Efficient Algorithm of Well-Localized Bases Construction for OFTDM Systems

Transcription

1 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS Efficient Algorithm of Well-Localized ases Construction for OFTD Systems Dmitry Petrov.V.Lomonosov oscow State University Physical Faculty Flotsaya str oscow ussia. dapetroff@gmail.com Abstract This article presents the research wor related to the development of enhanced modulation technique with Orthogonal Time-Frequency Division ultiplexing (OFTD). Utilization of well localized bases allows to improve considerably the efficiency and robustness against intercarrier (C) and intersymbol interference (S) of now existing wideband wireless digital communication systems. Until recently one of the main disadvantages of well-localized bases was the complexity of their construction algorithm. This fact restricted the area of their practical implementation. The algebraic procedure of orthogonal well-localized Weyl- Heisenberg basis construction is considered in the paper. Several basis orthogonality conditions are observed. They allow to formulate computationally efficient orthogonalization procedure on the base of fast Fourier transform. This approach can receive wide implementation in wideband mobile networs (Wi-ax) digital television (DV-T/H) and other telecommunication systems. ndex Terms: well-localized bases OFD OFTD orthogonalization wideband signals.. TODUCTO Construction of wireless high-speed digital communication systems often faces with the situation when real radio channels are time-frequency dispersive. Signal coming to the receiving point have multiple reflections form the media inhomogeneitis (city buildings moving objects atmospheric layers etc.). Among the examples of such channels are short and ultra short radio lines wideband mobile (mobile Wi-ax) and digital TV (DV-T/H) radio lines. ecause of time-frequency dispersion of the signal on the receiver side such effects as multipath propagation amplitude/phase fading Doppler spectrum spreading frequency offset are observed. Their influence results in intercarrier (C) and intersymbol (S) interference which worsen considerably receiving characteristics. n addition such interchannel interferences could not be compensated or filtrated by the ordinary digital processing methods. At present Orthogonal Frequency Division ultiplexing (OFD) is one of the most efficient and widely spread technologies of data transmission in such highly dispersive channels. The structure and the properties of OFD signals are determined by the basis used for its construction. asic functions are taen as the intervals of harmonics with frequencies multiple to some offset frequency o. Their amplitude spectrum has narrow main lobe and slowly decay tails lie sin x x function. This basis can be constructed by uniform shifts of rectangular impulse within the given frequency range. The signal is generated as the liner combination of real or complex information symbols (determined by the signal constellation: QA PSK ets.). Channel equalization is simplified because OFD may be 106

2 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS viewed as many slowly-modulated narrowband signals rather than one rapidly-modulated wideband signal. Physically the appearance of C and S in time-frequency dispersive channels is caused by the loss of orthogonally of disturbed basic functions. As a result the demodulation procedure in the receiver becomes nonimal. The reason of connection breas when a subscriber enlarges its velocity or when the signal/noise limit is exceed is synchronization upset or inaccurate assessment of channel parameters in the conditions of strong frequency dispersion (strong C). The low symbol rate of OFD systems maes the use of a guard interval between symbols affordable maing it possible to handle time-spreading and eliminate S. The effect of time dispersion can be completely compensated but with the loss of spectral and energy efficiency. evertheless the rectangular form of forming function is not imal from the point of C. n addition the level of out-of-band emission in the classical OFD systems is overrated. This is why the problem of intercarrier interference rests actual for OFD systems and in many cases doesn t have satisfactory decision. This article presents the research wor related to the development of enhanced modulation technique with Orthogonal Time-Frequency Division ultiplexing (OFTD) affording to cope with main disadvantages of classical OFD mentioned earlier. n OFTD systems special orthogonal well-localized bases are used to cope with channel time-frequency dispersion [1]. t can be shown that tightly paced orthogonal bases can be constructed only of several initializing function. So in contrast to OFD two forming function differing in faze are used for basis generation (generalized Weyl-Heisenberg basis). n addition there are used shifts not only in frequency domain (OFD) but also in time. These bases minimize the level of channel interference simultaneously in time and frequency domain but do not spoil system s spectral efficiency. Until recently one of the main disadvantages of well-localized bases was the complexity of their construction algorithm. This fact didn t allow to use such bases in portable devices. n this article several basis orthogonality conditions are observed. They allow to formulate practical and computationally efficient orthogonalization procedure on the base of fast Fourier transform. Thus the potential spectrum of implementations of OFTD technology can be considerably broaden. Orthogonal Weyl-Heisenberg basis J. WEYL-HESEEG ASES and transmitted OFTD signal s t in discrete time can be equivalently presented in the following form 1 L1 L1 s n c n c n n J 0 l0 l0 l l l l (1) n gnl j n l mod exp / exp (3) J l n ln (4) njgn l j n l mod () 107

