Two-dimensional differential demodulation for OFDM - Semantic Scholar
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2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 4, APRIL 2003
Two-Dimensional Differential Demodulation for OFDM Erik Haas, Member, IEEE, and Stefan Kaiser, Senior Member, IEEE
Abstract—In this paper, the possibility of combining differential demodulators in time direction and frequency direction for orthogonal frequency-division multiplexing (OFDM) frames is investigated. The proposed algorithm uses not only the direction in which the information is modulated, but also joins it with other symbols in time or frequency direction of the OFDM frame, resulting in an additional second direction that is differentially demodulated as well. In the special case that is investigated here, the second direction does not carry any modulated information. Nevertheless, it is shown that it can be used for the two-dimensional demodulation process without changing the transmitter. Therefore, the proposed algorithm is applicable for existing digital transmission systems, for example, digital audio broadcasting, as well as new systems. Index Terms—Demodulation, differential phase-shift keying (DPSK), multidimensional signal detection, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
P
ERFORMANCE and mobility are important issues in state-of-the-art communications systems. Communications devices are nowadays based on digital modulation/demodulation techniques. These techniques can be divided into coherent, differential coherent, and noncoherent ones. Coherent systems typically outperform differential coherent and noncoherent systems due to the use of pilot-symbol aided channel estimation and equalization. However, the implementation of respective algorithms also necessitates a large computational and energy-consuming effort. This can, especially for mobile communications equipment, be an unwanted effect. Furthermore, the insertion of the necessary pilot symbols leads to a reduction of the available data rate. In contrast to this, differential coherent systems are designed to work without channel estimation and equalization. The missing estimated absolute reference in conjunction with a differential combination of the received symbols leads to a degradation of the bit-error rate (BER) performance, when compared with the coherent system. Due to the high processing speed and specialization of modern integrated circuits, orthogonal frequency-division multiplexing (OFDM) [1], using the inverse fast Fourier transformation (IFFT) to modulate multiple subcarriers at the same time, has become an interesting option. Like with single-carrier signals, differential coherent modulation techniques are appliPaper approved by P. Hoeher, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received November 16, 2000; revised July 1, 2002; October 17, 2002; and October 23, 2002. This paper was presented in part at the IEEE GLOBECOM 2000, San Francisco, CA, November/December 2000. The authors are with the German Aerospace Center (DLR), Institute of Communications and Navigation, 82230 Wessling, Germany ([email protected]). Digital Object Identifier 10.1109/TCOMM.2003.810831
cable and an according standard is already in use for digital audio broadcasting (DAB) [2]. In the following, the special characteristics of differentially modulated OFDM signals are considered. To overcome the degrading effects of differential demodulation (DD), a variety of differential detection techniques has been discussed in the literature [3]–[7]. Two of these algorithms are based on multiple-symbol detection [3], [4], while another algorithm applies an additional decision feedback from several multiple-symbol detectors [5]. The third algorithm uses multiple differentially encoded amplitude levels in conjunction with decision feedback and filtering in the receiver [6]. Another algorithm applies “turbo decoding” by using the differential modulation scheme as an inner code with rate one [7]. While [3]–[5] and [7] focus on single-carrier modulation, [6] applies OFDM, but also performs the DD in only one direction, namely, the one of the modulated information. In this paper, a new algorithm for DD is proposed that makes use of the two-dimensional (2-D) structure of OFDM frames. For this purpose, the basic principle of 2-D DD is introduced in Section II. In Section III, it is used in conjunction with a special characteristic of DD to choose alternative solutions for the bit-sequence estimation. Section IV gives mathematical background information on how and why the algorithm works. In Sections V and VI, a criterion and according methods to evaluate the alternative demodulation solutions are suggested. Results for the algorithm with different channel models and synchronization misalignments are presented in Section VII. Finally, the resulting conclusions are drawn in Section VIII. II. BASIC IDEA FOR 2-D DIFFERENTIAL DEMODULATION As commonly used, for example, in the European DAB standard [2], differential modulation and DD, see Fig. 1, is only performed in one direction (here by OFDM symbols in time ) with the help of one reference symbol at the beginning of the frame. This means that only the closest neighbor in negative direction of time is used for demodulation. It is obvious that there are additionally seven other closest neighbors, see Fig. 2. It is shown that these neighbors can also be used for DD without changing the transmitter. The distance to all eight neighbors is defined to be one. 2-D DD is achieved by not necessarily choosing the closest path with distance one (here in negative direction of time ), but by using a detour path. A detour path has the same origin and destination as the direct path, but with distance larger than one, resulting from the combination of several DD steps. DD between two arbitrary symbols within the frame is possible, since all transmitted differentially modulated symbols, in
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each of the eight neighbors of one symbol in the center, now a DD step as shown in Fig. 2 and Fig. 1 is performed according to (1) is the current symbol under investigation, is one where is the symbol after the DD, of the eight neighbor symbols, is the conjugate complex of the expression, , are the the arguments of the symbols. amplitudes, and , If M-DPSK is considered, the information is included in the differential phase (2) This means that the information-carrying phase of the original one-dimensional (1-D) DD is (3) If now instead of the direct path, a path with several intermediate steps is chosen, each step in this path has a differential , where is phase . If those differenthe current iteration path length, here tial intermediate phases were summed up, the result would be
Fig. 1. Differential modulation and demodulation concepts.
(4)
Fig. 2.
