WIRELESS DIGITAL DEMODULATION SYSTEM VIA HIERARCHICAL MULTIRESOLUTION EMPIRICAL MODE DECOMPOSITION APPROACH

Transcription

1 Proeedings of APSIPA Annual Summit and onferene Deember 015 APSIPA 015 WIRELESS DIGIAL DEMODULAION SYSEM VIA HIERARHIAL MULIRESOLUION EMPIRIAL MODE DEOMPOSIION APPROAH Weihao Kuang, Bingo W. K. Ling, Zhijing Yang and Wai-Lok Woo Shool of Information Engineering, Guangdong University of ehnology Panyu Distrit, Guangzhou, Guangdong Provine, , hina. ABSRA his aer rooses a hierarhial multiresolution based emirial mode deomosition aroah for erforming a wireless digital demodulation. he waveform orresonding to eah digital symbol is reresented as the sum of the intrinsi mode funtions. Eah obtained intrinsi mode funtion of eah symbol is further deomosed via a disrete osine transform aroah. First, zeros are inserted in eah intrinsi mode funtion in the disrete osine transform domain. hen, the next level emirial mode deomosition is erformed in the time domain. he disrete osine transform oeffiients of the next level intrinsi mode funtions where the zeros are added are removed in the disrete osine transform domain. his ste is reeated and the obtained intrinsi mode funtions in eah level form a ditionary. he reeived signals orruted by the additive noises with unknown distributions and distorted by nonlinear hannels are reresented by both the weight vetors and the residue vetors based on the ditionary at eah level of deomosition. he deoding sheme is to find the orresonding symbol suh that the sum of the residue vetors in various levels of the deomosition is minimized. Exerimental results show that the deoding auray is higher than that of the onventional mathed filter aroah. 1. INRODUION Many devies suh as home alianes, offie equiments, mobile handsets and lato omuters are onneted together via wireless ommuniation networks. he ommon wireless ommuniation networks inlude Wifi, Zigbee, Bluetooth and ellular mobile networks [1]-[3]. o transmit signals through wireless hannels, signals are usually modulated to high frequeny bands [4]. his is beause high frequeny signals an transmit more far away than low frequeny signals. he reeived signals are then demodulated bak to the baseband signals. Hene, modulation and demodulation are usually emloyed in wireless ommuniation systems and they lay a very imortant role in our daily life [1]-[4]. he ommon digital demodulation system is via the mathed filter aroah [5], [6]. A reeived signal is rojeted to a bank of mathed filters. he symbol orresonding to the maximum absolute rojetion value is taken as the orresonding deoded symbol. Here, the waveforms reresenting the symbols have to be orthonormal to eah others. Nevertheless, the waveforms are usually distorted by nonlinear hannels (he hannel harateristis are nonlinear with reset to the transmitted signals.). Hene, the reeived waveforms in the ratial situations are not orthonormal to eah others. Also, the wireless hannel noise is required to be an additive white Gaussian distributed. However, this is also not the ase in many ratial situations. A hierarhial multiresolution based emirial mode deomosition of signals is imlemented via the disrete Fourier transform [9]. However, as the disrete Fourier transform is a omlex valued transform, the required omutational ower for the further roessing is high. In this aer, a hierarhial multiresolution based emirial mode deomosition of signals is imlemented via the disrete osine transform. As the disrete osine transform is a real valued transform, the required omutational ower for the further roessing is signifiantly redued. Moreover, this aer further alies the hierarhial multiresolution based emirial mode deomosition method for erforming the wireless digital demodulation. It is worth noting that the onventional mathed filter aroah is a linear aroah based on both the orthonormality among the waveforms and the additive white Gaussian distributed hannel noises. On the other hand, although the signals are reresented as the sum of the intrinsi mode funtions, the intrinsi mode funtions are nonlinear with reset to the signals. Hene, our roosed aroah is a nonlinear aroah whih does not require the orthonormality among the waveforms and the additive white Gaussian distributed hannel noises. herefore, our aroah would be more suitable for nonlinear hannels [7], [8]. Moreover, eah symbol in the onventional mathed filter aroah is reresented by a single inut single outut linear time invariant filter while eah symbol in our roosed aroah is reresented by a weighted vetor. As the deoding roess of the onventional mathed filter aroah is to omare the magnitudes of the rojeted values in the one dimensional sae while the deoding roess of our roosed aroah is find the solution of an otimization roblem in a high dimensional sae, the degree of freedom of our roosed aroah is muh higher than that of the mathed filter aroah APSIPA 1166 APSIPA AS 015

