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) 2k k E xk2 (n) Mk [n] Re Mk2 [n] Ck e j 2k 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 2k 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. 7 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 8 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. 10 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. 14 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) 17 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.