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On Discrete Fourier Spectrum of Randomly Modulated Signals

On Discrete Fourier Spectrum of Randomly Modulated Signals In this work the problem of characterization of Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal modulated by random fluctuations of amplitude and phase is investigated. It is assumed that the amplitude and phase of signal values at discrete time moments of observations are distorted by adding realizations of independent and identically distributed random variables. The obtained results of theoretical analysis of such distorted signal spectra show that only in the case of amplitude modulation the DFT spectrum of the modulated bounded signal can be similar to the original signal spectrum, although there occur random deviations. On the her hand, if phase modulation is present, then the DFT spectrum of the modulated bounded signal n only shows random deviations but also amplitudes of peaks existing in the original spectrum are diminished, and consequently similarity to the original signal spectrum can be significantly blurred. Keywords: Discrete Fourier Transform, signal spectrum, amplitude modulation, phase modulation, random fluctuations 1. INTRODUCTION The Discrete Fourier Transform (DFT) based periodogram is a widely used tool for analyzing time series that can be decomposed as a sum of monochromatic oscillations plus noise. Important applications of the periodogram include detection of hidden periodicities and estimation of unknown oscillation parameters (amplitude and frequency). For example, it is well known that very accurate frequency estimates of the sinusoidal components can be obtained from the local maxima of a periodogram (Walker 1971). If time series of the analyzed signal observation values at discrete time moments xt , t is available, then its corresponding DFT is computed as follows (Gasquet and Witomski 1999) 1 ~ (1) xk xt exp( i 2 kt / N ) , Nt 0 for k 0,1,..., . Frequently, the well-known Fast Fourier Transform procedures are used to perform the relevant calculations (Cooley and Tukey 1965), (Press et al. 1992), (Singleto969). Theoretical as well as numerical properties of the DFT are described in time series analysis textbooks (Blackledge 2003), (Bloomfield 2000), (Bremaud 2002), (Brillinger 1975), (Koopmans 1974). Certain statistical properties of spectrum estimation using the DFT are investigated in the work of (Foster 1996a,b) and some her aspects like periodogram smohing are considered in (Speed 1985). 144 The present work is a continuation of the author's previous publication (Popi ski 1997) on theoretical, statistical, as well as numerical properties of the DFT spectrum. It deals with the problem of applicability of such technique to spectrum estimation of signals which are subject to random amplitude and phase modulation. The proposed approach is justified by the fact that all signals usually referred to as "periodic" have some amplitude and phase variation from period to period. For example an active sonar system transmits a periodic pulse train to detect targets. The received pulses are n perfectly periodic due to random modulation of the pulses from scattering and attenuation (Hinich 2003). Also El Niño signals are recognized as amplitude and phase modulated (Allen and Robertso996). The assumed concept of random modulation modeling is described in section 2. In section 3 the theoretical results related to the modulated signal DFT spectrum are presented bh in the case of a noiseless signal and in the case of signal observations corrupted by random errors. 2. MODULATION MODELING Let us consider finite duration time series of complex-valued signal observations at discrete time moments Re[ ] i Im[ ] , t 0,1,..., . We assume that the analyzed signal is of deterministic character and comprises some regular oscillations. Such a signal can represent for example the monochromatic oscillation o0t A0 exp(i 0t i 0 ) , t with frequency 0 , constant amplitude A0 , and constant phase 0 . Now, let us assume that the signal values amplitudes and phases are distorted by random additive fluctuations at and t , t respectively, which are realizations of independent and identically distributed random variables. We also assume that amplitude distortions are independent of phase distortions. Hence, we deal with an amplitude and phase modulated signal satisfying the mathematical model rt (1 at ) exp(i t ) exp(i t ) at exp(i t ) , (2) where exp(i t ) and at exp(i t ) , t are realizations of independent and identically distributed random variables. Let the distribution of t be uniform on the interval , which gives immediately for t ( , ) ( t ~ U ( , ) ) with 0 1 sin( ) , m E exp(i t ) exp(iu )du 2 (3) 2 1 2 2 2 E exp(i t ) m E exp(i t ) m2 exp(iu ) du m 2 1 m 2 . s one can clearly see the mean value m satisfies 0 m 1 and is equal to zero only for , so that we also have 0 About the distribution of the real-valued random variable at we assume only that Ea at 0 and Ea at 2 (4) (if 0 there is only phase modulation of the signal), which further implies for a t ma Ea E (1 at ) exp(i t ) Ea (1 at ) E exp(i t ) m , (5) Ea E (1 at ) exp(i t ) ma Ea (1 at ) 2 E exp(i t ) 2 ma m2 , and obviously we have . For example, at can be uniformly distributed on the interval [ A, A] , where A 0 , and then A2 / 3 . 