Contents lists available at ScienceDirectSignal Processing

Signal Processing 106 (2015) 331341http://d0165-16

n CorrE-m

d.mandjournal homepage: www.elsevier.com/locate/sigproSynchrosqueezing-based time-frequency analysisof multivariate data

Alireza Ahrabian a, David Looney a, Ljubia Stankovi a,b, Danilo P. Mandic a,n

a Department of Electrical and Electronic Engineering, Imperial College London, United Kingdomb Department of Electrical Engineering, University of Montenegro, Montenegroa r t i c l e i n f o

Article history:Received 27 December 2013Received in revised form10 July 2014Accepted 5 August 2014Available online 13 August 2014

Keywords:Amplitude and frequency modulated signalSynchrosqueezing transformAnalytic signalMultivariate modulated oscillationsMultivariate signalTime-frequency analysisInstantaneous frequencyx.doi.org/10.1016/j.sigpro.2014.08.01084/& 2014 Elsevier B.V. All rights reserved.

esponding author.ail addresses: aa1106@ic.ac.uk (A. Ahrabian),ic@imperial.ac.uk (D.P. Mandic).a b s t r a c t

The modulated oscillation model provides physically meaningful representations of time-varying harmonic processes, and has been instrumental in the development of moderntime-frequency algorithms, such as the synchrosqueezing transform. We here extend thisconcept to multivariate signals, in order to identify oscillations common to multiple datachannels. This is achieved by introducing a multivariate extension of the synchrosqueez-ing transform, and using the concept of joint instantaneous frequency multivariate data.For rigor, an error bound which assesses the accuracy of the multivariate instantaneousfrequency estimate is also provided. Simulations on both synthetic and real world dataillustrate the advantages of the proposed algorithm.

& 2014 Elsevier B.V. All rights reserved.1. Introduction

Numerous observations in science and engineeringexhibit time-varying oscillatory behavior that is not possibleto characterize adequately by conventional Fourier analysis.This limitation was first addressed by the modulated oscilla-tion model [1], which characterizes time-varying signals asamplitude and frequency modulated oscillations, therebycapturing the changing oscillatory dynamics of the signal.The univariate modulated oscillation model has sincebecome a standard in analyzing time-varying signals, infields ranging from communication theory [2] to biome-dical engineering [3].

The notion of the univariate modulated oscillation hasbeen recently extended to both bivariate and trivariate time-varying signals [46]; in the bivariate case the modulatedoscillations are modeled as tracing an ellipse with the jointinstantaneous frequency capturing the combined frequenciesof the individual channels. This elliptic (ellipsoid) character-ization has found applications in oceanography [7], where theunderlying physical processes are well modeled as particlesin 2D and 3D spaces which trace elliptical trajectories. For anarbitrary number of channels, the modulated multivariateoscillation has been proposed in [8,9], whereby the under-lying model assumes one common oscillation that best fits allof the individual channel oscillations. For instance, the workin [9] identifies modulated multivariate oscillations using themultivariate extension of the wavelet ridge algorithm, a localoptimization technique that identifies local maxima withrespect to scale parameter in the wavelet coefficients, withthe objective of extracting the local oscillatory dynamics ofthe signal. It should be noted that interest in time-frequencyanalysis of multichannel data has also recently been growingwith multivariate data driven algorithms [10,11] that directlyexploit multichannel interdependencies.

A recent class of time-frequency techniques, referred to asreassignment methods [1214], aim to improve the read-ability (localization) of time-frequency representations [15].

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A. Ahrabian et al. / Signal Processing 106 (2015) 331341332The synchrosqueezing transform (SST) [16,17] belongs to thisclass, it is a post-processing technique based on the contin-uous wavelet transform that generates highly localized time-frequency representations of nonlinear and nonstationarysignals. Synchrosqueezing provides a solution that mitigatesthe limitations of linear projection based time-frequencyalgorithms, such as the short-time Fourier transform (STFT)and continuous wavelet transforms (CWT). The synchros-queezing transform reassigns the energies of these trans-forms, such that the resulting energies of coefficients areconcentrated around the instantaneous frequency curves ofthe modulated oscillations. As such, synchrosqueezing is analternative to the recently introduced empirical mode decom-position (EMD) algorithm [18]; it builds upon the EMD, bygenerating localized time-frequency representations while atthe same time providing a well understood theoretical basis.

