Transcript Complex Modulation via Complementary Correlation

Complex Modulation via Complementary Correlation:
A New Feature for Natural Data Modeling and Analysis
Les Atlas and Pascal Clark, Electrical Engineering, UW
SONAR and theory collaboration with Ivars Kirsteins, NUWC Newport
•
Estimation of complementary modulation in actual data
– SONAR
– Speech
•
Maximum-likelihood demodulation: an information-theoretic approach
– Impropriety and estimation of complex modulators
– Results for synthesized and known models
•
Discussion: Directions for Future Work
•
Funded by AFOSR (through 2011) and ONR (for SONAR)
1
Demodulated Analysis in Noise:
Decades’ Old DEMON Analysis
Some
Detection
Operation
Short-Time
Transform
Mid-Time
Second
Transform
“Some Detection
Operation” is
conventionally
assumed to be
incoherent, e.g.
magnitudesquared or Hilbert
envelope.
Acoustic Frequency (kHz)
20
15
10
5
0
0
2
4
6
8
DEMON frequency (Hz)
10
Changing to Discrete Time:
Problem Statement and Results
•
General DEMON signal model of k multiple bands of products of
complex modulators and carriers
x[n]   mk [n]  ck [n]  w[n]
•
•
•
•
k
As proven in: [L. Atlas, Q. Li, and J. Thompson, Proc. ICASSP 2004] and
subsequent papers:
– For speech, music, and other audio sounds, most acoustic frequencies,
the modulation envelope mk [ n] is not necessarily real and positive, as
assumed for previous incoherent detection.
Coherent carrier estimation was previously needed to estimate the
modulator, yet such estimation has been found to be difficult or
impossible.
We used complementary processing to understand above process,
where x[n], ck [n], and w[n] are random and mk [n] is to be estimated.
Succeeded in using complementary processing based estimators to find
new information about desired new sonar modulators mk [n]
.
–
Pre-processing was needed to exploit DEMON signals’ periodically-correlated properties.
(Clark et. al., ICASSP ’10; Clark et. al., ASA ’11; Kirsteins et. al., UAM ‘11.)
3
Why Complementary Processing?
•
Scalar Case: Given a zero-mean scalar Gaussian complex random
variable x  u  jv:
2
– The standard (Hermitian) variance is RxxH  E x  E x  x
where * is complex conjugation.
Difference is
   
very significant
C
2
H
– The new, complementary, variance is Rxx  E  x   E  x  x   Rxx with   1

– The complex correlation coefficient  is between x and x is a measure
of the degree of impropriety of x . Why?
• If x is “proper,” u and v are uncorrelated, and have identical variances, then
E  x  x  E  u  jv    u  jv   E u 2   E v 2   2 jE  u  v 


 E u 2   E v 2   2 j  0  0  0  0
C
• Thus, if x is proper, the complementary variance Rxx
vanishes.
 But, as we now find for sonar and speech signals, after multi-band and
PC-MLE processing, the complementary variance RxxC is significant or
verysignificant!
 Thus a better signal model can advantageously us our hypothesized
complementary part.
Justification for Complex Processing of Real Data
Physical analogy: Wind velocity data [Kuh and Mandic, ICASSP 2009]
North
(meters/sec)
j Imag{ z }
Real{ z }
East
(meters/sec)
Real-valued
audio
4
xk [n]
Hilbert
Transform
+
4
2
-2
Complex-valued
subband signal
0
-2
-2
0
2
Real{ zk[n] }
z k [ n]
“Improper”
2
0
-4
-4
j
“Proper”
Im( z )
x[n]
Bandpass
Filter
j Imag{
Im( zz)k[n] }
Our case:
E z 2  0
E z 2  0
Complementary variance:
4
-4
-4
-2
0
2
4
Real{ zk[n] }
5
Impropriety, As Manifest Within Real Signal Statistics
Spectral auto-correlation
(Conventional or
“Hermitian”)
Z k ( )
0
Spectral auto-convolution
(Complementary)
2k
k



E xk2 (n)  Mk [n]  Re Mk2 [n]  Ck  e j 2k n   2
2
Real-valued
(dashed lines)
Baseband
Hermitian subband envelope

E zk [n]
2
  M [n]
2

2
Sidebands
Complementary subband envelope
(nonzero only for improper subbands)
E zk2 [n]  M 2 [n]  Ck  e j 2k n
6
Multivariate Generalized Likelihood Ratio (GLR) for Impropriety *
•
“Diagonal” GLR: With subband basebanding and downsampling, assume the
subband is white and compute the GLR from diagonal covariance matrix estimates.
•
Periodic Covariance Estimation: Use DEMON blade rate T and treat each cycle as
an i.i.d. sample of a WSS vector process.
 J 1
2
R  diag  z  n  jT   ,
 j 0

 J 1 2

R  diag  z  n  jT 
 j 0

H
C
In effect we have a T-dimensional random
process averaged over J realizations.
Monte Carlo simulation:
0.4
Probability
Threshold for a
p-value of 0.05
H1(0.8)
H1(0.6)
0.3
H1(0.4)
0.2
H1(0.2)
H0
0.1
0
0
0.1
0.2
0.3
0.4
0.5
GLR (multivariate)
* Reference: Schreier and Scharf, SP Letters 2006.
0.6
0.7
0.8
0.9
Minimum impropriety
we can confidently
detect.
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SONAR: Impropriety Dependence on Time and Frequency
Frequency Sweep: Merchant 2, between 15 and 30 sec
0.9
Threshold for rejecting null hypothesis, p-value = 0.05
Proper
GLR
0.8
0.7
Impropriety
detected
0.6
Improper
0.5
300
400
500
600
700
800
900
1000

