Randomly Modulated Periodic Signals Melvin J. Hinich Applied Research Laboratories University of Texas at Austin hinich@mail.la.utexas.edu www.la.utexas.edu/~hinich
Rotating Cylinder Data Fluid Nonlinearity 5 5 Amplitude -5 - -5 -
Hourly Alberta Electricity Demand Electricity Demand ( 5/4/996-6/5/996) 7 6 5 kw Hours 4 3 69 337 55 673 84 Time 3
Canadian $/US $ Rates of Return.5 Canadian$/US$ // - //3..5 Rates of Return -.5 -. -.5 // /5/ 3// 4/6/ 5/3/ 7/5/ 8/9/ 9/3/ /8/ // /7/ /3/3 Days 4
Bassoon Note.8.6.4. Amplitude -. -.4 -.6 -.8 - -. Time 5
Flute Note.8.6.4 Amplitude. -. -.4 -.6 Time 6
.8 Sound from a Power Boat.6.4. Amplitude -. 3 4 5 6 -.4 -.6 -.8 -. -. Time 7
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
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
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
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 [ ( ) ( )]
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
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
Five Standard Deviations Harmonics Modulation σ = 5 ρ =.9 Frame=.5.5.5 -.5 - -.5 4
Three Standard Deviations Harmonics Modulation σ = 3 ρ =.9 Frame=.5 Amplitude.5 -.5-5
Two Standard Deviations Harmonics Modulation σ = ρ =.9 Frame =.5 Ampliutde.5 -.5-6
One Standard Deviation Randomly Modulated Pulses Harmonics Modulation σ = ρ =.9 Frame =.5.5 Amplitude -.5 7
No Correlation in the Modulation Four Randomly Modulated Pulses Frame = σ=5.5 Amplitude.5 -.5-8 -.5
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
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.
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 ( )
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 ( )
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
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
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
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 6.9 5.8.7 4.6.5 3.4.3.. 68. 4..9 8.8 6.7 5.4 4.5 3.9 3.4 3..8.5.3. Period in Hours 6
Canadian$/US$ Daily Data Spectra db Spectrum - US$/Canadian$ Coherence Spectrum Coherence Probability.995-86.8 3.3 6.98.68 8.9 7.8 6.3 5.9 4.56 4.6 3.66 3.34 3.6.83.63.46.3.7.5 7.99 Period in Days
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 [ ] ( ) ( ) ( ) ( ) 3 3 3 c, = Extxt+ xt+ xxx [ ] ( τ τ ) ( ) ( τ ) ( τ ) 8
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
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
Nonlinear Model - Uniform Input σ =.5, c =.5, ω =.π, δ =.5 8 6 4 - -4-6 Nonlinear Process with Uniform Input σ =.5, c =.5, ω =.π, δ =.5-8 - 3
Phase Plot of the Nonlinear Impulse Response 4 Nonlinear Impulse Response Phase Plot c =.5 δ =.5 3 x(t+) -3 - - 3 x(t) 4 - - -3 3
Phase Plot of the Linear Impulse Response.9 x(t+) Phase Plot of Linear Impulse Response c =.5, ω =.π, δ =.7.5.3. x(t) -.9 -.7 -.5 -.3 -...3.5.7.9 -. -.3 -.5 -.7 -.9 33
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.6.4. -. 3 4 5 6 7 8 9 3 4 5 Time -.4 -.6 -.8 34
Bispectrum - Nonlinear AR() Signal Bispectrum of Nonlinear AR() c=.5 δ=. f=..96-.97.97-.98.98-.99.99- Frequency inhz......3.4. Frequency in Hz 35