L ECTURE 2 P ROBABILITY R EVIEW AND R ANDOM P ROCESS 1

R EVIEW OF LAST LECTURE The point worth noting are : The source coding algorithm plays an important role in higher code rate (compressing data) The channel encoder introduce redundancy in data The modulation scheme plays important role in deciding the data rate and immunity of signal towards the errors introduced by the channel Channel can introduce many types of errors due to thermal noise etc. The demodulator and decoder should provide high Bit Error Rate (BER). 2

R EVIEW : L AYERING OF S OURCE C ODING Source coding includes Sampling Quantization Symbols to bits Compression Decoding includes Decompression Bits to symbols Symbols to sequence of numbers Sequence to waveform (Reconstruction) 3

R EVIEW : L AYERING OF S OURCE C ODING 4

R EVIEW : L AYERING OF C HANNEL C ODING Channel Coding is divided into Discrete encoder\Decoder Used to correct channel Errors Modulation\Demodulation Used to map bits to waveform for transmission 5

R EVIEW : L AYERING OF C HANNEL C ODING 6

R EVIEW : R ESOURCES OF A C OMMUNICATION S YSTEM Transmitted Power Average power of the transmitted signal Bandwidth (spectrum) Band of frequencies allocated for the signal Type of Communication system Power limited System Space communication links Band limited Systems Telephone systems 7

R EVIEW : D IGITAL COMMUNICATION SYSTEM Important features of a DCS: Transmitter sends a waveform from a finite set of possible waveforms during a limited time Channel distorts, attenuates the transmitted signal and adds noise to it. Receiver decides which waveform was transmitted from the noisy received signal Probability of erroneous decision is an important measure for the system performance 8

R EVIEW OF P ROBABILITY 9

S AMPLE S PACE AND P ROBABILITY Random experiment: its outcome, for some reason, cannot be predicted with certainty. Examples: throwing a die, flipping a coin and drawing a card from a deck. Sample space: the set of all possible outcomes, denoted by S. Outcomes are denoted by E’s and each E lies in S, i.e., E ∈ S. A sample space can be discrete or continuous. Events are subsets of the sample space for which measures of their occurrences, called probabilities, can be defined or determined. 10

T HREE A XIOMS OF P ROBABILITY For a discrete sample space S, define a probability measure P on as a set function that assigns nonnegative values to all events, denoted by E, in such that the following conditions are satisfied Axiom 1: 0 ≤ P(E) ≤ 1 for all E ∈ S Axiom 2: P(S) = 1 (when an experiment is conducted there has to be an outcome). Axiom 3: For mutually exclusive events E1, E2, E3,... we have 11

C ONDITIONAL P ROBABILITY We observe or are told that event E1 has occurred but are actually interested in event E2: Knowledge that of E1 has occurred changes the probability of E2 occurring. If it was P(E2) before, it now becomes P(E2|E1), the probability of E2 occurring given that event E1 has occurred. This conditional probability is given by If P(E2|E1) = P(E2), or P(E2 ∩ E1) = P(E1)P(E2), then E1 and E2 are said to be statistically independent. Bayes’ rule P(E2|E1) = P(E1|E2)P(E2)/P(E1) 12

M ATHEMATICAL M ODEL FOR S IGNALS Mathematical models for representing signals Deterministic Stochastic Deterministic signal: No uncertainty with respect to the signal value at any time. Deterministic signals or waveforms are modeled by explicit mathematical expressions, such as x(t) = 5 cos(10*t). Inappropriate for real-world problems??? Stochastic/Random signal: Some degree of uncertainty in signal values before it actually occurs. For a random waveform it is not possible to write such an explicit expression. Random waveform/ random process, may exhibit certain regularities that can be described in terms of probabilities and statistical averages. e.g. thermal noise in electronic circuits due to the random movement of electrons 13

14 E NERGY AND P OWER S IGNALS The performance of a communication system depends on the received signal energy: higher energy signals are detected more reliably (with fewer errors) than are lower energy signals. An electrical signal can be represented as a voltage v ( t ) or a current i ( t ) with instantaneous power p ( t ) across a resistor defined by OR

15 E NERGY AND P OWER S IGNALS In communication systems, power is often normalized by assuming R to be 1. The normalization convention allows us to express the instantaneous power as where x ( t ) is either a voltage or a current signal. The energy dissipated during the time interval (-T/2, T/2) by a real signal with instantaneous power expressed by Equation (1.4) can then be written as: The average power dissipated by the signal during the interval is:

16 E NERGY AND P OWER S IGNALS We classify x ( t ) as an energy signal if, and only if, it has nonzero but finite energy (0 < E x < ∞) for all time, where An energy signal has finite energy but zero average power Signals that are both deterministic and non-periodic are termed as Energy Signals

17 E NERGY AND P OWER S IGNALS Power is the rate at which the energy is delivered We classify x ( t ) as an power signal if, and only if, it has nonzero but finite energy (0 < P x < ∞) for all time, where A power signal has finite power but infinite energy Signals that are random or periodic termed as Power Signals

