1. Random Processes - MIT

1 Design Principles for Ocean Vehicles Prof. Techet Spring 2005 1. Random Processes A Random variable,()x , can be defined from a Random event, , by assigning values ix to each possible outcome, iA, of the event. Next define a Random process , ()xt ,, a function of both the event and time, by assigning to each outcome of a Random event, , 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.

2 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 . 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.

3 TIME AVERAGE (Temporal Mean) {}01()()TtlimiTiMxtxtdtxT == (8) TIME VARIANCE (Temporal Variance) {}201{()}[()()]TtlimiTiiVxtxt Mxt dtT = (9) TEMPORAL CROSS/AUTO CORRELATION This gives us the correlation or similarity in the signal and its time shifted version. {}{ }01()[()()][( )( )]TtlimttiTii iiRxt M xt xtM xtdtT = + + (10) is the correlation variable (time shift).

4 TiR|| is between 0 and 1. If tiR is large ( ( )1tiR ) then ( )ixt and ()ixt + are similar . For example, a sinusoidal function is similar to itself delayed by one or more periods. If tiR is small then ( )ixt and ()ixt + are not similar for example white noise would result in ( ) 0tiR =. EXPECTED VALUE: 111{( )}( )xtExtxfxtdx ==, (11) STATISTICAL VARIANCE: 2221111[()()]() ( )xx xtExttxfxtdx = = , (12) AUTO-CORRELATION: [][]{}12121122{()()}(){()} (){()}xRxt t ExtxtE xtExtxtExt ,= , , = , ,, , (13) Example: Roll the dice: 16k=: Assign to the event ( )kAt a Random process function.

5 ()coskoxta kt = (14) Evaluate the time statistics: MEAN: {()}tkMxt=10cos0 TlimToTaktdt = VARIANCE: {()}tkVxt=222120cosTlimaToTaktdt = CORRELATION:{()}tkRxt=210cos() cos(())TlimTooTaktktdt + = 22aocosk Looking at the correlation function then we see that if 2okt =/ 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 2 /, and cosine and sine are not correlated.

6 Now if we look at the STATISTICS of the Random process , for some time ott=, ()cos( ) ()kooo kxta kt y ,== (15) where k is the Random variable (123456k=,,,,,) and each event has probability, 1 6ip=/. EXPECTED VALUE: 6161{( )}cos()kkookEypxak t === VARIANCE: 622161{( )}cos()ookVyak t == CORRELATION: () {( )()}yyokokoRtEyt yt ,=,+, STATISTICS TIME AVERAGES In general the Expected Value does not match the Time Averaged Value of a function the statistics are time dependent whereas the time averages are time independent.

7 2. Stationary Random Processes A stationary Random process is a Random process , ()Xt ,, whose statistics (expected values) are independent of time. For a stationary Random process : 11(){()}()xtExtft =, 222111()( )[ ( )( )]xxxVttE xtt == =()()()xxxxRtRft ,== ()( 0)()Vt RtV f t=,= 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()yt ,: () cos(())oytat ,=+ (16) ()cos()ioiyt at =+ (17) where ( ) is a Random variable which lies within the interval 0 to 2 , with a constant, uniform PDF such that 12for(02 )()0elsef /; =.

8 (18) STATISTICAL AVERAGE: the statistical mean is not a function of time. 201{( )}cos()02oooEytatd =+= (19) STATISTICAL VARIANCE: Variance is also independent of time. 2() ( 0)2oaVtR === (20) STATISTICAL CORRELATION: Correlation is not a function of t, is a constant.

9 2202{()()}( )1cos() cos([])21cos2oooooo ooEytytRtatt da ,+,=,=+++= (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()0 TlimiTo ilimToiomytatdtTaTT ,=+=+= (22) TIME VARIANCE: 2(0)2ttaVR== (23) CORRELATION.

10 2021()cos()cos( [])1cos2 TtlimToioioRatt dtTa =+++= (24) STATISTICS = TIME AVERAGES Therefore the process is considered to be an ERGODIC Random process ! : This particular Random process will be the building block for simulating water waves. 3. ERGODIC Random Processes Given the Random process ()yt , it is simplest to assume that its expected value is zero. Thus, if the expected value equals some constant, { ()}oExtx ,=, where 0ox , then we can simply adjust the Random process such that the expected value is indeed zero:() ()oyt xt x ,=.