Differential neighborhood demodulation of one OFDM subcarrier.
general, come from the same transmission symbol alphabet. Thus, the demodulation between two arbitrary symbols delivers a symbol from the source alphabet, see Fig. 1. Due to stronger statistical correlations, especially in fading channel environments, only the eight surrounding symbols are regarded as relevant here. The detour path is obtained from hopping to one or several intermediate neighbors. The additional steps are chosen according to an algorithm that assures that the possibility of a bit error is decreased by choosing the detour path rather than the direct path. III. FRAME-WORM ALGORITHM FOR DETOUR PATH DETERMINATION An OFDM frame can be interpreted as a 2-D structure consisting of complex-valued symbols in the direction of frequency and time . Even though other modulation schemes are possible, the modulation scheme here is chosen to be -ary differential phase-shift keying (M-DPSK) [8] in the time direction. As discussed above, all transmitted symbols come from the same alphabet, here -ary phase-shift keying (M-PSK) symbols. For
remains unwhich means that the phase difference changed and a simple adding of the intermediate phases would not bring any advantage, but on the other hand, also would not . This offers influence the phase of the original path the possibility of taking a closer look at the intermediate steps individually. Each received differential phase can be divided and a phase error , into a valid phase that corresponds to the phase offset of the closest matching source-symbol phase (5) to the The common 1-D DD directly concludes from . phase of the source symbol In contrast to this, the new 2-D DD algorithm proposed , here decides for every individual intermediate phase on a possible phase of the source al, . Additionphabet , , ally, the phase errors , can be used and the amplitudes for the path criterion and the resulting estimation, or to obtain channel reliability information for the chosen path. The path criterion has to make sure that choosing a detour path rather than the direct path is more reliable, even given the fact that the detour path with its intermediate steps is “longer.” Therefore, at first, the direct path is considered the most reliable ). For and is chosen as the reference path (iteration length a certain criterion, now all paths or a subgroup of all possible paths up to a definable iteration path depth that reach the destination symbol are investigated. If, considering the criterion, one of the alternative paths is more reliable than the reference
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path, this alternative path becomes the new reference path, and so on. Considering the criterion, the most reliable path, therefore, . For the final decision about the has a length of possible phase of the source symbol, now the differential phase sum of the chosen path is calculated as follows: for L=1 for L=2
resulting in
(12) is the error function . where Since the result of (12) cannot be analytically integrated in further steps, the following simplification is introduced:
otherwise. (6) To get the final corresponding bit sequence of the demodulation, the information from the obtained path has to be exis differential quatertracted. Therefore, the phase sum nary phase-shift keying (DQPSK) demodulated. IV. MATHEMATICAL THEORY FOR PATH DETERMINATION It has already been mentioned that when a detour path is chosen, one has to make sure that this path is more reliable than the direct path. To clarify what is meant by a more reliable path, a closer look at the mathematical theory of DPSK is taken. Since the information is included in the phase alone, minimum distance in the receiver is equal to minimum phase distance to one of the possible transmitted symbols. Therefore, the 2-D density function for additive white Gaussian noise (AWGN)
(13) where (14) is the zero-order modified Bessel function of first kind. and is chosen since most of the distributed The use of . energy can be found in this distance due to and can be considered indepenThe distributions of of (2) is the substraction of two independent, so that dent random variables. Again, without loss in generality, it can has been the transbe assumed that the reference symbol as well, so that the joint distribution for mitted symbol (this is only one of possible sowith M-DPSK) can be written as the autolutions for correlation function (ACF)
(7) in the Cartesian in-phase/quadrature (I/Q) coordinate system ( , ) is transformed to the polar coordinate system ( , ) (8)
(15) The result for the approximation of (13) is
is the radius, is the angle, is where the standard deviation, and ( , ) is the mean value in Cartesian coordinates. Without loss in generality, one possible transcan be assumed to be mitted symbol . By substituting
(16) A further simplification step with
(9) (17) the 2-D probability density function (PDF) [8], [9] (10) . is the transmitted is obtained for the received symbol is the AWGN variance. signal energy, and So far, only the received symbols have been investigated. From (1), it can be seen that during the DD process, two of these symbols are multiplied with each other, so that the information arises. Since the amplitudes carrying phase difference and play no role in the decision process, they can be eliminated by integrating (10) over (11)
for small delivers the closed-form solution as shown in (18) at the bottom of the next page. Numerical integration of (15) and the approximation (18) match almost perfectly, as shown in [10]. Another numerical integration step delivers the commonly of -ary DPSK known values for the symbol-error rate [11] (19)
In the following, DQPSK modulation is chosen. Using from [11] and (19) with the approximation gives the standard 1-D exact and approximated representations of the
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Fig. 3. Conditional PDFs for DQPSK reception.
BER. At a BER of , the approximation is less than 0.3 dB below the exact signal-to-noise ratio (SNR). Investigation of all possible symbol combinations that result , delivers the in a transmitted symbol shown in Fig. 3 for the received differential PDFs , where are the source symbols before differsymbol ential modulation. for making the corThe logarithmic-likelihood ratio rect decision for a differential symbol, in spite of a wrong decision for the symbol in general, can be expressed by (20) at the bottom of the page. The exact representation and the close form log-likelihood ratio using the approximation (18) are shown in Fig. 4. of one First, a look is taken at the differential phase in Fig. 3. A possible differentially demodulated symbol is shown as well. If now the two log-likesecond symbol lihoods for the phases of these symbols are compared, it can be seen that the likelihood of a correct decision is larger for than for . If is the result of the direct path DD, and if it is possible to find a detour path with a criterion that matches the , one can assume that the requisite and delivers a possible probability of making a wrong decision can be decreased, and thus, the BER will decrease as well. This attempt is made in the following section.
