Lesson 1. Optimal signalbehandling LTH. September Statistical Digital Signal Processing and Modeling, Hayes, M:

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1 Lesson 1 Optimal Signal Processing Optimal signalbehandling LTH September 2013 Statistical Digital Signal Processing and Modeling, Hayes, M: John Wiley & Sons, ISBN Nedelko Grbic Mtrl from Bengt Mandersson Department of Electrical and Information Technology, Lund University Lund University

The sound of Signalbehandling s i g n s noise i harmonic signal How can this be generated as output from a linear filter? Determine the filter and the input signal.

2 The sound of Signalbehandling s i g n s noise i harmonic signal How can this be generated as output from a linear filter? Determine the filter and the input signal. LPC model of syntetic sound production pulse train speech output from pulse train white noise H 0 z LPC-model speech output from white noise waveform and spectra In syntetic speech production, the parameters often are updated every 5 milliseconds. 2

3 Chapter 2. Digital signal processing impulse response, convolution, system function, Fourier, z-transforms page 7-20 Matri description. page Hints. page 8-18, 21, 49. Chapter 3. Random processing, such as correlation functions, correlation matrices. Random variables page Random Processes page Hints. page 77, 79, 80, 85, 95, 99, 100, 101, 106 Chapter 4. Signal models, Deterministic and Stochastic approach. Padé, Prony page Shank page All-pole Modeling page 160,165 Linear prediction page not included 4.6 page Stochastic Models page Hints. page 130, 135, 138, 147, 148, , 195 Chapter 5. Levinson-Durbin recursion. page , page not included Hints. Table , figure 5.10 Chapter 6. Latttice FIR and IIR filters, only 6.2 and 6.4.1, page , 297, 298, page Chapter 7. Optimal filters. Linear prediction. Wiener filters. Specially FIR filters. FIR- Wiener filter page IIR- Wiener filter page Kalman filters page Hints. page , 354, 355, , 370 Chapter 8. Spectrum estimation. Nonparametric methods page , see chap 4, 8.6 page , Hints. page 394, 408 3

4 Digital Signal Processing application Radar 4

5 An application from the tet book Noise cancellation chapter 7, page 349 A signal is disturbed by additive noise v 1 n. Try to measure the noise vn from the source and estimate the noise v 1 n added to the signal. Then subtract the noise v 1 n from the received signal. Signal source Noise source vn vn sn sn+v 1 n sn Hz v 1 n Wiener filter Estimate of v 1 n 5

6 Optimal signal processing in Hay's book Chapter 2: Chapter 3: Brief review of digital signal processing. Brief review of random signals. noise vn white noise wn or impulse δn transmitted h gen n sn H gen z received signal n h receiver n H receiver z yn Estimate H gen z from properties of sn Determine H receiver z The filters H gen z and H receiver z are of type FIR IIR all-pole IIR Chapter 4, 5 and 6: Make a model H gen z from the properties of sn. Chapter 7: Chapter 8: Determine H receiver z. Estimation of spectra. 6

7 Chapter 2 Digital Signal Processing Difference equation p y n a k y n k b k n k k 1 q k 0 MATLAB: A=[ ]; B=[1 1]; y=filterb,a,; Convolution k y n h k n k impulse: n [ ] unit step: un [ ] System function H z Frequency function B z A z j j B e H e j A e 7

8 FIR, IIR filters FIR: length Circuit with impulse response with finite Eample y n n n 1, h n n n 1 IIR: Circuit with impulse response with infinite length Eample n y n 0.5 y n 1 n, h n 0.5 u n All-pole IIR-filters IIR-filters with poles only all zeroes in origin, Bz=constant Eample H z z 1 8

9 Solvning the convolution sum. y n h k n k k h n k k k y n h0 n h1 n 1 h2 n 2 Eample n [ ], h n [4 2 2] Method A: Vector notation n n 1 y n n. n N 1 T T h0 h1... h N 1. h Method B: Graphical solution Write k : h0 k : y h1 k : y Gives the output yn [ ] MATLAB: =[ ]; h=[4 2 2]; y=conv,h 9

10 Method C: Convolution matri Use matri notations n [ ], h n [4 2 2] h h h y 0 y 1 y 2 y 3 y 4 y X h y In Matlab: =[ ] ; X=convmt,3 h=[4 2 2]', y=x*h In signal processing, all vectors are column vectors 10

11 Properties of matrices The square matri A n n is: symmetrical if A A T T Hermitian if A A A H invertable if AA 1 I Toeplitz if all diagonals are identical 345 A Hermitian symmetrical Toeplitz if A A Toep[3, 2,1] orthogonal if T A A I 11

12 Linear equation page A is a A [ n m] matri b gives 1 A b if n m A invertable, 1 H H A A A b if n m overdetermined, more equations than variables. Described more in chapter 4 A H A A H 1 b if n m underdetermined, less equations than variables Eigenvalue: Av v, A I 0 eigenvalues, v eigenvectors A V V 1 with eigenvectors in the columns of V, eigenvalues in the diagonal of 12

