EC 4 STATISTICAL COMMUNICATION THEORY Insrucor: Dr. Heba A. Shaban Lecure # 4 RANDOM SIGNALS AND NOISE A random process is a process i.e. variaion in ime or one dimensional space whose behavior is no compleel predicable and can be characerizeded b saisical laws.
RANDOM PROCESSES Random signals canno be eplicil described prior o heir occurrence. Noises canno be described b deerminisic funcions of ime. However when observed over a long period a random signal or noise ehibi cerain regulariies ha can be described in erms of probabilisic descripion of a collecion of funcions of imes which is called a random process. 3 EAMPLES OF AN ENSEMBLE OF SAMPLE FUNCTIONS Boh and coninuous Selec a realizaion b means of random eperimen RV
RANDOM PHASED SINE WAVE uniform 0 ] DISTRIBUTION AND DENSITY FUNCTIONS Disribuion and probabili densi funcion of RV : F P{ } f d F d F Disribuion and probabili densi funcion of Sochasic Process : P{ } f F Time parameer is involved i.e. F and f will in general depend on ime 3
4 SECOND ORDER JOINT DISTRIBUTION AND PROBABILITY DENSITY FUNCTIONS Join disribuion and probabili densi funcion of wo RVs and Y: F d } P{ F F f F f Y Y d d d } P{ Y F Y Second order disribuion and probabili densi funcion of : Join disribuion and probabili densi funcion of wo Sochasic Processes and Y : } ' P{ ' Y F Y ' ' F f 8
HIGHER ORDER JOINT DISTRIBUTION AND PROBABILITY DENSITY FUNCTIONS Two processes and Y are saisicall independen iff For all arbirar choices of he ime parameers... N... N STATIONARITY Firs order saionari: f f Consequence: Saisical average ensemble averagei.e. independen of ime N-h order saionari: Saionari of cerain order N Saionari of all orders k N Sric-sense saionar: Saionar of an arbirar order 5
STATIONARITY Second order saionari AUTOCORRELATION FUNCTION E [ ] 6
STATIONARITY A random process is called sric-sense saionar SSS if is saisics are invarian o a shif of origin. A random process is called wide-sense saionar WSS if: Is mean is consan. Is auocorrelaion depends onl on he ime-difference τ and consequenl is auo-covariance also depends onl on ime difference τ. 3 AUTOCORRELATION FUNCTION Correlaion of RVs = and = For second order saionar process: E[ ] consan independen of Definiion Wide-sense saionari WSS Second order saionari: wide-sense saionari 7
TIME STATISTICS OF A RANDOM PROCESS s order CDF: s order PDF: mean: E F f s order saionari nd order PDF:f = consan nd order saionari Sric-sense saionari i i SSS ACF: R E Wide-sense saionari WSS: R R PROPERTIES OF R OF WSS PROCESSES 6 8
ERGODICITY Time average: RV Ergodic process defined b Process has o be wide-sense saionar ERGODICITY Time average: RV Ergodic process defined b Process has o be wide-sense saionar 9
ERGODIC PROCESS. A random process issaidobeergodic if is ime averages are he same for all funcions and equal o he corresponding ensemble averages. Tesing he ergodici of a random process is usuall ver difficul. A reasonable assumpion in he analsis of mos communicaion signals is ha a random waveform is ergodic in he mean and in he auocorrelaion. 9 WIDE-SENSE STATIONARITY AND ERGODICITY Wide-sense saionari WSS: Ergodici: 0
CROSS-CORRELATION FUNCTION CCF The cross-correlaion funcion R Y + of wo processes and Y is defined as and Y are called joinl wide-sense saionar if boh are individuall WSS and R Y + τ = R Y τ i.e. independen of If R Y + = 0 hen and Y are orhogonal Processes and Y are joinl ergodic if individuall ergodic and Processes have o be joinl WSS