Unstable Oscillations!

Transcription

1 Unstable Oscillations X( t ) = [ A 0 + A( t ) ] sin( ω t + Φ 0 + Φ( t ) ) Amplitude modulation: A( t ) Phase modulation: Φ( t ) S(ω) S(ω) Special case: C(ω) Unstable oscillation has a broader periodogram peak. Amplitude drifts -> sidelobes Phase drifts equivalent to frequency ω changing with time. C(ω) A( t ) = A sin Ω t + B cos Ω t Φ( t ) = α sin Ω t + β cos Ω t

2 Amplitude Modulation Set A 0 = 1 and Φ 0 = 0. Oscillation frequency ω with slow amplitude modulation at lower frequency Ω: χ ( ω ) X(t) = (1+ AsinΩt + BcosΩt ) sinω t = sinω t ω Ω ω + Ω A( ω ) + (A /)[cos(ω Ω)t cos(ω + Ω)t] + (B /)[sin(ω Ω)t + sin(ω + Ω)t] S( ω ) Note: Sidelobes at ω ± Ω. Sine amplitudes in phase. Cosine amplitudes anti-phased. C( ω )

3 Phase Modulation Oscillation frequency ω with slow phase modulation at lower frequency Ω: X(t) = sin(ω t +α sinωt + β cosωt) Note: sin(x + Δx) sin x + Δx cos x : X(t) sinω t + α cosω t = sinω t ( ) sinωt + ( β cosω t) cosωt + (α / )[ sin(ω Ω)t + sin(ω + Ω)t ] + (β / )[ cos(ω Ω)t + cos(ω + Ω)t ] Again, sidelobes at ω ± Ω but now with Sine amplitudes anti-phased. Cosine amplitudes in phase ω Ω ω + Ω χ ( ω ) A( ω ) S( ω ) C( ω )

4 Phase relations for sidelobes Both Amplitude and Phase Modulation: X(t) = (1+ A sinωt + B sinωt ) sin(ω t + α sinωt + β sinωt ) = sinω t + B α A(Ω) = C(ω Ω) C(ω + Ω) B(Ω) = S(ω Ω) + S(ω + Ω) α(ω) = S(ω Ω) + S(ω + Ω) β(ω) = C(ω Ω) + C(ω + Ω) sin(ω Ω) t + S(ω) A + β + B + α sin(ω + Ω) t A β Amplitude and Phase Modulation Spectra: cos(ω Ω) t cos(ω + Ω) t Amplitude modulation Phase modulation C(ω)

5 Sawtooth and Square Wave Sawtooth Square wave χ ( ω ) A( ω ) S( ω ) C( ω )

6 Non-sinusoidal Waveforms Fundamental frequency: ω 0 Harmonics at ω = k ω 0, k =,... modify the shape of the waveform. Fit any shape periodic function by including amplitudes for : sin(ω 0 t), cos(ω 0 t) sin(3ω 0 t), cos(3ω 0 t) etc X(t) = ˆX 0 + k=1 Ŝ k sin( kω 0 t) + Ĉk cos ( kω t 0 ) The harmonics are approximately orthogonal (for well-sampled data with uniform phase coverage). Add harmonics to the model until their values become poorly determined -- Occam s razor, simplest model that fits. Use e.g. the BIC to decide which terms to include/omit. Harmonics above the Nyquist frequency will be aliased, by folding back across ω Nyq, from ω to ω Nyq - (ω -ω Nyq ). " Â k = Ŝk + Ĉk, ˆφ k = atan( Ŝk, Ĉ k ) # $

7 Data gaps and aliasing Gap of length T gap Cycle-count ambiguity: How many cycles elapse in the gap between two data segments? χ ( ω ) Periodogram has sidelobes (aliases) spaced by Δω = π T gap Δf = 1 cycle T gap Sidelobes appear within a broader envelope determined by duration of data segments. A( ω ) S( ω ) C( ω )

8 Dynamic Power Spectrum For periodic oscillations with amplitude and phase that vary with time. Data: X i ±σ i at t=t i Model: µ(t) = X 0 (t)+ S(t) sin(ω t)+ C(t)cos(ω t) 3 Patterns: 1, s i = sin(ω t i ), c i = cos(ω t i ) Iterated Optimal Scaling: ˆX 0 (t) = Ŝ(t) = Ĉ(t) = ( X i Ŝ s i Ĉ c ) i w i (t), w i (t) ( X i ˆX 0 Ĉ c ) i s i w i (t), s i w i (t) ( X i ˆX 0 Ŝ s ) i c i w i (t), c i w i (t) Â (t) = Ĉ (t)+ Ŝ (t) Iterate ( patterns not orthogonal ). Like Running Optimal Average, but including Sin and Cos amplitudes in the fit to each time window. w i (t) = G(t t i) σ i G(t) = exp t Δ Time-resolution set by parameter Δ.

9 Dynamic Power Spectrum Phase modulation is equivalent to a wandering frequency. Badness-of-Fit: χ ( ω, t ) Probability: P ~ exp{ - χ / } Power density: A ( ω, t ) Note probability peak much sharper than power density peak.

10 Periodogram of a sinusoid + spike Single high value is sum of cosine curves all in phase at time t 0 : ω δ(t t 0 ) = cosω(t t 0 ) dω Raises the amplitude uniformly at all frequencies. χ ( ω ) A( ω ) S( ω ) C( ω )

11 Periodogram of a sinusoid + white noise White noise can be generated by sampling a Gaussian random number at each time. Equivalent to using a Gaussian random number for each sine and cosine amplitude. C k = 0 C k = X i = µ ±σ 1 cos (ω k t i ) σ i Note: Both white noise and a spike have flat periodograms. N i=1 σ N A k C k + S k ~ σ N χ A k = 4σ N N / Parseval's theorem: A k = σ k=1 χ ( ω ) A( ω ) S( ω ) C( ω )

12 model: µ(t) = µ 0 + Summary: Fourier Analysis k c k cosω k t + s k sinω k t even spacing: t i = t 0 + iδt T = N Δt i =1,,..., N Fourier frequencies: ω k = k Δω = π P k P k = T / k k = 0,1,..., K max = N / Nyquist frequency: ω Nyq = π / Δt = π P Nyq Orthogonal basis: C k = cosω k t Model: µ = µ o C 0 + ( c k C k + s k S k ) k=1 Badness-of-fit: χ = X µ Optimal fit: ˆµ 0 = X C 0 C 0 C 0 Power spectrum: S k = sinω k t ĉ k = X C k C k C k P(ω k )Δω = Âk ĉ k + ŝ k P Nyq = Δt ŝ k = X S k S k S k Exact fit possible by using N parameters to fit N data points. C 0 X Decomposes lightcurve into frequency components. S k C k