Noise Robust AM-FM Demodulation using Least- Squares Truncated Power Series Approximation

Transcription

1 oise Robust AM-FM Demodulation using Least- Squares Trunated ower Series Approximation Wooi-Boon Goh Shool of Computer Engineering, anyang Tehnologial University, Singapore ABSTRACT Abstrat - This paper desribes a parametri approah for demodulating multiomponent AM-FM sinusoidal signals. The parametri model onsist of several trunated power series whose oeffiients are estimated using linear least-square minimisation ast within an iterative parameter-substitution framewor. It is shown that the proposed tehnique for AM-FM estimation is relatively robust to baground noise ompared to other tehniques suh as the smoothed energy operator separation algorithm and the Hilbert transform method. Keywords Signal roessing, AM/FM Demodulation I. ITRODUCTIO The analysis of an AM-FM signal usually involves the estimation of the signal s time-varying amplitude envelope and instantaneous frequeny. AM-FM demodulation tehniques an be lassified into parametri and non-parametri methods. AM-FM parametri models are generally not popular beause the parameter estimation proess is non-linear in nature and diffiult to solve []. Grenier [], Muhopadhyay and Sirar [3] have proposed using time-dependent autoregressive moving average (TARMA) models and attempted to linearised the parameter estimation proess by expressing the time-dependent AR oeffiients in terms of some suitable basis funtions suh as the Fourier and Chebyshev basis funtions. roblemati issues suh as the optimum hoie of basis funtions and the estimation of the time-varying AR oeffiients under noisy onditions still remain [3]. On the other hand, nonparametri methods suh as the Hilbert transform method [4] and the energy separation algorithm [5]-[7] are popular and have been applied to speeh resonane modeling and musial instruments analysis [8]-[]. Quatieri et al. proposed a tehnique for AM-FM separation based on the relative amplitude envelope of the output of two linear transdution filters [6]. onparametri methods are generally simpler and omputationally more effiient but are not as noise robust ompared to parametri methods [3]. Additionally, many parametri methods [], [] an be readily extended to signals ontaining multiple AM-FM sinusoids. on-parametri methods wor-around this problem by applying appropriate bandpass filtering (e.g. Gabor filter) to the signal in order to extrat eah of the sinusoids before the demodulation algorithm is applied to eah extrated omponent in turn [5], [8]-[9]. But suh filters introdue amplitude distortions sine not all frequenies with the bandwidth of the filter are equally attenuated. This paper proposes a parametri method for demodulating multiomponent AM-FM sinusoidal signals. The proposed least-squares trunated power series approximation (L-STSA) model is developed in setion. In setion 3, the performane of the proposed parametri model is evaluated for different orders of L- STSA model and under various noise onditions. An appliation of the L-STSA model to singing voie (multiomponent) analysis is also presented. II. THE L-STSA AM-FM MODEL An AM-FM signal x( an represented by x( = A( os[ wn + θ( ] () where w is a fixed arrier frequeny with time-varying amplitude A(. The instantaneous frequeny w( is the derivative of the time-varying phase θ( given by w( = w + [ dθ ( dn] () Given a signal x( and the arrier frequeny w, the lassial AM-FM demodulation problem is to estimate the amplitude envelope A( and the instantaneous frequeny w(. A. The AM Model We start with a simple AM signal model given by ~ x( = A( os[ w n + ϕ ] (3) where the arrier frequeny w is nown to be present in the AM-FM signal x(. A( is the time-varying amplitude and ϕ is some phase value. We expand (3) into its odd and even quadrature omponents given by ~ xn ( ) = an ( ) os( w + bn ( ) sin( w (4) where a( and b( are given by an ( ) = A( os( ϕ ) and bn ( ) = An ( )sin( ϕ ) (5) Here we assume the funtions desribing the timevarying amplitude A( and phase θ( of the signal are analyti, whih means a derivative exist at all points on the funtion []. Analyti funtions an be desribed by a power series. For example os x is given by 4 6 osx = x! + x 4! x 6! +... (6)

