Stochastic Processes- IV

Transcription

1 !! Module 2! Lecture 7 :Random Vibrations & Failure Analysis Stochastic Processes- IV!! Sayan Gupta Department of Applied Mechanics Indian Institute of Technology Madras

2 Properties of Power Spectral Density (PSD) If X(t) is a real valued, weakly stationary random process, and if R XX ( ) is an even function, then the PSD function S XX (!) is real and even. This can be proved as follows: S XX (!) R XX ( )e i! d R XX ( ) cos(! )d R XX ( ) cos(! )d...() R XX ( ) 2 S XX (!) cos(! )d!....(2) 2

3 Properties of Power Spectral Density (PSD) The auto-psd of a real valued process is always greater or equal to zero, i.e., S XX (!) 8!...(3) The variance of the process is obtained as the area under the PSD curve.! The cross PSD function S XY (!) for the processes X(t) and Y (t) is generally complex and Hermitian.! The auto-psd of a process is an even function, i.e., S XX (!) S XX (!) 8!....(4) 3

4 Properties of Power Spectral Density (PSD) Now, we know from Eq. (.7) that R XX ( ) Di erentiating both sides of Eq. (.26), (5) we get d d R XX( ) d d i In general, it can be shown that S XX (!)exp[i! ] d!....(5) S XX (!)e i! d! S XX (!) d d e i! d!!s XX (!)e i! d!....(6) d j h i d j R XX ( ) (i) j! j S XX (!)e i! d!....(7) 4

5 Properties of Power Spectral Density (PSD) Now, the cross-covariance function between the process and its first time derivative can be derived as R X Ẋ ( ) d h i R XX ( ) d i!s XX (!)e i! d!....(8) Similarly, the auto-covariance function between the process derivative can be obtained as RẊẊ ( ) d 2 h i i d 2 R XX ( ) d d hr ( ) XẊ (i!) 2 S XX (!)e i! d!! 2 S XX (!)e i! d!....(9) 5

6 Properties of Power Spectral Density (PSD) But R X Ẋ ( ) S X Ẋ (!)ei! d!...() RẊẊ ( ) SẊẊ ( )ei d...() It follows that S X Ẋ (!) i!s XX(!)...(2) SẊẊ (!)!2 S XX (!)...(3) In general, we get S X j X k(!) ( )k i j+k S XX (!)....(4) 6

7 Properties of Power Spectral Density (PSD) Note that S XX (!) <X(!)X (!) > S X Ẋ (!) <X(!)Ẋ (!) >...(5) SẊX (!) < Ẋ(!)X (!) > SẊẊ (!) < Ẋ(!)Ẋ (!) > We also know that X(t) Ẋ(t) Thus, it follows RẊẊ ( ) < Ẋ(t)Ẋ(t + ) > X(!)e i!t d! i! X(!)e i!t d! (i!) 2 <X(!)X (!) > e i! d!...(6) i 2! 2 S XX (!)e i! d!! 2 S XX (!)e i! d!....(7) 7

8 PSD of Physical Processes The PSD of any stochastic process is even and is symmetric about the origin. Given below are some schematic diagrams of power spectral density functions. Now, we know that the area under the PSD curve is equal to the variance of the process. Thus, R XX () 2 Z S XX (!) d! S XX (!) d! + S XX (!) d! + S XX (!) d! 2 X. S XX (!) d! S XX (!) d!...(8) 8

9 PSD of Physical Processes But physical processes are usually defined only in terms of a -sided PSD, with the PSD defined along the positive axis of!. Thus, for physical processes, one might write S XX (!) d! Z + S XX (!) d! + S XX (!) d! 2 X. S XX (!) d!...(9) Eq. (.4) (9) is clearly wrong as this does not take care of the negative side of the PSD. Even though a physical process is usually defined only in the positive side, by the very definition of the PSD, there exists a mirror image about the origin. Thus, to get the actual variance from the PSD of a physical process, one needs to multiply the area of the -sided physical process by a factor of 2. 9

10 Narrowband Process Consider the PSD of the process shown in the above figure. Here, S XX (!) is defined within a narrow-band [! c b,! c + b]. Since S XX (!) is an even function, it exists in the negative side of the zero axis also; see the figure above. Thus, S XX (!) everywhere, except within a distance b of ±! c. Mathematically, the PSD can be expressed in terms of the Heaviside function as h i S XX (!) S nu! (! c b) h io U! (! c + b)....(2)

