1 Part 5 Response of Linear Systems 6.Linear Filtering of a Random Signals 7.Power Spectrum Analysis 8.Linear Estimation and Prediction Filters 9.Mean-Square Estimation

2 6. Linear Filtering of a Random Signal Linear System  Our goal is to study the output process statistics in terms of the input process statistics and the system function.

3 Deterministic System Deterministic Systems Systems with MemoryMemoryless Systems Linear-Time Invariant (LTI) systems Time-Invariant systems Linear systems Time-varying systems LTI system

4 Memoryless Systems The output Y(t) in this case depends only on the present value of the input X(t). i.e.,. Memoryless system Strict-sense stationary input Strict-sense stationary output. Memoryless system Wide-sense stationary input Need not be stationary in any sense. Memoryless system X(t) stationary Gaussian with Y(t) stationary,but not Gaussian with

5 Linear Time-Invariant Systems Time-Invariant System Shift in the input results in the same shift in the output. Linear Time-Invariant System A linear system with time-invariant property. LTI Impulse response of the system Impulse response Fig. 14.5

6 Linear Filtering of a Random Signal LTI arbitrary input By Linearity By Time-invariance

7 Theorem 6.1 Pf ：

8 Theorem 6.2 If the input to an LTI filter with impulse response h(t) is a wide sense stationary process X(t), the output Y(t) has the following properties: (a) Y(t) is a WSS process with expected value autocorrelation function (b) X(t) and Y(t) are jointly WSS and have I/O cross- correlation by (c) The output autocorrelation is related to the I/O cross-correlation by

9 Theorem 6.2 (Cont’d) Pf:

10 Example 6.1 X(t), a WSS stochastic process with expected value  X = 10 volts, is the input to an LTI filter with What is the expected value of the filter output process Y(t) ? Sol ： Ans: 2(e 0.5  1) V

11 Example 6.2 A white Gaussian noise process X(t) with autocorrelation function R W (  ) =  0  (  ) is passed through the moving- average filter For the output Y(t), find the expected value E[Y(t)], the I/O cross-correlation R WY (  ) and the autocorrelation R Y (  ). Sol ：

12 Theorem 6.3 If a stationary Gaussian process X(t) is the input to an LTI Filter h(t), the output Y(t) is a stationary Gaussian process with expected value and autocorrelation given by Theorem 6.2. Pf ： Omit it.

13 Example 6.3 For the white noise moving-average process Y(t) in Example 6.2, let  0 = 10  15 W/Hz and T = 10  3 s. For an arbitrary time t 0, find P[Y(t 0 ) > 4  10  6 ]. Sol ： Ans: Q(4) = 3.17  10  5

14 Theorem 6.4 The random sequence X n is obtained by sampling the continuous-time process X(t) at a rate of 1/T s samples per second. If X(t) is a WSS process with expected value E[X(t)] =  X and autocorrelation R X (  ), then X n is a WSS random sequence with expected value E[X n ] =  X and autocorrelation function R X [k] = R X (kT s ). Pf ：

15 Example 6.4 Continuing Example 6.3, the random sequence Y n is obtained by sampling the white noise moving-average process Y(t) at a rate of f s = 10 4 samples per second. Derive the autocorrelation function R Y [n] of Y n. Sol ：

16 Theorem 6.5 If the input to a discrete-time LTI filter with impulse response h n is a WSS random sequence, X n, the output Y n has the following properties. (a) Y n is a WSS random sequence with expected value and autocorrelation function (b) Y n and X n are jointly WSS with I/O cross-correlation (c) The output autocorrelation is related to the I/O cross- correlation by

17 Example 6.5 A WSS random sequence, X n, with  X = 1 and auto- correlation function R X [n] is the input to the order M  1 discrete-time moving-average filter h n where For the case M = 2, find the following properties of the output random sequence Y n : the expected value  Y, the autocorrelation R Y [n], and the variance Var[Y n ]. Sol ：

18 Example 6.6 A WSS random sequence, X n, with  X = 0 and auto- correlation function R X [n] =  2  n is passed through the order M  1 discrete-time moving-average filter h n where Find the output autocorrelation R Y [n]. Sol ：

19 Example 6.7 A first-order discrete-time integrator with WSS input sequence X n has output Y n = X n + 0.8Y n-1. What is the impulse response h n ? Sol ：

20 Example 6.8 Continuing Example 6.7, suppose the WSS input X n with expected value  X = 0 and autocorrelation function is the input to the first-order integrator h n. Find the second moment, E[Y n 2 ], of the output. Sol ：

21 Theorem 6.6 If X n is a WSS process with expected value  and auto- correlation function R X [k], then the vector has correlation matrix and expected value given by

22 Example 6.9 The WSS sequence X n has autocorrelation function R X [n] as given in Example 6.5. Find the correlation matrix of Sol ：

23 Example 6.10 The order M  1 averaging filter h n given in Example 6.6 can be represented by the M element vector The input is The output vector, then.