2 With this assumption, it is proposed that the quadrature omponents of xn ~ () be modeled as a general trunated power series (with entre at ) of order given by = n a( a, (7) b( = n b = = Given a signal x( of sample length, the modeled signal ~ xn ( ) in (4) an be estimated by minimising the mean square-error ε in (8) with respet to the (+) pairs of amplitude oeffiients (a to a, b to b ) and a predetermined arrier frequeny of (w = w ), as detailed in [3]. B. The AM-FM Model ε ~ A = { x ( x( } n= If the signal being modelled is a pure AM sinusoidal signal, the phase omponent ϕ of xn ~ ( ) in (3) would be a onstant. Sine our onern is with signals that are modulated both in amplitude and frequeny, then the instantaneous phase ϕ will be time-varying and an be desribed by a time-varying funtion θ$( given by ( = artan[ b( a( ] (8) ˆ θ (9) where a( and b( are the signal's quadrature amplitude omponents given in (7). Starting with an AM model, the time-varying funtions a( and b( are omputed for a fixed arrier frequeny w using a leastsquare trunated power series approximation (L-STSA) model of order. These values are then substituted into (9) to obtain a preliminary estimate of the time-varying instantaneous phase funtion θ$(. This unwrapped phase funtion θ$( is inorporated into the AM-FM signal model given in () by traing the π jumps in the phase values obtained in (9). The unwrapped phase is then modeled as an analyti funtion using a L-STSA of order Q in the same way the L-STSA quadrature amplitude funtions were omputed earlier. The Q-order phase funtion θ$( is then substituted into () as θ(. Lie before, the AM-FM signal model in () an be expanded into its odd and even omponents, given by [ ] ( ) [ ( )] xn ~ ( ) = an ( ) os wn ( n ) bn sin + θ + wn + θ n () Similar to the AM model, the time-varying funtions a( and b( of the AM-FM model are obtained from the (+) pairs of amplitude oeffiients (a to a, b to b ) using the least-square minimization. However, this time around, instead of using a fixed arrier frequeny of (w=w ), a time-varying frequeny of (w=[w n+θ(]) is used. To obtain a new expression for the instantaneous phase of the AM-FM signal model, () is expanded using suitable trigonometri identities to the form [ ]+ xn= ~ ( ) a( os( w os( θ ( ) sin( w sin( θ ( ) [ sin( w os( θ ( ) + os( w sin( θ ( ) ] () b( n ) Rearranging (), the quadrature omponents of xn= ~ ( ) [ a( os( θ ( ) + b( sin( θ( )] os( w + [ b( os( ( ) a( sin( θ ( )] sin( w ~ x( are θ () A new time-varying phase funtion obtained lie in (8) and is given by $( ) θ n an be ˆ b( os ( ) ( θ( ) + a( sin( θ( ) θ n = artan (3) a( os( θ( ) + b( sin( θ( ) This newly estimated phase funtion θ$( is then unwrapped, smoothened using a L-STSA model and substituted ba into the signal model in () to obtain new values of a( and b(. These new values of a( and b( in turn, allow the omputation of another timevarying phase funtion $( θ n ) using (3). This iterative parameter-substitution proess is repeated until the AM- FM signal model ~ x ( M n ) obtained in the Mth iteration losely resembles that in the (M+)th iteration. The termination riterion for this iterative proess ould be a simple mean-square error riterion given by [ n= ~ x n ~ x n M ( ) ( ) M+ ] ε M (4) where the error threshold ε M is some suitable small value that is obtained empirially. It is dependent on the average signal magnitude. Fixed iteration ounts ould also be used. For the signals presented in this paper, reasonable onvergene was obtained in iterations. The L-STSA parametri model ompletely desribed an AM-FM sinusoid as Q Q ~ xn ( ) na os wn n nb sin wn n = = = = = (5) where (a to a, b to b ) are the (+) amplitude oeffiients pairs and ( to ) are the (Q+) phase oeffiients. The proposed parametri model an desribe a single omponent AM-FM signal segment x( ompatly using (Q++3) real-valued oeffiients and a fixed-arrier frequeny omponent w. The amplitude envelope of the AM-FM signal x( is given by An ( ) = na nb + ½ (6) = = The time-varying frequeny w( in () is given by Q wn ( ) = w + ( + ) + n = (7) where the derivative of the analyti instantaneous phase funtion θ( an be obtained by differentiating the assoiated power series term by term []. If the phase funtion θ( is available, baward-differene will provide the instantaneous frequeny and it is given by wn ( ) = w + θ( θ( n ) (8)