3. MODULATED SIGNAL SPECTRA It follows immediately from (2) that in order to analyze the DFT spectrum of our model signal we shall consider the case of signals rt zt , t modulated by series of independent and identically distributed complex-valued random variables zt . Indeed, the case of zt exp(i t ) corresponds to phase modulation, the case of zt (1 at ) exp(i t ) to phase and amplitude modulation, and the case of real-valued modulation series zt at is needed for amplitude modulation analysis, respectively. In all three mentioned cases the random variables zt have constant mean value and variance according to (3), (4) and (5). In order to compute the DFT of the modulated signals of the form rt zt , t we apply the well-known convolution formula (Gasquet and Witomski 1999) ~ rk k j 0 ~ z ok j ~j j k ~ zj (6) for k 0,1,..., . Hence, if we want to analyze the DFT of the modulated signal ~ , it is necessary to rk characterize the statistical properties of the random modulating series DFT ~ j , z j 0,1,..., . Such a characterization is given in the following lemma. Lemma 1. If a series zt , t its DFT satisfies represents realizations of uncorrelated complex-valued random variables with constant mean value E z zt E z ~0 z , E z ~k z and finite variance E z zt 0 for kl 0 , then 1,..., , and E z ( ~k z where the symbol Proof. kl 1 N denes the Kronecker delta. E z ~k )( ~l z z E z ~l ) z The first property Ez ~0 follows immediately from the assumptions of the lemma and the z definition of the DFT. In order to prove the equality E z ~k 0 for k 1,..., , observe that z 1 1 1 E z zt exp( i 2 kt / N ) E z zt exp( i 2 kt / N ) exp( i 2 kt / N ) N t 0 Nt 0 N t 0 1 1 exp( i 2 k ) 0 . exp( i 2 kt / N ) N t 0 exp( i 2 k / N ) Also from the definition of the DFT and from the assumed zero correlation of the random variables considered we have 1 E z ( ~k E z ~k )( ~l E z ~l ) z z z z E z ( zt E z zt ) exp( i 2 kt / N ) ( z s E z z s ) exp(i 2 ls / N ) N2 t 0 s 0 E z ~k z 1 N2 1 N E z ( zt t 0 s 0 kl Ez zt )( z s E z z s ) exp( i 2 ( kt ls ) / N ) 1 N2 t 0 exp( i 2 (k l )t / N ) l )t / N ) 1 exp( i 2 (k l )) 1 exp( i 2 ( k l ) / N ) 0 since t 0 exp( i 2 ( k for k l, which proves the second property. Now, as we know the mean values of the random variables ~j , j z according to the above lemma, we easily obtain from (6) Ez ~ rk and further since E z ( ~k E z ~k )( ~l z z z yields for k ~ ok for E z ~l ) z kl (7) / N the same formula Ez ~ rk 2 Ez ~ rk Ez j 0 ~ z ok j ( ~j E z ~j ) z ~ k (z j 2 z E z ~j ) z (8) where k 0 the 2 t 0 last equality N j 0 follows from ~ ok 2 j 2 j k the N k 0 N well-known property of ~ 2 ok , the DFT ~ ok / N (Gasquet and Witomski 1999). From the Schwartz inequality (Bremaud 2002) we also have the following estimate related to rk rl the covariance of the random variables ~ and ~ Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez ~ rk Ez ~ rk Ez ~ rl 2 Ez ~ rl (9) . N where B Let us remark that for bounded signals satisfying B , t 0 is a real number, we immediately obtain from (8) and (9), 2 2 2 2 2 zB zB Ez ~ Ez ~ rk rk B2 Ez (~ Ez ~ )(~ Ez ~ ) rk rk rl rl , . (10) N2 k 0 N N In such a case the random variables ~ , k representing the spectrum of the rk modulated signal, have equal variances decreasing asymptically to zero as N (which implies that their covariances also decrease asymptically to zero). Bounded signals are of course of primary interest in this work since we intend to investigate spectra of regular oscillations of stationary character, modulated by random amplitude and phase fluctuations. In view of the equalities in (3), (4), (5) and lemma 1, the equalities (7), (8) and inequality (9) hold in the case of phase modulation series zt exp(i t ) with m and z2 1 m 2 , as well as in the case of amplitude and phase modulation series zt (1 at ) exp(i t ) with m and m 2 . Thus, taking into account the equality (7) and the inequalities in (10), we see that if a bounded signal is phase or amplitude and phase modulated according to our model, then its DFT spectrum will have random character. The modulated signal DFT ~ ~ rk rk spectrum has mean values E z rk ok , k with random deviations ~ Ez ~ imposed, i.e. random phase or amplitude and phase modulation diminishes the amplitudes of the original bounded signal spectrum by the factor 0 m 1 and corrupts the spectrum by additive deviations of random character, which have