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225 nips-2011-Probabilistic amplitude and frequency demodulation

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Author: Richard Turner, Maneesh Sahani

Abstract: A number of recent scientiﬁc and engineering problems require signals to be decomposed into a product of a slowly varying positive envelope and a quickly varying carrier whose instantaneous frequency also varies slowly over time. Although signal processing provides algorithms for so-called amplitude- and frequencydemodulation (AFD), there are well known problems with all of the existing methods. Motivated by the fact that AFD is ill-posed, we approach the problem using probabilistic inference. The new approach, called probabilistic amplitude and frequency demodulation (PAFD), models instantaneous frequency using an auto-regressive generalization of the von Mises distribution, and the envelopes using Gaussian auto-regressive dynamics with a positivity constraint. A novel form of expectation propagation is used for inference. We demonstrate that although PAFD is computationally demanding, it outperforms previous approaches on synthetic and real signals in clean, noisy and missing data settings. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Probabilistic amplitude and frequency demodulation Richard E. [sent-1, score-0.665]

2 uk Abstract A number of recent scientiﬁc and engineering problems require signals to be decomposed into a product of a slowly varying positive envelope and a quickly varying carrier whose instantaneous frequency also varies slowly over time. [sent-7, score-0.52]

3 Although signal processing provides algorithms for so-called amplitude- and frequencydemodulation (AFD), there are well known problems with all of the existing methods. [sent-8, score-0.154]

4 The new approach, called probabilistic amplitude and frequency demodulation (PAFD), models instantaneous frequency using an auto-regressive generalization of the von Mises distribution, and the envelopes using Gaussian auto-regressive dynamics with a positivity constraint. [sent-10, score-1.175]

5 We demonstrate that although PAFD is computationally demanding, it outperforms previous approaches on synthetic and real signals in clean, noisy and missing data settings. [sent-12, score-0.19]

6 1 Introduction Amplitude and frequency demodulation (AFD) is the process by which a signal (yt ) is decomposed into the product of a slowly varying envelope or amplitude component (at ) and a quickly varying sinusoidal carrier (cos(φt )), that is yt = at cos(φt ). [sent-13, score-1.147]

7 In its general form this is an ill-posed problem [1], and so algorithms must impose implicit or explicit assumptions about the form of carrier and envelope to realise a solution. [sent-14, score-0.119]

8 In this paper we make the standard assumption that the amplitude variables are slowly varying positive variables, and the derivatives of the carrier phase, ωt = φt − φt−1 called the instantaneous frequencies (IFs), are also slowly varying variables. [sent-15, score-0.607]

9 It has been argued that the subbands of speech are well characterised by such a representation [2, 3] and so AFD has found a range of applications in audio processing including audio coding [4, 2], speech enhancement [5] and source separation [6], and it is used in hearing devices [5]. [sent-16, score-0.31]

10 AFD is also of importance in neural signal processing applications. [sent-18, score-0.154]

11 Within each such band, both the amplitude of the oscillation and the precise center frequencies may vary with time; and both of these phenomena may reveal important elements of the mechanism by which the ﬁeld oscillation arises. [sent-20, score-0.408]

12 Because of these problems, the Hilbert method, which recovers an amplitude from the magnitude of the analytic signal, is still considered to be the benchmark despite a number of limitations [11, 12]. [sent-23, score-0.324]

13 In this paper, we show examples of demodulation of synthetic, audio, and hippocampal theta rhythm signals using various AFD techniques that highlights some of the anomalies associated with existing methods. [sent-24, score-0.33]

14 The limitations of these methods suggest an improved model (section 2) which we demonstrate on a range of synthetic and natural signals (sections 4 and 5). [sent-27, score-0.13]

15 1 Simple models for probabilistic amplitude and frequency demodulation In this paper, we view demodulation as an estimation problem in which a signal is ﬁt with a sinusoid of time-varying amplitude and phase, yt = ℜ (at exp (iφt )) + ǫt . [sent-29, score-1.542]

16 We are interested in the situation where the IF of the sinusoid varies slowly around a mean value ω . [sent-31, score-0.115]

17 In this case, the phase can be expressed in terms of ¯ the integrated mean frequency and a small perturbation, φt = ω t + θt . [sent-32, score-0.262]

18 ¯ Clearly, the problem of inferring at and θt from yt is ill-posed, and results will depend on the speciﬁcation of prior distributions over the amplitude and phase perturbation variables. [sent-33, score-0.589]

19 A simpler alternative is to generate the sinusoidal signal from a rotating two-dimensional phasor. [sent-35, score-0.184]

20 For example, re-parametrizing the likelihood in terms of the components x1,t = at cos(θt ) and x2,t = at sin(θt ), yields a linear likelihood function yt = at (cos(¯ t) cos(θt ) − sin(¯ t) sin(θt )) + ǫt = cos(¯ t)x1,t − sin(¯ t)x2,t + ǫt = wT xt + ǫt . [sent-36, score-0.183]