Fig. 4. Log-likelihood ratio
3(1') for SNR = 11 dB.
V. MINIMUM PHASE ERROR CRITERION of the intermediate steps is now The phase error used as a decision variable for the optimum path criterion. At of the difirst, the absolute value of the phase error rect path is calculated. This value is used as the reference value. Now, all possible paths up to the chosen maximum iteration depth that lead to the destination symbol are investigated. For each intermediate step of the current alternative path under is determined. If the abinvestigation, the phase error solute value of one of these phase errors is larger than the reference value, this alternative path is rejected. If, on the other hand, all individual absolute phase errors in the alternative path are smaller than the reference value, the maximum absolute value of the phase errors in this alternative path becomes the new reference value, and this path becomes the new optimum path. VI. PATH EVALUATION Theoretically, the amount of possible paths that can be evaluated is only limited by the frame size. Due to the fact that for each step, DD can be performed for all eight neighbors, the
(18)
(20)
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Fig. 5. Examples of BFM detour paths.
Fig. 6.
Examples of stream method detour paths.
complexity increases exponentially as . Measures have to be taken to reduce the amount of investigated paths to reasonable ones. A. Brute-Force Method The brute-force method (BFM) examines all possible paths up to the maximum predefined depth . In doing so, all impossible paths, i.e., paths that use a symbol of the frame that already has been used in this path before, are excluded from the search algorithm. A variety of possible paths is shown in Fig. 5. The complexity of this evaluation still increases exponentially with . Since all paths of length are investigated, the is included original 1-D direct path with iteration length is the as well. Therefore, the result for iteration depth common 1-D DD.
Fig. 7. BER performance for AWGN channel.
A. AWGN Channel B. Stream Method In order to reduce the exponentially increasing complexity of the brute-force path evaluation, it is desirable to reduce the amount of investigated paths to the most likely ones. Doing so, the investigated lengths of the most likely paths can be increased. One way to achieve this is to follow the streams within an OFDM frame. The heading of the stream is the result of frequency and/or time correlation due to fading. Following this heading is not sufficient; it is also necessary at some point to close the detour path by connecting two elements of parallel streams. There exist six simple solutions for the headings of a stream, two horizontal ones and four diagonal ones. Possible solutions for such detour paths are shown in Fig. 6. The original 1-D direct path is calculated as a reference first before the alternative paths are evaluated. VII. SIMULATION RESULTS The investigated transmission system uses OFDM frames with 24 OFDM symbols consisting of 128 subcarriers each. The information is DQPSK modulated in the time direction on the subcarrier level. The first OFDM symbol of the OFDM frame is a reference symbol that is used for the first differential step.
In Fig. 7, the performance of the OFDM system with 2-D DD using the minimum phase-error criterion in the AWGN channel is shown for the different path evaluation methods. The theoretical BER performances of DQPSK and coherent QPSK with AWGN are shown, as well [8], [11]. It can be seen that for , the system performs like brute-force iteration path depth , the required common 1-D DD. For iteration path depths SNR can be decreased significantly. For example, to achieve a , the SNR can be reduced by about 1.25 dB already BER of . The stream method does not for an iteration depth of perform as well as the BFM here, since it does not evaluate all possible paths and cannot take advantage of correlations in time or frequency direction. Additionally, the noise on all subcarriers at all different times is due to AWGN statistically independent, and the stream method does not make use of shorter paths that might give a more reliable result. For comparison, the simulation results with multiple-symbol differential detection from [3] are depicted, as well. It shows for 2-D that choosing a detour path with two steps, i.e., DD, is more advantageous than using the comparable observafor multiple-symbol detection [3]. The tion interval of and . Already outpersame applies for . It follows that analyzing multiple dimensions, if forms available, and treating each differential step independently from
HAAS AND KAISER: TWO-DIMENSIONAL DIFFERENTIAL DEMODULATION FOR OFDM
Fig. 8.
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BER performance for Rician channel.
other differential steps has an edge over the detection algorithm proposed in [3]. The 2-D DD algorithm, furthermore, makes it possible to eliminate bad paths during the search, and thus, reduces the overall computational complexity.
Fig. 9. Statistical distribution of stream method path length channel at E =N 20 dB.
=
L
for Rician
B. Fading Channels Fig. 8 shows the performance of the system in a critical aeronautical Rician fading channel environment [12]. The system parameters have been chosen as follows: subcarrier kHz, maximum Doppler frequency spacing kHz, line-of-sight path (LOS) at , scattered components with classical Jakes distribution [13] (positive dB, exponentially frequencies only), Rice factor s, s, decreasing delay spread with s, carrier lock on LOS. Again, it can guard interval be seen that already, for low SNRs, the performance of the system improves significantly with short iteration path depths of the BFM. For high SNRs, the BER can be decreased by approximately one decade. Using the stream method additionally improves the achievable gain compared to the depicted BFMs . This can be explained by the exploitation of the with streams of the underlying fading channel. The stream method is able to follow possible stream headings with longer path depths. Due to its reduced complexity, it is able to perform as , whose computational complexity fast as the BFM with . increases significantly for Fig. 9 shows the statistical distribution of the chosen path length with the stream method for the Rician channel dB . About the same amount of detour paths with length three is chosen as direct paths with length one. The distribution of the remaining longer paths decreases exponentially. That means that the algorithm chooses about one third of paths with length one, one third of paths with length three, and another third of paths with length equal or greater than five. Further investigations with the typical urban Rayleigh fading channel taken from [14] without a LOS component have shown , a gain of approximately 1.0 that at the uncoded BER of dB can be achieved for both methods, see Fig. 10. The used with classical Jakes maximum Doppler was distribution. The delay spread and all other parameters remain the same as above.
