Signal processing Frequency analysis

Transcription

1 Signal processing Frequency analysis Jean-Hugh Thomas Fourier series and Fourier transorm (h30 lecture+h30 practical work) 2 Sampling (h30+h30) 3 Power spectrum estimation o random processes (h30+h30)

2 Bibliography Théorie et traitement des signaux (F. de Coulon), Dunod, 984. Traitement numérique des signaux (M. Kunt), Presses polytechniques romandes, 984. Méthodes et techniques de traitement du signal et applications aux mesures physiques (J. Max), Tome Masson, 985. Techniques modernes de traitement numérique des signaux (M. Kunt), Presses polytechniques romandes, 99.

3 Bibliography Traitement numérique du signal une introduction (A.W.M. Van Den Enden, N.A.M. Verhoeckx), Masson 992. Signaux et systèmes linéaires (Y. Thomas), Masson, 994. Temps-réquence (P. Flandrin), Hermès, 993.

4 Bibliography Modern Spectral Estimation (S. M. Kay), Englewood Clis, NJ:Prentice Hall, 988. Discrete-Time Signal Processing (A. V. Oppenheim and R. W. Schaer) Englewood Clis, NJ:Prentice Hall, 989. Digital Signal Processing: Principles, Algorithms and Applications (J. G. Proakis and D. G. Manolakis), Upper Saddle River, NJ:Prentice Hall, 996. A wavelet tour o signal processing (S. Mallat), Academic Press,

5 Dirichlet conditions They guarantee that the Fourier series will be equal to the signal x(t) except at the values o t or which x(t) is discontinuous : The signal has a inite number o discontinuities in any period The signal contains a inite number o maxima and minima during any period. The signal is absolutely integrable in any period: x( t) dt< T 5

6 x(2t) A AT/2 0.5 (/2) =2k/Τ t -T/4 T/4 x(t) () A AT =k/τ t -T/2 x(t/2) T/2 2AT 2 (2) A =k/2τ t -T T 6

7 Continuous-time signals and discrete time signals Sampling 2 Reconstruction 7

8 Analog-to-Digital Conversion Sampling Quantization Coding 0 Analog signal Discrete-time signal Quantized signal Digital signal 8

9 x(t) Continuous time x(n) Discrete time Continuous amplitude t n x q (t) Discrete amplitude - x q (n) t n

10 Sampling () e (+ e ) -B B e () /T e e (- e ) - e e /2 e 0

11 Aliasing () -B e () /T e B - e e /2 e e () /T e - e e

12 Resulting spectrum e () /T e - e e /2 e ^ e () /T e - e /2 e /2 2

13 Sampling processing ^ x (t) x*(t) x (n T Anti-aliasing Sample and e ) ilter hold 3

14 Shannon interpolation {x(nt)} H() / e x(t) - e /2 e /2 Ideal lowpass ilter 4

15 Sampling () e (+ e ) -B B e () /T e e (- e ) - e e /2 e 5

16 Shannon interpolation x(0) x(t) x(t) x(2t) x(3t) t 0 T 2T 3T 6

17 h(t) A H() t p(t) /T P() t T -/T /T h e (t) A/T H e () t -/2T /2T 7

18 h e (t) A/T H e () t -/2T /2T x(t) t () T 0 =k/τ 0 -T 0 /2 T 0 /2 h e (t) x(t) kat 0 /T H e ()*() -T 0 /2 T 0 /2 t -/2T /2T 8

19 h e (t) x(t) kat 0 /T H e ()*() t -T 0 /2 T 0 /2 -/2T /2T T 0 p (t) t... P ()... -T 0 T 0 /T 0 T 0 ^ h(t) H(n) t N N 9

20 Spectral analysis Concepts, properties and requency content o random processes 2 Basic periodogram 3 Average periodogram 4 Welch periodogram 5 Correlogram 20

21 Examples o random signals Electrocardiogram Speech signal Rolling boat Electricity consumption Pressure in the combustion chamber o an engine 2

22 Random signals 3 experiments x (t) Random process (t) x 2 (t) x 3 (t) 22

23 Random process Random process Time Experiments 23

24 Ergodic signals Statistical mean values t Time mean values N E[ ( t) ] = lim x ( t) N i N i= 24

25 Ergodic signals [ ] lim 2 m = E ( t ) = x ( t ) dt T T T σ = = x t m dt 2 [ 2 ] T E ( ) 2 c ( t) lim ( ) T 2T T [ ] T ( τ ) = ( ) ( τ ) = lim x T c ( t) xc ( t τ ) dt 2T C E c t c t T T 25

26 Estimation Consistency o the estimate 2 Correlation estimates 26

27 Noise Estimation x System y ( ) g y $x x : Vector o unknown variables y : Vector o measured variables $x : Estimation 27