3 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS where cl e( al ) and cl m( al ) are real and imaginary parts of complex information QA symbols a l ; sn snt ; gn gnt and gn - initializing functions; J L - number of subcarriers L - any natural number unequal zero; - phase parameter. The system of basic functions J on the Hilbert space of discrete functions on J where * is the sign of complex conjugation. is orthogonal in terms of real scalar product defined 1 x n y n e x n y* n (5) n0 Orthogonality condition of basis J where is U U - square J in matrix form is e U* U (6) identity matrix; U U U is a bloc matrix with blocs matrixes constructed from columns of basic functions ψ T and l l l l 0... L 1. ψ T for all indexes l l l Equation (1) describes the modulation algorithm of OFD/OQA signal in discrete time. Appropriate demodulation algorithm has the form n n c sn n c s (7) l l l l.. WELL-LOCALZED WEYL-HESEEG ASES COSTUCTO t is necessary to solve the problem of orthogonal Weyl-Heisenberg basis synthesis to construct a robust OFD/OQA system. Ambiguity function for the basis s prototype function g t f Agt f gx t g* x t exp jfxdx has to possess maximum localization simultaneously in time and frequency domain. This criteria is analogous to orthogonal system finding which minimizes the left part of inequality responding to the Heisenberg uncertainty principle 4 gt () ( t) g() t dt ( f ) Fg() t df (8) 16 where Fg() t is the Fourier transform of function gt (). t is nown [] that equality in (8) is satisfied for Gaussian function 1 4 g0 t exp t. (9) 108

4 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS So prototype function g0 t is the best one from the point of localization. Unfortunately the basis constructed from Gaussian function as a prototype function (Gabor basis) is not orthogonal. n the article [1] it was proposed an orthogonalization algorithm. The following problem was solved: Problem 1. On the subset A of complex matrixes of size which satisfy orthogonality condition it is necessary to find an imal matrix * e( ) UU UA U which minimizes the following functional: U :min G U UA E (10) where G is the Gabor basis matrix and The algorithm consists of several successive steps: 1. Gabor basis matrix A tr( AA *) is a Frobenius norm. E G G G is constructed where 0 exp mod. G nl g nl j n 3. G n l jg0 n /l exp j n. mod 4. Extended real Gabor matrix is constructed eg 5. G. m G 6. Spectral decomposition of matrix GG * is performed: GG* SΛS * 7. The following matrixes are calculated 1 Σ Λ 8. The imal real matrix is calculated V SW * 9. atrix V is decomposed in two parts W G * SΣ V1 10. V V 11. The imal matrix of orthogonal Weyl-Heisenberg basis is constructed U V1 jv. Optimal prototype function gn is specified be the first gnu n1. column of imal matrix: This algorithm allows to solve Problem 1 and thus to find orthogonal Weyl-Heisenberg basis which has the best localization characteristics because it is the closest to Gabor basis under the Frobenius norm