However, despite all those efforts, multivariate time-frequency algorithms are still at their infancy. This is in starkcontrast with the developments in sensor technology whichhave made readily accessible multivariate data (3D inertialbody sensor and 3D anemometers). To this end, we develop amultivariate time-frequency algorithm based on SST thatgenerates a compact time-frequency representation of multi-channel signals, based on the principles developed in [8,9,19].

The organization of this paper is as follows: Section 2introduces the notion of modulated multivariate oscil-lations and joint instantaneous frequency. Section 3describes the SST, Section 4 presents the proposed multi-variate extension of the synchrosqueezing algorithm, andSection 5 verifies the algorithm through simulations.

2. Modulated multivariate oscillations

Signals containing single time varying amplitudes andfrequencies are readily described by the modulated oscil-lation model

xt at cos t 1where a(t) and t are respectively the instantaneousamplitude and phase, and are termed the canonical pair[2]. The application of the Hilbert transform to the originalsignal, yields the analytic signal x t in the formx t ate_t xt _Hfxtg 2whereHfg is the Hilbert transform operator, and _

ffiffiffiffiffiffiffiffi1

p.

The analytic signal x t is complex valued and admits aunique time-frequency representation for the signal x(t),based on the derivative of the instantaneous phase, t.

Recently, the concept of univariate modulated oscilla-tion has been extended to the multivariate case, in orderto model the joint oscillatory structure of a multichannelsignal, using the well understood concepts of joint instan-taneous frequency and bandwidth. Extending the repre-sentation in (2), for multichannel signal xt, we canconstruct a vector at each time instant t, to give a multi-variate analytic signal

x t

a1te_1ta2te_2t

aNte_N t

266664

377775 3where an(t) and nt represent the instantaneous ampli-tude and phase for each channel n 1;;N. The work in[9] proposed the joint instantaneous frequency (powerweighted average of the instantaneous frequencies of allthe channels) of multivariate data in the form:

x t I xH t

ddt

x t

Jx tJ2

Nn 1a

2nt0nt

Nn 1a2nt4

where the symbol I denotes the imaginary part of acomplex signal and 0nt is the instantaneous frequencyfor each channel.

Remark 1. It should be noted that for single channelmulticomponent signals, the weighted average instanta-neous frequency [20] has the same form as in (4) and isconsistent with the joint instantaneous frequency.

Both measures of instantaneous frequency overcome afundamental problem that arises when estimating instan-taneous frequency of multiple modulated oscillations, thatis, power imbalances between the components lead toinstantaneous frequency estimates that are outside thebounds of the individual instantaneous frequencies [21].

The joint instantaneous frequency captures the com-bined oscillatory dynamics of multivariate signals, whilethe joint instantaneous bandwidth xt captures thedeviations of the multivariate oscillations in each channelfrom the joint instantaneous frequency, and is given by

x t ddt

x t _x t x t

x t : 5

Therefore, the joint instantaneous bandwidth representsthe normalized error of the joint instantaneous frequencyestimate with respect to the rate of change of the multi-variate analytic signal x t. Inserting (3) into (5) results inthe expression for the squared instantaneous bandwidth:

2x t Nn 1a0nt2Nn 1a2nt

0ntxt2

Nn 1a2nt: 6

Remark 2. Observe that the instantaneous bandwidthdepends upon the rate of change of the instantaneousamplitudes for each channel, as well as the deviation ofthe instantaneous frequencies in each channel from thecombined joint instantaneous frequency. Large deviationsof the individual instantaneous frequencies from the jointinstantaneous frequency result in a large instantaneousbandwidth, implying that the multivariate signal wouldnot be well modeled as a multivariate modulatedoscillation.

It has been shown in [6] that the global moments of thejoint analytic spectrum can be expressed in terms of thejoint instantaneous frequency and bandwidth. The firstand second global moments are termed the joint meanfrequency and the joint global second central moments(multivariate bandwidth squared). As a result, given the

1 A detailed implementation of the synchrosqueezing transform canbe found in [22].

2 We have used linear frequency scales therefore is constant,however for logarithmic frequency scales, would vary with frequency[22].