3 III. EXERIMETAL RESULTS A. Order of L-STSA Model Trunated power series approximation of os x rder of power series fw = 5 Hz fw = 75 Hz fw = Hz Figure. RMS errors in modeling osx at different frequenies using a L-STSA model of various orders. Fig. shows the root-mean-squared (RMS) errors between the funtion f(=os(πnf w /f s ) and its trunated power series approximated equivalent given in (6) over 3 samples. Funtion f( was omputed with only even powers as the order of the power series was varied from 6 to 5. The frequeny f w of the sinusoidal funtion was set at 5, 75 and Hz, with the sampling frequeny f s = 5 Hz. As expeted, when the order of the trunated power series inreases, the auray of the estimated signal model improves as an be seen by the rapidly reduing RMS errors. However, it is also observed that for a given order, the RMS error inreases almost logarithmially with a linear inrease in the signal's frequeny. Frequeny (Hz) Figure. AM-FM estimates with L-STSA model of orders ( = Q = ). Original test signal. The extrated amplitude modulation and () instantaneous frequeny. Original signals are in solid lines and estimated signals are in dash lines (not visible as they are overlapping the solid lines). In order to observe the auray of the L-STSA AM- FM signal model, the RMS errors were measured for different orders of L-STSA models. These results were obtained for a sinusoidal AM-FM signal with arrier frequeny w = Hz, FM and AM frequenies of w fm = Hz and w am = Hz respetively (see Fig. a). Fig. 3a and 3b show the RMS errors in the amplitude envelope and instantaneous frequeny estimates for () model orders = Q from 6 to 5. Exept for lower order values, the error harateristis in Fig. 3 are similar to those observed in Fig., exhibiting a sharp drop in RMS error as the order is inreased. With AM and FM frequenies of Hz, and Q orders of would be suffiient to model the AM and FM signal with a RMS error of < % of the signal's pea-to-pea value (see Fig. b and ). As the AM/FM frequenies redue, the order of the L-STSA ould be redued almost logarithmially while maintaining the same auray. (Hz) AM error with various orders of L-STSA rder of L-STSA model (=Q) FM error with various orders of L-STSA rder of L-STSA model (=Q) Figure 3. RMS errors using L-STSA of different orders for the amplitude envelopes and instantaneous frequenies. B. oise sensitivity The performane of the L-STSA algorithm is ompared with two well-nown AM-FM demodulation algorithms, the smoothed energy operator separation algorithm (SEOSA) of otamianos [5] and the Hilbert transform method [4]. White-Gaussian noise of varying varianes were added to a 5-sampled disrete sinusoidal AM-FM signal with a arrier frequeny w = Hz, FM frequeny w fm = Hz and AM frequeny w am = Hz, similar to that shown in Fig. 6a. At various signal-to-noise ratios (SR), the amplitude envelope and instantaneous frequeny were estimated using the proposed L-STSA model approah, the Hilbert transform method and the energy separation approah (i.e. SEOSA). The robustness of the L-STSA approah in demodulating noisy AM-FM signals an be learly observed in results shown in Fig. 4 and Fig. 5, whih showed signifiantly lower RMS errors for both AM and FM estimates for a SR between db to 9 db. Even at a SR of db, the AM and FM estimates obtained using the L-STSA models were reasonably aurate, as an be observed in Fig. 6b and 6. Errors were observed mainly at the two ends of the analysis frame. This is to be expeted sine the signals are modeled using a power series. Small errors in the estimated oeffiients will result in large errors as the value of n in (6) quily inreases as one moves away

