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 explainingaway 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 slidingwindow
based methods, i.e., the tradeoff 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 superresolution property of the new method
again.
6. Sampling Rate, Aliasing, and Amplitude Conversation
Demonstrates Lombscargle and New method estimating
frequencies beyond half of the average sampling rate.
Demonstrates the amplitude conservation property
of the new algorithm while the Lombscargle and other singlefrequencymodel
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.
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.
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.



Spectrogram by LombScargle
X axis: Time (Sec.) Y axis: Frequency (Hz) 


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 LombScargle
and New methods




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







coupled with sparsification 






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












Spectral analysis for an unevenly sampled signal that contains 39 and
41 Hz sinusoids
(ac) LombScargle periodograms with sliding windows of
100,200,and 300 data points respectively, which illustrate the interference
of neighboring frequencies in LombScargle periodograms.
(d) Spectrogram by the new method with noninformative model parameters.
There is no interference between neighboring frequencies, which demonstrates
the superresolution 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.








LombScargle 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.