24 6.Linear Filtering of a Random Signals 7.Power Spectrum Analysis 8.Linear Estimation and Prediction Filters 9.Mean-Square Estimation

25 7. Power Spectrum Analysis Definition: Fourier Transform Definition: Power Spectral Density

26 Theorem 7.1 Pf ：

27 Theorem 7.2 Pf ：

28 Example 7.1 Sol ：

29 Example 7.2 A white Gaussian noise process X(t) with autocorrelation function R W (  ) =  0  (  ) is passed through the moving- average filter For the output Y(t), find the power spectral density S Y (f ). Sol ：

30 Discrete-Time Fourier Transform (DTFT) Definition : Example 7.3 : Calculate the DTFT H(  ) of the order M  1 moving-average filter h n of Example 6.6. Sol ：

31 Power Spectral Density of a Random Sequence Definition : Theorem 7.3 : Discrete-Time Winer-Khintchine

32 Theorem 7.4

33 Example 7.4 Sol ：

34 Example 7.5 Sol ：

35 Cross Spectral Density Definition :

36 Example 7.6 Sol ：

37 Example 7.7 Sol ：

38 Frequency Domain Filter Relationships Time Domain ： Y(t) = X(t)  h(t) Frequency Domain : W(f) = V(f)H(f) where V(f) = F{X(t)}, W(f) = F{Y(t)}, and H(f) = F{h(t)}. LTI system x(t)x(t) )()(.)()( )()()( thtxdtxh dxthty          

39 Theorem 7.5 Pf ：

40 Example 7.8 Sol ：

41 Example 7.9 Sol ：

42 Example 7.10 Sol ：

43 Theorem 7.6 Pf ：

44 I/O Correlation and Spectral Density Functions h(t)hnh(t)hn h(-t) h -n RX()RX() RX[k]RX[k] R XY (  ) R XY [k] RY()RY() RY[k]RY[k] H(f)H()H(f)H() H*(f) H*(  ) SX(f)SX(f) SX()SX() S XY (f) S XY (  ) SY(f)SY(f) SY()SY() Time Domain Frequency Domain

45 6.Linear Filtering of a Random Signals 7.Power Spectrum Analysis 8.Linear Estimation and Prediction Filters 9.Mean-Square Estimation

46 8. Linear Estimation and Prediction Filters Linear Predictor 1.Used in cellular telephones as part of a speech compression algorithm. 2.A speech waveform is considered to be a sample function of WSS process X(t). 3.The waveform is sampled with rate 8000 samples/sec to produce the random sequence X n = X(nT). 4.The prediction problem is to estimate a future speech sample, X n+k using N previous samples X n-M+1, X n-M+2, …, X n. 5.Need to minimize the cost, complexity, and power consumption of the predictor.

47 Linear Prediction Filters Use to estimate a future sample X=X n+k. We wish to construct an LTI FIR filter h n with input X n such that the desired filter output at time n, is the linear minimum mean square error estimate Then we have The predictor can be implemented by choosing.

48 Theorem 8.1 Let X n be a WSS random process with expected value E[X n ] = 0 and autocorrelation function R X [k]. The minimum mean square error linear filter of order M  1, for predicting X n+k at time n is the filter such that where is called as the cross-correlation matrix.

49 Example 8.1 X n be a WSS random sequence with E[X n ] = 0 and autocorrelation function R X [k]= (  0.9) |k|. For M = 2 samples, find, the coefficients of the optimum linear predictor for X = X n+1, given. What is the optimum linear predictor of X n+1, given X n  1 and X n. What is the mean square error of the optimal predictor? Sol ：

50 Theorem 8.2 If the random sequence X n has a autocorrelation function R X [n]= b |k| R X [0], the optimum linear predictor of X n+k, given the M previous samples is and the minimum mean square error is. Pf ：

51 Linear Estimation Filters Estimate X=X n based on the noisy observations Y n =X n +W n. We use the vector of the M most recent observations. Our estimates will be the output resulting from passing the sequence Y n through the LTI FIR filter h n. X n and W n are assumed independent WSS with E[X n ]=E[W n ]=0 and autocorrelation function R X [n] and R W [n]. The linear minimum mean square error estimate of X given the observation Y n is Vector From :

52 Theorem 8.3 Let X n and W n be independent WSS random processes with E[X n ]=E[W n ]=0 and autocorrelation function R X [k] and R W [k]. Let Y n =X n +W n. The minimum mean square error linear estimation filter of X n of order M  1 given the input Y n is given by such that

53 Example 8.2 The independent random sequences X n and W n have expected zero value and autocorrelation function R X [k]= (  0.9) |k| and R W [k]= (0.2)  k. Use M = 2 samples of the noisy observation sequence Y n = X n +W n to estimate X n. Find the linear minimum mean square error prediction filter Sol ：

54 6.Linear Filtering of a Random Signals 7.Power Spectrum Analysis 8.Linear Estimation and Prediction Filters 9.Mean-Square Estimation

55 9. Mean Square Estimation Linear Estimation ： Observe a sample function of a WSS random process Y(t) and design a linear filter to estimate a sample function of another WSS process X(t), where Y(t) = X(t) + N(t). Wiener Filter ： The linear filter that minimizes the mean square error. Mean Square Error ： LTI system Y(t)Y(t)

56 Theorem 9.1 ： Linear Estimation

57 Example 9.1 Sol ：