21 To complete the model, prior distributions can ω ω t be now be speciﬁed over xt . [sent-38, score-0.076]

22 (2) When the dynamical parameter tends to unity (λ → 1) and the dynamical noise variance to zero 2 (σx → 0), the dynamics become very slow, and this slowness is inherited by the phase perturbations and amplitudes. [sent-40, score-0.252]

23 This model is an instance of the Bayesian Spectrum Estimation (BSE) model [13] (when λ = 1), but re-interpreted in terms of amplitude- and frequency-modulated sinusoids, rather than ﬁxed frequency basis functions. [sent-41, score-0.128]

24 Now the full complex representation of the sinusoid is retained. [sent-45, score-0.06]

25 As before, the real part corresponds to the observed data, but the imaginary part is now treated explicitly as missing data, yt = ℜ (x1,t cos(¯ t) − x2,t sin(¯ t) + ix1,t sin(¯ t) + ix2,t cos(¯ t)) + ǫt . [sent-46, score-0.136]

26 ω ω ω ω (3) The new form of the likelihood function can be expressed in vector form, yt = [1, 0]zt + ǫt , using a new set of variables, zt , which are rotated versions of the original variables, zt = R(¯ t)xt where ω cos(θ) − sin(θ) . [sent-47, score-0.235]

27 (4) R(θ) = sin(θ) cos(θ) An auto-regressive expression for the new variables, zt , can now be found using the fact that rotation matrices commute, R(θ1 + θ2 ) = R(θ1 )R(θ2 ) = R(θ2 )R(θ1 ), together with expression for the dynamics of the original variables, xt (eq. [sent-48, score-0.125]

28 2), zt = λR(¯ )R(¯ (t − 1))xt−1 + R(¯ t)ǫt = λR(¯ )zt−1 + ǫ′ ω ω ω ω (5) t 2 where the noise is a zero mean Gaussian with covariance ǫ′ ǫ′T = R(¯ t) ǫt ǫT RT (¯ t) = σx I. [sent-49, score-0.096]

29 2 Problems with simple models for probabilistic amplitude and frequency demodulation BSE-PPV is used to demodulate synthetic and natural signals in Figs. [sent-53, score-0.845]

30 Perhaps most unsatisfactory is the fact that the IF estimates are often ill behaved, to the extent that they go negative, especially in regions where the amplitude of the signal is low. [sent-57, score-0.536]

31 It is easy to understand why this occurs by considering the prior distribution over amplitude and phase implied by our choice of prior distribution over xt (or equivalently over zt ), 1 λ at exp − 2 a2 + λ2 a2 ¯ p(at , φt |at−1 , φt−1 ) = t−1 + 2 at at−1 cos(φt − φt−1 − ω ) . [sent-58, score-0.632]

32 (6) 2 2πσx 2σx t σx Phase and amplitude are dependent in the implied distribution, which is conditionally a uniform distribution over phase when the amplitude is zero and a strongly peaked von Mises distribution [15] when the amplitude is large. [sent-59, score-1.196]

33 In some applications this may be desirable, but for signals like sounds it presents a problem. [sent-61, score-0.125]

34 Second, the same noiseless signal at different intensities will yield different estimated IF content. [sent-63, score-0.178]

35 When the phase-perturbations vary slowly (λ → 1), there is no correlation between successive IFs ( ωt ωt−1 − ωt ωt−1 → 0). [sent-66, score-0.055]

36 35 y a a ˆ y ˆ frequency /Hz Hilbert In the next section we will propose a new model for PAFD which addresses these problems, retaining the same likelihood function, but modifying the prior to include independent distributions over the phase and amplitude variables. [sent-75, score-0.654]

37 35 time /s time /s Figure 1: Comparison of AFD methods on a sinusoidally amplitude- and frequency-modulated sinusoid in broad-band noise. [sent-111, score-0.06]

38 The gray areas show the region where the true amplitude falls below the noise ﬂoor (a < σy ) and the estimates become less accurate. [sent-113, score-0.436]

39 2 PAFD using Auto-regressive and generalized von Mises distributions We have argued that the amplitude and phase variables in a model for PAFD should be independently parametrized, but that this introduces difﬁculties as the likelihood is highly non-linear in these variables. [sent-115, score-0.632]

40 25 3000 2000 1000 frequency /Hz 0 2500 2000 1500 1000 time /s Figure 2: AFD of a starling song. [sent-132, score-0.165]

41 The light gray bar indicates the problematic low amplitude region. [sent-134, score-0.37]

42 Bottom panels: IF estimates superposed onto the spectrum of the signal. [sent-135, score-0.081]

43 An important initial consideration is whether to use a representation for phase which is wrapped, θ ∈ (−π, π], or unwrapped, θ ∈ R. [sent-137, score-0.134]