A. Ahrabian et al. / Signal Processing 106 (2015) 331341 333joint analytic spectrum

S 1EJX J2; 7

where X is the Fourier transform of x t and E is thetotal energy of the Fourier coefficients given by

E 12

Z 10

JX J2 d; 8

this makes possible to express the joint global meanfrequency expressed as the first moment of the jointanalytic spectrum as

12

Z 10

S d: 9

The joint global second central moment (multivariatebandwidth squared) measures the spectral deviation ofthe joint analytic spectrum from the joint global meanfrequency, and is given by

2 12

Z 10

2S d: 10

Accordingly, the global moments of the analytic spectrumcan be related to the moments of joint instantaneousfrequency and bandwidth as

1E

Z 11

Jx t J2x t dt 11

2 1E

Z 11

Jx t J22x t dt; 12

where 2x t is the joint instantaneous second centralmoment, given by

2x t 2x txt2: 13

Remark 3. Observe that the multivariate bandwidthsquared, 2, depends on both the joint instantaneousbandwidth, 2x t, and the deviations of the joint instanta-neous frequency from the joint global mean frequency, .

3. Synchrosqueezing transform

The synchrosqueezing transform is a post-processingtechnique applied to the continuous wavelet transform inorder to generate localized time-frequency representa-tions of non-stationary signals. The continuous wavelettransform is a projection based algorithm that identifiesoscillatory components of interest through a series oftime-frequency filters known as wavelets. A wavelet tis a finite oscillatory function, which when convolved witha signal x(t), in the form

W a; b Z

a1=2tba

x t dt 14

gives the wavelet coefficients Wa; b, for each scale-timepair (a,b). In this way, the wavelet coefficients in (14) andcan be seen as the outputs of a set of scaled bandpassfilters. The scale factor a shifts the bandpass filters in thefrequency domain, and also changes the bandwidth ofthe bandpass filters. Therefore, the energy of the wavelettransform of a sinusoid at a frequency s will spread outaround the scale factor as =s, where is the centerfrequency of a wavelet, while the energy of the originalfrequency s is spread across as. Thus, the estimatedfrequency present in those scales is equal to the originalfrequency s. Consequently, the instantaneous frequencyxa; b can be estimated as

x a; b _Wa; b1Wa;b

b15

for each scale-time pair (a,b). The resulting wavelet coeffi-cients that contain the same instantaneous frequenciescan then be combined using a procedure referred to assynchrosqueezing (SST). For a set of wavelet coefficientsWa; b, the synchrosqueezing transform1 Tl; b is givenby

Tl; b ak : xak ;bl r=2

Wak; ba3=2 ak 16

where l are the frequency bins with a resolution2 of .The SST has been shown to reconstruct univariate modu-lated oscillations of the form in (1), as follows [16]

xb R R1 lTl;b

" #17

where R 12R10

n d

, is the normalization constantand is the Fourier transform of the mother wavelet t.

4. Multivariate time-frequency representation using theSST

In order to extend the SST to the analysis of multi-variate signals, recall that if the modulated oscillatorycomponents are known for each channel as in (3), thenwe can determine the joint instantaneous frequency,provided that the frequencies of the modulated oscilla-tions are sufficiently close together. With that insight, wepropose to first partition the time-frequency domain intoK frequency bands fkgk 1;;K . This makes it possible toidentify, a set of matched monocomponent signals from agiven multivariate signal. The instantaneous amplitudesand frequencies present within those frequency bands canthen be determined (yielding amplitude and frequencymodulated oscillations, similar to the intrinsic-mode func-tions (IMFs) of the EMD algorithm [18]).

4.1. Partitioning of the time-frequency domain

We propose to partition the time-frequency domain viaa multivariate extension of an adaptive frequency tilingtechnique first proposed in [19]; the underlying concept isto determine multivariate monocomponent signals basedon the multivariate bandwidth. The time-frequency planeis first partitioned into 2l equal-width frequency bands, forthe frequency range, l;m m=2l1; m1=2l1, where

0 1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16

3

2

1

0

Normalised Frequency

Leve

l

B1.0

B0.0

B1.1

B2.0 B2.1 B2.2 B2.3

B3.2B3.1B3.0 B3.3 B3.4 B3.5 B3.6 B3.7

Fig. 1. The partitioned frequency domain with the multivariate band-width given by Bl;m , where l corresponds to the level of the frequencyband and m is the frequency band index.