Fig. 10.
BER performance for typical urban Rayleigh fading channel.
C. Synchronization Misalignment Misalignment (MA) in the receiver synchronization is a source for performance degradations in digital communications systems, as well. Since timing synchronization is performed first in OFDM receivers [15], it is the most critical one as a possible source of errors. Timing misalignment causes intersymbol interference (ISI) if the fast Fourier transformation (FFT) of the OFDM demodulation is performed on a part of the dedicated symbol with an additional interfering part of a second symbol. The part of the interfering symbol is expressed by a MA percentage. The performance results with MAs of one and two percent are shown in Fig. 11 for the AWGN channel. It can be observed that 2-D DD techniques are more robust than the classical 1-D method. VIII. CONCLUSION A novel 2-D DD algorithm has been proposed that, for the most part, overcomes the degrading effects of DPSK demodulation compared to PSK demodulation. For the proposed scheme, neither a change in the transmitter nor knowledge about the transmitted information is necessary. Moreover, no information
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[10] E. Haas and S. Kaiser, “Analysis of two-dimensional differential demodulation for OFDM,” in Proc. IEEE Global Telecommunications Conf. (GLOBECOM’00), Nov./Dec. 2000, pp. 751–755. [11] R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun., vol. COM-30, pp. 1828–1841, Aug. 1982. [12] P. Hoeher and E. Haas, “Aeronautical channel modeling at VHF band,” in Proc. IEEE Vehicular Technology Conf. (VTC 1999-Fall), Sept. 1999, pp. 1961–1966. [13] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [14] COST 207, “Digital Land Mobile Radio Communications,” Office for Official Publications of the European Communities, Luxembourg, Final Tech. Rep., 1989. [15] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613–1621, Dec. 1997.
Fig. 11.
BER performance for receiver timing synchronization MA.
about the channel (in other words, no channel estimation) is required. The efficiency of the algorithm can be adjusted by the criterion and the chosen path-evaluation method. With the new algorithm, significant performance improvements for AWGN, fading channels, and synchronization timing MAs can be obtained with a minimum of additional computational complexity, compared with conventional 1-D DPSK demodulation. To obtain channel reliability information for soft-decision channel decoding, the calculated criterion value for the chosen path can be used. It has been shown that this value is directly connected to the reliability of the detected bits for each symbol. REFERENCES [1] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol., vol. COM-19, pp. 628–634, Oct. 1971. [2] Radio Broadcasting Systems; Digital Audio Broadcasting (DAB) to Mobile, Portable and Fixed Receivers, ETSI ETS 300 401, Feb. 1995. [3] D. Divsalar and M. K. Simon, “Multiple symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [4] D. Makrakis and K. Feher, “Optimal noncoherent detection of PSK signals,” Electron. Lett., vol. 26, no. 6, pp. 398–400, Mar. 1990. [5] F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of -ary DPSK signals,” IEEE Trans. Veh. Technol., vol. 44, pp. 203–210, May 1995. [6] H. Rohling and K. Brueninghaus, “High rate OFDM modem with quasicoherent DAPSK,” in Proc. IEEE Vehicle Technology Conf. (VTC’97), May 1997, pp. 2055–2059. [7] P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, pp. 837–843, June 1999. [8] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [9] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991.
M
Erik Haas (M’98) was born in Munich, Germany, in December 1971. He received the electrical engineering degree (Dipl.-Ing. Univ.) from the Technical University of Munich, Munich, Germany, in 1996, and the Ph.D. degree from the University of Essen, Essen, Germany, in 2002. Since 1996, he has been with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he is a Scientific Research Engineer in Communications and Navigation. His interests include multicarrier modulation techniques, channel modeling, aeronautical communications, and system implementation and testing in DSP hardware.
Stefan Kaiser (M’96–SM’02) received the Dipl.-Ing. and Ph.D. degrees in electrical engineering from the University of Kaiserslautern, Kaiserslautern, Germany, in 1993 and 1998, respectively. In 1993, he joined the Institute of Communications and Navigation of the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. From February to August 1998, he was a Visiting Researcher with the Telecommunications Research Laboratories (TRLabs), Edmonton, AB, Canada, working in the area of wireless communications. Since 1999, he has been the Leader of the Mobile Radio Transmission Group at the Institute of Communications and Navigation of the German Aerospace Center (DLR). His research interests include multicarrier communications, multiple-access schemes, and space–time processing for mobile radio applications. Dr. Kaiser is co-organizer of the international workshop series on Multicarrier Spread Spectrum (MC-SS), and co-editor of the book series Multicarrier Spread Spectrum & Related Topics (Amsterdam, The Netherlands: Kluwer 2000/2002). He is also Guest Editor of several special issues on multicarrier spread spectrum of the European Transactions on Telecommunications (ETT). He is the Vice Chair of the Communication Theory Symposium, ICC 2004. He is Organizer and Lecturer of the seminar series on Wireless LANs at the Carl-Cranz-Gesellschaft (CCG), Oberpfaffenhofen, Germany. He is a member of the VDE/ITG.
