13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Random Processes A random variable, ()xζ, can be defined from a Random event, ζ, by assigning values ix to each possible outcome, iA , of the event. Next define a Random Process, ()xtζ,, a function of both the event and time, by assigning to each outcome of a random event, ζ, a function in time, 1()xt , chosen from a set of functions, ( )ixt . 111222()()()nnnApxtApxtApxt→→⎛⎞⎜⎟→→⎜⎟⎜⎟⎜⎟→→⎝⎠MMM (6) This “menu” of functions, ( )ixt , is called the ensemble (set) of the random process and may contain infinitely many ( )ixt , which can be functions of many independent variables. EXAMPLE: Roll the dice: Outcome is iA , where 16i=: is the number on the face of the dice and choose some function ()iixtt= (7) to be the random process.3.1. Averages of a Random Process Since a random process is a function of time we can find the averages over some period of time, T , or over a series of events. The calculation of the average and variance in time are different from the calculation of the statistics, or expectations, as discussed in the previously. TIME AVERAGE (Temporal Mean) {}01() ()TtlimiT iMxt xtdtxT→∞==∫ (8) TIME VARIANCE (Temporal Variance) {}201{()} [() ()]TtlimiT i iVxt xt Mxt dtT→∞=−∫ (9) TEMPORAL CROSS/AUTO CORRELATION This gives us the “correlation” or similarity in the signal and its time shifted version. {} { }01() [() ()][( ) ( )]Tt lim t tiT i i i iRxt M xt xt M xt dtTτττ→∞=− +−+∫ (10) • τ is the correlation variable (time shift). • tiR|| is between 0 and 1. • If tiR is large (i.e. ( ) 1tiRτ→ ) then ( )ixt and ( )ixtτ+ are “similar”. For example, a sinusoidal function is similar to itself delayed by one or more periods. • If tiR is small then ( )ixt and ( )ixtτ+ are not similar – for example white noise would result in ( ) 0tiRτ= .EXPECTED VALUE: 11 1{()} ( )xtExt xfxtdxµ∞−∞==,∫ (11) STATISTICAL VARIANCE: 222111 1[() ()] ( ) ( )xx xtExt t x fxtdxσµ µ∞⎧⎫⎨⎬⎩⎭−∞=−=−,∫ (12) AUTO-CORRELATION: [][]{}12 1 2 1 1 2 2{( )( )} ( ) {( )} ( ) {( )}xRxtt Ext xt E xt Ext xt Extζζ ζ ζ ζ ζ,= , , = , − , , − , (13) Example: Roll the dice: 16k =: Assign to the event ( )kAt a random process function: () coskoxta ktω= (14) Evaluate the time statistics: MEAN: {()}tkMxt=10cos 0TlimToTaktdtω→∞=∫ VARIANCE: {()}tkVxt=222120cosTlimaToTaktdtω→∞=∫ CORRELATION:{()}tkRxt=210cos( )cos( ( ))TlimTooTaktktdtωωτ→∞+∫ = 22aocoskωτ Looking at the correlation function then we see that if 2oktωπ=/ then the correlation is zero – for this example it would be the same as taking the correlation of a sine with cosine, since cosine is simply the sine function phase-shifted by 2π/ , and cosine and sine are not correlated.Now if we look at the STATISTICS of the random process, for some time ott= , ()cos( ) ()ko oo kxtaktyζωζ,== (15) where k is the random variable ( 123456k=,,,,, ) and each event has probability, 1 6ip=/ . EXPECTED VALUE: 6161{()} cos( )kk ookEy px a k tζω===∑∑VARIANCE: 622161{()} cos( )ookVy a k tζω==∑ CORRELATION: (){()( )}yy o k o k oRt Eyt ytτζτζ,=,+, STATISTICS ≠ TIME AVERAGES In general the Expected Value does not match the Time Averaged Value of a function – i.e. the statistics are time dependent whereas the time averages are time independent. 2. Stationary Random Processes A stationary random process is a random process, ()Xtζ,, whose statistics (expected values) are independent of time. For a stationary random process: 11() {( )} ()xtExt ftµζ=,≠ 222111() ( ) [ ( ) ( )]xxxVt t E xt tσµσ⎧⎫⎨⎬⎩⎭== − =() () ()xx xxRtR ftττ,==≠ () ( 0) ()Vt Rt V ft=,=≠ The statistics, or expectations, of a stationary random process are NOT necessarily equal to the time averages. However for a stationary random process whose statistics ARE equal to the time averages is said to be ERGODIC.EXAMPLE: Take some random process defined by()ytζ,: () cos( ())oyt a tζωθζ,=+ (16) () cos( )ioiyt a tωθ=+ (17) where ( )θζ is a random variable which lies within the interval 0 to 2π, with a constant, uniform PDF such that 12 for(0 2 )()0elsefθπθπθ/; ≤≤⎧=⎨;⎩ (18) STATISTICAL AVERAGE: the statistical mean is not a function of time. 201{( )} cos( ) 02oooEyt a t dπζωθθπ=+=∫ (19) STATISTICAL VARIANCE: Variance is also independent of time. 2() ( 0)2oaVt Rτ=== (20) STATISTICAL CORRELATION: Correlation is not a function of t, τ is a constant. 2202{( )( )} ( )1cos( )cos( [ ] )21cos2oo ooo o ooEyt yt Rtat t daπζτζ τωθωτθθπωτ,+,=,=+++=∫ (21) Since statistics are independent of time this is a stationary process!Let’s next look at the temporal averages for this random process: MEAN (TIME AVERAGE): []01{( )} cos( )1sin( ) 0TlimiT oilimToiomyt a t dtTaTTζωθωθω→∞→∞,= +=+=∫ (22) TIME VARIANCE: 2(0)2ttaVR== (23) CORRELATION: 2021() cos( )cos( [ ] )1cos2TtlimToioioRat t dtTaτωθωτθωτ→∞=+++=∫ (24) STATISTICS = TIME AVERAGES Therefore the process is considered to be an ERGODIC random process! N.B.: This particular random process will be the building block for simulating water waves. 3. ERGODIC RANDOM PROCESSES Given the random process ( )ytζ, it is simplest to assume that its expected value is zero. Thus, if the expected value equals some constant, { ( )}oExt xζ,= , where 0ox ≠ , then we can simply adjust the random process such that the expected value is indeed zero:() ()oytxt xζζ,= , −.The autocorrelation function ()Rτ is then 0() {( )( )} ()1() ( )tTlimTiiREytyt Rytyt dtTτζτζ ττ→∞=,+,==+∫ (25) with CORRELATION PROPERTIES: 1. (0)R = variance 2σ== (RMS)20≥ 2. () ( )RRττ=− 3. (0) ( )RRτ≥| | EXAMPLE: Consider the following random process that is a summation of cosines of different frequencies – similar to water waves. 1() cos( ())Nnnnnyt a tζωψζ=,= +∑ (26) where ( )nψζ are all independent random variables in [0,2π] with a uniform pdf. This random process is stationary and ergodic with an expected value of zero. The autocorrelation ()Rτ is 21() cos( )2NnnnaRτωτ==∑