2 Proeedings of APSIPA Annual Summit and onferene Deember 015 APSIPA 015 he outline of this aer is as follows. he roosed digital demodulation sheme is formulated as an otimization roblem via a hierarhial mutliresolution based emirial mode deomosition aroah. he details are disussed in Setion. omuter numerial simulation results are resented in Setion 3. Finally, a onlusion is drawn in Setion 4.. PROBLEM FORMULAION Suose that there are symbols in a wireless digital ommuniation system and eah symbol is reresented by an N oint disrete time sequene. Let x for 0, be the vetors reresenting these sequenes. Assume that x for 0, are distorted by a nonlinear hannel. Let x for 0, be the vetors reresenting these distorted sequenes. Suose that x for 0, an be reresented by the sum of its intrinsi mode funtions. Let F be the matrix with its olumns being the vetors reresenting the intrinsi mode funtions of x for 0,. Obviously, F an be obtained via erforming the emirial mode deomosition on x for 0,. Define F F0 F 1. Here, the total number of rows of F is equal to the signal length and the total number of olumns is equal to the total number of intrinsi mode funtions. It is worth noting that the total number of intrinsi mode funtions is usually smaller than the length of the signal. Hene, F is a tall matrix. Moreover, the intrinsi mode funtions are usually linear indeendent. Hene, we an assume that F F is invertible. Let l for 0, be the weight vetor reresenting x using F. hat is, x Fl for 0,. hen, we have 1 l F F F x for 0,. It is well known in the wavelet theory that the frequeny bands at high sale wavelets are narrower than those at the low frequeny bands. Hene, fine details of the signals an be extrated out if the signals are reresented using high sales of wavelets. Similarly, if the intrinsi mode funtions an be further deomosed, then more fine information an be emloyed for the demodulation. Nevertheless, it is worth noting that the intrinsi mode funtions annot be further deomosed if the emirial mode deomosition is diretly alied to the intrinsi mode funtions. In order to address this diffiulty, the deomosition of the signal is erformed via a disrete osine transform aroah. he detail roedures are as follow. First, the disrete osine transforms of the intrinsi mode funtions are omuted. Seond, zeros are inserted in the intrinsi mode funtions in the disrete osine transform domain. hird, the inverse disrete osine transforms of these zero inserted intrinsi mode funtions are omuted. Fourth, the next level emirial mode deomosition of eah zero inserted intrinsi mode funtion is erformed in the time domain. Fifth, the disrete osine transforms of these next level intrinsi mode funtions are omuted. Sixth, the disrete osine transform oeffiients of the next level intrinsi mode funtions are removed where the removed oeffiients are loated exatly the same as that the zeros are added in the disrete osine transform domain. Seventh, the inverse osine transforms of these zero removed intrinsi mode funtions are omuted. As a result, the next level intrinsi mode funtions are obtained. Similarly, denote F ~ as the matrix with its olumns being the vetors reresenting the next level intrinsi mode funtions. Let ~ l for 1 0,, be the weight vetor reresenting x using F ~. hat is, x Fl for 0,. Similarly, we also assume that F F is invertible. Hene, we have 1 l F F F x for 0,, 1. In fat, more intrinsi mode funtions in higher levels of the deomosition an be obtained by reeating the above roedures. Now onsider the ase that the transmitted signals are orruted by the noises with an unknown distribution and distorted by a nonlinear hannel. Let y be the reeived signal. In