2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
580
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 4, APRIL 2003
Two-Dimensional Differential Demodulation for OFDM Erik Haas, Member, IEEE, and Stefan Kaiser, Senior Member, IEEE
Abstract—In this paper, the possibility of combining differential demodulators in time direction and frequency direction for orthogonal frequency-division multiplexing (OFDM) frames is investigated. The proposed algorithm uses not only the direction in which the information is modulated, but also joins it with other symbols in time or frequency direction of the OFDM frame, resulting in an additional second direction that is differentially demodulated as well. In the special case that is investigated here, the second direction does not carry any modulated information. Nevertheless, it is shown that it can be used for the two-dimensional demodulation process without changing the transmitter. Therefore, the proposed algorithm is applicable for existing digital transmission systems, for example, digital audio broadcasting, as well as new systems. Index Terms—Demodulation, differential phase-shift keying (DPSK), multidimensional signal detection, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
P
ERFORMANCE and mobility are important issues in state-of-the-art communications systems. Communications devices are nowadays based on digital modulation/demodulation techniques. These techniques can be divided into coherent, differential coherent, and noncoherent ones. Coherent systems typically outperform differential coherent and noncoherent systems due to the use of pilot-symbol aided channel estimation and equalization. However, the implementation of respective algorithms also necessitates a large computational and energy-consuming effort. This can, especially for mobile communications equipment, be an unwanted effect. Furthermore, the insertion of the necessary pilot symbols leads to a reduction of the available data rate. In contrast to this, differential coherent systems are designed to work without channel estimation and equalization. The missing estimated absolute reference in conjunction with a differential combination of the received symbols leads to a degradation of the bit-error rate (BER) performance, when compared with the coherent system. Due to the high processing speed and specialization of modern integrated circuits, orthogonal frequency-division multiplexing (OFDM) [1], using the inverse fast Fourier transformation (IFFT) to modulate multiple subcarriers at the same time, has become an interesting option. Like with single-carrier signals, differential coherent modulation techniques are appliPaper approved by P. Hoeher, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received November 16, 2000; revised July 1, 2002; October 17, 2002; and October 23, 2002. This paper was presented in part at the IEEE GLOBECOM 2000, San Francisco, CA, November/December 2000. The authors are with the German Aerospace Center (DLR), Institute of Communications and Navigation, 82230 Wessling, Germany ([email protected]). Digital Object Identifier 10.1109/TCOMM.2003.810831
cable and an according standard is already in use for digital audio broadcasting (DAB) [2]. In the following, the special characteristics of differentially modulated OFDM signals are considered. To overcome the degrading effects of differential demodulation (DD), a variety of differential detection techniques has been discussed in the literature [3]–[7]. Two of these algorithms are based on multiple-symbol detection [3], [4], while another algorithm applies an additional decision feedback from several multiple-symbol detectors [5]. The third algorithm uses multiple differentially encoded amplitude levels in conjunction with decision feedback and filtering in the receiver [6]. Another algorithm applies “turbo decoding” by using the differential modulation scheme as an inner code with rate one [7]. While [3]–[5] and [7] focus on single-carrier modulation, [6] applies OFDM, but also performs the DD in only one direction, namely, the one of the modulated information. In this paper, a new algorithm for DD is proposed that makes use of the two-dimensional (2-D) structure of OFDM frames. For this purpose, the basic principle of 2-D DD is introduced in Section II. In Section III, it is used in conjunction with a special characteristic of DD to choose alternative solutions for the bit-sequence estimation. Section IV gives mathematical background information on how and why the algorithm works. In Sections V and VI, a criterion and according methods to evaluate the alternative demodulation solutions are suggested. Results for the algorithm with different channel models and synchronization misalignments are presented in Section VII. Finally, the resulting conclusions are drawn in Section VIII. II. BASIC IDEA FOR 2-D DIFFERENTIAL DEMODULATION As commonly used, for example, in the European DAB standard [2], differential modulation and DD, see Fig. 1, is only performed in one direction (here by OFDM symbols in time ) with the help of one reference symbol at the beginning of the frame. This means that only the closest neighbor in negative direction of time is used for demodulation. It is obvious that there are additionally seven other closest neighbors, see Fig. 2. It is shown that these neighbors can also be used for DD without changing the transmitter. The distance to all eight neighbors is defined to be one. 2-D DD is achieved by not necessarily choosing the closest path with distance one (here in negative direction of time ), but by using a detour path. A detour path has the same origin and destination as the direct path, but with distance larger than one, resulting from the combination of several DD steps. DD between two arbitrary symbols within the frame is possible, since all transmitted differentially modulated symbols, in
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each of the eight neighbors of one symbol in the center, now a DD step as shown in Fig. 2 and Fig. 1 is performed according to (1) is the current symbol under investigation, is one where is the symbol after the DD, of the eight neighbor symbols, is the conjugate complex of the expression, , are the the arguments of the symbols. amplitudes, and , If M-DPSK is considered, the information is included in the differential phase (2) This means that the information-carrying phase of the original one-dimensional (1-D) DD is (3) If now instead of the direct path, a path with several intermediate steps is chosen, each step in this path has a differential , where is phase . If those differenthe current iteration path length, here tial intermediate phases were summed up, the result would be
Fig. 1. Differential modulation and demodulation concepts.
(4)
Fig. 2.