A. Ahrabian et al. / Signal Processing 106 (2015) 331341334l 0;; L, corresponds to the level of the frequency bands(L5 typically) and m 0;;2l1, is the index of thefrequency band.

The multivariate bandwidth Bl;m for a given frequencyband at level l and index m, is then calculated as shown inFig. 1. Within a given frequency band l;m, the multivariatebandwidth is split into two frequency subbands l1;2mand l1;2m1, as follows [19]: If the frequency band l;m contains a multivariatemonocomponent signal, then, Bl;m Bl1;2mBl1;2m1.3 A bound on the error of the estimated multivariate instantaneousfrequency is provided in the Appendix.If each frequency subband contains separate multi-variate monocomponents then, Bl;m4Bl1;2mBl1;2m1.

As a result, given a multivariate signal xt with N channelswith the SST coefficients for each channel given byTn;b, the multivariate bandwidth for a given frequencyband l;m [5,6], is obtained by first calculating the Fouriertransform of the inverse of the SST coefficients

l;m F R1 Al;m

Tn; b( )" #

n 1;;N18

where F fg is the Fourier transform operator, R thenormalization constant [16] and l;mARN a columnvector. The multivariate bandwidth is then determined viaEqs. (7)(10), as outlined in Section 2.

The rationale behind the adaptive frequency scales isthen as follows: if the initial multivariate bandwidth iscalculated for the entire signal at level l0, then thebandwidth is split based on the following condition:

Bl;m4Bl1;2ml1;2m1Bl1;2m1l1;2m1

l1;2m1l1;2m119

where

l1;2m T

b 1Amultil1;2mb2

l1;2m1 T

b 1Amultil1;2m1b2and Amultil1;2mb and Amultil1;2m1b correspond to the multi-variate instantaneous amplitudes for the respective fre-quency subbands, as defined by (23). The right hand sideof (19) factors the total energy of the frequency subbands,such that the subbands with negligible signal content arenot considered. The final set of adaptive frequency bands isgiven by fkgk 1;;K , where K is the number of oscillatoryscales and 14244K .Remark 4. For modulated oscillations separated in fre-quency, the proposed partitioning method provides arobust method for separating monocomponent signals.However, for closely spaced monocomponent functionsthat are separated in both time and frequency (i.e. twoparallel chirp signals), the method cannot resolve theseparate monocomponent signals.4.2. Multivariate time-frequency representation

For a multivariate signal xt with the corresponding SSTcoefficients for each channel Tn; b (the SST coefficientsTn;b have been normalized with the constant R ), and agiven a set of oscillatory scales fkgk 1;;K obtained using amultivariate extension of a method proposed in [19], theinstantaneous frequency nk b for each frequency band k isgiven by

nk b Ak jTn; bj2Ak jTn; bj2

20

and the instantaneous amplitude Aikb for each frequencyband as

Ank b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

AkjTn; bj2

r: 21

The following condition holds for the instantaneous frequen-cies calculated in each frequency band, nk b4 nk1b,that is, at each point in time the instantaneous frequenciesare well separated. The second step is to estimate themultivariate instantaneous frequency by combining, for agiven frequency band k, the instantaneous frequencies acrossthe N channels, using the joint instantaneous frequency in(4). As a result, the multivariate instantaneous frequencyband multik b is given by3

multik b Nn 1Ank b2

nk b

Nn 1Ank b222

while the instantaneous amplitude Amultik b for eachfrequency band becomes

Amultik b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN

n 1Ank b2

s: 23

Now that we have determined the joint instantaneousamplitude and frequency for each frequency band, it ispossible to generate the multivariate time-frequency

101Proposed Method

A. Ahrabian et al. / Signal Processing 106 (2015) 331341 335coefficients Tmultik ; b for each oscillatory scale k, as4

Tmultik ;b Amultik bmultik b 24where is the Dirac delta function and the multivariatetime-frequency coefficients for each oscillatory scale aregiven by Tmulti; b Tmultik ; bjk 1;;K . However, it shouldbe noted that phase information has been lost throughcalculating the instantaneous frequency, and so the originalmultivariate signal xt cannot be reconstructed. A summaryof the proposed method is shown in Algorithm 1.100

(B)

MPWDAlgorithm 1. Multivariate extension of the SST.1 Rat

io

1.