zero mean and known variance. In consequence, any peaks present in the spectrum of the original bounded signal may be less distinguishable in the modulated signal spectrum. Distortion of the bounded signal spectrum values close to 0 (i.e. if phase in the case of random phase modulation is small for ~ E ~ 2 0 . In the her extreme case of fluctuations are small), since then m z 1 and E z rk z rk we have m z 0 , and then the modulated signal spectrum will have purely stochastic character without any frequencies distinguished, so that any peaks present in the original signal spectrum will be completely blurred. In accordance with our model (2), if an original signal is only amplitude modulated, then rt rat , where rat at , t so we have ~ ok ~ , k 0,1,..., . rk ~ rak Hence, the spectrum of the amplitude modulated signal differs from the original signal rak spectrum only by the additive distortion terms ~ , k 0,1,..., . Having in view the ssumptions in (4) and putting zt at with z we can apply again lemma 1, which further yields E z ~ ok , as well as equality (8) and inequality (9) for random variables rk ~ ~ ~ E ~ , k 0,1,..., . Consequently, if the original signal is bounded, then by (10) rak rk z rk the distortion terms will have equal variances tending to zero as N , so that they will also have asymptically zero covariances. This means that they will n distort completely the original bounded signal spectrum on which they are superimposed. Some small peaks present in the original signal spectrum can be smohed due to amplitude modulation but possibly larger ones will be still distinguishable. In Figure 1. the above described effects of amplitude and/or phase signal modulation on the DFT spectrum of a model signal (sum of 7 monochromatic complex harmonics with constant amplitudes and phases) is shown. The random variables representing amplitude and phase fluctuations were distributed according to uniform distribution at ~ U ( A, A) and , ) with parameters A 1.0 and / 2 , respectively. Assume now that the time series of the original signal values , t corrupted by random errors t , according to the model ~U( is where yt (11) t t, are realizations of uncorrelated complex-valued random variables having zero mean 0 and finite second moment E | |2 Let us see what happens if the corrupted signal values are submitted to random modulation ryt zt yt , t of the same kind as considered above. For a modulating series zt , t satisfying the assumptions of lemma 1, we obtain on the basis of the DFT convolution formula (Gasquet and Witomski 1999) ~ ryk k j 0 ~ ~ yk j z j ~ yN j k ~ zj (12) 148 for k which by assertion of the lemma yields E z ~yk ~k , r y E E z ~yk r E ~k y ~ E (ok ~ )m k z ~ ok . (13) frequency Fig. 1. Spectra of a model series representing sum of 7 monochromatic oscillations (N=2048): original signal (black), phase modulated signal ( / 2 , blue), amplitude modulated signal ( A 1.0 , green), amplitude and phase modulated signal ( / 2 , A 1.0 , brown). Furthermore, simple calculation shows that and analogously as in (8) we obtain N which by (11) together with the equalities E lemma 1 finally gives E E z ~yk r E E z ~yk r yt ( ~k y 0, E t s ~ 2 ok ) , st , s, t and E yt E ~k 2 2 t (14) N for k as well as E E z ( ~yk E E z ~yk )( ~yl E E z ~yl ) r r r r E E z ~yk r E E z ( ~yk r E E z ~yk )(~yl r r 2 E E z ~yl r E E z ~yl ) r (15) E E z ~yk r E E z ~yl r N 0,1,..., . 150 Hence, we see that the presence of errors corrupting the original signal values does n change the character of the DFT spectrum of the modulated series ryt zt yt , t 0,1,..., . Indeed, the formulae (13), (14), (15) are analogous to (7), (8), (9), respectively, except for addition of 2 2 2 the term 2 ( z2 ) / N E z zt / N which now occurs because of non-zero second moment of the errors. For bounded signals to (10) , namely, B, t we can easily obtain the inequalities analogous 2 B2 , (16) 2 2 E Ez (~yk r 0,1,..., . E Ez ~yk )(~yl r r E Ez ~yl ) r Thus, our earlier assertions concerning the character of the modulated signal spectrum hold also in the case of a bounded signal corrupted by uncorrelated random errors which have zero mean and finite second moment. This means that only in the case of amplitude modulation the DFT spectrum of the modulated corrupted signal can resemble the one of the corrupted signal itself. 