Professor A G
Constantinides© 1
AGC
DSP
AGC
DSP
Power Spectral Estimation
The purpose of these methods is to obtain an approximate estimation of the power spectral density of a given real random process
)( jyy e
}{ ny
Professor A G
Constantinides© 2
AGC
DSP
AGC
DSP
Autocorrelation
The autocorrelation sequence is
The pivot of estimation is the Wiener-Khintchine formula (which is also known as the Einstein or the Rayleigh formula)
]}[][{][ nynmyEmyy
m
jmyy
jyy eme ][)(
Professor A G
Constantinides© 3
AGC
DSP
AGC
DSP
Classification The crux of PSD estimation is the determination of
the autocorrelation sequence from a given process. Methods that rely on the direct use of the given
finite duration signal to compute the autocorrelation to the maximum allowable length (beyond which it is assumed zero), are called Non-parametric methods
Methods that rely on a model for the signal generation are called Modern or Parametric methods.
Personally I prefer the names “Direct” and “Indirect Methods”
Professor A G
Constantinides© 4
AGC
DSP
AGC
DSP
Classification & Choice
The choice between the two options is made on a balance between “simple and fast computations but inaccurate PSD estimates” Vs “computationally involved procedures but enhanced PSD estimates”
Professor A G
Constantinides© 5
AGC
DSP
AGC
DSP
Direct Methods & Limitations
Apart from the adverse effects of noise, there are two limitations in practice Only one manifestation ,
known as a realisation in stochastic processes, is available
Only a finite number of terms, say , is available
]}[{ ny
12 N
Professor A G
Constantinides© 6
AGC
DSP
AGC
DSP
Assumptions
Assume to be Ergodic so that statistical expectations can
be replaced by summation averages Stationary so that infinite averages can be
estimated from finite averages Both of these averages are to be derived
from
]}[{ ny
]}[{ ny
Professor A G
Constantinides© 7
AGC
DSP
AGC
DSP
Windowing
Thus an approximation is necessary. In effect we have a new signal given by
where is a window of finite duration selecting a segment of signal from .
][][][ nynwnx
][nw
]}[{ ny
]}[{ nx
Professor A G
Constantinides© 8
AGC
DSP
AGC
DSP
The Periodogram The Periodogram is defined as
Clearly evaluations at
are efficiently computable via the FFT.
21
0][
1)(
N
n
jnjN enx
NeI
kNk 2
1
0
1
0][][
1)(
N
r
jrN
n
jnjN erxenx
NeI
Professor A G
Constantinides© 9
AGC
DSP
AGC
DSP
Limited autocorrelations
Let
which we shall call the autocorrelation sequence of this shorter signal.
These are the parameters to be used for the PSD estimation.
mN
nxx nxmnxm
1
0][][][
Professor A G
Constantinides© 10
AGC
DSP
AGC
DSP
PSD Estimator It can be shown that
The above and the limited autocorrelation expression, are similar expressions to the PSD. However, the PSD estimates, as we shall see, can be bad.
Measures of “goodness” are the “bias” and the “variance” of the estimates?
1
)1(][
1)(
N
Nm
jmxx
jN em
NeI
Professor A G
Constantinides© 11
AGC
DSP
AGC
DSP
The Bias
The Bias pertains to the question:
Does the estimate tend to the correct value as the number of terms taken tends to infinity?
If yes, then it is unbiased, else it is biased.
Professor A G
Constantinides© 12
AGC
DSP
AGC
DSP
Analysis on Bias
For the unspecified window case considered thus far, the expected value of the autocorrelation sequence of the truncated signal is
]}[][][][{
}][][{]}[{
1
0
1
0
nynwmnymnwE
nxmnxEmE
mN
n
mN
nxx
Professor A G
Constantinides© 13
AGC
DSP
AGC
DSP
Analysis on Bias
or
Thus
2
1
)1(
)(*)(21
]}[{1
)}({
jjyy
N
Nm
jmxx
jN
eWeN
emEN
eIE
mN
nyywwxx mmmE
1
0][][]}[{
Professor A G
Constantinides© 14
AGC
DSP
AGC
DSP
Analysis on Bias
The asterisk denotes convolution. The bias is then given as the difference
between the expected mean and the true mean PSDs at some frequency.
)()(*)(21
)()({
2kj
yyjj
yy
kjyy
kjN
eeWeN
eeIEB
Professor A G
Constantinides© 15
AGC
DSP
AGC
DSP
Example For example take a rectangular window
then ,
which, when convolved with the true PSD, gives the mean periodogram, ie a smoothed version of the true PSD.