44 It is therefore necessary to work with wrapped phases and a sensible starting point for a prior is thus the von Mises distribution, 1 p(θ|k, µ) = exp(k cos(θ − µ)) = vonMises(θ; k, µ). [sent-139, score-0.22]

45 Crucially for our purposes, the von Mises distribution can be obtained by taking a bivariate isotropic Gaussian with an arbitrary mean, and conditioning onto the unit-circle (this connects with BSE-PPV, see eq. [sent-142, score-0.204]

46 The Generalized von Mises distribution is formed in an identical way when the bivariate Gaussian is anisotropic [16]. [sent-144, score-0.19]

47 These constructions suggest a simple extension to time-series data by conditioning a temporal bivariate Gaussian time-series onto the unit circle at all sample times. [sent-145, score-0.188]

48 One of the attractive features of this prior is that when it is combined with the likelihood (eq. [sent-149, score-0.068]

49 1) the resulting posterior distribution over phase variables is a temporal version of the Generalized von Mises distribution. [sent-150, score-0.274]

50 That is, it can be expressed as a bivariate anisotropic Gaussian, which is constrained to the unit circle. [sent-151, score-0.122]

51 Having established a candidate prior over phases, we turn to the amplitude variables. [sent-153, score-0.359]

52 With one eye upon the fact that the prior over phases can be interpreted as product of a Gaussian and a constraint, we employ a prior of a similar form for the amplitude variables; a truncated Gaussian AR(τ ) process, τ T p(a1:T |λ1:τ , σ 2 ) ∝ λt′ at−t′ , σ 2 1(at ≥ 0) Norm at ; t=1 . [sent-154, score-0.432]

53 Moreover, when the phase variables are drawn from a uniform distribution (k1 = k2 = 0) it reduces to the convex amplitude demodulation model [17], which itself is a form of probabilistic amplitude demodulation [18, 19, 20]. [sent-157, score-1.288]

54 The AR prior over phases has also been used in a regression setting [21]. [sent-158, score-0.073]

55 3 Inference via expectation propagation The PAFD model introduced in the last section contains three separate types of non-linearity: the multiplicative interaction in the likelihood, the unit circle constraint, and the positivity constraint. [sent-159, score-0.109]

56 In this new form the constraints have been incorporated with the non-linear likelihood into the potential ψt , leaving a standard Gaussian dynamical potential πt (st , st−1 ). [sent-168, score-0.065]

57 Using EP we approximate the posterior distribution using a product of forward, backward and constrained-likelihood messages [24], T T ˜ αt (st )βt (st )ψt (a1,t , x1,t , x2,t ) = q(s1:T ) = qt (st ). [sent-169, score-0.08]

58 (11) t=1 t=1 The messages should be interpreted as follows: αt (st ) is the effect of πt (st−1 , st ) and q(st−1 ) on the belief q(st ), whilst βt (st ) is the effect of πt+1 (st , st+1 ) and q(st+1 ) on the belief q(st ). [sent-170, score-0.251]

59 The updates for the messages can be found by removing the messages from q(s1:T ) that correspond to the effect of a particular potential. [sent-173, score-0.16]

60 The deleted messages are then updated by moment matching the two distributions. [sent-175, score-0.08]

61 The updates for the forward and backward messages are a straightforward application of EP and result in updates that are nearly identical to those used for Kalman smoothing. [sent-176, score-0.08]

62 First, we integrate over the amplitude variable, which involves computing the moments of a truncated Gaussian and 5 is therefore computationally efﬁcient. [sent-179, score-0.324]

63 Second, we numerically integrate over the one dimensional phase variable. [sent-180, score-0.134]

64 The computational complexity of PAFD is O T (N + τ 3 ) where N are the number of points used to compute the integral over the phase variable. [sent-184, score-0.134]

65 For the experiments we used a second order process over the amplitude variables (τ = 2) and N = 1000 integration points. [sent-185, score-0.354]

66 In this case, the 16-32 forward-backward passes required for convergence took one minute on a modern laptop computer for signals of length T = 1000. [sent-186, score-0.08]

67 4 Application to synthetic signals One of the main challenges posed by the evaluation of AFD algorithms is that the ground truth for real-world signals is unknown. [sent-187, score-0.21]

68 In particular, we consider amplitude- and frequency-modulated 1 ¯ sinusoids, yt = at cos(θt ) where at = 1 + sin(2πfa t) and 2π dθ = f + ∆f sin(2πff t), which have dt ¯ been corrupted by Gaussian noise. [sent-190, score-0.076]

69 1 compares AFD of one such signal (f = 50Hz, fa = 8Hz, ff = 5Hz and ∆f = 25Hz) by the Hilbert, BSE-PPV and PAFD methods. [sent-192, score-0.208]