this ase, it is not guaranteed that y an be reresented as a linear ombination of the olumns of F or F ~ and we have the reonstrution errors. Denote r and l as the reonstrution error and the weight vetor by reresenting y using F, resetively. Similarly, denote ~ r and ~ l as the reonstrution error and the weight vetor by reresenting y using F ~, resetively. hat is, y Fl r and ~ y F ~ l ~ r. It is worth noting that a new set of intrinsi mode funtions are required to be omuted for a new signal based on the onventional emirial mode deomosition aroah. However, as the differene between y and x for a artiular value of 0, is only due to the resene of the additive noise and the differene due to the time varying hannel, the required omutational ower an be redued if the new signal is reresented by the linear ombination of the intrinsi mode funtions of the original signal lus an error signal without erforming the emirial mode deomosition again. he robustness of the emirial mode deomosition deends on the energy of the additive noise and the stationarity of the hannel. From the demodulation viewoint, we need to determine l and ~ l for a new reeived signal y. For a slow varying hannel and a low noise energy environment, both y Fl and ~ ~ y F l are small. herefore, the demodulation sheme an be formulated as the following otimization roblem: Problem ( P ) ~ ~ min y Fl y F l, ~ ~ l l l l,, l. subjet to l and 0,, 1 ~ APSIPA 1167 APSIPA AS 015

3 Proeedings of APSIPA Annual Summit and onferene Deember 015 APSIPA 015 * Denote as the otimal solution of Problem ( P ). hen, the * deoding rule is to assign y to. Denote X and X as the matries with their olumns being x and x for 0,. Let I be the identity matrix. Suose that the waveforms for reresenting the symbols are orthonormal to eah others. hen, we have X X I. If there is no noise orruted in the hannel and there is no distortion in the hannel, then only one element in X y is one and all other elements are zero. Hene, by loating the osition of the nonzero element in X y, the transmitted symbol an be deoded. his is the working rinial of the onventional mathed filter aroah. However, as the hannel is nonlinear, in general X X I. Even though there is no noise orruted in the hannel, it is no longer true that there is only one nonzero element in X y. herefore, the deoding error based on the onventional mathed filter aroah ould be very large. On the other hand, it is still true in the nonlinear hannel environment that both x Fl and x Fl for 0,. If there is no noise orruted to the hannel and the nonlinear hannel is stationary, then there is no deoding error based on our roosed aroah. As our roosed aroah an eliminate the deoding error due to the stationary nonlinear hannel harateristis, our roosed aroah in general an ahieve a lower deoding error omared to that of the mathed filter aroah. 3. NUMERIAL OMPUER SIMULAION RESULS In this aer, is hosen for the demonstration. his is beause the binary ommuniation system is the most ommon wireless digital ommuniation systems emloyed in ratial situations. o reresent a waveform by a disrete time sequene, the samling rate is at least higher than the Nyquist samling rate of the signal. Hene, N ould not be too small. In this aer, N 56 is hosen whih is large enough for most simle waveforms. o order to have a fair omarison to the mathed filter aroah, x for 0, should be hosen as the signals whih are orthonormal to eah others. Here, n x0 n sin and N n x1n os for n 0,, N 1 are hosen. his is N beause they are the most ommon orthonormal waveforms used in binary ommuniation systems. o demonstrate the effetiveness of our roosed method, the following three ases are onsidered. he first ase onsiders the situation that there is no deterministi distortion introdued by the hannel. he seond ase onsiders the situation that the deterministi distortion introdued by the hannel is linear and time varying. In this ase, the deterministi hannel distortion an be modeled by a matrix multiliation. Let the matrix be H. Here, we assume that there is no attenuation introdued by this deterministi distortion beause of the simliity reason. his imlies that H is unitary. he last ase onsiders the situation that the deterministi distortion introdued by the hannel is nonlinear but time invariant. Here, the deterministi