Journal of Computer and Communications, 2014, 2, 8792 Published Online July 2014 in SciRes. http://www.scirp.org/journal/jcc http://dx.doi.org/10.4236/jcc.2014.29012 How to cite this paper: Xu, G.L., Wang, X.T., Xu, X.G. and Shao, L.M. (2014) Amplitude and Phase Analysis Based on Signed Demodulation for AMFM Signal. Journal of Computer and Communications, 2, 8792. http://dx.doi.org/10.4236/jcc.2014.29012 Amplitude and Phase Analysis Based on Signed Demodulation for AMFM Signal Guanlei Xu1*, Xiaotong Wang2, Xiaogang Xu2, Limin Shao1 1Ocean Department, Dalian Navy Academy, Dalian, China 2Navigation Dep art men t, Dalian Navy Academy, Dalian, China Email: *xgl_86 @163.com Received May 2014 Abstract This paper proposes a new amplitude and phase demodulation schem e different from the tradi tional method for AMFM signals. The traditional amplitude demodulation assumes that the am plitude should be nonnegative, and the phase is obtained under the case of nonnegative ampli tude, which approximates the true amplitude and phase but dis tor ts the true amplitude and phase in some cases. In this paper we assume that the amplitude is signed (zero, positive or negative), and the phase is obtained under the case of signed amplitude by optim iza ti on, as is called signed demodulation. The main merit of the signed demodulation lies in the revelat ion of senseful p hysi cal meaning on phase and frequency. Experiments on the realworld data show the effici ency of the me th od. Keywords Amplitude Demodulation, Phase, Hilbert Transform, Signed Demodulation 1. Introduction In many signal processing fields such as communication, wireless navigation and machine systems, the modula tion and demodulation are often used to process the amplitude component and the phase component [1][10]. In fact, the basic problem in processing AMFM signals is d emodulation, i.e., estimation of the information stored in the amplitude and phase signals while given the composite signal. For monocomponent AMFM signals many successful demodulation approaches have existed, ranging from standard methods such as Hilbert transform demodulation [1] or phaselocked loops (PLL’s) to the recent energy separation algorithm (ESA) that tracks and demodulates the energy of the source producing the AMFM signal using instantaneous nonlinear differential operators [2][18]. While each of these monocomponent algorithms may have its advantages and disadvantages, they more or less offer a solution to the monocomponent AMFM demodulation problem. However, these me thods shown in above only process the positive amplitudes, i.e., they think that the amplitudes should be the ab solute values (or nonnegative). In other words, they fail to demodulate the signed amplitudes (or nonpositive AM component). *
G. L. Xu et al. In this paper, we derive a new scheme, different from the traditional method for AMFM signals that can ob tain the signed amplitude and accordingly senseful physical meaning phase and frequency by optimization. 2. Signed Demodulation 2.1. Traditional Demodulation of Amplitude and Phase For a real function, a direct and simple way to obtain the complex signal is via Hilbert transform [????]. A real function and its Hilbert transform are related to each other in such a way that they together create a so called strong analytic signal . The strong analytic signal can be written with the am plitude and the phase where the derivative of the phase can be identified as the instantaneous frequency. The Hilbert transform defined in the time domain is a convolution between the Hilbert transformer and the real function : ( )( ) ( ) ( )( ) 11 *d x xtH xtxtP tt ττ ππ τ ∞ −∞ ===− ∫ , (1) where the P in front of the integral denotes the Cauchy principal value which expanding the class of functions for which the integral in (1) exist and “*” denotes the convolution operator and “ ” denotes the Hilbert transform operator. In frequency domain, we have the following relation: ( )( )( )( ) sgn H X uXuuXu= + , (2) where ( ) 10 sgn 00 10 for u ufor u for u > = = −< and is the Fourier transform of the real function . We also see that is the inverse Fourier transform of . The biggest advantage of Hilbert transform is that one can directly obtain the amplitude and phase for AMFM components, e.g., a real signal , one can obtain the follows via Hilbert transform Amplitude: , (3) Phase: ( )( )( ) ( ) ( ) ( ) Im lntxtjH xt φ = + , (4) where “ ” denotes the imaginary part operator and “ ” is the natural logarithm operator. After Hilbert transform, turns to a complex signal , which is typical AMFM signal with the AM component and the FM component . In fact, the FM component is a special AMFM signal with the amplitude being 1. This process is called demodulation. However, there is one question: 1) why the Hilbert transform of is but not ? 2) why cannot we obtain the amplitude function rather than that nearly all the demodulation methods obtain? For the first question, Bedrosian’s theorem [15] has yielded the answer. Now let us review the Bedrosian’s theorem. Bedro sian’s theorem [15]: For two real functions and , if then , and if then , , then ( )( ) { } ( )( ) { } 12 12 Hxtx txtHx t= ⋅ , (5) where and are the Fourier transform of and respectively. Bedrosian’s theorem tells us that for two functio ns, and , of which is with low frequency and the other com ponent is with high frequency, then through Hilbert transform we have ( )( ) { } ( )( ) { } ( )( ) coscos sinHatt atHt att φ φφ = = . For the second question, we will answer it in the following few sections via the signed demodulation and some optimizations.