70 3 summarizes the results at different noise levels in terms of the signal to noise ratio (SNR) of the estimated variables 2 T T and the reconstructed signal, i. [sent-194, score-0.274]

71 4 demonstrates that PAFD can be used to accurately reconstruct missing sections of this signal, outperforming BSE-PPV. [sent-199, score-0.084]

72 100 SNR ˆ /dB a 80 SNR ω /dB ˆ Hilbert BSE−PPV PAFD 100 60 40 80 SNR y /dB ˆ 120 50 0 20 0 10 20 30 40 SNR signal /dB ∞ −50 10 20 30 40 SNR signal /dB ∞ 60 40 20 0 10 20 30 40 50 SNR signal /dB Figure 3: Noisy synthetic data. [sent-200, score-0.512]

73 5 Application to real world signals Having validated PAFD on simple synthetic examples, we now consider real-world signals. [sent-205, score-0.13]

74 Birdsong is used as a prototypical signal as it has strong frequency-modulation content. [sent-206, score-0.154]

75 2 shows that PAFD can track the underlying frequency modulation even though there is noise in the signal which causes the other methods to fail. [sent-209, score-0.376]

76 In the ﬁrst, spectrally matched noise is added to the signal and the IFs and amplitudes are reestimated and compared to those derived from the clean signal. [sent-211, score-0.225]

77 In the second test, regions of the signal are removed and the model’s predictions for the missing regions are compared to the estimates derived from the clean signal (see ﬁg. [sent-214, score-0.489]

78 The EEG signal typically contains broadband noise and so a conventional analysis applies a band-pass ﬁlter before using the Hilbert method to estimate the IF. [sent-218, score-0.211]

79 Although this improves the estimates markedly, the noise component cannot be completely eradicated which leads to artifacts in the IF estimates (see Fig. [sent-219, score-0.099]

80 TOP: SNR of estimated variables as a function of gap duration in the input signal. [sent-244, score-0.074]

81 Solid markers indicate the examples shown in the bottom rows of the ﬁgure. [sent-246, score-0.073]

82 Light gray regions indicate missing sections of the signal. [sent-248, score-0.155]

83 40 30 SNR ω /dB ˆ SNR ˆ /dB a 20 Hilbert BSE−PPV PAFD 35 25 20 15 15 10 5 0 −5 −10 10 10 15 20 25 30 35 10 SNR signal /dB 15 20 25 30 35 SNR signal /dB Figure 5: Noisy bird song experiments. [sent-249, score-0.308]

84 SNR of estimated variables as compared to those estimated from the clean signal, as a function of the SNR of the input signal. [sent-250, score-0.116]

85 6 Conclusion Amplitude and frequency demodulation is a difﬁcult, ill-posed estimation problem. [sent-254, score-0.341]

86 We have developed a new inferential solution called probabilistic amplitude and frequency demodulation which employs a von Mises time-series prior over phase, constructed by conditioning a bivariate Gaussian auto-regressive distribution onto the unit circle. [sent-255, score-0.976]

87 The construction naturally leads to an expectation propagation inference scheme which approximates the hard constraints using soft local Gaussians. [sent-256, score-0.055]

88 5 0 10 5 envelopes 2 frequency /Hz frequency /Hz frequency /Hz frequency /Hz envelopes Figure 6: Missing natural data experiments. [sent-293, score-0.84]

89 TOP: SNR of estimated variables as a function of gap duration in the input signal. [sent-294, score-0.074]

90 Solid markers indicate the examples shown in the bottom rows of the ﬁgure. [sent-296, score-0.073]

91 Light gray regions indicate missing sections of the signal. [sent-298, score-0.155]

92 The left hand side shows estimates derived from the raw EEG signal, whilst the right shows estimates derived from a band-pass ﬁltered version. [sent-302, score-0.093]

93 The gray areas show the region where the true amplitude falls below the noise ﬂoor (a < σy ), where conventional methods fail. [sent-303, score-0.427]

94 We have demonstrated the utility of the new method on synthetic and natural signals, where it outperformed conventional approaches. [sent-304, score-0.074]

95 Coherent modulation spectral ﬁltering for single-channel music source separation. [sent-341, score-0.061]

96 On the upper cutoff frequency of the auditory critical-band envelope detectors in the context of speech perception. [sent-357, score-0.291]

97 On the dichotomy in auditory perception between temporal envelope and ﬁne structure cues (L). [sent-369, score-0.119]

98 On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for deﬁning amplitude and frequency. [sent-373, score-0.324]

99 Statistical inference for single- and multi-band probabilistic amplitude demodulation. [sent-420, score-0.396]

100 Expectation propagation for approximate inference in dynamic bayesian networks. [sent-444, score-0.055]

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