• DEPARTMENT OF ELECTRONICS ENGINEERING

NON PARAMETRIC METHODS FOR PSE

PRESENTED BY: BHAVIKA JETHANI (2) SUPRIYA ASUTKAR (8)

BHUSHAN GADGE (9) ROHIT NANDANWAR(10)

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Advanced Digital Signal Presentation

Topics to be covered :

Non-parametric methods of Power Spectrum

Estimation.

The Bartlett Method

The Welch Method

The Blackman and Tukey Method

Comparison of performance of Non-periodic

Power Spectral Estimation Methods

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• The well known form of power density spectrumestimate is called as periodogram

• Periodogram is not a consistent estimate of truePower Density Spectrum.

• That means, it does not converge to the true powerdensity spectrum.

• So the emphasis of classical Non-parametric Methodsis on obtaining a consistent estimate of powerspectrum through some averaging and smoothingoperations performed directly on the periodogram ordirectly on the autocorrelation.

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NON-PARAMETERIC METHODS FOR POWER SPECTRAL ESTIMATION

The power spectrum methods described

are the classical methods developed by Bartlett(1948)

,Blackman and Tukey (1958) ,and Welch (1967),

These methods make no assumption about how the

data were generated and hence are called

nonparametric.

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Since the estimates are based entirely on

a finite record of data ,the frequency resolution of

these methods is , at best , equal to the spectral width

of the rectangular window of length N, which is

approximately 1/N at the -3dB points. We shall be

more precise in specifying the frequency resolution of

the specific methods. All the estimation techniques

described in this presentation “Decrease the frequency

resolution in order to reduce the variance in the

spectral estimate”.

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THE BARTLETT METHOD : AVERAGING PERIODOGRAMS

------ (1)

------ (2)

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Finally , we average the periodograms for the K segments

to obtain the bartlett power spectrum estimate.

The statistical properties of this estimate are easily obtained.

The mean value is

------ (3)

------ (4)

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as :

where ,

is the frequency characteristics of the Bartlett window .

------ (5)

------ (6)

------ (7)

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In return for this reduction in resolution , we have reduced the

variance. The variance of the Bartlett estimate is

In general, the variance of the estimate does not delay to

zero as M tends to infinity. The variance for the Bartlett window is as

follows :

------ (8)

------ (9)

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Averaging the periodogram.

Welch made two modification in Bartlett method.

First he allows data segment to overlap.

Second is to window the data segments prior to computing the periodogram

Xi(n)= x(n+iD), n=0,1,. . . . . M-1i = 0,1 . . . . . N-1

The Welch Method:

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Where U is a normalization factor for the power in the window function is selected as

The Welch power spectrum estimate is the average of these modified periodogram, that is

……..(1)

………(2)

……..(3)11

…..(4)

……..(5)

………(6)

……..(7)12

……(8)

……(9)

……(10)

…….(11)

……..(12)13

BLACKMAN AND TUKEY METHOD:SMOOTHING THE PERIODOGRAM

•In this method , the sample auto correlation sequence is windowed first and then Fourier transformed to yield the estimate of the power spectrum.

•For values of data points of m approaching N, the variance of these estimates is very high

•Thus Blackman Tukey Estimate is -

----- (1)

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•Where w(n) is window function having length (2M-1) and is zero for m≥ M

•Frequency domain equivalent expression can be given as-

Where Pxx(f) is periodogram

•The effect of windowing the autocorrelation is to smooth the periodogramestimate.

----- (2)

CONT...

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•This ensures that ≥ 0 for |f| ≤ 1/2

•However some of the window function do not satisfy this condition and may result in negative spectrum estimates

•The expected value of Blackman – Tukey power spectrum estimate is-

CONT...

W(f) ≥ 0, |f| ≤ 1/2

•The window sequence w(n) should be symmetric (even) about m=0

----- (3)

----- (4)16

CONT...

----- (5)

Putting eqn (5) in (6)

•The expected value of the Blackman –Tukey power spectrum estimate is-

....In time domain

----- (6)

----- (7)

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•Where Barlett window-

CONT...

•W(n) should be narrower than Wb(m) to smooth the Periodogram

----- (8)

----- (9)

----- (10)

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•Variance of Blackman Tukey power spectrum estimate is

CONT...

•Assuming that the random process is Gaussian

----- (11)

----- (12)

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Substituting eqn (12) in eqn (11)

CONT...

•First term is square of the mean of Pxx(f) which is to be subtracted

•For N » M, [sinπ(θ+α)N ] and [sinπ(θ+α)N] will be relatively narrowTherefore-

----- (13)

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CONT...

•The variance of Pxx becomes-

•Where below term is assumed as-

----- (14)

----- (15)

----- (16)

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•When w(f) is narrowed compare to true power spectrum further approximate as-

CONT...