R ANDOM V ARIABLE Functions whose domain is a sample space and whose range is a some set of real numbers is called random variables. Type of RV’s Discrete E.g. outcomes of flipping a coin etc Continuous E.g. amplitude of a noise voltage at a particular instant of time 18

19 R ANDOM V ARIABLES Random Variables All useful signals are random, i.e. the receiver does not know a priori what wave form is going to be sent by the transmitter Let a random variable X ( A ) represent the functional relationship between a random event A and a real number. The distribution function F x ( x ) of the random variable X is given by

R ANDOM V ARIABLE A random variable is a mapping from the sample space to the set of real numbers. We shall denote random variables by boldface, i.e., x, y, etc., while individual or specific values of the mapping x are denoted by x (w). 20

R ANDOM PROCESS A random process is a collection of time functions, or signals, corresponding to various outcomes of a random experiment. For each outcome, there exists a deterministic function, which is called a sample function or a realization. Sample functions or realizations (deterministic function) Random variables time (t) Real number 21

R ANDOM P ROCESS A mapping from a sample space to a set of time functions. 22

R ANDOM P ROCESS CONTD Ensemble : The set of possible time functions that one sees. Denote this set by x(t), where the time functions x1(t, w1), x2(t, w2), x3(t, w3),... are specific members of the ensemble. At any time instant, t = tk, we have random variable x(tk). At any two time instants, say t1 and t2, we have two different random variables x(t1) and x(t2). Any realationship b/w any two random variables is called Joint PDF 23

C LASSIFICATION OF R ANDOM P ROCESSES Based on whether its statistics change with time: the process is non-stationary or stationary. Different levels of stationary: Strictly stationary: the joint pdf of any order is independent of a shift in time. Nth-order stationary: the joint pdf does not depend on the time shift, but depends on time spacing 24

C UMULATIVE D ISTRIBUTION F UNCTION ( CDF ) cdf gives a complete description of the random variable. It is defined as: FX(x) = P(E ∈ S : X(E) ≤ x) = P(X ≤ x). The cdf has the following properties: 0 ≤ FX(x) ≤ 1 (this follows from Axiom 1 of the probability measure). Fx(x) is non-decreasing: Fx(x1) ≤ Fx(x2) if x1 ≤ x2 (this is because event x(E) ≤ x1 is contained in event x(E) ≤ x2). Fx(−∞) = 0 and Fx(+∞) = 1 (x(E) ≤ −∞ is the empty set, hence an impossible event, while x(E) ≤ ∞ is the whole sample space, i.e., a certain event). P(a < x ≤ b) = Fx(b) − Fx(a). 25

P ROBABILITY D ENSITY F UNCTION The pdf is defined as the derivative of the cdf: fx(x) = d/dx Fx(x) It follows that: Note that, for all i, one has pi ≥ 0 and ∑ pi = 1. 26

C UMULATIVE J OINT PDF J OINT PDF Often encountered when dealing with combined experiments or repeated trials of a single experiment. Multiple random variables are basically multidimensional functions defined on a sample space of a combined experiment. Experiment 1 S1 = {x1, x2, …,xm} Experiment 2 S2 = {y1, y2, …, yn} If we take any one element from S1 and S2 0 <= P(xi, yj) <= 1 (Joint Probability of two or more outcomes) Marginal probabilty distributions Sum all j P(xi, yj) = P(xi) Sum all i P(xi, yj) = P(yi) 27

28 E XPECTATION OF R ANDOM V ARIABLES (S TATISTICAL AVERAGES ) Statistical averages, or moments, play an important role in the characterization of the random variable. The first moment of the probability distribution of a random variable X is called mean value mx or expected value of a random variable X The second moment of a probability distribution is mean- square value of X Central moments are the moments of the difference between X and mx, and second central moment is the variance of x. Variance is equal to the difference between the mean- square value and the square of the mean

Contd The variance provides a measure of the variable’s “randomness”. The mean and variance of a random variable give a partial description of its pdf. 29

T IME A VERAGING AND E RGODICITY A process where any member of the ensemble exhibits the same statistical behavior as that of the whole ensemble. For an ergodic process: To measure various statistical averages, it is sufficient to look at only one realization of the process and find the corresponding time average. For a process to be ergodic it must be stationary. The converse is not true. 30

G AUSSIAN ( OR N ORMAL ) R ANDOM V ARIABLE (P ROCESS ) A continuous random variable whose pdf is: μ and are parameters. Usually denoted as N(μ, ). Most important and frequently encountered random variable in communications. 31

C ENTRAL L IMIT T HEOREM CLT provides justification for using Gaussian Process as a model based if The random variables are statistically independent The random variables have probability with same mean and variance 32

CLT The central limit theorem states that “The probability distribution of Vn approaches a normalized Gaussian Distribution N(0, 1) in the limit as the number of random variables approach infinity” At times when N is finite it may provide a poor approximation of for the actual probability distribution 33

34 A UTOCORRELATION Autocorrelation of Energy Signals Correlation is a matching process; autocorrelation refers to the matching of a signal with a delayed version of itself The autocorrelation function of a real-valued energy signal x ( t ) is defined as: The autocorrelation function R x (  ) provides a measure of how closely the signal matches a copy of itself as the copy is shifted  units in time. R x (  ) is not a function of time; it is only a function of the time difference  between the waveform and its shifted copy.