10

lizat

ion

Tmu

funGiven a multivariate signal xt with N channels, applythe SST channel-wise in order to obtain the coefficientsTn; b.2oca2.10 5 0 5 10 15 20103

10LDetermine a set of partitions along the frequency axis forthe time-frequency domain, and calculate the instanta-neous frequency nk b and amplitude Aikb for eachfrequency bin k, as shown in Eqs. (20) and (21)respectively.Input SNR (dB)3. Calculate the multivariate instantaneous frequencymultik b and amplitude Amultik b according to Eqs. (22)and (23)respectively.14.10 5 0 5 10 15 20103

102

101

100

10

Input SNR (dB)

Loca

lizat

ion

Rat

io (B

)

Proposed MethodMPWDDetermine the multivariate synchrosqueezed coeffi-cients Tmulti; b.

5. Simulations

The performance of the proposed multivariate exten-sion of the synchrosqueezed transform was evaluated onboth synthetic and real-world signals. The synthetic datawere of sinusoidal oscillations in varying levels of noise, aswell as frequency- and amplitude- modulated oscillationsin noise. The real-world simulations were conducted onvelocity data collected from a freely drifting oceanographicfloat (used by oceanographers to analyze ocean currents)and Doppler shift signatures of a robotic device collectedfrom two Doppler radar systems.10 5 0 5 10 15 20103

102

101

100

101

Input SNR (dB)

Loca

lizat

ion

Rat

io (B

)

Proposed MethodMPWD5.1. Sinusoidal oscillation in noise

The first set of simulations provides a quantitativeevaluation the proposed multivariate time-frequency algo-rithm based on the synchrosquezing transform against themultivariate pseudo Wigner distribution (MPWD) algo-rithm (as outlined in Appendix B). The quantitative per-formance index was a modification of a measure proposedin [24], given by

BR R

t;ARjTFRt;j dtdR Rt;=2RjTFRt;j dtd

25

where the symbol TFRt; denotes the time-frequencyrepresentation, and R is the instantaneous frequency path4 The multivariate extension of the synchrosqueezed transformlti; b follows a similar form to the ideal time-frequency distributionction ITFt; 2jAtj20t [23].of the desired signal. We first considered a bivariatesinusoidal oscillation in varying levels of white Gaussiannoise

yst cos 2ft

cos 2f t

" #

n1tn2t

" #Fig. 2. A comparison between the localization ratios B, for both theproposed method and the MPWD, evaluated for a bivariate oscillationwith the following joint instantaneous frequencies: (a) 10.5 Hz,(b) 40.5 Hz, and (c) 100.5 Hz. A window length of 1001 samples wasused for the MPWD.

A. Ahrabian et al. / Signal Processing 106 (2015) 331341336where f 10;40;100 Hz are the set of frequencies pre-sent, and n1t and n2t are independent white Gaussiannoise realizations and 1 Hz corresponds to a frequencydeviation between the channels. The resulting jointinstantaneous frequency between the channels is givenby f jt 10:5;40:5;100:5 Hz. The values of the localiza-tion ratio B are shown in Fig. 2. It can be seen that theSamples

Freq

uenc

y (H

z)

200 400 600 800 1000 1200 1400 1600 18000

5

10

15

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25

30

0

0.5

1

1.5

2

Samples

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uenc

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z)

200 400 600 800 1000 1200 1400 1600 18000

5

10

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0

0.5

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1.5

2

Samples

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uenc

y (H

z)

200 400 600 800 1000 1200 1400 1600 18000

5

10

15

20

25

30

0

0.5

1

1.5

2

Fig. 3. The time-frequency representations for both the proposed method (lefinput SNR of (a) 10 dB, (b) 5 dB and (c) 0 dB. The window length used for MPWproposed multivariate time-frequency algorithm had ahigh localization ratio, as compared to the MPWD, parti-cularly when the SNR of the input signal is relatively high.The localization ratio of the MPWD remained largelyunchanged for different sinusoids, while the localizationratio for the proposed method decreases as the frequencyof the sinusoid increases.Samples

Freq

uenc

y (H

z)

500 1000 1500 2000 25000

5

10

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t panels) and the MWPD (right panels) for a bivariate AM/FM signal, withD was 681 samples.