22
)2/sin()2/sin(
)(
N
eW j
Professor A G
Constantinides© 16
AGC
DSP
AGC
DSP
Example
Note that the main lobe of the window has a width of
and hence as
we have
at every point of continuity of the PSD.
N/2N
)()}({lim jyy
jN
NeeIE
Professor A G
Constantinides© 17
AGC
DSP
AGC
DSP
Asymptotically unbiased
Thus is an asymptotically unbiased estimator of the true PSD.
The result can be generalised as follows.
)( jN eI
Professor A G
Constantinides© 18
AGC
DSP
AGC
DSP
Windows & Estimators
For the window to yield an unbiased estimator it must satisfy the following:
1) Normalisation condition
2) The main lobe width must decrease as 1/N
1
0
2 ][N
nNnw
Professor A G
Constantinides© 19
AGC
DSP
AGC
DSP
The Variance
The Variance refers to the question on the “goodness” of the estimate:
Does its variance of the estimate decrease with N? ie does the expression below tend to zero as N tends to infinity?
22 )})({(}))({()}(var{ jN
jN
jN eIEeIEeI
Professor A G
Constantinides© 20
AGC
DSP
AGC
DSP
Analysis on Variance
If the process is Gaussian then (after very long and tedious algebra) it can be shown that
where
AeIEeI jN
jN 2)})({()}(var{
2)(*)( )()()(
21
deWeWeN
A jjjyy
Professor A G
Constantinides© 21
AGC
DSP
AGC
DSP
Analysis
Hence it is evident that as the length of data tends to infinity the first term remains unaffected, and thus the periodogram is an inconsistent estimator of the PSD.
Professor A G
Constantinides© 22
AGC
DSP
AGC
DSP
Example
For example for the rectangular window taken earlier we have
where
CeIEeI jN
jN 2)})({()}(var{
21
)1(][
sin
)(sin
N
Nmyy mN
mNC
Professor A G
Constantinides© 23
AGC
DSP
AGC
DSP
Decaying Correlations
If has for then for we can write above
From which it is apparent that
ny 0yy 0mm
0mN
22
sinsin
1)()}(var{
NN
eeI jyy
jN
elsewheree
eeI j
yy
jyyj
N
,0
)(2
)()}(var{ 2
2
Professor A G
Constantinides© 24
AGC
DSP
AGC
DSP
Variance is large
Thus even for very large windows the variance of the estimate is as large as the quantity to be estimated!
Professor A G
Constantinides© 25
AGC
DSP
AGC
DSP
Smoothed Periodograms Periodograms are therefore inadequate for
precise estimation of a PSD. To reduce variance while keeping estimation
simplicity and efficiency, several modifications can be implemented
a) Averaging over a set of periodograms of (nearly) independent segments
b) Windowing applied to segments c) Overlapping the windowed segments for
additional averaging
Professor A G
Constantinides© 26
AGC
DSP
AGC
DSP
Welch-Bartlett Procedure
Typical is the Welch-Bartlett procedure as follows.
Let be an ergodic process from which we are given data points for the signal .
1) Divide the given signal into blocks each of length .
2) Estimate the PSD of each block 3) Take the average of these estimates
nyM ][ny
NMK /N
Professor A G
Constantinides© 27
AGC
DSP
AGC
DSP
Welch-Bartlett Procedure
Step 2 can take different forms for different authors.
For the Welch-Bartlett case the periodogram is suggested as
21
0][
1)(
N
n
jnr
jrN
emxN
eI
Professor A G
Constantinides© 28
AGC
DSP
AGC
DSP
Welch-Bartlett Procedure
where the segment is a windowed portion of
And is the overlap. (Strictly the Bartlett case has a
rectangular window and no overlap).
][nxrny
)]([][][ 0NNrnynwnxr
0N
Professor A G
Constantinides© 29
AGC
DSP
AGC
DSP
Comments FFT-based Spectral estimation is limited by
a) the correlation assumed to be zero beyond the measurement length and
b) the resolution attributes of the DFT. Thus if two frequencies are separated by
then a data record of length
is required.(Uncertainty Principle)
/2N
Professor A G
Constantinides© 30
AGC
DSP
AGC
DSP
Narrowband Signals The spectrum to be estimated is some cases
may contain narrow peaks (high Q resonances) as in speech formants or passive sonar.
The limit on resolution imposed by window length is problematic in that it causes bias.
The derived variance formulae are not accurate
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