Spectrally Concentrated Impulses for Digital Multicarrier Modulation

Transcription

1 Spectrally Concentrated Impulses for Digital ulticarrier odulation Dipl.-Ing. Stephan Pfletschinger and Prof. Dr.-Ing. Joachim Speidel Institut für Nachrichtenübertragung, Universität Stuttgart, Pfaffenwaldring 47, Stuttgart, Germany Abstract The design of optimum s for digital orthogonal multicarrier Offset-QA (C-OQA) is presented. The s are designed such that intercarrier and intersymbol interference are exactly zero and the maximum energy of the is concentrated around the subcarrier. This solution is obtained by solving a nonlinear optimization problem with constraints and by expanding the into a discrete prolate spheroidal sequence. The design method and optimum s are presented. With this technique an almost flat spectrum in the pass-band around the subcarrier frequency is achieved, and the out-of-band spectral parts are more than 0 db below the values of a rectangular, which is used by conventional Orthogonal Frequency Division ultiplexing (OFD) with Inverse Discrete Fourier Transform (IDFT). Thus, the presented C-OQA provides much lower spectral overlap between modulated subcarriers and is more robust against frequency-selective noise. This will be of advantage for channels which suffer from heavy frequency selective interference like the CaTV return channel.. Introduction ulticarrier modulation schemes first appeared in the literature in the late 50 s, but only in the last decade they have found widespread use in many applications. OFD is now being used in many wireless (DAB, DVB-T, radio LAN) as well as in wired applications (ADSL). Its main advantages over single carrier systems are the immunity against impulsive noise and multipath fading and the lack of a complex equalizer. ost applications of OFD use DFT-processing and a cyclic prefix as a guard interval to combat intersymbol- (ISI) and intercarrier-interference (ICI). This is a very attractive choice from an implementation point of view. DFT-processing inherently incorporates shaping with a rectangular at the transmitter and the receiver with different filter lengths due to the guard interval. However, the drawbacks are the violation of the matchedfilter-criterion at the receiver due to the guard interval and the slowly decreasing spectra of the modulated subcarriers according to a sin( f ) f function. As a consequence of the second point, the modulated subcarriers exhibit a considerable spectral overlap (while maintaining orthogonality) which leads to the effect that even frequency-selective noise, e.g. amateur radio, not only affects the subcarrier centered at this frequency band, but also to a certain extend the adjacent subcarriers. Furthermore, the guard interval reduces the spectral efficiency as the cyclic prefix contains no user data. During the last years there has been an increasing interest in shaping [], [. By designing proper shapes different from the rectangular, the main spectral parts of the modulated subcarriers can be well localized in the frequency domain, to minimize overlap and sensitivity to frequency-selective interferers. This paper presents new optimum shapes for digital multicarrier offset-qa (C-OQA). As an optimization criterion, the signal energy in the frequency band around one subcarrier is maximized, resulting in a minimum spectral overlap between all modulated subcarriers.. System odel Our optimization is based on the system model in Fig. which is the baseband equivalent to a C-OQA system with N subcarriers. It can be considered as a straightforward extension of a single carrier OQA. The incoming symbols D ν [ k], which may represent one bitstream after serial-to-parallel-conversion, arrive at symbol rate N T S. They are first upsampled by the factor before their real and imaginary parts are filtered with or gn [, respectively. For even carrier index ν, the real part of the input symbol D ν [ k] is fed into an shaping filter with response and the imaginary part of D ν [ k] is processed by gn [, and vice versa for ν odd. As we will see later, the reason for introducing the offset (delay ) is to reduce the number of constraints for the orthogonality condition by factor one half. The filter output signal is modulated with the complex exponential function w ν n, where w exp( jπ ), giving a subcarrier spacing of ω π T S. The number N of subcarriers is always assumed to be even. In Fig., k denotes the discrete time index at symbol rate T S while n corresponds to the higher sampling rate T A T S. If the upsampling factor is chosen as an integer multiple of the number of subcarriers N, the multipliers can be integrated into the filter bank. For N, the system is referred to as critically sampled or 9. Dortmunder Fernsehseminar; Sept. 00; Elektronische edien: Technologien, Systeme, Anwendungen; ITG-Fachbericht 67; VDE Verlag 00; S