28 Consistency o the estimate Bias o the estimate Variance-covariance matrix = E $ E $ $ E $ Σ [ $ ] b = E x [( [ ]) ( [ ]) T ] Variance (One dimension case) [( [ ]) 2 $ $ ] 2 σ = E E Mean quadratic error [( ) 2 ] E $ x = σ + b Consistent estimate 2 2 lim bn = lim σ 2 N = 0 N N 28

29 Time axis length o R τ? τ > 0 Correlation estimate x n 0 N- x n+τ τ -N 0 N- Time τ < 0 τ N-- τ τ N-- τ Time 29

30 Correlation estimate x n Range or n? or a given τ 0 N- x n+τ n Summation o N+τ samples τ 0 τ τ + N - n 30

31 Correlation estimate x n Range o n? τ 0 x n+τ 0 N- For a given τ n n Summation o N-τ samples τ τ + N - 3

32 Correlation estimate Rˆ Rˆ ' ' τ τ τ N τ = n n+ τ N n= 0 N + τ N = + n n+ τ n= τ τ 0 τ < 0 Bias b ( ˆ ' ) = 0 R τ Variance depends on Consistent estimate N ( ) N τ 2 32

33 Correlation estimate $R $R τ τ N τ = n τ n+ τ N n= 0 N = n n+ τ τ < N n = τ 0 0 Bias ( $ ) b R τ = Variance depends on τ N R τ Consistent N estimate 33

34 Measurement process Anti-aliasing iltering Sampling Windowing Signal x(t) Spectrum () DFT 34

35 Correlation estimate (reminder) $R $R τ τ N τ = n τ n+ τ N n= 0 N = n n+ τ τ < N n = τ 0 0 Bias ( $ ) b R τ = Variance depends on τ N R τ Consistent estimate N 35

36 Basic periodogram x ( n) DFT NT s. 2 s$ ( ) x ( n) Fourier transorm o the biased estimate o the autocorrelation o the windowed signal rˆbiased DFT s$ ( ) 36

37 Periodogram Sˆ ( ) ( ) 2 = NT s Non consistent estimate Bias b ( ( )) N k Sˆ = R e 2iπ k N k k = N + Variance ( ( )) S$ S ( ) σ 2 2 N 37

38 Average periodogram K rectangular windows L points x 0 x L- x 2L-2 x N- Bartlett (948) No overlap 38

39 Average periodogram K i = 0 s$ B ( ) = s$ ( i ) ( ) K T ˆ ( ) s i L n= ( ) = e x ( n) L 0 i e 2 jπ n 2 Segment i=0,...k- x ( n ) = i x ( n + i L ) i=0,...k- n=0,...l- 39

40 Average periodogram K i = 0 S$ B ( ) = S$ ( i ) ( ) K Consistent estimate N ( ( )) k b S$ B = L R ( k ) e k = N + 2iπ k Bias σ K ( ) ( B S$ ( )) σ S$ ( i ) ( ) 2 2 N L Variance 40

41 Modiied periodogram M windows D points x 0 x D- x 2D-2 x N- L points x L- x D+L-2 Welch (967) Overlap : L - D + points (M - ) D + L - M + = N 4

42 Welch periodogram i=0,...m- Segment $ ( ) $ ( ) ( ) s M s W i i M = = U L w n n L = = 2 0 ( ) ( ) ( ) ) ( ˆ = = L n n j i e W e n w n x LU T s π

43 Welch periodogram M i = 0 S$ W ( ) = S$ ( i ) ( ) M Consistent estimate Ε [ ( ) ] Sˆ W = S ( ) S ( ) * W Statistical average σ M ( ) ( W S$ ( )) σ S$ ( i) ( ) 2 2 N L Variance 43

44 Correlogram BT ( ) M S$ = R$ ( k) w( k) e k = M+ 2iπ k x ( n) $r DFT $s ( ) Blackman et Tukey (958) 44

45 Correlogram Ε [ ( ) ] Sˆ BT = S ( ) W ( ) * B Statistical average Variance σ 2 N [ ( ) ] ( ) M Sˆ BT S 2 2 w ( m) N m= M Consistent estimate 45

46 Transmission o a random signal into a linear system n H(z) Y n Mean o the output? Intercorrelation between the output and the input? Autocorrelation o the output? Power Spectrum Density o the output? 46

47 Ε [ ] Υ n Statistical values = Χ h Mean R = R h Y τ n Intercorrelation τ τ τ Autocorrelation R = h h R Υ τ τ Χ τ 2 PSD S Y ( ) = Η( ) S ( ) Χ τ 47

48 White noise with variance V N Power Spectrum Density t Y t ( ) = + = + + = N i i i N i i i z a z b z H 0 48 = i ) ( = + = + + = N i it j i N i it j i N Y s s e a e b V S π π