4 from the entre of the analysis frame, where n =. The results also show that the SEOSA approah outperforms the Hilbert transform method at high noise levels, whih is onsistent with that observed in [6] sine the arrier frequeny is reasonably high at Hz. (log sale).... AM erformane Comparison at Various oise Levels Signal to oise Ratio (db) L-STSA Hilbert Transform SEOSA Figure 4. RMS errors (in log sale) of amplitude envelopes estimated at various SR, using the L-STSA models of order ( = Q = ), Hilbert transform and SEOSA methods. (Hz) (log sale) FM erformane Comparison at Various oise Levels L-STSA Hilbert Transform SEOSA Signal to oise Ratio (db) Figure 5. RMS errors (in log sale) of instantaneous frequenies estimated at various SR, using the L-STSA models of order ( = Q = ), Hilbert transform and SEOSA methods. Frequeny (Hz) Figure 6. AM-FM estimates with L-STSA model of orders ( = Q = ) in noisy signals. Test signal with SR = db. The amplitude and frequeny. Original signals in solid lines and estimated signals in dash lines. C. Computational Load Fig. 7 shows that the omputational load of the L- STSA algorithm inreases linearly with signal segment length. However, the inrease in omputational load as the L-STSA model order inreases in not linear sine the number of elements in the oeffiient matrix, whose () inverse has to be omputed grows by O(n ), where n is the order of the trunated power series model used. Computational load (flops) Computational load at various and Q orders,q=5,q=,q=5,q= Signal sample size Figure 7. Computational load at various -Q orders. Measured by the flops shown at the end of a Matlab routine exeution. D. Singing voie analysis Fig. 8 shows a multiomponent AM-FM signal whih is a short segment of a soprano's voie. The vibrato present in signal an be observed in spetrogram and it an be desribed by two parameters, namely the rate of vibrato and the depth of vibrato. Seashore [4] found that the mean vibrato frequeny registered for 9 singers was 6.6 Hz, varying between extremes of 5.9 to 7.8 Hz. The average depth of vibrato was ±48 ents, varying between extremes of ±98 to ±3 ents ( ent is the interval between two tones having the frequeny ratio : / ). The proposed tehnique was applied to the multiomponent signal whih onsist of the diphthong /ou/ (as in boat), sung at the ey F5 (698 Hz). x ( n Frequeny (Hz) x Analysis Frame Time(ses) artial #5 Fundamental Time (ses) Figure 8. Digitized waveform of a soprano's voie, whih was sampled from [5] and its orresponding spetrogram. The extrated AM-FM sinusoid using polynomial models of orders =3, Q= (see Fig. 9). Different partials of the multiomponent signal were extrated by starting the iterative estimation proess using different estimated arrier frequeny w in (). The mean fundamental frequeny was estimated to be about 7

5 Hz, whih is lose to the frequeny of the ey F5. The estimated vibrato frequeny is 6.67 Hz and the vibrato depth is approximately ±84 ents. These values are within the limits observed in [4]. Fig. 9a and 9b show the vibrato parameters are relatively onsistent between the higher partials and fundamental. Fig. 9 and 9d show the estimated amplitude envelope of the fundamental frequeny and partial #5, whih though greatly redued in energy, still maintains a similar time-varying profile. Instantaneous frequeny (Hz) x Fundamental artial # () Instantaneous frequeny (Hz) Fundamental artial # Figure 9. Time-varying AM-FM plots extrated from a singing voie. Fundamental frequeny and partial #5, after iterations with w = 684 Hz and w = 3344 Hz, respetively. envelope of () the fundamental omponent, shown overlaid over the original waveform (red) and (d) partial #5. IV. COCLUSIOS This paper presents a parametri approah for demodulating multiomponent AM-FM sinusoidal signals. The quadrature amplitude omponents of an AM-FM sinusoidal signal are modeled using a