Differential neighborhood demodulation of one OFDM subcarrier.
general, come from the same transmission symbol alphabet. Thus, the demodulation between two arbitrary symbols delivers a symbol from the source alphabet, see Fig. 1. Due to stronger statistical correlations, especially in fading channel environments, only the eight surrounding symbols are regarded as relevant here. The detour path is obtained from hopping to one or several intermediate neighbors. The additional steps are chosen according to an algorithm that assures that the possibility of a bit error is decreased by choosing the detour path rather than the direct path. III. FRAME-WORM ALGORITHM FOR DETOUR PATH DETERMINATION An OFDM frame can be interpreted as a 2-D structure consisting of complex-valued symbols in the direction of frequency and time . Even though other modulation schemes are possible, the modulation scheme here is chosen to be -ary differential phase-shift keying (M-DPSK) [8] in the time direction. As discussed above, all transmitted symbols come from the same alphabet, here -ary phase-shift keying (M-PSK) symbols. For
remains unwhich means that the phase difference changed and a simple adding of the intermediate phases would not bring any advantage, but on the other hand, also would not . This offers influence the phase of the original path the possibility of taking a closer look at the intermediate steps individually. Each received differential phase can be divided and a phase error , into a valid phase that corresponds to the phase offset of the closest matching source-symbol phase (5) to the The common 1-D DD directly concludes from . phase of the source symbol In contrast to this, the new 2-D DD algorithm proposed , here decides for every individual intermediate phase on a possible phase of the source al, . Additionphabet , , ally, the phase errors , can be used and the amplitudes for the path criterion and the resulting estimation, or to obtain channel reliability information for the chosen path. The path criterion has to make sure that choosing a detour path rather than the direct path is more reliable, even given the fact that the detour path with its intermediate steps is “longer.” Therefore, at first, the direct path is considered the most reliable ). For and is chosen as the reference path (iteration length a certain criterion, now all paths or a subgroup of all possible paths up to a definable iteration path depth that reach the destination symbol are investigated. If, considering the criterion, one of the alternative paths is more reliable than the reference
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path, this alternative path becomes the new reference path, and so on. Considering the criterion, the most reliable path, therefore, . For the final decision about the has a length of possible phase of the source symbol, now the differential phase sum of the chosen path is calculated as follows: for L=1 for L=2
resulting in
(12) is the error function . where Since the result of (12) cannot be analytically integrated in further steps, the following simplification is introduced:
otherwise. (6) To get the final corresponding bit sequence of the demodulation, the information from the obtained path has to be exis differential quatertracted. Therefore, the phase sum nary phase-shift keying (DQPSK) demodulated. IV. MATHEMATICAL THEORY FOR PATH DETERMINATION It has already been mentioned that when a detour path is chosen, one has to make sure that this path is more reliable than the direct path. To clarify what is meant by a more reliable path, a closer look at the mathematical theory of DPSK is taken. Since the information is included in the phase alone, minimum distance in the receiver is equal to minimum phase distance to one of the possible transmitted symbols. Therefore, the 2-D density function for additive white Gaussian noise (AWGN)
(13) where (14) is the zero-order modified Bessel function of first kind. and is chosen since most of the distributed The use of . energy can be found in this distance due to and can be considered indepenThe distributions of of (2) is the substraction of two independent, so that dent random variables. Again, without loss in generality, it can has been the transbe assumed that the reference symbol as well, so that the joint distribution for mitted symbol (this is only one of possible sowith M-DPSK) can be written as the autolutions for correlation function (ACF)
(7) in the Cartesian in-phase/quadrature (I/Q) coordinate system ( , ) is transformed to the polar coordinate system ( , ) (8)
(15) The result for the approximation of (13) is
is the radius, is the angle, is where the standard deviation, and ( , ) is the mean value in Cartesian coordinates. Without loss in generality, one possible transcan be assumed to be mitted symbol . By substituting
(16) A further simplification step with
(9) (17) the 2-D probability density function (PDF) [8], [9] (10) . is the transmitted is obtained for the received symbol is the AWGN variance. signal energy, and So far, only the received symbols have been investigated. From (1), it can be seen that during the DD process, two of these symbols are multiplied with each other, so that the information arises. Since the amplitudes carrying phase difference and play no role in the decision process, they can be eliminated by integrating (10) over (11)
for small delivers the closed-form solution as shown in (18) at the bottom of the next page. Numerical integration of (15) and the approximation (18) match almost perfectly, as shown in [10]. Another numerical integration step delivers the commonly of -ary DPSK known values for the symbol-error rate [11] (19)
In the following, DQPSK modulation is chosen. Using from [11] and (19) with the approximation gives the standard 1-D exact and approximated representations of the
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Fig. 3. Conditional PDFs for DQPSK reception.
BER. At a BER of , the approximation is less than 0.3 dB below the exact signal-to-noise ratio (SNR). Investigation of all possible symbol combinations that result , delivers the in a transmitted symbol shown in Fig. 3 for the received differential PDFs , where are the source symbols before differsymbol ential modulation. for making the corThe logarithmic-likelihood ratio rect decision for a differential symbol, in spite of a wrong decision for the symbol in general, can be expressed by (20) at the bottom of the page. The exact representation and the close form log-likelihood ratio using the approximation (18) are shown in Fig. 4. of one First, a look is taken at the differential phase in Fig. 3. A possible differentially demodulated symbol is shown as well. If now the two log-likesecond symbol lihoods for the phases of these symbols are compared, it can be seen that the likelihood of a correct decision is larger for than for . If is the result of the direct path DD, and if it is possible to find a detour path with a criterion that matches the , one can assume that the requisite and delivers a possible probability of making a wrong decision can be decreased, and thus, the BER will decrease as well. This attempt is made in the following section.
