FGN 1.0 PPL 1.0 FD

Transcription

1 FGN PPL FD f f Figure SDFs for FGN, PPL and FD processes (top to bottom rows, respectively) on both linear/log and log/log aes (left- and right-hand columns, respectively) Each SDF S X ( ) is normalized such that S X () = The table below gives the parameter values for the various plotted curves (Adapted from Figure 282, Percival and Walden, 2, copyright Cambridge University Press) process thick solid dotted dashed thin solid FGN H = H =7 H =9 H =9 PPL α = α = α = 8 α = 9 FD δ = δ =2 δ =4 δ =4

2 FGN H = PPL α = FD δ = FGN H =9 PPL α = 9 FD δ = t Figure 2 Simulated realizations of FGN, PPL and FD processes The thick (thin) solid curves in Figure show the SDFs for the top (bottom) three series these SDFs differ markedly only at high frequencies We formed each simulated X,,X usingthe circulant embedding method, which does so by transforminga realization of a portion ε,,ε 23 of a white noise process To illustrate the similarity of FGN, PPL and FD processes with comparable H, α and δ, we used the same ε t to create all si series Although the top (bottom) three series appear to be identical, estimates of their SDFs show high frequency differences consistent with their theoretical SDFs (Adapted from Figure 283, Percival and Walden, 2, copyright Cambridge University Press)

3 δ t Simulated X t (a) (b) δ t Simulated X t (c) 2 23 Time, t (d) 2 23 Time, t Figure 3 Four eamples of time series simulated from TVFD processes The upper panel of each plot shows the sequence δ t, t =,,23, used to generate the simulated time series in the lower panel All four simulated series were created usingthe circulant embeddingmethod on the same set of 248 independent deviates from a standard Gaussian distribution (Figure, Percival and Constantine, 22)

4 (a) (b) f (Hz) (c) (d) 2 3 f (Hz) Figure 4 Periodogram (a) and direct spectral estimates (b,c,d) of noise simulated from composite FD process with four components, namely, δ = 2,, and (corresponding to α = 4, 3, 2 and ) The true SDF is the solid curve in each plot, while the dots are the spectral estimates The data tapers used in the direct spectral estimates are Slepian tapers (ie, discrete prolate spheroidal sequences) with the resolution bandwidth W set via (b) NW =, (c) NW = 2 and (d) NW = 4 The simulated series has length N =

5 k = 3 k = 2 k = k = 24 t 24 t Figure Sinusoidal tapers (left-hand column) as applied to a simulated FD time series with δ = 4 (top plot, right-hand column), resulting in tapered series (right-hand column, second to bottom rows)

6 3 2 K = f (db) K = 2 K = 3 K = f (db) Figure 6 Direct spectral estimates formed usingthe kth sinusoidal taper, k =,,3 (left-hand column, top to bottom row), alongwith multitaper estimates formed by averaging K =,,4 of these direct spectral estimates (right-hand column, top to bottom row)

7 t t Figure 7 Row vectors Wn T of the discrete wavelet transform matri W based on the Haar wavelet for N = 6 and n = to 7 (top to bottom on left plot) and n = 8 to (right plot)

8 scale 64 scale 32 scale 6 scale 8 scale 4 scale 2 scale t (days) 32 τ (lag) Figure 8 Haar DWT coefficients for clock 7 and sample autocorrelation sequences (ACSs)

9 Figure 9 Haar wavelet filters for scales τ j =2 j, j =, 2,,7

10 Figure D(4), C(6) and LA(8) wavelet filters for scales τ j =2 j, j =, 2,,7

11 Figure Haar scalingfilters for scales λ J =2 J, J =, 2,,7

12 Figure 2 D(4), C(6) and LA(8) scalingfilters for scales λ J =2 J, J =, 2,,7

13 2 22 scale 64 scale 32 scale 6 scale 8 scale 4 scale 2 scale t (days) Figure 3 Haar MODWT coefficients for clock 7

14 X t 7 t Y t (τ ) 6 X () X (2) t t t (days) Figure 4 Plot of differences in time {X t } as kept by clock 7 (a cesium beam atomic clock) and as kept by the time scale UTC(USNO) maintained by the US Naval Observatory, Washington, DC (top plot); its first backward difference {X () t } (middle); and its second backward difference {X (2) t } (bottom) In the middle plot, Y t (τ ) denotes the τ average fractional frequency deviates (given in parts in 3 ) these are proportional to X () t

15 o + o o o o o + o o o o o o o o o o τ j t (days) τ j t (days) Figure Square roots of wavelet variance estimates for atomic clock time differences {X t } based upon the unbiased MODWT estimator and the followingwavelet filters: Haar ( s in left-hand plot, through which a least squares line has been fit), D(4) (circles in left- and righthand plots) and D(6) (pluses in left-hand plot) The right-hand plot also shows 9% confidence intervals for the unknown wavelet variances

16 2 3 4 o o o o o o o o o o o o o o o o τ j t (days) τ j t (days) Figure 6 Square roots of wavelet variance estimates for atomic clock one day average fractional frequency deviates {Y t (τ )} based upon the unbiased MODWT estimator and the followingwavelet filters: Haar ( s in left-hand plot) and D(4) (circles in left and right-hand plots)