hannel distortion funtion is modeled by a olynomial funtion. his is beause aylor series an be emloyed for modeling a wide lass of nonlinear funtions. Let the order, the D gain and the roots P of the olynomial be P, for 0,, P 1, G and resetively, as well as the olynomial funtion be h. P1 hat is, x n h x n G x n 0 for 0,. Here, for 0,, P 1 are assumed to be uniformly disturbed between -1 and 1 beause the dynamial ranges of these sinusoidal waveforms are between -1 and 1. Besides, G 1.3 is hosen beause the D gain of this nonlinear distortion funtion is aroximately reserved. Moreover, P 54 is hosen beause more terms in the olynomial an ahieve a more aurate aroximation of a nonlinear funtion. On the other hand, the hannel is also orruted by an additive random noise. Denote the random noise vetor be v. Hene, we have x x for 0, and y x v for some 0, for the first ase, x Hx for 0, and y Hx v for some 0, for the seond ase, and y x v for some 0, for the last ase. In this aer, two tyes of random noises are onsidered. hey are the Gaussian disturbed random noise and the Rayleigh distributed random noise. hey are hosen beause they are ommonly emloyed for wireless ommuniation hannels. For the Rayleigh distributed noise, it is in the form of R e, where is the arameter of the Rayleigh distribution ontrolling the noise energy. For the Gaussian distributed noise, it is in the form of 1 G e, where and are the mean and the standard deviation of the distribution, resetively. Here, ontrols the noise energy and 0 is hosen beause of the simliity reason. In the hierarhial multiresolution based emirial mode deomosition algorithm, only two levels of the deomositions are erformed beause of the simliity reason. Hene, we only have F and F ~. Also, the total numbers of zeros to be inserted and removed are exatly equal to the length of the signal. Also, they are inserted and removed at the highest frequeny band APSIPA 1168 APSIPA AS 015

4 Proeedings of APSIPA Annual Summit and onferene Deember 015 APSIPA 015 Define the signal to noise ratio as SNR E x x. Here, v v E is the exetation oerator. Figure 1 and Figure lot the bit error rates against the signal to noise ratios for the hannel without any deterministi distortion but with the Gaussian disturbed noise and the Rayleigh disturbed noise, resetively. From Figure 1 and Figure, we an see that our roosed method ahieves the same erformanes as those of the mathed filter aroah for both the Gaussian disturbed noise and the Rayleigh disturbed noise. Sine the mathed filter aroah ahieves the otimal solutions for both the Gaussian disturbed noise and the Rayleigh disturbed noise for the hannel without any deterministi distortion, this imlies that our roosed method also ahieves the otimal solutions for the hannel without any deterministi distortion for both the Gaussian disturbed noise and the Rayleigh disturbed noise. Figure 3 and Figure 4 lot the bit error rates against the signal to noise ratios for the hannel with the linear time varying deterministi distortion as well as with the Gaussian disturbed noise and the Rayleigh disturbed noise, resetively. It an be seen from Figure 3 and Figure 4 that the bit error rates for the mathed filter aroah of both the Gaussian disturbed noise and the Rayleigh disturbed noise dro very slowly as the signal to noise ratios inrease. his imlies that the mathed filter aroah is not good for the hannel with the linear time varying deterministi distortion. his is beause X Hx is no longer equal to the vetor with the unique nonzero element loated at the th element. On the other hand, our roosed method still erforms very well for the hannel with the linear time varying deterministi distortion for both the Gaussian disturbed noise and the Rayleigh disturbed noise. his is beause the linear time varying harateristi of the hannel is aroximated by its nonlinear harateristi. Finally, Figure 5 and Figure 6 lot the bit error rates against the signal to noise ratios for the hannel with the nonlinear time invariant deterministi distortion as well as with the Gaussian disturbed noise and the Rayleigh disturbed noise, resetively. Similarly to the above, it an be seen from Figure 5 and Figure 6 that the bit error rates of both the Gaussian disturbed noise and the Rayleigh disturbed noise for the mathed