ECE 5345 Random Processes - Example Random Processes copyright Robert J. Marks II Example RP’s Example Random Processes Gaussian Recall Gaussian pdf f X (x) 1 2 n/2 1/ 2 1 T 1 (xm) K (xm) e 2 |K | Let Xk=X(tk) , 1 k n. Then if, for all n, the corresponding pdf’s are Gaussian, then the RP is Gaussian. The Gaussian RP is a useful model in signal processing. copyright Robert J. Marks II Flip Theorem Let A take on values of +1 and -1 with equal probability Let X(t) have mean m(t) and autocorrelation RX Let Y(t)=AX(t) Then Y(t) has mean zero and autocorrelation RX What about the autocovariances? copyright Robert J. Marks II Multiple RP’s X(t) & Y(t) Independence (X(t1), X(t2), …, X(tk )) is independent to (Y(1), Y( 2), …, Y( j )) …for All choices of k and j and all sample locations copyright Robert J. Marks II Multiple RP’s X(t) & Y(t) Cross Correlation RXY(t, )=E[X(t)Y()] Cross-Covariance CXY(t, )= RXY(t, ) - E[X(t)] E[Y()] Orthogonal: RXY(t, ) = 0 Uncorrelated: CXY(t, ) = 0 Note: Independent Uncorrelated, but not the converse. copyright Robert J. Marks II Example RP’s Multiple Random Process Examples Example X(t) = cos(t+), Y(t) = sin(t+), Both are zero mean. Cross Correlation=? p.338 copyright Robert J. Marks II Example RP’s Multiple Random Process Examples Signal + Noise X(t) = signal, N(t) = noise Y(t) = X(t) + N(t) If X & N are independent,RXY=? Note: also, var Y = var X + var N SNR var X var N copyright Robert J. Marks II p.338 Example RP’s Multiple Random Process Examples (cont) Discrete time RP’s X[n] Mean Variance Autocorrelation Autocovariance Discrete time i.i.d. RP’s Bernoulli RP’s Binomial RP’s Binary vs. Bipolar Random Walk p.341-2 copyright Robert J. Marks II p.340 Autocovariance of Sum Processes n Sn X [ k ] X[k]’s are iid. k 1 E [ S n ] nX var[ S n ] n var( X ) Autocovariance=? copyright Robert J. Marks II Autocovariance of Sum Processes E ( S n n X )( S k k X ) C S ( n , k ) E ( S n S n )( S k S k ) n k E ( X i X ) ( X j 1 i 1 j X ) When i=j, the answer is var(X). Otherwise, zero. How many cases are there where i = j? min( n , k ) C S ( n , k ) min( n , k ) var( X ) copyright Robert J. Marks II Autocovariance of Sum Processes For Bernoulli sum process, var( X ) pq C S ( n , k ) min( n , k ) pq For Bipolar case var( X ) 4 pq C S ( n , k ) 4 min( n , k ) pq copyright Robert J. Marks II Continuous Random Processes Poisson Random Process Place n points randomly on line of length T t T Choose any subinterval of length t. The probability of finding k points on the subinterval is Pr[ k n k nk t po int s ] p q ;p T k copyright Robert J. Marks II Continuous Random Processes Poisson Random Process (cont) The Poisson approximation: For k big and p small… Pr[ k k n k nk np ( np ) points ] p q e k! k nt / T ( nt / T ) e k! copyright Robert J. Marks II k Continuous Random Processes The Poisson Approximation… For n big and p small (implies k << n since p k/n<<1) k n k nk np ( np ) p q e k! k Here’s why… k n n! n ( n 1)( n 2 )...( n k 1) n k! k! k k ! ( n k )! q nk (1 p ) nk n (1 p ) ( e copyright Robert J. Marks II p n ) Continuous Random Processes Poisson Random Process (cont) Pr[ k k n k nk ( np ) np points ] p q e k! k e nt / T ( nt / T ) k k! Let n such that =n/T = frequency of points remains constant. Pr[ k points on interval t] e t (t ) k! copyright Robert J. Marks II k Continuous Random Processes Poisson Random Process (cont) Pr[ k points on interval t] e t ( t ) k! This is a Poisson process with parameter occurrences per unit time Examples: Modeling Popcorn Rain (Both in space and time) Passing cars Shot noise Packet arrival times copyright Robert J. Marks II t k Continuous Random Processes Poisson Counting Process X(t ) Poisson Points Pr[ X ( t ) k ] e copyright Robert J. Marks II t ( t ) k! k Continuous Random Processes Recall for Poisson RV with parameter a Pr[ X k ] e a ( a ) k! k X var( X ) a Poisson Counting Process Expected Value is thus E [ X ( t )] t copyright Robert J. Marks II Continuous Random Processes The Poisson Counting Process is independent increment process. Thus, for t and j i, Pr[ X ( t ) i , X ( ) j ] Pr[ X ( t ) i , X ( ) X ( t ) j i ] Pr[ X ( t ) i ] Pr[ X ( ) X ( t ) j i ] t i e t i! ( t) j i ( t ) e ( j i )! copyright Robert J. Marks II Continuous Random Processes t Autocorrelation: If > t R X ( t , ) E X ( t ) X ( ) E X ( t ) X ( ) X ( t ) X ( t ) 2 E X ( t ) X ( ) X ( t ) E X ( t ) 2 E X ( t ) E X ( ) X ( t ) E X ( t ) t ( t ) t t 2 t t 2 2 2 R X ( t , ) t min( t , ) copyright Robert J. Marks II Continuous Random Processes Autocovariance of a Poisson sum process C X ( t , ) R X ( t , ) E X ( t ) E X ( ) t min( t , ) t 2 min( t , ) copyright Robert J. Marks II Continuous Random Processes Other RP’s related to the Poisson process Random telegraph signal X(t ) Poisson Points copyright Robert J. Marks II Poisson Random Processes Random telegraph signal E [ X ( t )] e 2 | t | C X ( t , ) e 2 |t | PROOF… copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t>0, E [ X ( t )] 1 Pr X ( t ) 1 ( 1) Pr[ X ( t ) 1] Pr number of points on ( 0 ,t) is even Pr number of points on ( 0 ,t) is odd copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t>0, Pr number of points on ( 0 ,t) is even 2 4 ( t ) ( t ) t e 1 ... 2! 4! e t cosh t copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t>0. Similarly… Pr number of points on ( 0 ,t) is odd 3 5 ( t ) ( t ) t e t ... 3! 5! e t sinh( t ) copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t>0. Thus E [ X ( t )] 1 Pr X ( t ) 1 ( 1) Pr[ X ( t ) 1] Pr number of points on ( 0 ,t) is even Pr number e e t cosh( 2t of points on ( 0 ,t) is odd t ) sinh( t ) ;t 0 For all t… E [ X ( t )] e copyright Robert J. Marks II 2 | t | Poisson Random Processes Random telegraph signal. For t > , 1 X() -1 1 X(t) -1 Pr[ X ( t ) X ( ) 1] Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t > , Pr X ( t ) 1 | X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr number e Thus… of points on ( , t ) is even ( t ) cosh ( t ) Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 cosh ( t ) e ( t ) cosh( ) e copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t > , Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 cosh ( t ) e t 1 cosh( ) And… Pr X ( t ) 1, X ( ) 1 -1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 cosh ( t ) e t sinh( ) copyright Robert J. Marks II X() 1 -1 X(t) Poisson Random Processes Random telegraph signal. For t > . Onward… Pr X ( t ) 1 | X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr number e ( t ) of points on ( , t ) is odd sinh ( t ) copyright Robert J. Marks II Poisson Random Processes Random telegraph signal. For t > . Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 sinh ( t ) e t 1 cosh( ) X() And… Pr X ( t ) 1, X ( ) 1 -1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 sinh ( t ) e t sinh( ) copyright Robert J. Marks II 1 -1 X(t) Poisson Random Processes Random telegraph signal. For t > . Pr X ( t ) 1, X ( ) 1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 sinh ( t ) e t 1 cosh( ) X() And… Pr X ( t ) 1, X ( ) 1 -1 Pr X ( t ) 1 | X ( ) 1 Pr X ( ) 1 sinh ( t ) e t sinh( ) copyright Robert J. Marks II 1 -1 X(t) Poisson Random Processes Random telegraph signal. For t > . 1 X() -1 1 -1 R X ( t , ) E X ( t ) X ( ) 1 Pr X ( t ) X ( ) 1 1 Pr X ( t ) X ( ) 1 sinh cosh ( t ) e t cosh( ) sinh ( t ) e ( t ) e t cosh( X(t) ) cosh ( t ) e t t In general…C X ( t , ) R X ( t , ) X ( t ) X ( ) e copyright Robert J. Marks II ) sinh( ) sinh( 2 |t | Continuous Random Processes Other RP’s related to the Poisson process Poisson point process, Z(t) Let X(t) be a Poisson sum process. Then Z(t ) d dt X ( t ) ( t Sn ) Z(t ) n pp.352 Poisson Points copyright Robert J. Marks II Continuous Random Processes Other RP’s related to the Poisson process Shot Noise, V(t) Z(t) h(t) V(t) V ( t ) h( t S n ) n V(t ) pp.352 Poisson Points copyright Robert J. Marks II Continuous Random Processes Wiener Process Assume bipolar Bernoulli sum process with jump bilateral height h and time interval E[X(t)]=0; Var X(n) = 4npqh 2 = nh 2 Take limit as h 0 and 0 keeping = h 2 / constant and t = n . Then Var X(t) t By the central limit theorem, X(t) is Gaussian with zero mean and Var X(t) = t We could use any zero mean process to generate the Wiener process. p.355 copyright Robert J. Marks II Continuous Random Processes Wiener Processes: =1 copyright Robert J. Marks II Continuous Random Processes Wiener processes in finance S= Price of a Security. = inflationary force. If there is no risk…interest earned is proportional to investment. dS ( t ) S ( t ) dt dS t dt S Solution is S ( t ) S 0 e With “volatility” , we have the most commonly used model in finance for a security: dS ( t ) S ( t ) dt S ( t ) dV ( t ) V(t) is a Wiener process. copyright Robert J. Marks II