A. Ahrabian et al. / Signal Processing 106 (2015) 331341 3375.2. Amplitude and frequency modulated signal analysis

We next considered a multicomponent bivariate AM/FM signal yt corrupted by noise

yt s1ts2ts3ts4t

" #

n1tn2t

" #

where n1t and n2t are independent white Gaussiannoise realizations and the signal components s1t; s2t;s3t; s4t are given bys1t 10:5 cos 2t cos 2t20s2t 10:5 cos 2t cos 2t20s3t cos 210t3:5 cos ts4t cos 210t3:5 cos t:0 200 400 60.08

0.06

0.04

0.02

0

0.02

0.04

0.06

Samp

Vel

ocity

Samples

Nor

mal

ized

Fre

quen

cy

200 400 600 800 10000

0.05

0.1

0.15

0

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 4. Time-frequency analysis of real world float drift data. (a) The time dorepresentation of float data using the proposed multivariate extension of the SSTwas used for the MPWD.

Table 1Localization ratios, B, for both the proposed algorithm and the MPWD.

SNR (dB) Proposed method MPWD

10 0.208 0.0375 0.105 0.0250 0.04 0.014Therefore, the components s1t and s2t were AM signals,with an amplitude modulation index of 0.5. A frequencydeviation, 0:3 Hz, was introduced to the carrier of s2t(this is analogous to a frequency bias that may arisebetween sensors during data acquisition). The informationbearing components s3t and s4t of the bivariate signalyt were sinusoidally modulated FM signals, while s4talso had a frequency deviation of 0:3 Hz.

Fig. 3 shows the time-frequency representations usingboth the proposed method and the MPWD, in processingthe bivariate AM/FM multicomponent signal yt, over arange of input SNRs. Observe from Fig. 3(a) that for aninput SNR of 10 dB, that the proposed method localizesthe energy of the oscillations along the instantaneousfrequency frequencies that correspond to the componentsof yt. However as the noise power increased, the perfor-mance of the proposed method degraded as the jointinstantaneous frequency estimator is sensitive to noise.On the other hand, the MPWD is less localized at higherSNRs, while the performances of both methods convergefor lower SNRs. Table 1 shows the localization ratio B forboth techniques, illustrating that as the SNR decreases thelocalization ratio B for the proposed algorithm convergeswith that of the MPWD, implying that while localized the00 800 1000les

LattitudeLongitude

Samples

No

rma

lize

d F

req

ue

ncy

200 400 600 800 10000

0.05

0.1

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main waveforms of bivariate float velocity data. (b) The time-frequency(left panel) and the MPWD algorithm (right panel). A window length 501

0 500 1000 1500 20000.2

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Fig. 5. Time-frequency analysis of Doppler radar data. (a) The time domain waveforms of both the high gain and low gain Doppler radar data. (b) The time-frequency representation of Doppler radar data using the proposed multivariate extension of the SST (left panel) and the MPWD algorithm (right panel).A window length 1063 was used for the MPWD.

A. Ahrabian et al. / Signal Processing 106 (2015) 331341338proposed method is not accurately representing the com-ponents instantaneous frequencies.5.3. Float drift data

The real world data was collected from a freely driftingoceanographic float, used by oceanographers to studyocean current drifts.5 The latitude and the longitude ofthe float was recorded, and the resulting drift velocity inboth the latitude and longitude were processed as abivariate signal. The drift velocities along the latitudeand longitude (shown in Fig. 4(a)) contain a time-varyingoscillation that is common to both channels, howeverthese oscillations are not in phase. Also the noise in bothchannels had different characteristics. Fig. 4(b) illustratesthat the common oscillatory dynamics of the float driftdata that is frequency modulated is effectively localized5 The float drift data was obtained from the Jlab toolbox, and isavailable at http://www.jmlilly.net.using the proposed method, while the multivariate pseudoWigner distribution had poorer localization.5.4. Doppler speed estimation

