Randomly Modulated Periodic Signals

Transcription

1 Randomly Modulated Periodic Signals Melvin J. Hinich Applied Research Laboratories University of Texas at Austin

2 Rotating Cylinder Data Fluid Nonlinearity 5 5 Amplitude

3 Hourly Alberta Electricity Demand Electricity Demand ( 5/4/996-6/5/996) kw Hours Time 3

4 Canadian $/US $ Rates of Return.5 Canadian$/US$ // - //3..5 Rates of Return // /5/ 3// 4/6/ 5/3/ 7/5/ 8/9/ 9/3/ /8/ // /7/ /3/3 Days 4

5 Bassoon Note Amplitude Time 5

6 Flute Note Amplitude Time 6

7 .8 Sound from a Power Boat.6.4. Amplitude Time 7

8 Definition of a RMP xt ( ) A signal is called a randomly modulated periodicity with period T=Nδ if it is of the form n f s + u ( t ) cos f t + ( ) ( π ) K k k n k n µ = + N ( ) ( ) k sk uk( tn) sin π = + fktn k k T = tn = nδ Eu ( t ) = Eu ( t ) = k n k n for each k =,, K where K T δ 8

9 Random Modulations The vector of the K modulations u t = u ( t ), u t : k =, L, K { } ( ) ( ) n k n k n are jointly dependent random processes for all < t < L < t < T m 9

10 Finite Dependence Condition needed to ensure that averaging over frames yields asymptotically gaussian estimates { } { ' ' u( t ), L, u( t ) & u( t ), L, u( t )} m are independently distributed if r t + D< t D m ' for some & and all t < L< t & t < L< t ' m ' n

11 Fourier Series for Components Thus xt ( ) = st ( ) + ut ( ) where n n n K n = + k k n + k k n k= s( t ) s N s cos π f t s sin π f t [ ( ) ( )] K ( n) = kcos k n + ksin k n k= ut N u π ft u π ft [ ( ) ( )]

12 Signal Plus Noise s( tn) is the mean of x( t) u t has a periodic jo distribution { ( )} int n The modulation is part of the signal It is not measurement noise

13 Artificial Data Examples + u t cos f t + ( σ ( )) ( π ) K k n k n = + ( ( )) ( ) k σuk tn sin π = + fktn xt ( ) s N n u t = ρu t T + e t ( ) ( ) ( ) k n k n n u t = ρu t T + e t ( ) ( ) ( ) k n k n n 3

14 Five Standard Deviations Harmonics Modulation σ = 5 ρ =.9 Frame=

15 Three Standard Deviations Harmonics Modulation σ = 3 ρ =.9 Frame=.5 Amplitude

16 Two Standard Deviations Harmonics Modulation σ = ρ =.9 Frame =.5 Ampliutde

17 One Standard Deviation Randomly Modulated Pulses Harmonics Modulation σ = ρ =.9 Frame =.5.5 Amplitude -.5 7

18 No Correlation in the Modulation Four Randomly Modulated Pulses Frame = σ=5.5 Amplitude

19 Block Data into Frames The data block is divided into M frames of length T T is chosen by the user to be the period of the periodic component The t-th observation in the mth frame is x ( m ) T + nδ n =,, N ( ) 9

20 Frame Rate Synchronization The frame length T is chosen by the user to be the hypothetical period of the randomly modulated periodic signal. If T is not an integer multiple of the true period then coherence is lost.

21 Signal Coherence Spectrum γ x ( k) = s k sk + σ u ( k) The signal-to to-noise ratio is x = σ k u ρ ( k) s ( k) ρ x k ( ) γ ( k ) x = γ k x ( )

22 Estimating Signal Coherence xˆ t : n =,, N is the mean frame { ( ) } n averaged over the M frames N Xˆ ( k) = xˆ ( tn)exp( i π fktn) ˆ γ σˆ ( k) n= = Xˆ ( k) x ˆ X( k) + σˆ u ( k) M u( ) = m( ) m= k M X k Xˆ k ( )

23 Statistical Measure of Modulation SNR Z k σˆ ( ) = = M N M N Xˆ k σˆ ˆ ρ x x ( ) k ( ) k ( ) ( k) Xˆ ( k) ( k) x = +σˆu 3

24 If the each Z k Z k Chi Squared Statistic = M N ρ ˆ x k ( ) ( ) mod ulation is stationary the distribution of is approximately ( ) ( ) they are independently distributed. λ k = M N ρ x k ( ) χ λ k & 4

25 s t ( ) - Spectrum of the Variance a stationary random process Fourier Series Expansion K s t = a + a cos π f t + b sin π f t ( ) ( ) ( ) n k k n k k n k= k= Var a Var b S f Spectrum ( ) = ( ) ( ) - k k k K σ s K = k= S f ( ) k 5

26 Power & Signal Coherence Spectra - Demand Power & Signal Coherence Spectra of the Residuals from an AR() Fit of the Alberta Electricity Hourly Spot Demand db Spectrum Coherence Signal Coherence Period in Hours 6

27 Canadian$/US$ Daily Data Spectra db Spectrum - US$/Canadian$ Coherence Spectrum Coherence Probability Period in Days

28 Bicorrelations of a Random Signal c t, t, t = E x t x t x t xxx [ ] ( ) ( ) ( ) ( ) 3 3 If { xt ( )} is stationary then c t, t, t = E x t+ t t x t+ t t x t xxx [ ] ( ) ( ) ( ) ( ) c, = Extxt+ xt+ xxx [ ] ( τ τ ) ( ) ( τ ) ( τ ) 8

29 S f, f xxx ( ) The Bispectrum c τ, τ exp iπ fτ + f τ dτ dτ xxx = [ ] ( ) ( ) If the noise is gaussian then S f, f = xxx ( ) 9

30 Example of a Simple Nonlinear Model xt + a xt xt + a xt xt ( ) ( ) ( ( )) ( ) ( ) ( ) n n n n n = σu t ( ) ( ( )) n =, n ( ( )) cxt n a x t e a xt = a xt cosω xt ( ( )) ( ) ( ) ( ( )) n n n ( ) c x t = c + δ x t ω ( ( )) ( ) n n ( ) xt = ω + δx t ( ( )) ( ) n n 3

31 Nonlinear Model - Uniform Input σ =.5, c =.5, ω =.π, δ = Nonlinear Process with Uniform Input σ =.5, c =.5, ω =.π, δ =

32 Phase Plot of the Nonlinear Impulse Response 4 Nonlinear Impulse Response Phase Plot c =.5 δ =.5 3 x(t+) x(t)

33 Phase Plot of the Linear Impulse Response.9 x(t+) Phase Plot of Linear Impulse Response c =.5, ω =.π, δ = x(t)

34 Correlations Functions of x(t) & x(t) Correlation Exponential Input σ =.5, c =.5, ω =.π, δ =.5.8 x(t) Correlations x(t) Correlations Time

35 Bispectrum - Nonlinear AR() Signal Bispectrum of Nonlinear AR() c=.5 δ=. f= Frequency inhz Frequency in Hz 35