EE4601 Communication Systems

Transcription

1 EE4601 Communication Systems Week 4 Ergodic Random Processes, Power Spectrum Linear Systems 0 c 2011, Georgia Institute of Technology (lect4 1)

2 Ergodic Random Processes An ergodic random process is one where time averages are equal to ensemble averages. Hence, for all g(x) and X E[g(X)] = g(x)p X(t)(x)dx = lim 2T g[x(t)]dt T = < g[x(t)] > T 1 For a random process to be ergodic, it must be strictly stationary. However, not all strictly stationary random processes are ergodic. A random process is ergodic in the mean if T < X(t) >= µ X and ergodic in the autocorrelation if < X(t)X(t+τ) >= φ XX (τ) 0 c 2011, Georgia Institute of Technology (lect4 2)

3 Example (cont d) Recall the random process X(t) = Acos(2πf c t+θ) where A and f c are constants, and Θ is a uniformly distributed random phase. p Θ (θ) = The time average mean of X(t) is 1 < X(t) >= lim T 2T 1/(2π), 0 θ 2π 0, elsewhere T T Acos(2πf ct+θ)dt = 0 In this example µ X (t) =< X(t) >, so the random process X(t) is ergodic in the mean. N.B. Make sure you understand the difference between the time average and ensemble average. 0 c 2011, Georgia Institute of Technology (lect4 3)

4 Example (cont d) The time average autocorrelation of X(t) is < X(t)X(t+τ) > = lim T = lim T 1 2T A 2 4T T T A2 cos(2πf c t+2πf c τ +θ)cos(2πf c t+θ)dt T = A2 2 cos(2πf cτ) T [cos(2πf cτ)+cos(4πf c t+2πf c τ +2θ)]dt The random process X(t) is ergodic in the autocorrelation. It follows that the random process X(t) in this example is ergodic in the mean and autocorrelation. 0 c 2011, Georgia Institute of Technology (lect4 4)

5 Example Consider the random process shown below. X (t) = a P = 1/4 1 1 X (t) = 0 P = 1/2 2 2 X (t) = -a 3 3 P = 1/4 0 c 2011, Georgia Institute of Technology (lect4 5)

6 Example (cont d) For this example, the ensemble and time average means are, respectively, X(t) = Hence, X(t) is not ergodic in the mean. µ X = E[X(t)] = 0 a with probability 1/4 0 with probability 1/2 a with probability 1/4 The ensemble and time average autocorrelations are φ XX (τ) = E[X(t)X(t+τ)] = a 2 (1/4)+0(1/2)+( a) 2 (1/4) = a 2 /2 X(t)X(t+τ) = a 2 with probability 1/2 0 with probability 1/2 Hence, X(t) is not ergodic in the autocorrelation. 0 c 2011, Georgia Institute of Technology (lect4 6)

7 Example (cont d) Note that E[ X(t) ] = µ X E[ X(t)X(t+τ) ] = φ XX (τ) Because of this property X(t) and X(t)X(t+τ) are said to provide unbiased estimates of µ X and φ XX (τ), respectively. 0 c 2011, Georgia Institute of Technology (lect4 7)

8 Power Spectral Density The power spectral density (psd) of a random process X(t) is the Fourier transform of its autocorrelation function, i.e., Φ XX (f) = = φ XX (τ) = φ XX(τ)e j2πfτ dτ Φ XX(f)e j2πfτ df. We have seen that φ XX (τ) is real and even. Therefore, Φ XX ( f) = Φ XX (f) meaning that Φ XX (f) is also real and even. The total power (ac + dc), P, in a random process X(t) is P = E[X 2 (t)] = φ XX (0) = a famous result known as Parseval s theorem. Φ XX(f)df 0 c 2011, Georgia Institute of Technology (lect4 8)

9 Example X(t) = Acos(2πf c t+θ) where A and f v are constants and We have seen before that p Θ (θ) = 1 2π, π θ π 0, elsewhere Hence, φ XX (τ) = A2 2 cos(2πf cτ) Φ XX (f) = A2 2 F[cos(2πf cτ)] = A2 4 (δ(f f c)+δ(f +f c )) 0 c 2011, Georgia Institute of Technology (lect4 9)

10 Properties of Φ XX (f) 1. Φ XX (0) = φ XX (τ)dτ Φ XX (f)df = dc power 3. φ XX (0) = Φ XX (f)df = total power 4. Φ XX (f) 0 for all f. Power is never negative. 5. Φ XX (f) = Φ XX ( f) (even function) if X(t) is a real random process. 6. Φ XX (f) is always real. 0 c 2011, Georgia Institute of Technology (lect4 10)

11 Discrete-time Random Processes Consider a discrete-time real random process X n, that consists of an ensemble of sample sequences {x n }. The ensemble mean of X n is defined as µ Xn = E[X n ] = The ensemble autocorrelation of X n is φ XX (n,k) = E[X n X k ] = x nf Xn (x n )dx n X nx k f Xn,X k (x n,x k )dx n dx k For a wide-sense stationary discrete-time real random process, we have µ Xn = µ X, n φ XX (n,k) = φ XX (n k) From Parseval s theorem, the total power in the process X n is P = E[X 2 n ] = φ XX(0) 0 c 2011, Georgia Institute of Technology (lect4 11)

12 Power Spectrum of Discrete-time RP The power spectrum of the real wide-sense stationary discrete-time random process X n is the discrete-time Fourier transform of the autocorrelation function, i.e., Φ XX (f) = φ XX (n) = n= 1/2 φ XX (n)e j2πfn 1/2 Φ XX(f)e j2πfn df Observe that the power spectrum Φ XX (f) is periodic in f with a period of unity. In other words Φ XX (f) = Φ XX (f +k), for k = ±1,±2,... This is a characteristic of any discrete-time sequence. For example, one obtained by sampling a continuous-time random process X n = x(nt s ), where T s is the sample period. 0 c 2011, Georgia Institute of Technology (lect4 12)