11 Narrowband Process The auto-correlation function of the narrowband process can be expressed as R XX ( ) Z!c +b! c b S XX (!)e i! d! S e i! d! + Z!c +b! c b Z!c +b S cos! +isin!! c b Z!c +b S! c b cos! +isin! Z!c +b 2S cos! d!! c b S e i! d! d! n sin(!c + b) sin(! c b) 2S 4S cos(! c ) sin(b ) 4S b cos(! c ) sin(b ) b d! + o...(2)

12 Narrowband Process Thus, R XX () lim! R XX( ) 4S b 2 X...(22) It follows that R XX ( ) X 2 sin b cos(! c ) b...(23) If b<<! c, then the oscillation of sin(b ) is slower than cos(! c ). Thus, the term sin(b )/(b ) behaves like an envelope function. This is explained in the figure below. 2

13 Broadband Process The most narrowband process is typically a process which can be represented as a sinusoid function, example, X(t) a cos!t. The PSD of such a process is characterized by a dirac-delta function. At the other extreme of the sinusoid function is what is called a white-noise process, that contains frequencies across the entire spectrum. The PSD of such processes are characterized by a straight line (see the above figure). It can be shown that such a process is so erratic that X(t) and X(s) are independent of each other, for any t and for any s. This implies that the correlation length of such a process is infinitely small. Deriving the auto-correlation function, it can be shown that R XX ( ) 2 S ( )....(24) This leads to the terminology of such processes as delta-correlated processes. 3

14 Measures of bandwidth Processes whose PSD are characterized by a larger spread along the spectrum are typically referred to as broadband processes. White noise processes are extreme broadband processes. More discussions on such processes would be considered later in this course. It has been shown that the behavior of processes depend on the spectral content. In fact, depending on the bandwidth of processes, several approximations can be made to predict the behavior of such processes. It is therefore essential to derive some measures on the bandwidth of processes that can be used to decide on the applicability of approximations derived in the literature, for predicting behavior of processes. Vanmarcke (972) noted that the PSD has the following similarities with a probability density function: for!, S XX (!), the integral of S XX (!) is bounded as long as the process is not a deltacorrelated process. This is similar to a pdf, where the area under the pdf is unity. 4

15 Measures of bandwidth The spread of a probability density function is characterized by a non-dimenshionless number, namely, the coe cient of variation, expressed as c.o.v µ,...(25) where, is the standard deviation and µ is the mean value of the random variable. A random variable with small variance is narrower in shape. For a random variable with the same mean value but with a higher spread has a higher variance. Thus, the co.v. of the second random variable would be greater than the first. Thus, c.o.v. provides a measure of the spread of a random variable. Vanmarcke used a similar analogy to represent the measure of the bandwidth of a process. Recognizing that the mean and the standard deviation, in Eq. (.46) are the first and second moments of the pdf, a similar definition was applied to find the spectral moments. Z S XX (!) d! zeroeth spectral moment Z!S XX (!) d! first spectral moment Z! 2 S XX (!) d! second spectral moment 2...(26)...(27)...(28) 5

16 Measures of bandwidth By analogy, the mean and the variance of the PSD can be expressed as (noting that the zeroeth moment of the pdf is unity) mean variance 2 2 Thus, the measure of bandwidth, s, (analogous to c.o.v.) can be expressed as s p ( 2 / ) ( / ) 2 / s (29) It can be shown that for a narrowband process, S!, and s> for any other process. The drawback of this definition of measure of bandwidth is that it does not give a proper definition for a broadband process. This necessitated a modification of the definition of the bandwidth measure. 6

17 Measures of bandwidth Assume p 2,...(3) such that, (s 2 + ) 2....(3) Here, is a bandwidth parameter, apple apple, and for a perfectly narrowband process and for a perfectly broadband process. Other versions of the above measure can also be defined. In general, where, m m p, 2m Z! m S XX (!) d!. m...(32)...(33) 7

18 Measures of bandwidth The most popularly used definition is where, It can be shown that 2 2 X 2Ẋ 2 2Ẍ 4 2 Z Z Z X 2Ẋ Ẍ S XX (!) d!,! 2 S XX (!) d!,! 4 S XX (!) d!, X Ẍ ( )...(34)...(35)...(36)...(37)...(38) i.e., 2 is equal to the negative of the correlation coe and its second time derivative. cient between the process 8