t N Property: u t = δ τ δ t dt = x(0) dτ TSKS01 Digital Communication - Lecture 1 14

15 Special Outputs General case: x(t) Energy-free system y(t) Energy-free system: No transients, constant input à constant output Impulse response: δ(t) Energy-free system h(t) TSKS01 Digital Communication - Lecture 1 15

16 Linear Systems Definition: Let x " (t) and x K t be input signals to a one-input energy-free system, and let y " t and y K t be the corresponding output signals. If y t = a " y " t + a K y K (t) is the output corresponding to the input x t = a " x " t + a K x K t, for any x " (t), x K t, a ", and a K, then the system is referred to as linear. A system that is not linear is referred to as non-linear. Examples: y(t) = x(t 1) is linear K y t = x t is non-linear TSKS01 Digital Communication - Lecture 1 16

17 Time-Invariant and LTI Systems Time-invariant system Definition: Let x(t) be the input to an energy-free system and let y(t) be the corresponding output. If y(t + τ) is the output for the input x(t + τ) for any x(t), t, and τ, then the system is referred to as time-invariant. Otherwise the system is referred to as time-varying. Examples: y(t) = x(t 1) is time-invariant y t = cos 2πft x(t) is time-varying LTI system Definition: A system that is both linear and time-invariant is referred to as a linear time-invariant (LTI) system TSKS01 Digital Communication - Lecture 1 17

18 Convolution Definition: The convolution of the signals a(t) and b(t) is denoted by (a b)(t) and is defined as a b \ t = e a τ b t τ dτ. The convolution is a commutative operation: a b t = b a t TSKS01 Digital Communication - Lecture 1 18

19 Output of an LTI Systems Theorem: Let x(t) be the input to an energy-free LTI system with impulse response h(t), then the output of the system is y t = x h t. Proof: Let H a(t) denote the output for an arbitrary input a(t), then y t = H x(t) = H e Linear \ = e x τ Time inv \ = e x τ \ x τ H δ t τ dτ δ t τ dτ h t τ dτ = x h t TSKS01 Digital Communication - Lecture 1 19

20 Frequency Domain x(t) X(f) t f \ Fourier transform: F x(t) = x t e ]JKLMN dt Exists if \ x t dt < Inverse transform: F ]" \ X(f) = X f e JKLMN df Common notation Spectrum of x(t): X(f) Amplitude spectrum: X f Phase spectrum: arg X(f) Image from: TSKS01 Digital Communication - Lecture 1 20

21 LTI Systems: Time and Frequency Domain Property: Let A f = F a(t) and B f = F b(t) be the Fourier transforms of the signals a(t) and b(t), then F (a b)(t) = A f B f. Property: The input and output of LTI systems are related as x(t) X(f) h t H(f) y t = (x h)(t) Y f = X f H(f) TSKS01 Digital Communication - Lecture 1 21

22 Example: Channel is an LTI Filter Time delay: τ " Time delay: τ + Time delay: τ K Time delay: τ r Impulse response: h t = δ t τ " + δ t τ K + δ t τ + + δ(t τ r ) Only time-invariant for a limited time period (coherence time) TSKS01 Digital Communication - Lecture 1 22

23 Probability, Stochastic Variable, and Events Total probability: Pr Ω u = 1 Probability of event A: Pr {A} [0,1] Sample space: Ω ω Stochastic variable Joint probability: Pr {A, B} Conditional probability: Pr A B = { {},~} { {~} X ω Event A Event B Measureable sample space: Ω u = X ω : for some ω Ω TSKS01 Digital Communication - Lecture 1 23

24 Probabilities and Distributions Probability distribution function: F X (x) = Pr {X x} [0,1] Probability density function (PDF): f X (x) = d dx F X(x) Properties: F X (x) is non-decreasing, F X (x) 0 and f X (x) 0 for all x, \ f u (x) dx = 1, Pr x 1 < X x 2 = f u x dx TSKS01 Digital Communication - Lecture 1 24

25 Expectation and Variance Expectation (mean): E X = e \ xf u x dx For discrete variables: E X = ˆ x Pr{X = x } For functions of a variable: E Y = e Y = g X \ \ yf Œ y dy = E g(x) = e g x f u x dx Quadratic mean (power): Variance \ E X K = e x K f u x dx Var X = E X E X K Common notation: = E X K E X K m u = E X, m Œ = E Y σ u K = Var X σ u is called the standard deviation TSKS01 Digital Communication - Lecture 1 25

26 Gaussian/Normal Distribution, N(m, σ K ) Probability density f u x = 1 σ 2π e] ] K Probability distribution F u x = is complicated Q x = 1 F u x = Q-function \ " KL e] dt Mean: m Variance: σ K Can be computed from table f u x dx for N(0,1) TSKS01 Digital Communication - Lecture 1 26