13 Linear Systems φ Φ X( t) Y( t) XX XX ( τ) ( f) h ( t) H( f) φ Φ YY YY τ ( ) f ( ) 0 c 2011, Georgia Institute of Technology (lect4 13)

14 Linear Systems Suppose that the input to the linear system h(t) is a wide sense stationary random process X(t), with mean µ X and autocorrelation φ XX (τ). The input and output waveforms are related by the convolution integral Hence, The output mean is µ Y = Y(t) = h(τ)x(t τ)dτ. Y(f) = H(f)X(f). h(τ)e[x(t τ)]dτ = µ X h(τ)dτ = µ XH(0). This is just the mean (dc component) of the input signal multiplied by the dc gain of the filter. 0 c 2011, Georgia Institute of Technology (lect4 14)

15 Linear Systems The output autocorrelation is φ YY (τ) = E[Y(t)Y(t+τ)] = E [ h(β)x(t β)dβ h(α)x(t+τ α)dα ] = h(α)h(β)e[x(t β)x(t+τ α)]dβdα = h(α)h(β)φ XX(τ α+β)dβdα = h(α) h(β)φ XX(τ +β α)dαdβ { } = h(β)φ XX(τ +β)dβ h(τ) = h( τ) φ XX (τ) h(τ). Taking transforms, the output psd is Φ YY (f) = H (f)φ XX (f)h(f) = H(f) 2 Φ XX (f). 0 c 2011, Georgia Institute of Technology (lect4 15)

16 Cross-correlation and Cross-covariance If X(t) and Y(t) areeach widesense stationaryand jointlywidesense stationary, then φ XY (t,t+τ) = E[X(t)Y(t+τ)] = φ XY (τ) µ XY (t,t+τ) = µ XY (τ) = φ XY (τ) µ x µ y The crosscorrelation function φ XY (τ) has the following properties. 1. φ XY (τ) = φ YX ( τ) 2. φ XY (τ) 1 2 [φ XX(0)+φ YY (0)] 3. φ XY (τ) 2 φ XX (0)φ YY (0) if X(t) and Y(t) have zero mean. 0 c 2010, Georgia Institute of Technology (lect4 17)

17 Example Consider the linear system shown in the previous example. The crosscorrelation between the input process X(t) and the output process Y(t) is φ YX (τ) = E[Y(t)X(t+τ)] = E = [ h(α)x(t α)dαx(t+τ) ] h(α)e[x(t α)x(t+τ)]dα The cross power spectral density is = h(α)φ XX(τ +α)dα = h( τ) φ XX (τ) Φ YX (f) = H (f)φ XX (f) Note also that φ YX ( τ) = φ XY (τ) 0 c 2011, Georgia Institute of Technology (lect4 17)

18 Example R X(t) C Y(t) 0 c 2011, Georgia Institute of Technology (lect4 18)

19 Example The transfer function of the filter is H(f) = 1 1+j2πfRC Suppose that φ XX (τ) = e α τ. What is φ YY (τ)? We have where Hence, Φ YY (f) = Φ YY (f) = H(f) 2 Φ XX (f) H(f) 2 = Φ XX (f) = 1 1+(2πfRC) 2 2α α 2 +(2πf) (2πfRC) 2 2α α 2 +(2πf) 2 0 c 2011, Georgia Institute of Technology (lect4 19)

20 Example Do you remember partial fractions? Now you need them! We write A Φ YY (f) = α 2 +(2πf) + B 2 1+(2πfRC) 2 and solve for A and B. We have Clearly, A(1+(2πfRC) 2 )+B(α 2 +(2πf) 2 ) = 2α A+Bα 2 = 2α A(2πfRC) 2 +B(2πf) 2 = 0 From the second equation where β = 1/(RC). A = B (RC) 2 = Bβ2 0 c 2011, Georgia Institute of Technology (lect4 20)

21 Example Then using the first equation Also, Finally, Φ YY (f) = β2 β 2 α 2 B = 2α α 2 β 2 A = Bβ 2 = 2αβ2 α 2 β 2 Now take inverse Fourier transforms to get 2α α 2 +(2πf) + αβ 2 α 2 β 2 φ YY (τ) = β2 β 2 α 2 e α τ + αβ α 2 β 2 e β τ 2β β 2 +(2πf) 2 0 c 2011, Georgia Institute of Technology (lect4 21)

22 Discrete-time Random Processes Consider a wide-sense stationary discrete-time random process X n that is input to a discrete-time linear time invariant filter with impulse response h n. The transfer function of the filter is H(f) = n= h n e j2πfn The output of the filter is the convolution sum It follows that the output mean is Y k = n= h n X k n µ Y = E[Y k ] = n= = µ X h n E[X k n ] h n n= = µ X H(0) 0 c 2011, Georgia Institute of Technology (lect4 22)

23 Discrete-time Random Processes The autocorrelation function of the output process is φ YY (k) = E[Y n Y n+k ] = i= j= = i= j= h i h j E[X n i h j X n+k j ] h i h j φ XX (k j +i)] By taking the discrete-time Fourier transform of φ YY (k) and using the above relationship, we can obtain Φ YY (f) = Φ XX (f) H(f) 2 Again, note in this case that Φ YY (f) is periodic in f with a period of unity. 0 c 2011, Georgia Institute of Technology (lect4 23)