Fig. 4. Log-likelihood ratio
3(1') for SNR = 11 dB.
V. MINIMUM PHASE ERROR CRITERION of the intermediate steps is now The phase error used as a decision variable for the optimum path criterion. At of the difirst, the absolute value of the phase error rect path is calculated. This value is used as the reference value. Now, all possible paths up to the chosen maximum iteration depth that lead to the destination symbol are investigated. For each intermediate step of the current alternative path under is determined. If the abinvestigation, the phase error solute value of one of these phase errors is larger than the reference value, this alternative path is rejected. If, on the other hand, all individual absolute phase errors in the alternative path are smaller than the reference value, the maximum absolute value of the phase errors in this alternative path becomes the new reference value, and this path becomes the new optimum path. VI. PATH EVALUATION Theoretically, the amount of possible paths that can be evaluated is only limited by the frame size. Due to the fact that for each step, DD can be performed for all eight neighbors, the
(18)
(20)
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Fig. 5. Examples of BFM detour paths.
Fig. 6.
Examples of stream method detour paths.
complexity increases exponentially as . Measures have to be taken to reduce the amount of investigated paths to reasonable ones. A. Brute-Force Method The brute-force method (BFM) examines all possible paths up to the maximum predefined depth . In doing so, all impossible paths, i.e., paths that use a symbol of the frame that already has been used in this path before, are excluded from the search algorithm. A variety of possible paths is shown in Fig. 5. The complexity of this evaluation still increases exponentially with . Since all paths of length are investigated, the is included original 1-D direct path with iteration length is the as well. Therefore, the result for iteration depth common 1-D DD.
Fig. 7. BER performance for AWGN channel.
A. AWGN Channel B. Stream Method In order to reduce the exponentially increasing complexity of the brute-force path evaluation, it is desirable to reduce the amount of investigated paths to the most likely ones. Doing so, the investigated lengths of the most likely paths can be increased. One way to achieve this is to follow the streams within an OFDM frame. The heading of the stream is the result of frequency and/or time correlation due to fading. Following this heading is not sufficient; it is also necessary at some point to close the detour path by connecting two elements of parallel streams. There exist six simple solutions for the headings of a stream, two horizontal ones and four diagonal ones. Possible solutions for such detour paths are shown in Fig. 6. The original 1-D direct path is calculated as a reference first before the alternative paths are evaluated. VII. SIMULATION RESULTS The investigated transmission system uses OFDM frames with 24 OFDM symbols consisting of 128 subcarriers each. The information is DQPSK modulated in the time direction on the subcarrier level. The first OFDM symbol of the OFDM frame is a reference symbol that is used for the first differential step.
In Fig. 7, the performance of the OFDM system with 2-D DD using the minimum phase-error criterion in the AWGN channel is shown for the different path evaluation methods. The theoretical BER performances of DQPSK and coherent QPSK with AWGN are shown, as well [8], [11]. It can be seen that for , the system performs like brute-force iteration path depth , the required common 1-D DD. For iteration path depths SNR can be decreased significantly. For example, to achieve a , the SNR can be reduced by about 1.25 dB already BER of . The stream method does not for an iteration depth of perform as well as the BFM here, since it does not evaluate all possible paths and cannot take advantage of correlations in time or frequency direction. Additionally, the noise on all subcarriers at all different times is due to AWGN statistically independent, and the stream method does not make use of shorter paths that might give a more reliable result. For comparison, the simulation results with multiple-symbol differential detection from [3] are depicted, as well. It shows for 2-D that choosing a detour path with two steps, i.e., DD, is more advantageous than using the comparable observafor multiple-symbol detection [3]. The tion interval of and . Already outpersame applies for . It follows that analyzing multiple dimensions, if forms available, and treating each differential step independently from
HAAS AND KAISER: TWO-DIMENSIONAL DIFFERENTIAL DEMODULATION FOR OFDM
Fig. 8.
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BER performance for Rician channel.
other differential steps has an edge over the detection algorithm proposed in [3]. The 2-D DD algorithm, furthermore, makes it possible to eliminate bad paths during the search, and thus, reduces the overall computational complexity.
Fig. 9. Statistical distribution of stream method path length channel at E =N 20 dB.
=
L
for Rician
B. Fading Channels Fig. 8 shows the performance of the system in a critical aeronautical Rician fading channel environment [12]. The system parameters have been chosen as follows: subcarrier kHz, maximum Doppler frequency spacing kHz, line-of-sight path (LOS) at , scattered components with classical Jakes distribution [13] (positive dB, exponentially frequencies only), Rice factor s, s, decreasing delay spread with s, carrier lock on LOS. Again, it can guard interval be seen that already, for low SNRs, the performance of the system improves significantly with short iteration path depths of the BFM. For high SNRs, the BER can be decreased by approximately one decade. Using the stream method additionally improves the achievable gain compared to the depicted BFMs . This can be explained by the exploitation of the with streams of the underlying fading channel. The stream method is able to follow possible stream headings with longer path depths. Due to its reduced complexity, it is able to perform as , whose computational complexity fast as the BFM with . increases significantly for Fig. 9 shows the statistical distribution of the chosen path length with the stream method for the Rician channel dB . About the same amount of detour paths with length three is chosen as direct paths with length one. The distribution of the remaining longer paths decreases exponentially. That means that the algorithm chooses about one third of paths with length one, one third of paths with length three, and another third of paths with length equal or greater than five. Further investigations with the typical urban Rayleigh fading channel taken from [14] without a LOS component have shown , a gain of approximately 1.0 that at the uncoded BER of dB can be achieved for both methods, see Fig. 10. The used with classical Jakes maximum Doppler was distribution. The delay spread and all other parameters remain the same as above.
