Power spectral density estimation using MULTITAPER METHOD

1. SPECTRAL ESTIMATION using MTM
By-Pankaj Kumar & Nikhil Singh

2. Spectral estimation is generally done by two
methods:-
• Parametric :- This method use an a priori,
parameter dependent model of the process
that generated the time series under analysis
eg:-MEM,MA,ARMA etc.
• Non-parametric :-This method does not use
an a priori i.e. do not rely on the estimation of
parameters describing the distribution of
variables of interest in the population eg:-
MTM, periodogram,tapered periodogram etc.

3. MULTITAPER METHOD
Select data taper for a chosen frequency
bandwidth
Each data taper is multiplied element wise to
signal
Estimation of power spectrum at each
component frequency
Averaging over all the tapered spectra for the
estimation of power spectra.
Generating multiple
power spectra from
the same signal by
orthogonal(Slepian)
tapers and then
finally averaging
these individually
tapered spectra to
get the total power
spectral estimate.
1
2
3
4

4. Step -1 of MTM
• Tapers are discrete set of eigen functions that minimize
leakage outside the bandwidth p .
• The optimal tapers belong to a family of Discrete Prolate
Spheroidal Sequence(DPSS) and we choose them in such a
way to minimize leakage.
• Rayleigh frequency:
=1/(N t)
where N=number of samples
t =sampling interval .
• In practice only the first 2p-1 tapers provide small spectral
leakage. Thus the number of tapers K used should be less
than 2p-1.

5. Taper demonstration using MatLab
CODE

6. Step-1 of MTM
λ= 0< λ<1
where F is the sampling frequency, |W|<F/2
• DPSS is derived from the time frequency concentration relationship. This
ratio determines which index limited sequence has largest proportion of
energy in band [W,-W].
• The choice of bandwidth 2p and number of tapers K thus represents
the classical tradeoff between spectral resolution and the stability or
variance reduction properties.
• The time half bandwidth product is NW where N is the length of the
sequence and [–W,W] is the effective bandwidth of the sequence.

7. Step-2 of MTM
Assume the time series X(t):
X(t)=B + ἠ(t)
Where B is the amplitude of a sinusoid
f is the frequency
ἠ(t) is the white noise.
Z(t)= ∑X(t) (t)
where t=1,2,3…….,N

8. Step-3 of MTM
=DFT of Z(t)
The k-th eigen spectrum is:-

9. Step-4 of MTM
The above relation is power spectrum of signal
X(t) using MTM. But there are several other
MTM just to improve the resolution.

10. MTM Spectrum using MatLab
CODE

11. • The high resolution multitaper spectrum is the
weighted sum of the K eigen spectra;
where is weighting of taper to reduce the
variance.

12. Advantages of MTM
• Power spectral density obtained from the
signal’s Fourier transform is a biased estimate
of the true spectral content and it can be
resolved by averaging over many realizations
of the same event.
• But in case when a single sample of process is
present then in that case estimation of
spectral properties will not be reliable.

13. Continued….
• It reduces the bias of estimation which was
more pronounced due to spectral leakage in
classical methods.
• Unlike MLM ,it can ascertain the presence of
pure sinusoids in fairly high noise background.

14. APPLICATIONS OF MTM
MTM is used in :-
• Analysis of atmospheric and oceanic data.
• Paleoclimate proxy data.
• Geochemical tracer data.
• Seismological data.
• Signal reconstruction

15. REFERENCES
• Ghil et al. (2001) Advanced spectral methods for
climatic time series
• http://in.mathworks.com