Periodogram of a sinusoid + spike Single high value is sum of cosine curves all in phase at time t 0 :

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1 Periodogram of a sinusoid + spike Single high value is sum of cosine curves all in phase at time t 0 : ( ) ±σ X(t) = µ + Asin(ω 0 t)+ Δ δ t t 0 N =100 Δ =100 χ ( ω ) Raises the amplitude uniformly at all A frequencies. k = Δ N = A( ω ) S k = Δsin(ω kt 0 ) /σ N i=1 sin (ω k t i ) /σ C k = Δcos(ω kt 0 ) /σ N i=1 cos (ω k t i ) /σ # A k = S k + C k = % Δ $ N Note : sin = cos =1/ & ( ' = Δ N sin(ω k t 0 ) = Δ N cos(ω k t 0 ) S( ω ) C( ω )

2 Periodogram of a Sinusoid + White Noise White Noise is generated by sampling a Gaussian random number at each time. Or, use a Gaussian random number for each sine and cosine amplitude. C k C k = N i=1 X i ~ G(µ,σ ) ( X i µ )cos(ω k t i ) /σ i N i=1 = Var [ C k ] = cos (ω k t i ) /σ i N i=1 1 cos (ω k t i ) /σ i C k = 0 = σ N A k C k + S k ~ σ N χ A k = 4σ N N / Parseval's theorem: A k = σ k=1 ( ) C 0 = µ, C k, S k ~ G 0, σ N χ ( ω ) A( ω ) Note: Both white noise and a spike have flat periodograms. N =100 µ =100 σ =10 A k 4σ N = 4 S( ω ) C( ω )

3 Unstable or Quasi-Periodic Oscillations X( t ) = [ A 0 + A( t ) ] sin( ω 0 t + Φ 0 + Φ( t ) ) Amplitude modulation: A( t ) Phase modulation: Φ( t ) S(ω) S(ω) Special case: Unstable oscillation has a broader periodogram peak. Amplitude drifts -> sidelobes C(ω) 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

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

5 Phase Modulation Oscillation frequency ω 0 with slow phase modulation at lower frequency Ω : X(t) = sin(ω 0 t +α sinωt + β cosωt ) Note: sin(x + Δx) sin x + Δx cos x : ( ) cosω 0 t X(t) sin(ω 0 t)+ α sinωt + β cosωt = sin(ω 0 t)+ (α / )[ sin(ω t )+ sin(ω + t )] + (β / )[ cos(ω t )+ cos(ω + t)] Again, sidelobes at ω ± =ω 0 ± Ω but now with Sine amps anti-phased: Cosine amps in phase: S + = S C + =C ω 0 ω ω + S + S χ ( ω ) A( ω ) S( ω ) C( ω ) C C +

6 Phase relations for sidelobes Both Amplitude and Phase Modulation: X(t) = (1+ A sinωt + B sinωt ) sin(ω 0 t +α sinωt + β sinωt ) sin(ω 0 t) + S sin(ω t) + C cos(ω t) Amplitude and Phase Modulation Spectra: A(Ω) = C (Ω) C + (Ω) B(Ω) = S (Ω)+ S + (Ω) α(ω) = S + (Ω) S (Ω) β(ω) = C + (Ω)+ C (Ω) + S + sin(ω + t) + C + cos(ω + t) ω ± ω 0 ± Ω, S ± = B ±α, C ± = β A S(ω) Amplitude modulation Phase modulation C(ω)

7 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.

8 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.

9 Example of Dynamic Power Spectra Observation of Gravitational Waves from a Binary Black Hole Merger Abbott et al. (016) Phys Rev L 116, Nobel Prize : R.Weiss (MIT), K.Thorne, B.Barish (Caltech)

10 Example : Dynamic Power Spectrum "Gravitational Waves and Gamma Rays from a Binary Neutron Star Merger: GW and GRB A. Abbott et al. (017) ApJL 848, L13. Published 16 Oct 017.

11 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

12 Wavelet Analysis - Wavelet Basis Functions Fourier basis isolates in frequency but not in time. Delta basis isolates in time but not in frequency. Wavelet basis isolates in both frequency and time. W k j (x) = W [ k (x j) ] j = 0,..., ( k 1) Exact fit possible by using N parameters to fit N data points. Many wavelet shapes possible. e.g. Mexican Hat wavelet: W (x) (1 x ) exp( x / ) W 00 At each new level, k -> k+1: Double the wavelet frequency. Double the number of wavelets. Complete orthogonal basis. (Used e.g. for data compression) W 10 W 11 W 0 W 1 W W 3

13 Various Wavelet Shapes Haar wavelet Morlet wavelet Mexican Hat wavelet Meyer wavelet Coiflet wavelet Shannon wavelet

14 Spline Decomposition - Red Noise Fit and subtract sequence of cubic splines. Orthogonal spline basis. Red noise power spectrum Power law d Power dln(ω) ω Red Noise: Random walk

15 Spline Decomposition - White Noise Fit and subtract sequence of cubic splines. Orthogonal spline basis. White noise power spectrum. d Power dln(ω) ω 0 White Noise: Independent noise at each time