Subband center frequency (Hz)
Frequency Sweep: Merchant 2, between 35 and 50 sec
0.9
GLR
0.8
Improper
0.7
Impropriety
detected
0.6
0.5
300
Less improper
400
500
600
700
800
Subband center frequency (Hz)
900
1000

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Speech: Massive Impropriety Detected!
Generalized likelihood ratio test [Schreier and Scharf, 2006],
using three averaging bandwidths, 25, 12.5, and 6.25 Hz, and
Monte-Carlo null-rejection threshold:
Impropriety GLR
25 Hz
12.5 Hz
6.25 Hz
0.5
0
0
0.2
0.4
0.6
Time (sec)
0.8
1
Signal spectrogram
6000
Frequency (Hz)
More improper
1
Null rejection
threshold for the
weakest
estimator (pvalue = 0.05)
4000
2000
0
0
0.2
0.4
0.6
Time (sec)
0.8
1
Note:
Impropriety most
significant during
voiced speech.
Maximum-Likelihood Demodulation I
•
Hypothesis: Natural signals (such as SONAR cavitation noise and human
speech) show elaborate spectral and temporal modulations.
•
Possible underlying causes:
– Product model?
– Linear time-varying system?
•
As an example, we define a subband product model in the informationtheoretic sense:
zk [n]  mk [n]  ck [n]
Observed random process
(nonstationary)
Modulator
Latent random
process (stationary)
where each component is generally complex-valued.
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Maximum-Likelihood Demodulation II
•
For a Gaussian process, impropriety is necessary and sufficient for
the existence of a complex envelope (in max. likelihood sense).
•
Second-order statistics describe a complex Gaussian process:
R [n]  E  zk [n]  z [n]
H
k
*
k
RkC [n]  E zk [n]  zk [n]
Zeroth lag
assumes
whiteness
•
Complementary covariance RkC [n] is identically zero for a proper
process.
•
Hermitian covariance RkH [n] is necessarily real and non-negative. Hence
the proper MLE yields a real modulator.
•
RkH [n] is generally complex. Hence the improper MLE can yield a
complex modulator.
11
Phase Disambiguation with Improper MLE
Estimation of complex modulator phase (hence linear demodulation)
requires full knowledge of proper and improper statistics!
Log-likelihood for proper
ck [ n]
10
Total
phase
ambiguity
mk
0
-5
-5
-5
0
Re( m )
5
+/- phase
ambiguity
5
0
-10
-10
ck [ n]
10
5
Im( m )
(Infinite
solutions)
Log-likelihood for improper
10
-10
-10
mk
-5
0
Re( m )
Complex plane for a single modulator sample in time.
(Blue signifies highest likelihood[s].)
5
(Two
identically
optimal
solutions)
10
12
Synthetic Demodulation Demonstration
Harmonic, 180 Hz
(induces subband impropriety)
w[n]
3-Hz and 6-Hz modulators
in every subband
c[n]
X
White, real Gaussian
LTV
m[k,n
Acoustic freq. (kHz)
Conventional DEMON
Spectrum
8
6
6
4
4
2
2
Spurious modulator lines at 0, 9, 12 Hz.
Max.
Likelihood
demodulation
Linear ComplexModulation Spectrum
8
-10 -5
0
5
10
Modulation frequency (Hz)
x[n]
-10 -5
0
5
10
Modulation frequency (Hz)
0 dB
–20
Modulator lines only where they should be!
Directions for Future Work
•
It’s there, but how to physically interpret impropriety and high-frequency
nonstationarity?
•
Simplest approach: Estimate impropriety via double frequency terms from a
square-law. Can also be expanded it to widely linear/quadratic processing,
based upon recent work of Schreier and Scharf (several papers) and Fang
and Atlas, IEEE Trans SP, 1995.
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Backup Slides
15
Statistical Impropriety Tests for Periodically-Modulated Subbands
Generalized Likelihood Ratio (GLR) [Schreier and Scharf, SP Letters 2006]:
N 1
GLR(z)   1  ki2 
i 0
N-dimensional
random process
Canonical correlations
between z and z*
The GLR is invariant to linear transformations:
z  Mc
Observed
signal
carrier
(random)
modulator matrix
(diagonal)
GLR(z)  GLR(c)
Hence we can test the modulated signal z(n)
for impropriety in the carrier c(n).
16
Effect of SNR on the Impropriety GLR Statisic
Probability
SNR = Inf
H1(0.8)
0.4
H1(0.6)
H1(0.4)
0.2
0
Threshold for a
p-value of 0.05
0
0.1
0.2
0.3
H1(0.2)
0.4
0.5
GLR (multivariate)
0.6
H0
0.7
0.8
0.9
0.7
0.8
0.9
0.8
0.9
Probability
SNR = +3 dB
0.4
H1(0.8)
0.2
0
0
0.1
0.2
0.3
H1(0.6)
H1(0.4)
0.4
0.5
GLR (multivariate)
0.6
Probability
SNR = -3 dB
0.4
0.2
0
H1(0.8)
H1(0.6)
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0
0.1
0.2
0.3
0.4
0.5
GLR (multivariate)
0.6
0.7
Acoust. Freq.
PC-MLE
(synchronous, quadratic)
Interesting side-note: General
region correspond to improper
cross-subband correlations…
haven’t looked into this yet!
Mod. Freq.
Hermitian subband
DEMON spectra
(synchronous, quadratic)
Complementary (improper)
subband DEMON spectra
(synchronous, quadratic)
Optimization routines and system constraints
(modulation bandwidth, Viterbi sign-flips)
Acoust. Freq.
System Estimate
(synchronous, linear, underspread)
18
Mod. Freq.