Graz University of Technology – SPSC Laboratory Lesson 3: Fading Memory Nonlinearities Nonlinear Signal Processing – SS 2017 Christian Knoll Signal Processing and Speech Communication Laboratory Graz University of Technology May 19, 2017 NLSP SS 2017 May 19, 2017 Slide 1/17
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Graz University of Technology – SPSC Laboratory
Lesson 3: Fading Memory Nonlinearities
Nonlinear Signal Processing – SS 2017
Christian KnollSignal Processing and Speech Communication Laboratory
Graz University of Technology
May 19, 2017
NLSP SS 2017 May 19, 2017 Slide 1/17
Graz University of Technology – SPSC Laboratory
Session contents
◮ Today:◮ Volterra Series as representation of fading-memory NL◮ System identification using different inputs◮ Time series modelling (Homework)
◮ Next time:◮ Higher-order statistics and spectral analysis
NLSP SS 2017 May 19, 2017 Slide 2/17
Graz University of Technology – SPSC Laboratory
Volterra series – Definition
◮ A finite Volterra series of order p and memory length M:
◮ z [n] = x [n] + y [n], where x [n], y [n] jointly stationary andstatistically independent:
czr (·) = cxr (·) + cyr (·)Czr (·) = Cx
r (·) + Cyr (·)
NLSP SS 2017 May 19, 2017 Slide 11/17
Graz University of Technology – SPSC Laboratory
HOSA – Pros and Cons
Pros
◮ Analysis of nonlinearities
◮ Cumulants are additive for independent processes
◮ Gaussian noise: HOS zero (blind to Gaussian noise)
Cons
◮ Difficult to estimate from finite length data
◮ Influence of window
◮ Once you have them, how do you interpret them?
NLSP SS 2017 May 19, 2017 Slide 12/17
Graz University of Technology – SPSC Laboratory
HOSA – Example
◮ Example for a third-order cumulant
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
−0.02
−0.01
0
0.01
0.02
0.03
NLSP SS 2017 May 19, 2017 Slide 13/17
Graz University of Technology – SPSC Laboratory
HOSA – Example
◮ Example for a Bispectrum
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
NLSP SS 2017 May 19, 2017 Slide 13/17
Graz University of Technology – SPSC Laboratory
HOSA – Example
◮ Example for a PSD
0 0.5 1 1.5 2 2.5 3−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
ω, [rad/sample]
PS
D e
stim
ate
C2x(ω)
C2y(ω)
NLSP SS 2017 May 19, 2017 Slide 13/17
Graz University of Technology – SPSC Laboratory
HOSA – Matlab (1)
◮ HOSA toolbox, free, included in download-file
◮ HOSA toolbox manual is a great ressource (Matlab-central)!
◮ You will need:◮ cumest.m used to estimate cumulants◮ rpiid.m used to help generating input processes◮ gabrrao.m used to calculate window for 2D-FFT◮ viscumul3.m and◮ visbispec3.m for visualization
NLSP SS 2017 May 19, 2017 Slide 14/17
Graz University of Technology – SPSC Laboratory
HOSA – Matlab (2)
◮ cumest.m for third order cumulant calculates just one slice ofthe 2D correlation function
for k = -MaxLag : MaxLag
c3(:, k+MaxLag+1) = cumest(#, #, #, #, #, #, k);
end
◮ Bispectrum calculation: use fftshift(fft2( c3 .* w ))
to have a familiar picture
◮ Window w[n] obtained from gabrrao.m, optimal smoothingwindow, minimum bias in estimation
NLSP SS 2017 May 19, 2017 Slide 15/17
Graz University of Technology – SPSC Laboratory
Higher order statistics and spectral analysis - Problems (1)
◮ Limited amount of data leads to higher variance of estimators
◮ Problem even for ACF → Grows exponentially with order
◮ This makes visual interpretation much harder:◮ When is a Bispectrum zero?◮ When is a Bicoherence function flat?
NLSP SS 2017 May 19, 2017 Slide 16/17
Graz University of Technology – SPSC Laboratory
Higher order statistics and spectral analysis - Problems (2)
◮ Linear system w. K = 10, driven by white noise (500 samples)
−50 −40 −30 −20 −10 0 10 20 30 40 50−0.2
0
0.2
0.4
0.6
0.8
1
1.2
k
R
xx[k]
NLSP SS 2017 May 19, 2017 Slide 17/17
Graz University of Technology – SPSC Laboratory
Higher order statistics and spectral analysis - Problems (2)
◮ Linear system w. K = 10, driven by white noise (500 samples)
−50 −40 −30 −20 −10 0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
k
R
xx[k]
Ryy,lin
[k]
NLSP SS 2017 May 19, 2017 Slide 17/17
Graz University of Technology – SPSC Laboratory
Higher order statistics and spectral analysis - Problems (2)
◮ Linear system w. K = 10, driven by white noise (500 samples)
−50 −40 −30 −20 −10 0 10 20 30 40 50−0.2
0
0.2
0.4
0.6
0.8
1
1.2
k
R
xx[k]
NLSP SS 2017 May 19, 2017 Slide 17/17
Graz University of Technology – SPSC Laboratory
Higher order statistics and spectral analysis - Problems (2)
◮ Linear system w. K = 10, driven by white noise (500 samples)