1 spectrum estimation dr. hassanpour payam masoumi mariam zabihi advanced digital signal processing...
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Spectrum Estimation
Dr. HassanpourPayam MasoumiMariam ZabihiAdvanced Digital Signal Processing SeminarDepartment of Electronic EngineeringNoushirvani University
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Course Outlines
•Introduction
Fourier Series and Transform
Time/Frequency Resolutions
Autocorrelation & spectrum estimation
•Non-parametric Methods
Periodogram
Modified Periodogram
Bartlett’s Method
Welch’s Method
Blackman-Tukey Method
•Parametric Methods
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Fourier Series and Transform
Fourier basis functions:
real and imaginar parts of a complex sinusoid
vector representation of a complex exponential.
tjke 0
Re
Im
t
)sin( 0tk )cos( 0tk
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Fourier Series:
k
tjkkectx 0)(
2
20
0
0
0)(1 T
T
tjkk dtetx
Tc
k=…,-1,0,1,…
kt
)(tx )(kc
,
n0T
ffT0offon TTT 0 0
1
T
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t k
dtetxfX ftj 2)()(
)(tx )( fX
dfefXtx ftj 2)()(
,
Fourier Transform:
ffT0
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Discrete Fourier Transform (DFT)
)0(X)1(X
)1( Nx )1( NX
)0(x
)1(x
1
0
2
)()(N
m
N
kmj
emxkX
,....1,0,1....,,)()(1
0
2
kekXmxN
m
N
kmj
DFT
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Autocorrelation & Spectrum estimation
Autocorrelation:
k
jkx
jkx ekreP )()(
)()}(*)(12
1{lim krnxknx
N x
N
NnN
Power spectrum :
Spectrum estimation is a problem that involves
estimating from finite number of noisy
measurements of x(n).
)( tjx eP
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Nonparametric methods
•Peroidogram
•Modified periodogram
•Bartlett method
•Welch method
•Blackman-Tukey method
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The periodogram
kN
nx nxknx
Nkr
1
0
)(*)(1
)(ˆ
k
jkx
jkper ekreP )(ˆ)(ˆ
Estimated autocorrelation:
Estimated power spectrum or periodogram:
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2|)(|1
)()(1
)(ˆ jN
jN
jN
jper eX
NeXeX
NeP
The periodogram cont.
)(nx
)()()( nxnwnx RN
x N N N Nn
1 1r̂ (k) x (n k)x (n) x (k) x ( k)
N N
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The periodogram of white noise
)(nx
DFT )(kX N22 |)(|
1)(ˆ kX
NeP N
Nkjper
: white noise with a variance , length N=32 2
)(nxN2|.|
1
N
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The estimated autocorrelation sequence
White noise power spectrum
The periodogram of white noise cont.
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Periodogram of sinusoid in noise
)()sin()( 0 nvnAnx
)(2
1)( 0
22 AeP vj
x
0
2v
2
2
1A
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Periodogram of sinusoid in noise cont.
)1(win )1()(ˆ1
1 NzeP jX
)(nx
2|.|1
N
)(1 nz)(1 ny
)1()(ˆ2
2 NzeP jX
2|.|1
N
)(2 nz)(2 ny
)1()(ˆ NzeP Lj
XL2|.|
1
N
)(nzL)(nyL
)2(win
)(Lwin
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Periodogram Bias
)()(2
1)}(ˆ{
j
Bj
xj
per eWePePE
{}E)(ˆ krx )()(1 1
0
krN
kNkr
N x
kN
nx
B
N | k || k | 0
w (k) N0 o.w
jkBx
jper ekwkrePE _)()()}(ˆ{
)()}(ˆ{lim jx
jper
NePePE
Thus, the bias is deference between estimated and actual Power spectrum.
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)()sin()( 0 nvnAnx
2
4
1A
0
Periodogram of sinusoid in noise cont.
)]()([4
1)}(ˆ{ )()(22 00 j
Bj
Bvj
per eWeWAePE
2
kNk
20
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x(n) 1sin(0.4 n ) v(n) Example:
128N 512N
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Periodogram Resolution
)()sin()sin()( 221111 nvnAnAnx
)(2
1)(
2
1)( 2
221
21
2 AAeP vj
x
)]()([4
1
)]()([4
1)}(ˆ{
)()(22
)()(21
2
22
11
jB
jB
jB
jBv
jper
eWeWA
eWeWAePE
NeP j
per
289.0)](ˆ[Res
Set equal to the width of main lobe of the spectral windowat it’s half power or 6dB point.