G. L. Xu et al. 2.2. The Proposed Signed Demodulation Method The first work is to o b tain the signed amplitude function out of the positive amplitude function via taking abso lute value of the complex signal . However, it is not hard to see that a function and its Hil bert transform are not absolutely orthogonal (even though they are orthogona l in principle) because of trunca tions in numerical calculations and the boundary effects. Therefore, for a real function with its complex function after Hilbert transform has no more than the following relations: and . (6) Therefore, taking absolute value of the complex signal , we have . Thus, we only can obtain an approximation (i.e. ) of even if we know the exact si gns of . Hence, we have the following method. The process of signed demodulation: 1) For the signal with lowfrequency component and highfrequency component , obtain the complex signal ( )( )( ) { } 12h x tHxtx t= via Hilbert transform, then find all the zero positions (indeed these positions make be the local minima) in , and M is the total num ber of zero positions; 2) Obtain the highfrequency signal by ; 3) Estimate the amplitud e function by ( )( )( ) ( )( ) 1 ifis evenodd ifis oddeven h h xt m xt xt m =− , where and ; 4) Reconstruct the highfrequency signal by ( )( )( ) ( ) ( ) 2 12 sgn h xtxt xtxt= ⋅⋅ where ( ) 10 sgn 00 10 s ss s > = −< . 3. Experiment and Discussion Here we have , , , . Now we use the traditional demodulation method and our signed demodulation to demodulate signal and give the comparison (see Figure 1). The first row ((a) (b) (c)) is the composed two signals with lowfrequency and highfrequency respectively. The second row ((d) (e) (f)) is the demodulated amplitude, the highfrequency signal and the phase respectively by traditional method. The third row ((g) (h) (i)) is the demodulated amplitude, the highfr equency signal and the phase respectively by our method. (a) (b)
G. L. Xu et al. (c) (d) (e) (f) (g) (h)
G. L. Xu et al. (i) Figure 1. The comparison of two methods for demodulation of amplitude and phase. (a) The lowfrequency signal; (b) The high frequency signal; (c) The composed signal by (a) × (b); (d) Demodulated amplitude by traditional method; (e) Demo dulated highfrequency signal by traditional method; (f) The phase of (e); (g) Demodulated amplitude by our method; (h) Demodulated high frequency signal by our method; (i) The phase of (h). Clearly, our demodulation method gives more rational physical sense. We allow our amplitude to be negative, under such case we obtain the rational phase in (i) (compared with (f)). 4. Conclusion This paper proposes a new amplitude and phase demodulation scheme different from the traditional method for AMFM signals. We assume that the amplitude is signed (zero, positive or negative), and the phase is obtained under the case of signed amplitude by optimizatio n, as is called signed demodulation. The main merit of the signed demodulation lies in the revelation of senseful physical meani ng on phase and frequency. Experiments on the realworld data show the ef ficienc y of the method. Acknowledgem ents This work is sponsored by NSFCs (Grant No. 61002052, 61273262, 61250006). References [1] Potamianos, A. and Maragos, P. (1994) A Comparison of the Energy Operator and Hilbert Transform Approaches for Signal and Speech Demodulation. Signal Processing, 37, 95120. http://dx.doi.org/10.1016/01651684(94 )90 16 94 [2] Marago s, P., Kaiser, J.F. and Quatieri, T.F. (1993) On Amplitude and Frequency Demodulation Using Energy Opera tors. IEEE Transactions on Signal Processing, 41, 15321550. http://dx.doi.org/10.1109/78.212729 [3] Marago s, P., K aiser, J.F. and Quatieri, T.F. (1993) Energy Separations in Signal Modulations with Application to Speech An alys is. IEEE Transactions on Signal Processing, 41, 30243051 . http://dx.doi.org/10.1109/78.277799 [4] Gupta, S.C. (1975) PhaseLocked Loops. Proceedings of the IEEE, 63, 291306. http://dx.doi.org/10.1109/PROC.1975.9735 [5] Lindsey, W.C. and Chie, C.M. (1981) A Survey of Digital PhaseLocked Loops. Proceedings of the IEEE, 41043 1. [6] Hawkes, H.W. (1991) Study of Adjacent and CoChannel FM Interference. IEE Proceedings I (Communications, Speech and Vision), 138 , 31932 6. [7] Xu, G.L., Wang, X.T. and Xu, X.G. (20 09 ) TimeVarying FrequencyShifting Signal Assisted Empirical Mode De composition Method for AMFM Signals. Mechanical Systems and Signal Processing, 23, 24582469. http://dx.doi.org/10.1016/j.ymssp.2009.06.006 [8] Brad ley, J.N. and Kirlin, R.L. (1993) PhaseLocked Loop Cancellation of Interfering Tones. IEEE Transactions on Signal Processing, 41 , 391394. http://dx.doi.org/10.1109/TSP.1993.193161
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