4. SUMMARY The properties of the DFT spectrum examined in this work are helpful for understanding the possible changes such a spectrum undergoes in the case of random amplitude and phase modulation of an original signal. Our modulation model includes additive distortions of stochastic nature in the amplitudes and phases of original signal values at observation moments. For bounded signals of deterministic character (like a sum of monochromatic oscillations with constant amplitudes and phases) it is proved that occurrence of random phase modulation of the signal can completely change the character of its DFT spectrum. Namely, the phase modulated signal spectrum can show purely stochastic character without any frequencies distinguished, like the spectrum of white noise. In the case of random amplitude modulation of such a signal it is shown that the modulated signal spectrum can still resemble the spectrum of the original signal, although small peaks can be significantly smohed. Similar conclusions can be deduced also in the case of a deterministic signal which is corrupted at the moments of observation by uncorrelated random errors with zero mean and finite second moment. It is worth remarking that our conclusions agree with observations of Ni and Huo (2007), concerning the importance of phase and amplitude information in signal and image reconstruction. The concept of phase randomization used for obtaining multivariate surrogate time series (Mammen and Nandi 2008) with distribution similar to observed series is also related to the subject considered here. Hinich (2003) used similar approach to amplitude modulation modeling, as the one applied in this work, to derive statistics for detecting randomly modulated pulses in noise. The Singular Spectrum Analysis (SSA) method has an important property, first ned by Vautard and Ghil (1989), that it may be used directly to identify modulated oscillations in the presence of noise. Allen and Robertson (1996) proposed a generalization of the "Monte Carlo SSA" algorithm which allows for objective testing for the presence of modulated oscillations at low 151 signal-to-noise ratios in multivariate data. They demonstrated the application of the test to the analysis of interannual variability in tropical Pacific sea-surface temperatures. Acknowledgements. This research work was supported by the Polish Ministry of Science and Higher Education through the grant No. N N526 160136 under leadership of Dr Tomasz Niedzielski at the Space Research Centre of Polish Academy of Sciences. REFERENCES Allen M.R., Robertson A.W. (1996) Distinguishing Modulated Oscillations from Coloured Noise in Multivariate Datasets, Climate Dynamics, Vol. 12, No. 11, 775­784. Blackledge J.M. (2003) Digital Signal Processing, Horwood Publishing, Chichester, West Sussex, England. Bloomfield P. (2000) Fourier Analysis of Time Series: An Introduction, Wiley, New York. Bremaud P. (2002) Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis, Springer Verlag Inc., New York. Brillinger D.R. (1975) Time Series ­ Data Analysis and Theory, Holt, Rinehart and Winston Inc., New York. Cooley J.W. and Tukey J.W. (1965) An Algorithm for the Machine Calculation of Complex Fourier Series, Mathematics of Computation, Vol. 19, 297-301. Foster G. (1996a) Time Series by Projection I: Statistical Properties of Fourier Analysis, The Astronomical Journal, Vol. 111, No. 1, 541-554. Foster G. (1996b) Time Series by Projection II: Tensor Methods for Time Series Analysis, The Astronomical Journal, Vol. 111, No. 1, 555-566. Gasquet C., Witomski P. (1999) Fourier Analysis and Applications ­ Filtering, Numerical Computation, Wavelets, Springer Verlag Inc., New York. Hinich M.J. (2003) Detecting Randomly Modulated Pulses in Noise, Signal Processing, Vol. 83, Issue 6, 1349-1352. Koopmans L.H. (1974) Spectral Analysis of Time Series, Academic Press, New York. Mammen E. and Nandi S. (2008) Some Theoretical Properties of Phase-Randomized Multivariate Surrogates, Statistics, Vol. 42, No. 3, 195-205. Ni X. and Huo X. (2007) Statistical Interpretation of the Importance of Phase Information in Signal and Image Reconstruction, Statistics and Probability Letters, Vol. 77, Issue 4, 447-454. Popi ski W. (1997) On Consistency of Discrete Fourier Analysis of Noisy Time Series, Artificial Satellites ­ Journal of Planetary Geodesy, Vol. 32, No. 3, 131-142. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T. (1992) Numerical Recipes ­ The Art of Scientific Computing, Cambridge University Press, Cambridge. Singleton R.C. (1969) An Algorithm for Computing the Mixed Radix Fast Fourier Transform, IEEE Transactions on Audio and Electroacoustics, Vol. AU-17, No. 2, 93-103. Speed T.P. (1985) Some Practical and Statistical Aspects of Filtering and Spectrum Estimation, In Price J. F. (Editor), Fourier Techniques and Applications, Plenum Press, New York, 101-118. 152 Vautard R., Ghil M. (1989) Singular Spectrum Analysis in Nonlinear Dynamics with Applications to Paleoclimatic Time Series, Physica D, Vol. 35, 395­424. Walker A.M. (1971) On the Estimation of a Harmonic Component in a Time Series with Stationary Independent Residuals, Biometrika, Vol. 58, No. 1, 21­36. Received: 2010-12-11, Reviewed: 2011-02-04, Accepted: 2011-02-07. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Artificial Satellites de Gruyter

On Discrete Fourier Spectrum of Randomly Modulated Signals