27 Example of the Q Function Q(1.96) ]K TSKS01 Digital Communication - Lecture 1 27

28 Multi-Dimensional Stochastic Variables Distribution: F u,,u š x ",, x = Pr {X " x ",, X x } Density: f u,,u š x ",, x = œ š œ <<< œ š F u,,u š x ",, x Vector notation: X = (X ",, X ), x = (x ",, x ), F u x, f u x Mutual independence: F u x = Ÿ F u (x ) " Marginal distribution and density f u x = Ÿ f u (x ) " Covariance (pairwise): Cov X, X J = E X X J E X E X J Pairwise uncorrelated if X and X J if Cov X, X J = 0 for all i, j TSKS01 Digital Communication - Lecture 1 28

29 Example: Jointly Gaussian Variables Definition: X = (X ",, X ) is jointly Gaussian, denoted as N(m, Λ u ), if f u x = 1 2π det (Λ u ) e] " K ] ] where m = E X, Λ u = λ "," λ ",, λ J = Cov X, X J. λ," λ, If X = (X ",, X ) are pairwise uncorrelated à λ J = 0 for i j à f u x = 1 e ] ] K±, = Ÿ f 2π u x λ "," λ, " à Independence TSKS01 Digital Communication - Lecture 1 29

30 Bayes Theorem Thomas Bayes Joint and conditional probability: Pr{X = x, Y = y} = Pr X = x Y = y Pr{Y = y} = Pr {Y = y X = x} Pr{X = x} f u,œ x, y = f u Œ x y f Œ y = f Œ u y x f u x Bayes theorem (discrete): Pr {X = x Y = y} = { {Œ ² u } { {Œ ²} Pr {X = x} (continuous): f u Œ x y = M ³ ² M ³ ² f u x (X discrete, Y cont.): Pr {X = x Y = y} = M ³ ² M ³ ² Pr {X = x} TSKS01 Digital Communication - Lecture 1 30 Image from:

31 Example: Additive Noise Channel X W Y + Detector X Input: X 1, +1 with Pr X = +1 = Pr X = 1 = 1/2 Output: Y = X + W f (w) 1/2 W is a continuous stochastic variable w How should we best detect X from Y? TSKS01 Digital Communication - Lecture 1 31

32 Stochastic Process X " (ω ", t) Sample space: Ω ω " t ω K X K (ω K, t) ω + t X + (ω +, t) t Stochastic process = Stochastic time-continuous signal (N = ) TSKS01 Digital Communication - Lecture 1 32

33 Examples of Stochastic Processes Example 1: Finite number of realizations: X(t) = sin (t + φ), φ {0, π/2, π, 3π/2}. Example 2: Infinite number of realizations: X(t) = A sin (t), A~N 0, TSKS01 Digital Communication - Lecture 1 33

34 Examples of Stochastic Processes cont d Example 3: Infinite number of realizations: cos (πt), t < 1/2 X t = ½ A ½ g(t k), g t = V 0, elsewhere Each A k is independent and N(0,1) One realization: TSKS01 Digital Communication - Lecture 1 34

35 Sampling of Stochastic Process: N Time Instants Vector notation Time instants: t = (t ",, t ) Stochastic variable: X(t ) = X(t " ),, X(t ) Realizations: x = (x ",, x ) N time instants Distribution: F u(n ) x = Pr {X(t " ) x ",, X(t ) x } Density: f u(n ) x = œš œ œ š F u(n ) x Sampling of stochastic process à Stochastic variable TSKS01 Digital Communication - Lecture 1 35

36 Ensemble Averages Observe many realizations and make an average functions of time Expectation (mean): m u t = E X(t) \ = e xf u(n) x dx Quadratic mean (power): E X K (t) = e \ x K f u(n) x dx Auto-correlation function (ACF): \ r u t ", t K = E X t " X(t K ) = \ e e x " x K f u N,u N x ", x K Symmetry: r u t ", t K = r u t K, t " Power: r u t, t = E X K (t) dx " dx K Variance K σ u(n) = E X(t) m u t = E X K (t) m u K t K Special case: A is stochastic var. X t = g(t, A) r u t ", t K = \ g(t ", A)g(t K, A)f } a da TSKS01 Digital Communication - Lecture 1 36

37 Strict-Sense Stationarity Stationarity is statistical invariance to a shift of the time origin. Definition: Consider time instants t = (t ",, t ) and shifted time instances u = t + Δ = (t " + Δ,, t + Δ). The process X(t) is said to be strict-sense stationary (SSS) if F u(n ) x = F u(á ) x holds for all N and all choices of t and Δ. Equivalence: F u(n ) x = F u(á ) x Û f u(n ) x = f u(á ) x TSKS01 Digital Communication - Lecture 1 37

38 Wide-Sense Stationarity Definition: A stochastic process X(t) is said to be wide-sense stationary (WSS) if Mean satisfies m u t = m u t + Δ for all Δ. ACF satisfies r u t ", t K = r u t " + Δ, t K + Δ for all Δ. Interpretation: Constant mean, ACF only depends on time difference τ = t " t K Notation: Mean m u ACF r u τ TSKS01 Digital Communication - Lecture 1 38

