SIRP%20MODELING

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SIRP MODELING of IFO DATA
Research-Team Prof. Innocenzo Pinto Prof.
Maurizio Longo Prof. Stefano Marano Prof.
Vincenzo Matta Dr. Roberto Conte
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The WavesGroup Preliminary and Confidential,
March 2006
Preamble...

• In what follows, we present a possible analytical
  model accounting for
• the nongaussian/nonstationary features of real
  IFO data.
• The elected model is referred to as SIRP
  (Spherical Invariant Random Process)
• We show how to
• Identify the SIRP-nature of a given data-stream
• Capitalize on this model to simulate the noise
• Easily implement an optimal Neyman-Pearson
  detector
• Partial evidences of the effectiveness of the
  SIRP model
• for real IFO noise have been obtained by our
  workgroup for TAMA data,
• and are obviously not included in this
  presentation.

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The WavesGroup Preliminary and Confidential,
March 2006

• Outline
• Data Conditioning
• Conditioned Data Testing/Characterization
• SIRP Modeling
• Noise Simulation
• Detection in SIRP noise

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The WavesGroup Preliminary and Confidential,
March 2006
Summary of data conditioning
a) Preparation
-Identify frequency band (window) W of
interest -Compute smoothed estimate of PSD in W
from data -Apply band-pass filter (FIR) to
extract spectral band of interest -Equalize
(whiten) filtered data within band of interest
b) Narrowband features identification (spectral
domain)
-Assume ?? distribution in each frequency
bin. -Compute threshold for the presence of tones
in each bin Kay test, S.Kay, Modern Spectral
Estimation, Prentice-Hall, 1984, ch. 2 -Tag all
bins where level exceeds the above threshold
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The WavesGroup Preliminary and Confidential,
March 2006
Summary of data conditioning (cont.d)
c) Remove narrowband feaures (time domain)
-Use sliding time-window (215 time-bins) to
estimate toneparameters (frequency, phase,
amplitude) in each tagged bin. -Subtract
estimated tone from next-neighboring 210 time-bin
chunk
d) Remove non-narrowband features (frequency
domain)
-Remove by brute force using stop-band (FIR)
filter.(Further physical insights into their
nature is needed for doing better.)
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The WavesGroup Preliminary and Confidential,
March 2006
Summary of conditioned datatesting/characterizati
on
a) Preliminary test for (non)-Gaussianity
-Compare CDF of all data before and after data
conditioning (earlier described) with
theoretical colored Gaussian noise obtained by
N(0,1) noise pass-band filtered in W.
b) Decorrelate data by subsampling
-Estimate correlation length of conditioned
data from 2nd zero of autocorrelation
function. -Subsample data according to the above.
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The WavesGroup Preliminary and Confidential,
March 2006
Summary of conditioned datatesting/characterizati
on (cont.d)
c) Split data into consecutive chunks of
width between 210 to 215 time bins.
-Estimate of local variance in each
chunk -Compute of Kolmogorov distance between
actual data and Gaussian with locally
computed variance, in each chunk. -Perform KS
test of Gaussianity in each chunk. -Report
fraction of chunks passing test vs. chunk
length. -Check for outliers in chunks which do
not pass KS test.
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The WavesGroup Preliminary and Confidential,
March 2006
Breath of the Gaussian r.v.
Exogenous model
x(k)s(k) w(k) w(k) Complex White Gaussian
Process s(k) stationary sequence, which
decorrelates on a time scale much longer than
y(k)
Admissible solutions - Weibull Distributions -
K-Distributions - Rayleigh (obviously!!!!)
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The WavesGroup Preliminary and Confidential,
March 2006
Exogenous model the SIRP
Exogenous model does not specify the higher order
statistics
SIRP MODEL
Spherically Invariant Random Process is a
degenerate exogenous process in the special
case of constant s(k). The SIRP process is a
Gaussian Process modulated by a random variable
s, rather than a random process s(k).
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The WavesGroup Preliminary and Confidential,
March 2006
SIRP main features
Every N-Dimensional Vector from a SIRP is a SIRV
(Spherically Invariant Random Vector).
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The WavesGroup Preliminary and Confidential,
March 2006
Some useful SIRP properties
Theorem 1 Invariance under linear
transformations
We can obtain colored SIRV starting from white
SIRV, i.e. SIRV with identity covariance matrix,
by linear transformation (filtering).
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The WavesGroup Preliminary and Confidential,
March 2006
Some useful SIRP properties (cont.d)
Theorem 2 Goldman characterization theorem
X X1,,XN is a white SIRV with characteristic
pdf fs(s) and zero mean If and only if
expressing X in generalized spherical coordinates
(3)
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The WavesGroup Preliminary and Confidential,
March 2006
Goldman theorem at work
Flavours of spherical invariance property in R3.
Left panel theta-distribution right panel
phi-distribution.
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The WavesGroup Preliminary and Confidential,
March 2006
SIRP Identification Modeling

1. Check spherically-invariant features

-Use Goldman characterization theorem J.
Goldman, IEEE Trans. IT-22, 52-59, 1976. Check
in N2 and N3 dimensions.
b) Identify SIRP model
-Model radial distribution. -Closed form
solution for R K(v,ß) -Extract Rangaswamys
function hN N3 and hence any N, in view of
known recursion formulas). M. Rangaswamy, IEEE
Trans. AES-29, 111-123, 1993 E. Conte and M.
Longo, IEE Proc. F-134, 191-197, 1987.
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The WavesGroup Preliminary and Confidential,
March 2006
K-distribution Fit
If R K(v,ß)
Integral equation
Admits closed form solution (assuming E(s2)1,
no loss of generality)
Thus, we can simulate the noise !!
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The WavesGroup Preliminary and Confidential,
March 2006
Detection in SIRP noise
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The WavesGroup Preliminary and Confidential,
March 2006
Detection in SIRP with K-Radial Distribution
Optimum Log-Likelihood ratio test
If noise has K-radial distribution and obeys the
SIRP model
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The WavesGroup Preliminary and Confidential,
March 2006
Optimum Detector Block Diagram
Distances q0 r and q1 r u are
computed and warped through the
Zero-Memory-Non-Linearity then the difference
between the outputs is computed and compared to
the detection threshold T. E. Conte ,M. Longo
et al., IEE Proc. F,Vol-138, No.2, pp 131-138,
1991.
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The WavesGroup Preliminary and Confidential,
March 2006
Conclusions
Perspectives
-Construct IFO-noise simulator.
-Implement optimum Neyman-Pearson Detector
in identified SIRP noise
Work in Progress
-Comparison between optimum NP detector in
identified SIRP noise and locally-tailored
standard matched filter (in terms of
ROCs) -Detection strategies for signals whose
time spread is wider than coherence time of
SIRP.