13 Optimisation minimizing: page 49 If z real: f z z 2 d d 2 d 2 f z z 2 z; z 0 dz dz dz gives z 0 as min imum; If z is comple: 2 f z z z z z is the conjugate of z z Derivate with respect to treating the other as a constant. or z separately while d dz d dz d z z z z dz d z z z z dz 2 2 Setting this derivatives equal to zero gives the same minimum page 49. This is used sometimes in the tetbook. 13

14 Eample on circuits A n Xz z -1 yn Yz 0.5 y n 0.5 y n 1 n 1 Y z 0.5 z 1 Y z z 1 X z B n Xz z -1 z -1 yn Yz C Lattice filters n Xz Г 1 z -1 Г 1 z -1 Г 2 Г 2 yn Yz FIR-lattice filter n, Xz yn, Yz -Г 2 -Г 1 Г 2 z -1 Г 1 z -1 IIR-lattice filter 14

15 Correlation functions deterministic Autocorrelation function r l n n l r l n Cross-correlation function r l y n n l y n r l l l ry l y l l Relation between input and output r l h l r l y r l r l r l y h 15

16 Eample on correlation, echo 1 2 y= r 1 r y r 1y 16

17 Eample of correlation, delay in mobile phones GSM Input signal to the GSM phone Output signal after GSM Crosscorrelation In Matlab: ry=corrinput,output 17

18 Chapter 3 Discrete-Time Random Processes Random variables 3.2 page Probability density function f X Probability distribution function: F X Epected value mean: Mean-square value: m E{ } f X d 2 2 E{ } f X d Variance: Var E m m f d [ ] {[ ] } [ ] X General: y g ; E{ y} E{ g } g f X d Relation: Var[ ] E{[ m] } E{ X } m Correlation. Dependency between random variables and y Correlation: r y E{ y} 18

19 Covariance: c y E {[ m ][ y m y]} 19

20 Stochastic processes 3.3 page 74 Wide-sense stationary processes, WSS Eample A: Sinusoids with random phase n Asin 0 n, is a random variable and n is a random process. Eample B: Noise white noise, colored noise. Eample C: Speech signals. The autocorrelation sequence and the cross-correlation sequence and their Fourier transforms are important in this course. Autocorrelation sequence: Cross-correlation sequence. { r } m E k k m { r } y m E k y k m Estimation of the autocorrelation sequence ergodic processes 1 r { } m E k k m k k m N sum over N values 20

21 Interpreting of autocorrelation sequence: Signal Autocorrelation sequence Sinusoid: White noise. Colored noise Speech signal: Vowel. 21

22 22 Optimal Signal Processing Properties of autocorrelation sequence page 83 Wide-sense stationary processes, WSS Definition: k r k n n E n k n E k n n E k n n E k r Symmetry: r k r k Mean-square value: 2 0 [ ] 0 r E n positive Maimum value: 0 r r k Non-stationary processes For signals that are not wide-sense stationary processes, not WSS, we have to use the definitions see chapter 4 } {, } {, * * l k y E l k r l k E l k r y

23 Correlation matri WSS [ N 1] T R H E[ ] r0 r 1 r 2 r p r 1 r 0 r 1 r p 1 r 2 r 1 r 0 r p 2 r p r p 1 r p 2 r 0 Properties of the correlation matri Hermitian Toeplitz Toeplitz if real-valued process Eigenvalues are real and non-negative Estimate of the correlation function N 1 1 rˆ k n n k N n0 Estimate of the cross-correlation function rˆ y k 1 N N 1 n0 n y n k 23

24 Power spectrum of random process page 94: Wide-sense stationary processes, WSS n is a wide sense stationary random process WSS, n real-valued, hn real with autocorrelation r k The Fourier transform and the z-transform are given by: The Fourier transform of r k The Z-transform of r : j P e r k e k : P z r k z k j k Properties Symmetry real processes j j : P e P e Positive: Total power: j P e 0 1 j r0 P e d 2 24

25 Filtering of random processes, 3.4 page 99, 100, 101: n r k hn He j yn r y k Input-output relation y n n h n k h n k k Autocorrelation function for the output r k E{ y n y n k} h l r m l k h m y l m Cross correlation functions r k E{ y n n k} h l r k l y l r y k E{ n y n k} l h l r k l 25

26 Using convolution and power spectra Define r h k h l h l k h k h k Correlation functions r k r k h k h k r k r k y h r k r k h k r y y k r k h k Spectra j j j P e P e H e y j j j P e P e H e P y y e j P e j H e j 2 1 Py z P z H z H z 26

27 Spectral factorization 3.5 page 104 n is a WSS process with autocorrelation r k. We assume that the process are generated from white noise vn filtered in a filter with system function Qz, Then, vn is called the innovation process of the process n. white noise vn Qz our process n 1/Qz white noise vn r v k 2 r k k v 2 P z v 0 0 r k 2 P z 0 Q z Q 1/ z Can we find the filter Qz from n and r k? Is Qz stable and causal? Is 1/Qz stable and causal? 27

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