39
1 Spectrum Estimation Dr. Hassanpour Payam Masoumi Mariam Zabihi Advanced Digital Signal Processing Seminar Department of Electronic Engineering Noushirvani University
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1
Spectrum Estimation
Dr. HassanpourPayam MasoumiMariam ZabihiAdvanced Digital Signal Processing SeminarDepartment of Electronic EngineeringNoushirvani University
2
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
3
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
4
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
5
t k
dtetxfX ftj 2)()(
)(tx )( fX
dfefXtx ftj 2)()(
,
Fourier Transform:
ffT0
6
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
7
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
8
Nonparametric methods
•Peroidogram
•Modified periodogram
•Bartlett method
•Welch method
•Blackman-Tukey method
9
The periodogram
kN
nx nxknx
Nkr
1
0
)(*)(1
)(ˆ
k
jkx
jkper ekreP )(ˆ)(ˆ
Estimated autocorrelation:
Estimated power spectrum or periodogram:
10
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
11
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
12
The estimated autocorrelation sequence
White noise power spectrum
The periodogram of white noise cont.
13
Periodogram of sinusoid in noise
)()sin()( 0 nvnAnx
)(2
1)( 0
22 AeP vj
x
0
2v
2
2
1A
14
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
15
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.
16
)()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
17
x(n) 1sin(0.4 n ) v(n) Example:
128N 512N
18
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.
19
Example:
128N 512N
1 2x(n) 1sin(0.4 n ) 1sin(0.45 n ) v(n)
20
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
21
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
22
1 2x(n) 0.1sin(0.2 n ) 1sin(0.3 n ) v(n) Example:
N=128Rectangular Window
N=128Hamming Window
23
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
24
Bartlett’s method (periodogram averaging)
kienxL
ePL
n
jni
jiper ...,,2,1;)(
1)(ˆ
21
0
....
PointsL PointsL PointsL
)(1 nx )(2 nx )(nxk
25
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
26
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
27
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
28
Welch’s method (M-periodogram averaging)
....
1,.....,1,0;)()( LniDnxnxi Overlap = L-D
29
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
304
512
k
N
hamming,%50
128,512
overlap
LN
)()25.0sin(3)2.0sin(1)( 21 nvnx Example:
31
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
32
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:
33
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
34
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.
35
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
36
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
37
)()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
38
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
39
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***