Fig. 10.
BER performance for typical urban Rayleigh fading channel.
C. Synchronization Misalignment Misalignment (MA) in the receiver synchronization is a source for performance degradations in digital communications systems, as well. Since timing synchronization is performed first in OFDM receivers [15], it is the most critical one as a possible source of errors. Timing misalignment causes intersymbol interference (ISI) if the fast Fourier transformation (FFT) of the OFDM demodulation is performed on a part of the dedicated symbol with an additional interfering part of a second symbol. The part of the interfering symbol is expressed by a MA percentage. The performance results with MAs of one and two percent are shown in Fig. 11 for the AWGN channel. It can be observed that 2-D DD techniques are more robust than the classical 1-D method. VIII. CONCLUSION A novel 2-D DD algorithm has been proposed that, for the most part, overcomes the degrading effects of DPSK demodulation compared to PSK demodulation. For the proposed scheme, neither a change in the transmitter nor knowledge about the transmitted information is necessary. Moreover, no information
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[10] E. Haas and S. Kaiser, “Analysis of two-dimensional differential demodulation for OFDM,” in Proc. IEEE Global Telecommunications Conf. (GLOBECOM’00), Nov./Dec. 2000, pp. 751–755. [11] R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun., vol. COM-30, pp. 1828–1841, Aug. 1982. [12] P. Hoeher and E. Haas, “Aeronautical channel modeling at VHF band,” in Proc. IEEE Vehicular Technology Conf. (VTC 1999-Fall), Sept. 1999, pp. 1961–1966. [13] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [14] COST 207, “Digital Land Mobile Radio Communications,” Office for Official Publications of the European Communities, Luxembourg, Final Tech. Rep., 1989. [15] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613–1621, Dec. 1997.
Fig. 11.
BER performance for receiver timing synchronization MA.
about the channel (in other words, no channel estimation) is required. The efficiency of the algorithm can be adjusted by the criterion and the chosen path-evaluation method. With the new algorithm, significant performance improvements for AWGN, fading channels, and synchronization timing MAs can be obtained with a minimum of additional computational complexity, compared with conventional 1-D DPSK demodulation. To obtain channel reliability information for soft-decision channel decoding, the calculated criterion value for the chosen path can be used. It has been shown that this value is directly connected to the reliability of the detected bits for each symbol. REFERENCES [1] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol., vol. COM-19, pp. 628–634, Oct. 1971. [2] Radio Broadcasting Systems; Digital Audio Broadcasting (DAB) to Mobile, Portable and Fixed Receivers, ETSI ETS 300 401, Feb. 1995. [3] D. Divsalar and M. K. Simon, “Multiple symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [4] D. Makrakis and K. Feher, “Optimal noncoherent detection of PSK signals,” Electron. Lett., vol. 26, no. 6, pp. 398–400, Mar. 1990. [5] F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of -ary DPSK signals,” IEEE Trans. Veh. Technol., vol. 44, pp. 203–210, May 1995. [6] H. Rohling and K. Brueninghaus, “High rate OFDM modem with quasicoherent DAPSK,” in Proc. IEEE Vehicle Technology Conf. (VTC’97), May 1997, pp. 2055–2059. [7] P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, pp. 837–843, June 1999. [8] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [9] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991.
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Erik Haas (M’98) was born in Munich, Germany, in December 1971. He received the electrical engineering degree (Dipl.-Ing. Univ.) from the Technical University of Munich, Munich, Germany, in 1996, and the Ph.D. degree from the University of Essen, Essen, Germany, in 2002. Since 1996, he has been with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he is a Scientific Research Engineer in Communications and Navigation. His interests include multicarrier modulation techniques, channel modeling, aeronautical communications, and system implementation and testing in DSP hardware.
Stefan Kaiser (M’96–SM’02) received the Dipl.-Ing. and Ph.D. degrees in electrical engineering from the University of Kaiserslautern, Kaiserslautern, Germany, in 1993 and 1998, respectively. In 1993, he joined the Institute of Communications and Navigation of the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. From February to August 1998, he was a Visiting Researcher with the Telecommunications Research Laboratories (TRLabs), Edmonton, AB, Canada, working in the area of wireless communications. Since 1999, he has been the Leader of the Mobile Radio Transmission Group at the Institute of Communications and Navigation of the German Aerospace Center (DLR). His research interests include multicarrier communications, multiple-access schemes, and space–time processing for mobile radio applications. Dr. Kaiser is co-organizer of the international workshop series on Multicarrier Spread Spectrum (MC-SS), and co-editor of the book series Multicarrier Spread Spectrum & Related Topics (Amsterdam, The Netherlands: Kluwer 2000/2002). He is also Guest Editor of several special issues on multicarrier spread spectrum of the European Transactions on Telecommunications (ETT). He is the Vice Chair of the Communication Theory Symposium, ICC 2004. He is Organizer and Lecturer of the seminar series on Wireless LANs at the Carl-Cranz-Gesellschaft (CCG), Oberpfaffenhofen, Germany. He is a member of the VDE/ITG.