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Example:
128N 512N
1 2x(n) 1sin(0.4 n ) 1sin(0.45 n ) v(n)
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Properties of the periodogram
Bias:
Resolution:
Variance:
21
0
)()(1
)(ˆ
n
n
jnR
jper enwnx
NeP
)()(2
1)}(ˆ{
j
Bj
xj
per eWePePE
NeP j
per
289.0)](ˆ[Res
)()}(ˆ{ 2 jx
jper ePePVar
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Modified Periodogram2
2 )()(1
|)(|1
)(ˆ
n
jnR
jN
jper enwnx
NeX
NeP
Would there be any benefit in replacing the rectangular window
with other windows? (for example triangular window)
2)(*)(
2
1)(ˆ
j
Rj
xj
per eWePN
eP 2)1(
)2sin(
)2sin()(,
Njj
R eN
eW
2
)()(1
)(ˆ
n
jnR
jM enwnx
NUeP
deWN
nwN
U jN
n
221
0
)(2
1)(
1
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1 2x(n) 0.1sin(0.2 n ) 1sin(0.3 n ) v(n) Example:
N=128Rectangular Window
N=128Hamming Window
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Properties of the M-periodogram
Bias:
Resolution: window dependent
Variance:
2)()(
2
1)}(ˆ{
jj
xj
M eWePNU
ePE
)()}(ˆ{ 2 jx
jM ePePVar
2
)()(1
)(ˆ
n
jnR
jM enwnx
NUeP
21
0
)(1
N
n
nwN
U
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Bartlett’s method (periodogram averaging)
kienxL
ePL
n
jni
jiper ...,,2,1;)(
1)(ˆ
21
0
....
PointsL PointsL PointsL
)(1 nx )(2 nx )(nxk
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Properties of Bartlett’s method
Bias:
Resolution:
Variance:
1
0
21
0
)(1
)(ˆk
i
L
n
jnjB eiLnx
NeP
)()(2
1)}(ˆ{
j
Bj
xj
B eWePePE
NkeP j
B
289.0)](ˆ[Res
)(1
)}(ˆ{ 2 jx
jper eP
kePVar
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1 2x(n) 1sin(0.25 n ) 3sin(0.45 n ) v(n) Example:
1
512
k
N
4
512
k
N
16
512
k
N
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1 2x(n) 1sin(0.25 n ) 3sin(0.45 n ) v(n) Example:
4
128
k
N
4
512
k
N
4
1024
k
N
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Welch’s method (M-periodogram averaging)
....
1,.....,1,0;)()( LniDnxnxi Overlap = L-D
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Properties of Welch’s method
1
0
21
0
)(1
)(ˆk
i
L
n
jnjW eiDnx
KLUeP
1
0
21
0
)( )(1
,)(ˆ1)(ˆ
L
n
k
i
jiM
jW nw
LUeP
LeP
Bias
Resolution Window dependent
Variance
2)()(
2
1)}(ˆ{
jj
xj
B eWePLU
ePE
overlapwithePN
LePVar j
xj
per %50)(16
9)}(ˆ{ 2
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512
k
N
hamming,%50
128,512
overlap
LN
)()25.0sin(3)2.0sin(1)( 21 nvnx Example:
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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n) Resolution:
hamming,%50
64,512
overlap
LN
hamming,%50
128,512
overlap
LN
hamming,%50
256,512
overlap
LN
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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n)
rRectangula,%50
128,512
overlap
LN
Bartlett,%50
128,512
overlap
LN
windowing:
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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n) windowing:
Hanningoverlap
LN
,%50
128,512
Hamming,%50
128,512
overlap
LN
Blackman,%50
128,512
overlap
LN
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Blackman-Tukey’s method (Periodogram smoothing)
•Note: Bartlett & Welch are design to reduce the variance if the priodogram
by averaging and modified it.
•Periodogram is computed by taking the Fourier transform of a consistent
estimate of the auto correlation sequence.
•For any finite data record of length N, the variance of will be large
for values of k that are close to N. for example:
)(ˆ krx
1)0()1(1
)1(ˆ nklagatxNxN
Nrx
•In Bartlett & Welch, the variance is decreased by reducing the variance
of autocorrelation estimate by averaging.
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Blackman-Tukey’s method cont.
•In the Blackman-Tukey method, the variance is decreased by applying a
window to in order to decrease the contribution of the unreliable
estimates to the periodogram.
Specifically, the Blackman-Tukey spectrum estimation is:
)(ˆ krx
M
Mk
jkx
jBT ekwkreP )()(ˆ)(ˆ
•For example, if w(k) is a rectangular window extending from –M to M
with M<N-1 , then having the largest variance are set to zero and
consequently, the power spectrum estimation will have a smaller variance.
)(ˆ krx
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Properties of B-T’s method
Bias
Resolution Window dependent
Variance
)()(2
1)}(ˆ{
jj
xj
BT eWePePE
M
Mk
jkx
jBT ekwkreP )()(ˆ)(ˆ
M
M
jx
jBT kw
NePePVar )(
1)()}(ˆ{ 22
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)()25.0sin(1)2.0sin(3)( 21 nvnx windowing:
Hanning
MN 128,512
rRectangula
128,512 MN
Bartlett
MN 128,512
Blackman
128,512 MN
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Performance comparisons
•We can summarized the performance of each technique in terms of two criteria.
(I) Variability (which is a normalized variance)
(II) Figure of merit
)}(ˆ{
)}(ˆ{2
j
x
jx
ePE
ePVar
•That is approximately the same for all of the nonparametric methods
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Summeryvariability Resolution Figure of merit
Periodogram
Bartlett
Welch
BlackmanTukey
1
k
1
k
1
8
9
N
M
3
2
N
289.0
Nk
289.0
L
228.1
M
264.0
N
289.0
N
289.0
N
272.0
N
243.0
***50% overlap and the Bartlett window***