2 D 0 [ k] D ν [ k] D N [ k] shaping gn [ gn [ gn [ modulation w νn w ( N )n maximally decimated [3]. Note that the filters are operating at the (faster) sampling rate T A. OFD without pulse shaping, i.e. the well-known implementation with FFTprocessing, can be represented in this system model by defining as rectangular of duration without delaying the real or the imaginary part. An efficient implementation of an C-OQA system using FFT-processing and polyphase filters can be found in [4]. The receiver exhibits a corresponding structure. For the analysis of the intersymbol and interchannel interference we assume an ideal channel. The multicarrier signal for an arbitrary input sequence is given by x mc [ n] ( N ) w ln ( D' l [ k] + jd'' l + [ k]w n )gn [ k] l 0 k D' l + [ k]w n + ( + jd'' l [ k] )g n ---- k where D' ν and D'' ν denote the real and the imaginary part of D ν, respectively. We now define the elementary r νµ, [ n] as the response at the receiver side in branch µ to a transmitted unit in branch ν, [5]: r νµ, [ n] y µ [ n] if D i [ k] δ[ i ν] δ[ k] () νµi,, 0,,, N, δ[ k] is the Kronecker delta. For the C-OQA system the elementary s are obtained after some calculation as r νµ, [ n] x mc [ n] equivalent baseband channel () (3) for zero interference : gn [ + with k Z, d { N +,, N } This condition is often referred to as the extended Nyquist criterion as it not only defines ISI-free but also ICI-free transmission. It is also called the orthogonality condition for OFD. Equations (4) imposes a condition on the shaping filter in Eq. (3). To simplify the investigation of Eq. (3), we introduce some assumptions which are convenient for practical filter implementation: with d ν µ, νµ, 0,,, N. Strictly speaking, we would have to define an analogous condition for the imaginary part, but it can be shown that for the above Based on these s we can now define the condition system model both conditions are equivalent. w µn w ( N ) n receiver filters gn [ + y µ [ n] gn [ + y 0 [ n] Dˆ 0 [ k] y N [ n] Fig. : Block diagram for a digital multicarrier offset-qa system. The shaping filters alternately delay the real part or the imaginary part of their input signals by half a symbol period. g n i ( νod) ---- π d( n i) g i + ( µod ) ---- i + j g n i ( νod) ---- π d( n i) sin g i + (( µ + )od) ---- i r νµ Dˆ µ [ k] Dˆ N [ k] (4) (5) 0 for n L, (6) that is, the is even and of length L. Inserting (5) into (3), we conclude after some calculation that Im{ r νµ, [ k ] } 0 holds, i.e. at the sampling instants the elementary s take on real values. Thus we get from (3): r νµ, [ k] i The index range χ { 0,, N } is sufficient, as both sides of (8) are even in χ. Now, we make use of assumption (6). The time-shifted, [ k ] δ[ ν µ, k] g k i ( νod) ---- g[ n] as example ν even,µ odd π d i g i + ( µod ) ---- It can be shown, that r νµ, [ k] in (7) satisfies condition (4), if d ν µ is odd. For d χ even, we obtain from (7) after some calculation: rkχ [, ] r νµ, [ k ] k k 4π (8) g n g n χn! δχk [, ] n (7) 9. Dortmunder Fernsehseminar; Sept. 00; Elektronische edien: Technologien, Systeme, Anwendungen; ITG-Fachbericht 67; VDE Verlag 00; S