The second real world data example consists of abivariate Doppler radar signal, collected from both a highgain and low gain Doppler radar system (the Doppler radaroperating frequency was f c 10:587 GHz). A Doppler shiftsignature was then collected from a robotic device movingat a constant speed towards both the high gain and lowgain Doppler radars [25]. The speed chosen for this workwas 0.065 m/s and the corresponding Doppler shift fre-quency,6 was f d 4:567 Hz. From Fig. 5(a), observe fromthe output of both Doppler radar systems, the amplitudeincreasing as the robotic device approaches the radar. Alsonote that the power from the output of the high gain radar6 The Doppler shift frequency, fd, is related to the speed of an objectby f d 2f cv=c, where v is the speed of the object and c is the speedof light.

http://www.jmlilly.net

A. Ahrabian et al. / Signal Processing 106 (2015) 331341 339is significantly higher than the output of the low gainradar. The multivariate time-frequency representationsusing both the proposed method and the MPWD is shownin Fig. 5(b). Observe that the proposed method localizesthe Doppler shift frequency more effectively, where itshould be noted that the speed of the robotic devicebetween the samples 400600, and the decelerationbetween the samples 16001800, can clearly be identified.Finally the localization ratio for the proposed multivariatetime-frequency7 method is 0.38 while for the MPWD 0.11,implying that the proposed method has a higher energyconcentration around the Doppler shift frequency, fd.

6. Conclusion

We have proposed a multivariate extension of the syn-chrosqueezing transform in order to identify oscillationscommon to the data channels within a multivariate signal.For each channel, the instantaneous frequencies of thesynchrosqueezed coefficients are determined for each oscilla-tory scale, and the resulting multivariate instantaneous fre-quency is then found by calculating the joint instantaneousfrequency of each oscillatory scale across the channels. Theperformance of the proposed algorithm has been illustratedboth analytically, in terms of an error bound, and throughsimulations on synthetic and real-world signals. Finally, whilethe algorithm has shown potential in generating a localizedmultivariate time-frequency representation, further work isrequired for the operation in highly noisy environments.Acknowledgments

We wish to thank the anonymous reviewers for theirvaluable comments and suggestions.

Appendix A

This Appendix presents a bound on the error of themultivariate instantaneous frequency estimate multib(shown in (20)), based on error bounds for the univariatesynchrosqueezing transform [16]. We first present a briefoverview of the main results presented in [16] and proceedto derive a bound for the estimated multivariate instanta-neous frequency.

A real-valued oscillatory function of the form f t At cos t, can be considered an intrinsic mode type(IMT) function, with accuracy , if the following conditionson A and hold

AAC1R \ L1R; AC2RinftAR

0t40; suptAR

0to1

jA0tj; jtjrj0tj; 8tAR

A signal f(t) that satisfies the above constraints, has itscorresponding wavelet transform given by Wf a; b, and7 It should be noted that the localization ratio of the proposedmethod and the univariate SST of the high gain channel are equal, asthe instantaneous frequency of interest in both cases is f d 4:567 Hz.the Fourier transform of the wavelet function having acompact support in 1;1. The synchrosqueezingtransform with accuracy and threshold ~ (where~ 1=3 is the threshold for which jWf a; bj4 ~) is thendetermined via [16]

Sf ; ~ b; Za:jWf a;bj4 ~

Wf a; b 1h

f a; b

a3=2 da

26where h(t) is a window function which satisfies,Rht dt 1. The following error bounds have been deter-

mined in [16] Given a scale band Z fa; b: ja0b1jog, thenfor each scale-time pair a;bAZ and jWf a; bj4 ~, itfollows that

jf a; b0bjr ~; 27which implies that the SST coefficients are concen-trated along the instantaneous frequency 0b. For all of bAR, there exists a constant C, such that as-0, the inverse of the synchrosqueezing transformalong the vicinity of the instantaneous frequencycurve, 0b, results in the following error bound:

lim-0

1R

Z:jf a;bjo ~

Sf ; ~ b; d !