Popiński, Waldemar
Artificial Satellites , Volume 45 (3) – Jan 1, 2010
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Abstract

In this work the problem of characterization of Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal modulated by random fluctuations of amplitude and phase is investigated. It is assumed that the amplitude and phase of signal values at discrete time moments of observations are distorted by adding realizations of independent and identically distributed random variables. The obtained results of theoretical analysis of such distorted signal spectra show that only in the case of amplitude modulation the DFT spectrum of the modulated bounded signal can be similar to the original signal spectrum, although there occur random deviations. On the her hand, if phase modulation is present, then the DFT spectrum of the modulated bounded signal n only shows random deviations but also amplitudes of peaks existing in the original spectrum are diminished, and consequently similarity to the original signal spectrum can be significantly blurred. Keywords: Discrete Fourier Transform, signal spectrum, amplitude modulation, phase modulation, random fluctuations 1. INTRODUCTION The Discrete Fourier Transform (DFT) based periodogram is a widely used tool for analyzing time series that can be decomposed as a sum of monochromatic oscillations plus noise. Important applications of the periodogram include detection of hidden periodicities and estimation of unknown oscillation parameters (amplitude and frequency). For example, it is well known that very accurate frequency estimates of the sinusoidal components can be obtained from the local maxima of a periodogram (Walker 1971). If time series of the analyzed signal observation values at discrete time moments xt , t is available, then its corresponding DFT is computed as follows (Gasquet and Witomski 1999) 1 ~ (1) xk xt exp( i 2 kt / N ) , Nt 0 for k 0,1,..., . Frequently, the well-known Fast Fourier Transform procedures are used to perform the relevant calculations (Cooley and Tukey 1965), (Press et al. 1992), (Singleto969). Theoretical as well as numerical properties of the DFT are described in time series analysis textbooks (Blackledge 2003), (Bloomfield 2000), (Bremaud 2002), (Brillinger 1975), (Koopmans 1974). Certain statistical properties of spectrum estimation using the DFT are investigated in the work of (Foster 1996a,b) and some her aspects like periodogram smohing are considered in (Speed 1985). 144 The present work is a continuation of the author's previous publication (Popi ski 1997) on theoretical, statistical, as well as numerical properties of the DFT spectrum. It deals with the problem of applicability of such technique to spectrum estimation of signals which are subject to random amplitude and phase modulation. The proposed approach is justified by the fact that all signals usually referred to as "periodic" have some amplitude and phase variation from period to period. For example an active sonar system transmits a periodic pulse train to detect targets. The received pulses are n perfectly periodic due to random modulation of the pulses from scattering and attenuation (Hinich 2003). Also El Niño signals are recognized as amplitude and phase modulated (Allen and Robertso996). The assumed concept of random modulation modeling is described in section 2. In section 3 the theoretical results related to the modulated signal DFT spectrum are presented bh in the case of a noiseless signal and in the case of signal observations corrupted by random errors. 2. MODULATION MODELING Let us consider finite duration time series of complex-valued signal observations at discrete time moments Re[ ] i Im[ ] , t 0,1,..., . We assume that the analyzed signal is of deterministic character and comprises some regular oscillations. Such a signal can represent for example the monochromatic oscillation o0t A0 exp(i 0t i 0 ) , t with frequency 0 , constant amplitude A0 , and constant phase 0 . Now, let us assume that the signal values amplitudes and phases are distorted by random additive fluctuations at and t , t respectively, which are realizations of independent and identically distributed random variables. We also assume that amplitude distortions are independent of phase distortions. Hence, we deal with an amplitude and phase modulated signal satisfying the mathematical model rt (1 at ) exp(i t ) exp(i t ) at exp(i t ) , (2) where exp(i t ) and at exp(i t ) , t are realizations of independent and identically distributed random variables. Let the distribution of t be uniform on the interval , which gives immediately for t ( , ) ( t ~ U ( , ) ) with 0 1 sin( ) , m E exp(i t ) exp(iu )du 2 (3) 2 1 2 2 2 E exp(i t ) m E exp(i t ) m2 exp(iu ) du m 2 1 m 2 . s one can clearly see the mean value m satisfies 0 m 1 and is equal to zero only for , so that we also have 0 About the distribution of the real-valued random variable at we assume only that Ea at 0 and Ea at 2 (4) (if 0 there is only phase modulation of the signal), which further implies for a t ma Ea E (1 at ) exp(i t ) Ea (1 at ) E exp(i t ) m , (5) Ea E (1 at ) exp(i t ) ma Ea (1 at ) 2 E exp(i t ) 2 ma m2 , and obviously we have . For example, at can be uniformly distributed on the interval [ A, A] , where A 0 , and then A2 / 3 . 