filter aroah are saturated. Here, saturated means that the robability of error er symbol is equal to 0.5. his imlies that the mathed filter aroah basially fails for erforming the demodulation for the hannel with the nonlinear time invariant deterministi distortion. On the other hand, it an be seen from Figure 5 and Figure 6 that our roosed method still erforms very well for the hannel with the nonlinear time invariant deterministi distortion for both the Gaussian disturbed noise and the Rayleigh disturbed noise. Existing mathed filter aroah Figure 1. hannel without any deterministi distortion but with additive Gaussian distributed noise. Existing mathed filter aroah Figure. hannel without any deterministi distortion but with additive Rayleigh distributed noise. Existing mathed filter aroah Figure 3. hannel with linear time varying deterministi distortion and additive Gaussian distributed noise. Existing mathed filter aroah Figure 4. hannel with linear time varying deterministi distortion and additive Rayleigh distributed noise APSIPA 1169 APSIPA AS 015

5 Proeedings of APSIPA Annual Summit and onferene Deember 015 APSIPA 015 Existing mathed filter aroah Figure 5. hannel with nonlinear time invariant deterministi distortion and additive Gaussian distributed noise. Existing mathed filter aroah Figure 6. hannel with nonlinear time invariant deterministi distortion and additive Rayleigh distributed noise. 4. ONLUSION his aer rooses a hierarhial multiresolution based emirial mode deomosition aroah for erforming the demodulation for wireless digital ommuniation systems. he waveform orresonding to eah digital symbol is reresented as the sum of the intrinsi mode funtions. Eah obtained intrinsi mode funtion of eah symbol is further deomosed in higher levels. he obtained intrinsi mode funtions in eah level form a ditionary. he reeived signals orruted by the additive noises with unknown distributions and distorted by nonlinear hannels are reresented by both the weight vetors and the residue vetors based on the ditionary at eah level of deomosition. he deoding sheme is to find the orresonding symbol suh that the sum of the residue vetors in various levels of the deomosition is minimized. Sine the emirial mode deomosition aroah an ature the nonlinear harateristis of the hannel, the exerimental results show that the deoding auray based on our roosed method is higher than that based on the onventional mathed filter aroah. networks for smart grid, IEEE Wireless ommuniations, vol. 0, no. 3, , 013. [] X. Li,. Jiang and Q. Zhang, Binary linear multiast network oding on ayli networks: riniles and aliations in wireless ommuniation networks, IEEE Journal on Seleted Areas in ommuniations, vol. 7, no. 5, , 009. [3] S. Misra, J. Mahaatro and M. Mahadevaa, Random room mobility model and extra-wireless body area network ommuniation in hosital buildings, IE Networks, vol. 4, no. 1, , 014. [4] F. du Burk, J. P. Wallerand and A. N. Gonharov, High-frequeny modulation transfer tehnique for ultra-high resolution setrosoy of I, IEEE ransations on Instrumentation and Measurement, vol. 50, no., , 001. [5] Y.. Eldar, A. V. Oenheim and D. Egnor, Orthogonal and rojeted orthogonal mathed filter detetion, Signal Proessing, vol. 84, no. 4, , [6] P. L. Lee, H.. hang and. Y. Hsieh, A brainwave-atuated small robot ar using ensemble emirial mode deomosition-based aroah, IEEE ransations on Systems, Man and ybernetis, Part A: Systems and Humans, vol. 4, no. 5, , 01. [7] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih, Q. Zheng, N. Yen,. ung and H. Liu, he emirial mode deomosition and the Hilbert setrum for nonlinear and nonstationary time series analysis, Proeedings of the Royal Soiety London A: Mathematial, Physial and Engineering Sienes, vol. 454, , [8] Z. Yang, L. Yang and. Qing, A method to eliminate riding waves aearing in the emirial AM/FM demodulation, Digital Signal Proessing, vol. 18, no. 4, , 008. [9] W. Kuang, Z. Yang, B. W. K. Ling,. Y. F. Ho and Q. Dai, Nonlinear and adative undeimated hierarhial multiresolution analysis for real valued disrete time signals via emirial mode deomosition aroah, Digital Signal Proessing, vol. 45, , 015. REFERENES [1] Q. D. Ho, Y. Gao and. Le-Ngo, hallenges and researh oortunities in wireless ommuniation APSIPA 1170 APSIPA AS 015