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ELEC 303 – Random Signals Lecture 20 – Random processes Dr. Farinaz Koushanfar ECE Dept., Rice University Nov 11, 2010 Lecture outline • • • • • • Basic concepts Random processes and linear systems Power spectral density of stationary processes Power spectra in LTI systems Power spectral density of a sum process Gaussian processes RP and linear systems • When a RP passes a linear time-invariant system the output is also a RP • Assuming a stationary process X(t) is input, the linear time-invariant system with the impulse response h(t), output process Y(t) • Under what condition the output process would be stationary? • Under what conditions will the input/output jointly stationary? • Find the output mean, autocorrelation, and crosscorrelation X(t) h(t) Y(t) Linear time invariant systems • If a stationary RP with mean mX and autocorrelation function RX() • Linear time invariant (LTI) system with response h(t) • Then, the input and output process X(t) and Y(t) will be jointly stationary with X(t) h(t) Y(t) The response mean • Using the convolution integral to relate the output Y(t) to the input X(t), Y(t)=X()h(t-)d This proves that mY is independent of t X(t) h(t) Y(t) Cross correlation • The cross correlation function between output and the input is This shows that RXY(t1,t2) depends only on =t1-t2 Output autocorrelation • The autocorrelation function of the output is This shows that RY and RXY depend only on =t1-t2, Output process is stationary, and input/output are jointly stationary Power spectral density of a stationary process • If the signals in the RP are slowly varying, then the RP would mainly contain the low frequencies in its power concentration • If the signal changes very fast, most of the power will be concentrated at high frequency • The power spectral density of a RP X(t) is denoted by SX(f) showing the strength of the power in RP as a function of frequency • The unit for SX(f) is Watts/Hz Wiener-Khinchin theorem • For a stationary RP X(t), the power spectral density is the Fourier transform of the autocorrelation function, i.e., Example 2 • Randomly choose a phase ~ U[0,2] • Generate a sinusoid with fixed amplitude (A) and fixed freq (f0) but a random phase • The RP is X(t)= A cos(2f0t + ) • From the previous lecture, we know Example 3 • X(t)=X • Random variable X~U[-1,1] • In this case • Thus, • For each realization of the RP, we have a different power spectrum Power spectral density • The power content of a RP is the sum of the powers at all frequencies in that RP • To find the total power, need to integrate the power spectral density across all frequencies • Since SX(f) is the Fourier transform of RX(), then RX() will be the inverse Fourier transform of SX(f), Thus • Substituting =0, we get Example 4 • Find the power in the process of example 2 Translation to frequency domain • For the LTI system and stationary input, find the translation of the relationships between the input/output in frequency domain • Compute the Fourier transform of both sides to obtain • Which says the mean of a RP is its DC value. Also, phase is irrelevant for power. Only the magnitude affects the power spectrum, i.e., power dependent on amplitude, not phase Example 5 • • • • If a RP passes through a differentiator H(f)=j2f Then, mY=mX H(0) = 0 Also, SY(f) = 42 f2 SX(f) Cross correlation in frequency domain • Let us define the cross spectral density SXY(f) • Since RYX() = RXY(-), we have • Although SX(f) and SY(f) are real nonnegative functions, SXY(f) and SYX(f) can generally be complex functions Example 6 • Randomly choose a phase ~ U[0,2] • Generate a sinusoid with fixed amplitude (A) and fixed freq (f0) but a random phase • The RP is X(t)= A cos(2f0t + ) • The X(t) goes thru a differentiator H(f)=j2f Example 7 • X(t)=X • Random variable X~U[-1,1] • If this goes through differentiation, then SY(f) = 42 f2 ((f)/3) = 0 SXY(f) = -j2f ((f)/3) = 0 Power spectral density of a sum process • Z(t) = X(t)+Y(t) • X(t) and Y(t) are jointly stationary RPs • Z(t) is a stationary process with RZ() = RX() + RY() + RXY() + RYX() • Taking the Fourier transform from both sides: SZ(f) = SX(F) + SY(f) + 2 Re[SXY(f)] • The power spectral density of the sum process is the sum of the power spectral of the individual processes plus a term, that depends on the cross correlation • If X(t) and Y(t) are uncorrelated, then RXY()=mXmY • If at least one of the processes is zero mean, RXY()=0, and we get: SZ(f) = SX(F) + SY(f) Example 8 • • • • • • • X(t)=X Random variable X~U[-1,1] Z(t) = X(t) + d/dt X(t), then SXY(f) = jA2f0 /2 [(f+f0) - (f-f0)] Thus, Re[SXY(f)] = 0 SZ(f)= SX(f)+SY(f) = A2(1/4+2f02)[(f+f0)+(f-f0)] Gaussian processes • Widely used in communication • Because thermal noise in electronics is produced by the random movement of electrons closely modeled by a Gaussian RP • In a Gaussian RP, if we look at different instances of time, the resulting RVs will be jointly Gaussian: Definition 1: A random process X(t) is a Gaussian process if for all n and all (t1,t2,…,tn), the RVs {X(ti)}, i=1,…,n have a jointly Gaussian density function. Gaussian processes (Cont’d) Definition 2: The random processes X(t) and Y(t) are jointly Gaussian if for all n and all (t1,t2,…,tn), and (1,2,…,m)the random vector {X(ti)}, i=1,…,n, {Y(j}, j=1,…,m have an n+m dimensional jointly Gaussian density function. • It is obvious that if X(t) and Y(t) are jointly Gaussian, then each of them is individually Gaussian • The reverse is not always true • The Gaussian processes have important and unique properties Important properties of Gaussian processes • Property 1: If the Gaussian process X(t) is passed through an LTI system, then the output process Y(t) will also be a Gaussian process. Y(t) and X(t) will be jointly Gaussian processes • Property 2: For jointly Gaussian processes, uncorrelatedness and independence are equivalent