3. MODULATED SIGNAL SPECTRA It follows immediately from (2) that in order to analyze the DFT spectrum of our model signal we shall consider the case of signals rt zt , t modulated by series of independent and identically distributed complex-valued random variables zt . Indeed, the case of zt exp(i t ) corresponds to phase modulation, the case of zt (1 at ) exp(i t ) to phase and amplitude modulation, and the case of real-valued modulation series zt at is needed for amplitude modulation analysis, respectively. In all three mentioned cases the random variables zt have constant mean value and variance according to (3), (4) and (5). In order to compute the DFT of the modulated signals of the form rt zt , t we apply the well-known convolution formula (Gasquet and Witomski 1999) ~ rk k j 0 ~ z ok j ~j j k ~ zj (6) for k 0,1,..., . Hence, if we want to analyze the DFT of the modulated signal ~ , it is necessary to rk characterize the statistical properties of the random modulating series DFT ~ j , z j 0,1,..., . Such a characterization is given in the following lemma. Lemma 1. If a series zt , t its DFT satisfies represents realizations of uncorrelated complex-valued random variables with constant mean value E z zt E z ~0 z , E z ~k z and finite variance E z zt 0 for kl 0 , then 1,..., , and E z ( ~k z where the symbol Proof. kl 1 N denes the Kronecker delta. E z ~k )( ~l z z E z ~l ) z The first property Ez ~0 follows immediately from the assumptions of the lemma and the z definition of the DFT. In order to prove the equality E z ~k 0 for k 1,..., , observe that z 1 1 1 E z zt exp( i 2 kt / N ) E z zt exp( i 2 kt / N ) exp( i 2 kt / N ) N t 0 Nt 0 N t 0 1 1 exp( i 2 k ) 0 . exp( i 2 kt / N ) N t 0 exp( i 2 k / N ) Also from the definition of the DFT and from the assumed zero correlation of the random variables considered we have 1 E z ( ~k E z ~k )( ~l E z ~l ) z z z z E z ( zt E z zt ) exp( i 2 kt / N ) ( z s E z z s ) exp(i 2 ls / N ) N2 t 0 s 0 E z ~k z 1 N2 1 N E z ( zt t 0 s 0 kl Ez zt )( z s E z z s ) exp( i 2 ( kt ls ) / N ) 1 N2 t 0 exp( i 2 (k l )t / N ) l )t / N ) 1 exp( i 2 (k l )) 1 exp( i 2 ( k l ) / N ) 0 since t 0 exp( i 2 ( k for k l, which proves the second property. Now, as we know the mean values of the random variables ~j , j z according to the above lemma, we easily obtain from (6) Ez ~ rk and further since E z ( ~k E z ~k )( ~l z z z yields for k ~ ok for E z ~l ) z kl (7) / N the same formula Ez ~ rk 2 Ez ~ rk Ez j 0 ~ z ok j ( ~j E z ~j ) z ~ k (z j 2 z E z ~j ) z (8) where k 0 the 2 t 0 last equality N j 0 follows from ~ ok 2 j 2 j k the N k 0 N well-known property of ~ 2 ok , the DFT ~ ok / N (Gasquet and Witomski 1999). From the Schwartz inequality (Bremaud 2002) we also have the following estimate related to rk rl the covariance of the random variables ~ and ~ Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez ~ rk Ez ~ rk Ez ~ rl 2 Ez ~ rl (9) . N where B Let us remark that for bounded signals satisfying B , t 0 is a real number, we immediately obtain from (8) and (9), 2 2 2 2 2 zB zB Ez ~ Ez ~ rk rk B2 Ez (~ Ez ~ )(~ Ez ~ ) rk rk rl rl , . (10) N2 k 0 N N In such a case the random variables ~ , k representing the spectrum of the rk modulated signal, have equal variances decreasing asymptically to zero as N (which implies that their covariances also decrease asymptically to zero). Bounded signals are of course of primary interest in this work since we intend to investigate spectra of regular oscillations of stationary character, modulated by random amplitude and phase fluctuations. In view of the equalities in (3), (4), (5) and lemma 1, the equalities (7), (8) and inequality (9) hold in the case of phase modulation series zt exp(i t ) with m and z2 1 m 2 , as well as in the case of amplitude and phase modulation series zt (1 at ) exp(i t ) with m and m 2 . Thus, taking into account the equality (7) and the inequalities in (10), we see that if a bounded signal is phase or amplitude and phase modulated according to our model, then its DFT spectrum will have random character. The modulated signal DFT ~ ~ rk rk spectrum has mean values E z rk ok , k with random deviations ~ Ez ~ imposed, i.e. random phase or amplitude and phase modulation diminishes the amplitudes of the original bounded signal spectrum by the factor 0 m 1 and corrupts the spectrum by additive deviations of random character, which have zero mean and known variance. In