35 A UTOCORRELATION symmetrical in  about zero maximum value occurs at the origin autocorrelation and ESD form a Fourier transform pair, as designated by the double-headed arrows value at the origin is equal to the energy of the signal

36 AUTOCORRELATION OF A PERIODIC (POWER) SIGNAL The autocorrelation function of a real-valued power signal x ( t ) is defined as: When the power signal x ( t ) is periodic with period T 0, the autocorrelation function can be expressed as:

37 A UTOCORRELATION OF POWER SIGNALS symmetrical in  about zero maximum value occurs at the origin autocorrelation and PSD form a Fourier transform pair, as designated by the double-headed arrows value at the origin is equal to the average power of the signal The autocorrelation function of a real-valued periodic signal has properties similar to those of an energy signal:

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S PECTRAL D ENSITY 40

SPECTRAL DENSITY The spectral density of a signal characterizes the distribution of the signal’s energy or power, in the frequency domain This concept is particularly important when considering filtering in communication systems while evaluating the signal and noise at the filter output. The energy spectral density (ESD) or the power spectral density (PSD) is used in the evaluation. Need to determine how the average power or energy of the process is distributed in frequency. 41

S PECTRAL D ENSITY Taking the Fourier transform of the random process does not work 42

43 ENERGY SPECTRAL DENSITY Energy spectral density describes the energy per unit bandwidth measured in joules/hertz Represented as  x (t), the squared magnitude spectrum  x (t) =|x(f)| 2 According to Parseval’s Relation Therefore The Energy spectral density is symmetrical in frequency about origin and total energy of the signal x(t) can be expressed as

44 P OWER S PECTRAL D ENSITY The power spectral density (PSD) function G x (f) of the periodic signal x(t) is a real, even ad nonnegative function of frequency that gives the distribution of the power of x(t) in the frequency domain. PSD is represented as (Fourier Series): PSD of non-periodic signals: Whereas the average power of a periodic signal x(t) is represented as:

N OISE 45

46 N OISE IN THE C OMMUNICATION S YSTEM The term noise refers to unwanted electrical signals that are always present in electrical systems: e.g. spark-plug ignition noise, switching transients and other electro-magnetic signals or atmosphere: the sun and other galactic sources Can describe thermal noise as zero-mean Gaussian random process A Gaussian process n ( t ) is a random function whose value n at any arbitrary time t is statistically characterized by the Gaussian probability density function

47 WHITE NOISE The primary spectral characteristic of thermal noise is that its power spectral density is the same for all frequencies of interest in most communication systems A thermal noise source emanates an equal amount of noise power per unit bandwidth at all frequencies—from dc to about Hz. Power spectral density G(f) Autocorrelation function of white noise is The average power P of white noise if infinite

W HITE N OISE 48

W HITE N OISE Since Rw( T ) = 0 for T = 0, any two different samples of white noise, no matter how close in time they are taken, are uncorrelated. Since the noise samples of white noise are uncorrelated, if the noise is both white and Gaussian (for example, thermal noise) then the noise samples are also independent. 49

The effect on the detection process of a channel with Additive White Gaussian Noise (AWGN) is that the noise affects each transmitted symbol independently Such a channel is called a memoryless channel The term “additive” means that the noise is simply superimposed or added to the signal—that there are no multiplicative mechanisms at work A DDITIVE W HITE G AUSSIAN N OISE (AWGN) 50

R ANDOM P ROCESSES AND L INEAR S YSTEMS If a random process forms the input to a time-invariant linear system, the output will also be a random process 51

D ISTORTION LESS T RANSMISSION Remember linear and non-linear group delays in DSP 52

53 DISTORTION LESS TRANSMISSION What is required of a network for it to behave like an ideal transmission line? The output signal from an ideal transmission line may have some time delay and different amplitude as compared with the input It must have no distortion—it must have the same shape as the input For idea distortion less transmission

54 I DEAL D ISTORTION L ESS T RANSMISSION The overall system response must have a constant magnitude response The phase shift must be linear with frequency All of the signal’s frequency components must also arrive with identical time delay in order to add up correctly The time delay t 0 is related to the phase shift and the radian frequency  = 2  f by A characteristic often used to measure delay distortion of a signal is called envelope delay or group delay, which is defined as

55 BANDWIDTH OF DIGITAL DATA Baseband signals Signals containing frequencies ranging from 0 to some frequency fs Bandpass or Passband Signals Signals containing frequencies ranging from fs1 to some frequency fs2

N OTE 56 Chapter 1 from Bernard Sklar Chapter 1 from Simon Haykin Appendix 1 from Digital Communication, Simon Haykin for Probability Periodic, Non-periodic Signals Analog and Digital Signals Ideal Filters Realizable filters Chapters/Topics from different books Topics to be covered on your own

R EFERENCES Bernard Sklar University of Saskatchewan Communication System, Simon Haykin MIT open source lectures (Robert Gallager) 57