Bayesian Spectrum Estimation of Unevenly Sampled Nonstationary Data


Yuan Qi, Thomas P. Minka, and Rosalind W. Picard
Email contact: yuanqi@media.mit.edu

Matlab Code: Download

1. Comparison with Classical Spectrum Estimation Algorithms
  Demonstrates the new algorithm's accuracy in frequency estimation.
Data Set A.
Data Set B.

2. Estimation of Unevenly Sampled Fast Decaying Amplitude of Sinusoid Wave
    Demonstrates the new algorithm's accuracy in amplitude estimation.
   Also, shows the explaining-away effect of the joint estimation.
Spectrograms
Amplitude Estimates

3. Estimation of Unevenly Sampled  Frequency Modulated Signal
    Manifests the effectiveness of using sparsification with the new algorithm.
    Illustrates the limitation of popular sliding-window based methods, i.e., the trade-off between time and frequency resolutions.

4. Estimation of Evenly Sampled Chirp Signal with Missing Data
   Demonstrates the new algorithm's ability to track a quadratic chirp signal.

5.Estimation Ambiguity
   Demonstrates the influence of the model parameters.
   Illustrates the ambiguity of the spectrogram estimation.
   Shows the super-resolution property of the new method again.

6. Sampling Rate, Aliasing, and Amplitude Conversation
    Demonstrates Lomb-scargle and New method estimating frequencies beyond half of the average sampling rate.
    Demonstrates the amplitude conservation property of the new algorithm while the Lomb-scargle and other single-frequency-model based methods do not have this property. This property can be used to do aliasing detection, i.e., differentiating real aliases from a symmetric spectrum of a signal without aliasing.

1. Comparison with Classical Spectrum Estimation Algorithms

Comparison of different spectral estimation algorithms on evenly sampled stationary data.
A.
 

Welch

Burg

Music

Multitaper

New

The signal is the sum of 19, 20, and 21 Hz real sinusoid waves with amplitudes 0.5, 1, and 1 respectively. The variance of the additive white noise is 0.1. The signal is evenly sampled 128 times at 50 Hz.

B.
 

Welch

Burg

Music

Multitaper

New

The signal is the sum of 19, 20, and 21 Hz real sinusoid waves, all with amplitude 1, and white noise with variance 0.001. The signal is evenly sampled at 50 Hz over 3 seconds.

2. Estimation of Unevenly Sampled Fast Decaying Amplitude of Sinusoid Wave

A. Spectrograms by the Lomb-Scargle and New methods
 
Spectrogram by Lomb-Scargle
X axis: Time (Sec.)
Y axis: Frequency (Hz)
New
New with Smoothing

Estimated spectrograms for an unevenly sampled signal that contains one 125 Hz sinusoid modulated with an exponentially fast decaying amplitude.
B. Amplitude Estimate by the Lomb-Scargle and New methods
 
True Amplitudes
 Lomb-Scargle
 New
 New with smoothing

True and estimated amplitudes for the  unevenly sampled signal that contains one 125 Hz sinusoid modulated with an exponentially fast decaying amplitude.

3. Estimation of Unevenly Sampled  Frequency Modulated Signal

 
 Lomb-Scargle periodogram with a window size of 100 points
 Lomb-Scargle periodogram with a window size of 200 points

 
Spectrogram by the new method
Spectrogram by the new method
 coupled with sparsification
Spectral analysis for an unevenly sampled signal whose frequency jumps from 20 Hz to 40 Hz at the sampling time -0.833 second, and then jumps  from 40 Hz to 60 Hz at 0.833 second.

4. Estimation of Evenly Sampled Chirp Signal with Missing Data

Lomb-Scargle Spectrogram (The dotted curve is the ground truth)
Spectrogram by the new method
Spectrogram by the new method with smoothing

 Spectral Analysis for a evenly sampled quadratic chirp signal with 10%  missing data.

5. Estimation Ambiguity

(a) Lomb-Scargle periodograms with a sliding window of 100 points
(b)Lomb-Scargle periodograms with a sliding window of 200 points
(c)Lomb-Scargle periodograms with a sliding window of 300 points
(d) Spectrogram by the new method with noninformative model parameters. 
(e) Spectrogram by the new method with informative model parameters. 
(f) Dotted curve: estimated amplitudes of 39 and 41 Hz by the new method with a noninformative model; solid curve: the estimated amplitude of 40 Hz by the new method with an informative model.

Spectral analysis for an unevenly sampled signal that contains 39 and 41 Hz sinusoids
(a-c)  Lomb-Scargle periodograms with  sliding windows of 100,200,and 300 data points respectively, which illustrate the interference of neighboring frequencies in Lomb--Scargle  periodograms.
(d) Spectrogram by the new method with noninformative model parameters. There is no interference between neighboring frequencies,  which demonstrates the super-resolution property of this new method
(e) Spectrogram by the new method with informative model parameters. This figure tries to illustrate the underlying ambiguity for spectral analysis.
(f) Dotted curve: estimated amplitudes of 39 and 41 Hz by the new method with a noninformative model; solid curve: the estimated amplitude of 40 Hz by the new method with an informative model.

6. Sampling Rate and Aliasing

(a) Lomb-Scargle, even
(b)New, even
(c) Lomb-Scargle, uneven
(d)New, uneven

Lomb-Scargle periodogram and the spectra estimated by the new method for a signal x= sin(2*pi*39*t) + sin(2*pi*41*t), sampled 100 times over 2 seconds, with samples either evenly or randomly (unevenly) spaced.