----- (17)

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PERFORMANCE CHARACTERISTICS

OF NON-PARAMETRIC POWER SPECTRAL ESTIMATORS

In this section the Quality of three methods i.e. Bartlett, Welch and Blackman and Tukey power spectral estimate is being compared.

QUALITY is defined as ratio of mean square to variance of power spectrum

estimate.

Lets take example of periodogram:

periodogram has mean and variance as

----- (1)

----- (2)

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As indicated earlier periodogram is asymptotically unbiased estimate of power spectrum

but it is not consistent also since variance does not tends to zero as N tends to infinity.

The fact that Qp is fixed and independent of data length N is another indication of poor quality of the estimate

----- (3)

Substituting eqn (2) and (3) in eqn (1)

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The mean and variance of Bartlett power spectrum estimate is

1. Bartlett power spectrum estimate:

----- (4)

----- (5)

----- (put (4) & (5) in (1)

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2.Welch power spectral estimate :

The mean and variance of power spectrum estimate is

----- (6)

----- (7)

Put (6) and (7) in (1)

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3.BLACKMAN-TUKEY POWER SPECTRAL ESTIMATE :

The mean and variance of this estimate is

----- (8)

----- (9)

----- (10)

Put 8 , 9 , 10 in 1

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SUMMARY OF QUALITY OF POWER SPECTRAL ESTIMATE

CONCLUSION:

1. Welch and Blackman-Tukey power spectrum estimate is somewhat better

than Bartlett

2. However the difference in their performance is relatively small.

3. The main point is that Quality factor increases as N length of data increases.

4. This characteristic behavior is not shared by periodogram.

5. Furthermore Quality factor depends on product of length N and freq

resolution ∆f

6.For desired level of quality ∆f can be decreased(freq resolution increased)by

increasing length N of data and vice versa.

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COMPUTATIONAL REQUIREMENT OF POWER SPECTRAL ESTIMATE

The other important aspect of nonparametric power spectrum estimate is their

computational requirement. For this comparison we assume the estimates are

based on fixed amount of data and specified resolution of ∆f the radix 2FFT

algorithm is assumed in all the computation. We shall count only the number of

complex multiplication required to compute the power spectrum estimate.

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2 N additional computation is required for windowing.

Blackman-Tukey Power Spectral Estimate

We cant use N point DFT for its computation because its maximum value limits

to 1024 point DFT for which we required 2M point DFT and one 2M point

IDFT hence we are using the FFT algorithm.

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Conclusion

• There is additional M computation required for

Fourier transform of the windowed

autocorrelation sequence but still the number

of computation Is increased by a small amount

• We conclude that Welch method requires a little

more computational power than do the other 2

methods

• Bartlett requires the smallest number of

computation

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Ex.. A freq resolution is 0.09 and N=100 samples . Determine the quality factor, recorded length and no. of computation requirements for Bartlett , Welch and Blackman – Tukey methods.

Solution:- Given- Δf = 0.09N=100

1] Barlett method:-Quality factor (QB ) =1.11 N ΔF

= 1.11 X 100 X0.09= 9.99

Recorded length(M)=0.9/Δf=0.9/0.09=10

No. of FFT’s=N/M=100/10=10

No. of computations =N/2 log2 0.9/Δf=100/2 log2 0.9/0.09=166

L = 1.1NΔf x M= 99.9

BARLETT METHOD

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2] Welch method :-

Quality factor (Qw)=0.78N ΔF (Non-overlapping)= 0.78 X 100 X0.09= 7.02

Recorded length(m)=1.28/Δf=1.28/0.09=14.22

No. of FFT’s=2N/M OR =1.56 N ΔF=2 X100/14.22 =1.56 X 100 X 0.09=14.06 =14.04

No. of computations =N X log2 X 1.28/Δf (Non – overlapping) =100 log 2 x 1.28/0.09=383

QB=1.39NΔF (50% overlapping)=1.39x100x0.09=12.51

Total no. of computations= N Log2 x5.12/Δf (50% overlapping)=583

Welch method

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3]Blackman- Tukey method :-

Quality factor (QB)=2.34N ΔF (Non-overlapping)= 2.34 X 100 X0.09= 21.06

Recorded length(2M)=1.28/Δf=1.28/0.09=14.22

M=7.11

No. of FFT’s=N/M =2 X100/14.22 =14.06

No. of computations =N X log2 X 1.28/Δf =100 log 2 x 1.28/0.09=383

BLACKMAN TUKEY METHOD

References:1. Digital signal processing

Fourth edition by JOHN G. PROAKIS , DIMITRIS G. MANOLAKIS

2. Statistical spectral analysis

A non - probabilistic theory , Prentice Hall

3. Statistical Digital Signal Processing And Modelling ,Monson H. Hayes

4. Digital signal processing and its application by Ramesh Babu.

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