consequence, any peaks present in the spectrum of the original bounded signal may be less distinguishable in the modulated signal spectrum. Distortion of the bounded signal spectrum values close to 0 (i.e. if phase in the case of random phase modulation is small for ~ E ~ 2 0 . In the her extreme case of fluctuations are small), since then m z 1 and E z rk z rk we have m z 0 , and then the modulated signal spectrum will have purely stochastic character without any frequencies distinguished, so that any peaks present in the original signal spectrum will be completely blurred. In accordance with our model (2), if an original signal is only amplitude modulated, then rt rat , where rat at , t so we have ~ ok ~ , k 0,1,..., . rk ~ rak Hence, the spectrum of the amplitude modulated signal differs from the original signal rak spectrum only by the additive distortion terms ~ , k 0,1,..., . Having in view the ssumptions in (4) and putting zt at with z we can apply again lemma 1, which further yields E z ~ ok , as well as equality (8) and inequality (9) for random variables rk ~ ~ ~ E ~ , k 0,1,..., . Consequently, if the original signal is bounded, then by (10) rak rk z rk the distortion terms will have equal variances tending to zero as N , so that they will also have asymptically zero covariances. This means that they will n distort completely the original bounded signal spectrum on which they are superimposed. Some small peaks present in the original signal spectrum can be smohed due to amplitude modulation but possibly larger ones will be still distinguishable. In Figure 1. the above described effects of amplitude and/or phase signal modulation on the DFT spectrum of a model signal (sum of 7 monochromatic complex harmonics with constant amplitudes and phases) is shown. The random variables representing amplitude and phase fluctuations were distributed according to uniform distribution at ~ U ( A, A) and , ) with parameters A 1.0 and / 2 , respectively. Assume now that the time series of the original signal values , t corrupted by random errors t , according to the model ~U( is where yt (11) t t, are realizations of uncorrelated complex-valued random variables having zero mean 0 and finite second moment E | |2 Let us see what happens if the corrupted signal values are submitted to random modulation ryt zt yt , t of the same kind as considered above. For a modulating series zt , t satisfying the assumptions of lemma 1, we obtain on the basis of the DFT convolution formula (Gasquet and Witomski 1999) ~ ryk k j 0 ~ ~ yk j z j ~ yN j k ~ zj (12) 148 for k which by assertion of the lemma yields E z ~yk ~k , r y E E z ~yk r E ~k y ~ E (ok ~ )m k z ~ ok . (13) frequency Fig. 1. Spectra of a model series representing sum of 7 monochromatic oscillations (N=2048): original signal (black), phase modulated signal ( / 2 , blue), amplitude modulated signal ( A 1.0 , green), amplitude and phase modulated signal ( / 2 , A 1.0 , brown). Furthermore, simple calculation shows that and analogously as in (8) we obtain N which by (11) together with the equalities E lemma 1 finally gives E E z ~yk r E E z ~yk r yt ( ~k y 0, E t s ~ 2 ok ) , st , s, t and E yt E ~k 2 2 t (14) N for k as well as E E z ( ~yk E E z ~yk )( ~yl E E z ~yl ) r r r r E E z ~yk r E E z ( ~yk r E E z ~yk )(~yl r r 2 E E z ~yl r E E z ~yl ) r (15) E E z ~yk r E E z ~yl r N 0,1,..., . 150 Hence, we see that the presence of errors corrupting the original signal values does n change the character of the DFT spectrum of the modulated series ryt zt yt , t 0,1,..., . Indeed, the formulae (13), (14), (15) are analogous to (7), (8), (9), respectively, except for addition of 2 2 2 the term 2 ( z2 ) / N E z zt / N which now occurs because of non-zero second moment of the errors. For bounded signals to (10) , namely, B, t we can easily obtain the inequalities analogous 2 B2 , (16) 2 2 E Ez (~yk r 0,1,..., . E Ez ~yk )(~yl r r E Ez ~yl ) r Thus, our earlier assertions concerning the character of the modulated signal spectrum hold also in the case of a bounded signal corrupted by uncorrelated random errors which have zero mean and finite second moment. This means that only in the case of amplitude modulation the DFT spectrum of the modulated corrupted signal can resemble the one of the corrupted signal itself. 