A b e_b

rC ~

28The multivariate instantaneous frequency estimate is thenthe power weighted average of the instantaneous ampli-tudes and frequencies of the multivariate signal accordingto (4). Therefore, a bound on the error of the multivariateinstantaneous frequency depends upon the channel-wiseerrors when estimating the instantaneous amplitudeand frequency of a modulated oscillation using the syn-chrosqueezed transform. Based upon the univariate SSTerror bounds with the instantaneous frequency 0nband amplitude An(b) (where n is the channel index)and scale bands for each channel given by Zn fa; b:ja0nb1jog, with jWna;bj4 ~, the instantaneousfrequency for each channel is bounded by

0nb ~rna; br0nb ~ 29with the bound on the corresponding error bound for theinstantaneous amplitude, obeying

AnbC ~o AnboAnbC ~ 30where

An b lim-0

1R

Z:jna;bjo ~

Sf ; ~ ;n b; d

:

For the estimate of the multivariate instantaneous fre-quency (determined using (22)), the objective is to deter-mine an error bound on jmultibmultibj, in the form:

jmulti b multi b j Nn 1A

2nbna; b

Nn 1A2nb

multi b

A. Ahrabian et al. / Signal Processing 106 (2015) 331341340 Nn 1A

2nbna; bmultib

Nn 1A2nb

:

31

Using the following property, jNn 1ynjrNn 1jynj,Eq. (31) can be written as follows:

Nn 1A2nbna;bmultib

Nn 1A2nb

o N

n 1

A2nbna; bmultib

A2lower

N

n 1

A2nb

A2lower

na;bmultibA2lower

:

where A2lower Nn 1AnbC ~2. Finally, using inequal-ities (30) and (29), we have

N

n 1

A2nb

A2lower

na; bmultibA2lower

o N

n 1

AnbC ~2A2lower

0nbmultib ~A2lower

32Therefore, the error bound in (32) is dependent upon thedifferences of the individual channel-wise instantaneousfrequencies from the calculated multivariate instantaneousfrequency. However, if the instantaneous frequencieswithin each channel are equal, from (32) this implies thatthe bound is a multiple of the threshold ~.Appendix B

This Appendix derives a multivariate extension for theWigner distribution which naturally estimates the jointinstantaneous frequency for a multivariate signal. Given amultivariate analytic signal x t, the Wigner distributionis defined by

WD ; t Z 11

xH t2

x t

2

e j d: 33

and its inverse as

xH t2

x t

2

12

Z 11

WD ; t ej d

where xH t is the Hermitian transpose of a vector x tdefined in (3).

The central frequency of the Wigner distribution of amultivariate signal x t, for a given instant t, is given by

t R11WD; t dR11 WD; t d

: 34Using the inverse Wigner distribution we can now rewrite(34) as

t djd

xH t2

x t2 h i

j 0

xH t2

x t

2

j 0

12jxH tx0 tx0H tx t

xH tx t:

For the multivariate signal components xnt ante_nt,the instantaneous frequency of a multivariate signal istherefore of the form

t Nn 1a

2nt0nt

Nn 1a2ntx t :

In a similar way, the instantaneous bandwidth (4)follows from

2x t R11 xt2WD; t dR1

1 WD; t d

with

12

Z 11

2WD ; t d d2

d2xH t

2

x t2 h i

j 0:

This analysis can be generalized to the Cohen class ofdistributions and general time-scale representations,including the spectrogram and the scalogram as theenergetic forms of the short-time Fourier transform andthe wavelet transform, respectively [26]. Finally, in orderto implement the MWD, we used an multivariate exten-sion of the pseudo Wigner distribution [23], where awindow function is used to evaluate (33).

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Synchrosqueezing-based time-frequency analysis of multivariate dataIntroductionModulated multivariate oscillationsSynchrosqueezing transformMultivariate time-frequency representation using the SSTPartitioning of the time-frequency domainMultivariate time-frequency representation

SimulationsSinusoidal oscillation in noiseAmplitude and frequency modulated signal analysisFloat drift dataDoppler speed estimation

ConclusionAcknowledgmentsReferences