non-parametric methods for power spectrum estimation which includes bartlett method, welch method , blackman and tukey methods and also the comparision of all these methods

1. • DEPARTMENT OF ELECTRONICS ENGINEERING
NON PARAMETRIC METHODS FOR PSE
PRESENTED BY:
BHAVIKA JETHANI (2) SUPRIYA ASUTKAR (8)
BHUSHAN GADGE (9) ROHIT NANDANWAR(10)
1

2. 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
2

3. • The well known form of power density spectrum
estimate is called as periodogram
• Periodogram is not a consistent estimate of true
Power Density Spectrum.
• That means, it does not converge to the true power
density spectrum.
• So the emphasis of classical Non-parametric Methods
is on obtaining a consistent estimate of power
spectrum through some averaging and smoothing
operations performed directly on the periodogram or
directly on the autocorrelation.
3

4. 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.
4

5. 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”.
5

6. THE BARTLETT METHOD : AVERAGING PERIODOGRAMS
------ (1)
------ (2)
6

7. Finally , we average the periodograms for the K segments
to obtain the bartlett power spectrum estimate.
------ (3)
The statistical properties of this estimate are easily obtained.
The mean value is
------ (4)
7

8. as :
where ,
------ (5)
------ (6)
is the frequency characteristics of the Bartlett window .
------ (7)
8

9. In return for this reduction in resolution , we have reduced the
variance. The variance of the Bartlett estimate is
------ (8)
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 :
------ (9)
9

10. The Welch Method:
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-1
i = 0,1 . . . . . N-1
10

11. ……..(1)
Where U is a normalization factor for the power in the window
function is selected as
………(2)
The Welch power spectrum estimate is the average of these
modified periodogram, that is
……..(3) 11

12. …..(4)
……..(5)
………(6)
……..(7)
12

13. ……(8)
……(9)
……(10)
…….(11)
……..(12)13

14. 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)
14

15. CONT...
•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
----- (2)
•The effect of windowing the autocorrelation is to smooth the periodogram
estimate.
15

16. CONT...
•The window sequence w(n) should be symmetric (even)
about m=0
W(f) ≥ 0, |f| ≤ 1/2
•This ensures that ≥ 0 for |f| ≤ 1/2
----- (3)
•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-
----- (4)
16

17. CONT...
----- (5)
Putting eqn (5) in (6)
----- (6)
•The expected value of the Blackman –Tukey power spectrum estimate is-
----- (7)
....In time domain
17

18. CONT...
•Where Barlett window-
•W(n) should be narrower than Wb(m) to smooth the Periodogram
----- (8)
----- (9)
----- (10)
18

19. CONT...
•Variance of Blackman Tukey power spectrum estimate is
•Assuming that the random process is Gaussian
----- (11)
----- (12)
19

20. 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 narrow
Therefore-
----- (13)
20

21. CONT...
•The variance of Pxx becomes-
•Where below term is assumed as-
----- (14)
----- (15)
----- (16)
21

22. CONT...
•When w(f) is narrowed compare to true power spectrum further approximate as-
----- (17)
22

23. 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)
23

24. ----- (3)
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.
Substituting eqn (2) and (3) in eqn (1)
The fact that Qp is fixed and independent of data length N is another indication of
poor quality of the estimate
24

25. 1. Bartlett power spectrum estimate:
The mean and variance of Bartlett power spectrum estimate is
----- (4)
----- (5)
----- (put (4) & (5) in (1)
25

26. 2.Welch power spectral estimate :
The mean and variance of power spectrum estimate is
----- (7)
----- (6)
Put (6) and (7) in (1)
26

27. 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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28. 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 Δf
6.For desired level of quality Δf can be decreased(freq resolution increased)by
increasing length N of data and vice versa.
28

29. 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 Δf 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.
29

30. 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.
30

31. 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
31

32. 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.
32
Solution:- Given- Δf = 0.09
N=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

35. 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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