4. SUMMARY The properties of the DFT spectrum examined in this work are helpful for understanding the possible changes such a spectrum undergoes in the case of random amplitude and phase modulation of an original signal. Our modulation model includes additive distortions of stochastic nature in the amplitudes and phases of original signal values at observation moments. For bounded signals of deterministic character (like a sum of monochromatic oscillations with constant amplitudes and phases) it is proved that occurrence of random phase modulation of the signal can completely change the character of its DFT spectrum. Namely, the phase modulated signal spectrum can show purely stochastic character without any frequencies distinguished, like the spectrum of white noise. In the case of random amplitude modulation of such a signal it is shown that the modulated signal spectrum can still resemble the spectrum of the original signal, although small peaks can be significantly smohed. Similar conclusions can be deduced also in the case of a deterministic signal which is corrupted at the moments of observation by uncorrelated random errors with zero mean and finite second moment. It is worth remarking that our conclusions agree with observations of Ni and Huo (2007), concerning the importance of phase and amplitude information in signal and image reconstruction. The concept of phase randomization used for obtaining multivariate surrogate time series (Mammen and Nandi 2008) with distribution similar to observed series is also related to the subject considered here. Hinich (2003) used similar approach to amplitude modulation modeling, as the one applied in this work, to derive statistics for detecting randomly modulated pulses in noise. The Singular Spectrum Analysis (SSA) method has an important property, first ned by Vautard and Ghil (1989), that it may be used directly to identify modulated oscillations in the presence of noise. Allen and Robertson (1996) proposed a generalization of the "Monte Carlo SSA" algorithm which allows for objective testing for the presence of modulated oscillations at low 151 signal-to-noise ratios in multivariate data. They demonstrated the application of the test to the analysis of interannual variability in tropical Pacific sea-surface temperatures. Acknowledgements. This research work was supported by the Polish Ministry of Science and Higher Education through the grant No. N N526 160136 under leadership of Dr Tomasz Niedzielski at the Space Research Centre of Polish Academy of Sciences. REFERENCES Allen M.R., Robertson A.W. (1996) Distinguishing Modulated Oscillations from Coloured Noise in Multivariate Datasets, Climate Dynamics, Vol. 12, No. 11, 775­784. Blackledge J.M. (2003) Digital Signal Processing, Horwood Publishing, Chichester, West Sussex, England. Bloomfield P. (2000) Fourier Analysis of Time Series: An Introduction, Wiley, New York. Bremaud P. (2002) Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis, Springer Verlag Inc., New York. Brillinger D.R. (1975) Time Series ­ Data Analysis and Theory, Holt, Rinehart and Winston Inc., New York. Cooley J.W. and Tukey J.W. (1965) An Algorithm for the Machine Calculation of Complex Fourier Series, Mathematics of Computation, Vol. 19, 297-301. Foster G. (1996a) Time Series by Projection I: Statistical Properties of Fourier Analysis, The Astronomical Journal, Vol. 111, No. 1, 541-554. Foster G. (1996b) Time Series by Projection II: Tensor Methods for Time Series Analysis, The Astronomical Journal, Vol. 111, No. 1, 555-566. Gasquet C., Witomski P. (1999) Fourier Analysis and Applications ­ Filtering, Numerical Computation, Wavelets, Springer Verlag Inc., New York. Hinich M.J. (2003) Detecting Randomly Modulated Pulses in Noise, Signal Processing, Vol. 83, Issue 6, 1349-1352. Koopmans L.H. (1974) Spectral Analysis of Time Series, Academic Press, New York. Mammen E. and Nandi S. (2008) Some Theoretical Properties of Phase-Randomized Multivariate Surrogates, Statistics, Vol. 42, No. 3, 195-205. Ni X. and Huo X. (2007) Statistical Interpretation of the Importance of Phase Information in Signal and Image Reconstruction, Statistics and Probability Letters, Vol. 77, Issue 4, 447-454. Popi ski W. (1997) On Consistency of Discrete Fourier Analysis of Noisy Time Series, Artificial Satellites ­ Journal of Planetary Geodesy, Vol. 32, No. 3, 131-142. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T. (1992) Numerical Recipes ­ The Art of Scientific Computing, Cambridge University Press, Cambridge. Singleton R.C. (1969) An Algorithm for Computing the Mixed Radix Fast Fourier Transform, IEEE Transactions on Audio and Electroacoustics, Vol. AU-17, No. 2, 93-103. Speed T.P. (1985) Some Practical and Statistical Aspects of Filtering and Spectrum Estimation, In Price J. F. (Editor), Fourier Techniques and Applications, Plenum Press, New York, 101-118. 152 Vautard R., Ghil M. (1989) Singular Spectrum Analysis in Nonlinear Dynamics with Applications to Paleoclimatic Time Series, Physica D, Vol. 35, 395­424. Walker A.M. (1971) On the Estimation of a Harmonic Component in a Time Series with Stationary Independent Residuals, Biometrika, Vol. 58, No. 1, 21­36. Received: 2010-12-11, Reviewed: 2011-02-04, Accepted: 2011-02-07.

Journal

Artificial Satellites – de Gruyter

Published: Jan 1, 2010

References

  • Distinguishing Modulated Oscillations from Coloured Noise in Multivariate Datasets
    M. Allen