5 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS Unfortunately this algorithm is based on the matrix orthogonal decomposition. This operation is computationally inefficient so we need some other algorithm to use it in real systems. n fact the structure of Weyl-Heizenberg basis is determined by its forming function. So basis construction algorithm can be simplified by using gn criteria in instead of basis orthogonality conditions. Consider a particular case when -symmetry gn g n * of symmetry is mod V. ASS OTHOGOALTY CODTOS gn function s orthogonality gn is a real function and has a property of conjugate. Optimal value if faze parameter corresponding to this type []. Under these conditions it can be shown that the basis [3] E glm gl m J g n n n l m l l m m 4 g n g n l exp j mn mj ; l J lm L is orthogonal in term of ordinary scalar production if and only if the the Weyl-Heizenberg basis constracted on the base of the same function gn is also orthogonal but in terms of real scalar production. The following theorem is valid: Theorem 1. ecessary and sufficient condition of E J (and also J orthogonality in time domain is the following equality (11) ) basis L1 g n r g n r l 0. l n J l JL r 0 (1) The next step is to come from the orthogonality of basis E J to the orthogonality of Wiener basis. Definition 1. The set of functions f0 f1... f 1 of linear space V ( is the space of -periodical complex discrete functions) is called Wiener basis if descrete-periodical J g J form the basis of analog of Wiener theorem is valid for them: functions 1 0 space V if and only if g can be represented in the following form J 1 g a f and all a are nonzero. Thus the existence of Wiener basis is necessary and sufficient for the existence of function g whose vector of shifts is the basis of space V. gn g g gn g nj... gnj1j Definition. Transform which allows to pass from the vector of shifts to the orthogonal Wiener basis is called Wiener transform and can be written in the following form T 0 110

6 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS The inverse Wiener transform is j n g n r exp r. L r0 L L1 g n L1 1 L Theorem. ecessary and sufficient condition of E J (and also J 0 orthogonality in time domain is the following equality n n L n. ) basis 4. (13) V. EFFCET OTHOGOALZATO ALGOTH Orthogonality condition (13) allows to formulate an effective orthogonalization procedure. Firstly a real conjugate -symmetrical function gon n J should be chosen. To receive an orthogonal well-localized basis n also have to possess a good localization. For go example Gaussian function has the best time-frequency localization. Secondly the Wiener basis is constructed in the following from: n L1 where o n n n n L n n n j n g nr exp r L. t is obvious that the calculation of r0 L n in every point n J comes to a discrete Fourier transform. Thus computationally efficient fast Fourier algorithm can be implemented times. t is easy to test directly that Wiener basis (14) satisfies orthogonality condition(13). So inverse Wiener transform can be used to receive the required forming function gn of Weil- Heisenberg basis. V. ODELG ESULTS Two algorithms (on base on orthogonal decomposition and on base on FFT) of orthogonal well-localized bases were compared. t was discovered that these methods have the same output i.e. forming function constructed from the same go n (Gaussian function) match perfectly. Graphs of ideally localized Gabor basic function basic function centre of the interval close enough. (14) go n and imal Weyl-Heisenberg gn are shown in Fig.1. For clearness basic function are transmitted in the J. t is apparent that basic functions gn and g n are ACKOWLEDGET 0 The author would lie to than professor V.P. Volchov from oscow state University of Telecommunications and nformatics and professor A.. ogolubov from.v. Lomonosov oscow State University for invaluable help and support during the preparation of this wor. 111

7 6 TH SEA OF FSH-USSA UVESTY COOPEATO TELECOUCATOS Gbor basis Optimal basis 0.5 asic Functions Samples n Figure 1. Excellently localized Gaussian function and received imal forming function EFEECES [1] Volchov V.P.: Signal bases with good time-frequency localization. Electrosvyaz no. pp. 1-5 (007) [] Gabor D.: Theory of communication. J. nst. Elect. Eng. vol.93 no. 3 pp London (1946) [3] Volchov V.P. Petrov D.A.: Orthogonal Weyl-Heizenberg basis imization for digital communication systems based on OFD/OQA. Scientific vedomosti elsu nformatics Series 1(56) Vol. 9/1 pp [4] Petrov D. A.: Orthogonality Criterions of Well-Localized ases. umerical ethods and Programming (Scientific Journal) Vol. 10 Section 1. umerical methods and applications pp (009) 11