39 Gaussian Processes Recall: X = (X ",, X ) is jointly Gaussian, denoted as N(m, Λ u ), if f u x = 1 2π det (Λ u ) e] " K ] ] Definition: A stochastic process is called Gaussian if X(t ) = X(t " ),, X(t ) is jointly Gaussian for any t = (t ",, t ) Theorem: A Gaussian process that is wide-sense stationary is also stationary in the strict sense TSKS01 Digital Communication - Lecture 1 39

40 Repetition EXAMPLES TSKS01 Digital Communication - Lecture 1 40

41 Example: Linear system Is the system y t = x(t 1) linear? Consider x(t) = a " x " t + a K x K (t) Then: y t = a " x " t 1 + a K x K t 1 = a " y " t + a K y K (t) Linear combination of the outputs of x " t and x K t : Linear system K Is the system y t = x t linear? Consider x(t) = a " x " t + a K x K (t) Then: y t = a " x " t + a K x K (t) K = a K " x K " t + a K K x K K t + a " a K x " t x K t a " x K " t + a K x K K (t) Not a linear combination of the outputs of x " t and x K t : Non-linear TSKS01 Digital Communication - Lecture 1 41

42 Example: Time-invariant system Is the system y t = x(t 1) time-invariant? Note: y t + τ = x t + τ 1 = x( t + τ 1) Time-shifted output is given by equally time-shifted input: Time-invariant Is the system y t = cos 2πft x(t) time-invariant? Note: y t + τ = cos 2πf(t + τ) x(t + τ) cos 2πft x(t + τ) Time-shifted output is not given by the time-shifted input: Time-varying TSKS01 Digital Communication - Lecture 1 42

43 Example: Stochastic variables Ω = HH, HT, TH, TT Ω u = 200, 100, +400 Determine distribution and density: F u x = 0, x < , 200 x < , 100 x < 400 1, x 400 Game based on tossing two fair coins: Two heads +400 Two tails 100 One tail, one head , ω = HH, X ω = Â 100, ω = TT, 200, ω {HT, TH} = " u x " u x " u(x 400) K r r f u x = Å Å F X(x) = " δ x " δ x " δ(x 400) K r r TSKS01 Digital Communication - Lecture 1 43

44 Example (cont.): Stochastic variables f u x = " δ x " δ x " δ(x 400) K r r E X = e \ Determine mean and variance: x 1 2 δ x δ x δ(x 400) dx 4 = 200 < < < 1 4 = 25 E X K = 200 K < K < K < 1 4 = Var X = E X K E X K = TSKS01 Digital Communication - Lecture 1 44

45 Example: Determine mean and variance of different distributions Uniform distribution: 1 f u x = Â, b a a x < b 0, elsewhere Exponential distribution: f u x = 1 c e] /Ê u(x) Mean: Variance: Mean: Variance: ÆÇÈ K È]Æ "K c c K Binary distribution: f u x = pδ x a + 1 p δ(x b) Mean: p a + 1 p b Variance: p 1 p b a K TSKS01 Digital Communication - Lecture 1 45

46 Example: Uncorrelated Independent The stochastic variable X is +1, 0, and 1 with equal probability Mean value: m u = " " " + 1 = 0 Consider the stochastic variable Y = X K. Are X and Y correlated? Cov X, Y = E XY m u m Π= E X + 0 = = 0 Uncorrelated! Are X and Y independent? No, since Y = X K TSKS01 Digital Communication - Lecture 1 46

47 Example: Stochastic processes X(t) = A sin (t), for some stochastic variable A, with F } (a). F u N = x = Pr X t x = Pr A sin t x Pr A x sin (t) = F x }, t: sin t > 0 sin (t) Pr 0 x = u(x), t: sin t = 0 Pr A x sin (t) = 1 F x }, t: sin t < 0 sin (t) f u N x = d dx F u N x = 1 sin (t) f x } t: sin t > 0 sin (t) δ(x), t: sin t = 0 1 sin (t) f x }, t: sin t < 0 sin t TSKS01 Digital Communication - Lecture 1 47

48 Example: Stochastic processes (cont.) X(t) = A sin (t), for some stochastic variable A, with F } (a). Determine mean and ACF: m u t = e E{X K (t)} = e \ \ xf u(n) x dx x K f u(n) x dx \ = e a \ = e a sin t K sin t f } a da = sin t m } f } a da = sin K t E A K K σ u(n) = E X K (t) m u K t = sin K t E A K m } K = sin K t σ } K \ r u t ", t K = e a sin t " a sin t K f } a da = sin t " sin t K E A K TSKS01 Digital Communication - Lecture 1 48

49 Example: Filtering of Stochastic Process Let X(t) be a WSS process with r u t = e ] Ï LTI- system 1, f 1 X(t) H(f) Y(t) H f = V 0, elsewhere Determine the output power E{Y K (t)}: \ E Y K t = r Œ 0 = e R Œ f df = Calculator = 2 arctan (2π) π \ = e H f K R u f df = " e R u f df ]" TSKS01 Digital Communication - Lecture 1 49

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