pair of trunated power series. Based on the instantaneous phase derived from the modeled quadrature omponents, an iterative framewor was proposed in whih the timevarying instantaneous frequeny of the signal is progressively estimated through repeated omputation and substitution of amplitude and phase parameters. The trunated power series oeffiients are estimated using linear least-squares minimization, resulting in an algorithm that is more robust to additive white Gaussian noise ompared to demodulation algorithms suh as SEOSA and Hilbert transform. The proposed tehnique is also able to extrat AM- FM parameters of sinusoids within multiomponent signals as was seen in the vibrato analysis performed on a segment of singing voie. The inherent problems of non-parametri time-frequeny distributions suh as time-frequeny resolution has been alleviated using the proposed parametri modeling tehnique, as the vibrato depths of both the low frequeny fundamental and its high frequeny partial (partial #5) were estimated with high preision. This was also done without the need for distortion induing bandpass filtering pre-proessing. (d) Unfortunately, the trunated power series model onstrains the proposed tehnique to sinusoidal signals that are ontinuously differentiable in both its amplitude and frequeny variations. Signals with singularities in the form of sharp transients are problemati. In suh ases, it is possible to view these signals as pieewiseanalyti and the proposed analysis an still be applied to segmented regions of the signal whih do not straddle singular points. Some means of onset detetion ould be used in the pre-proessing stage. One example is the onset map proposed by Brown and Cooe [6], whih is derived using physiologially-motivated filters. REFERECES [] B. Friedlander and J. Franos, "Estimation of amplitude & phase parameters of multiomponent signals", IEEE Trans. Signal roessing, vol. 43, pp , 995. [] Y. Grenier, "Time-dependent ARMA modeling of nonstationary signals", IEEE Trans. Aoust., Speeh, Signal roessing, vol. 3, pp , Aug [3] S. Muhopadhyay and. Sirar, "arametri modelling of nonstationary signals: a unified approah", Signal roessing, vol. 6, pp. 35-5, 997. [4] S. L. Hahn, Hilbert transforms in signal proessing. Boston, MA: Arteh House, 996. [5] A. otamianos and. Maragos A Comparison of the Energy Operator and the Hilbert Transform approah to Signal and Speeh Demodulation, Signal roessing, vol. 37(), pp. 95-, 994. [6] T. F. Quatieri, T. E. Hanna and G. C. O'Leary, AM-FM separation using auditory-motivated filters", IEEE Trans. Speeh Audio roessing, vol. 5, pp , Sep [7]. Maragos, J. F. Kaiser and T. F. Quatieri, "On amplitude and frequeny demodulation using the energy operators", IEEE Trans. Signal roessing, vol. 4, pp , 993. [8]. Maragos, J.F. Kaiser and T.F. Quatieri, "Energy separation in signal modulations with appliation to speeh analysis", IEEE Trans. Signal roessing, vol. 4, pp , 993. [9] H.M. Hanson,. Maragos, and A. otamianos, A system for finding speeh formants and modulations via energy separation", IEEE Trans. Speeh Audio roessing, vol., pp , 994. [] R. B. Sussman and M. Kahrs, Analysis and resynthesis of musial instrument sounds using energy separation", ro. IEEE- ICASS'96, pp. 997-, 996. [] J. S. Marques and L. B. Almeida, "Frequeny-varying sinusoidal modeling of speeh", IEEE Trans. Aoust., Speeh, Signal roessing, vol. 37, pp , 989. [] J. D. aliouras, Complex variables for sientists and engineers. ew Yor: Mamillan ublishing Co., 975. [3] W.B.Goh and K.Y.Chan, '' Modulated Sinusoidal Modeling using Least-square Infinite Series Approximation with Appliations to Timbre Analysis,'' ro. IEEE-ICASS'98, vol. 6, pp , 998. [4] C.E. Seashore, syhology of musi, MGraw-Hill, 967. [5] K. Battle, Grae, CD Album produed by D. Frost and S. aul, Sony Musi Entertainment In., 997 [6] G. J. Brown and M. Cooe, "hysiologially-motivated signal representation for omputational auditory sene analysis", in Visual representation of speeh signals, Chihester, UK: John Wiley & Sons, 993.