3 s gn [ k and gn [ + k in (8) overlap as a function of n for L < k < L, and as both sides of (8) are even in k due to (5), it is sufficient to consider only the range k 0,, L. Thus, we obtain from (8) r[ k, χ] ( L k) k k 4π (9) g n g n χn! δχk [, ] n ( L k) + with χ 0,, N and k 0,, L (9) is an equation system with LN equations and L independent variables, namely g[ 0],, gl [ ]. In a critically sampled multicarrier system ( N ), as it is the case with most of today s OFD systems, we find that the number of independent variables equals the number of constraints (9), i.e. there is no degree of freedom left for shaping. In order to get some freedom for designing the spectral properties of we have to increase the sampling rate. As for practical reasons, it is convenient (although not necessary) that the upsampling factor be an integer multiple of the number of carriers. For the following we choose N (0) As mentioned earlier, OFD systems with DFT-processing use critical sampling and therefore they have to leave a certain number of subcarriers unmodulated in order to avoid aliasing. In contrast to that, in our system in Fig., all subcarriers can be used. Please note that the spectral efficiency is not affected by the choice of as the required channel bandwidth is given by ω B N ω N π T S, independent of. 3. Concentration of the Spectrum In order to achieve a good spectral concentration of the modulated subcarriers, it is desirable to have as much signal energy as possible inside the frequency band ω η ω () where ω π T S denotes the subcarrier spacing. Ideally, for η, all signal energy would be inside the frequency band and there would be no spectral overlap between adjacent modulated subcarriers. For η, each modulated subcarrier would only overlap with its two neighbors. Both cases can be realized only if we allow for an infinite length. For η the solution is a rectangular spectrum with the corresponding time function gt () sinc πt, with sinc( x) T S T S sin( x) x for x for x. Without the offset, at this point we would have twice as much equations. and for η we could use e.g. the well-known squareroot-raised-ine (srrc) with roll-off-factor α : gt () 4 ( πt T S ) πt S ( 4t T S ) In [] an optimization procedure was developed for an analog system model which achieves optimal s for η and an length of T S and 4T S. In this paper we focus on a digital implementation and an optimization for one subcarrier interval ( η ). The signal energy inside the frequency band defined by () is given by E η T A η ω Ge jωt A ( ) dω π LN η ω LN η gm [ ] sinc ηπ ( n m) N N n LN + m LN + () where Gz ( ) denotes the double-sided z-transform of and the second term is obtained by applying Parselval s theorem. E η has to be maximized with respect to, n 0,, LN under the constraints (9). 3. Discrete Prolate Spheroidal Sequences For the maximization of the signal energy E η we expand the into a series of indexlimited sequences. For this purpose the so called discrete prolate spheroidal sequences (dpss) [6], [7], also named Slepian sequences, are especially well suited because they are the set of indexlimited orthogonal sequences with the most concentrated spectra. The dpss v k [ n] are defined by N p W sinc( πw( n m) )v k [ m] m 0 λ k v k [ n] (3) with n Z, k { 0,, N p }, and depend on the parameters N p and W. The λ k are the eigenvalues for which holds: λ0 λ λn, and p > 0 N p λ k k 0 WN p (4) The dpss show the interesting property that they are orthogonal on both the interval 0,, N p and on,, : N p v i [ n] v [ j n ] λ i v i [ n] v [ j n ] δ[ i j] (5) n 0 n The dpss are alternately symmetric and antisymmetric with respect to n ( N p ), i.e. the sequences with even 9. Dortmunder Fernsehseminar; Sept. 00; Elektronische edien: Technologien, Systeme, Anwendungen; ITG-Fachbericht 67; VDE Verlag 00; S

4 index show an even symmetry while the sequences with odd index exhibit an odd symmetry: v k [ n] ( ) k v k [ N p n] (6) We can now expand the into a series with the dpss as basis functions, taking into account the symmetry properties of both sequences. g [ n] gn [ LN] v i [ n] i 0 (7) with g [ n]v i [ n] and N p LN + (8) LN n 0 Putting (7) into (), the signal energy as a function of the expansion coefficients becomes Using the definition (3) of the dpss with W η ( 4N ) and the orthogonality relation (5) we obtain (9) The constraints (9), expressed with the expansion coefficients, are Equation (9) and (0) form a nonlinear optimization problem for the expansion coefficients : the signal energy (9) has to be maximized under the condition that the constraints (0) have to be satisfied. This can be done by a numerical procedure [8] which uses a sequential quadratic programming (SQP) method. LN LN LN LN LN η E η a N i a j v i [ n] v j [ m] sinc ηπ ( n m) N i 0 j 0 n 0 m 0 r[ k, χ] i 0 j 0 n kn E η LN λi i 0 LN LN ( L k)n π (0) a j v i [ n kn]v j [ n + kn] χn! N δχk [, ] i 5 0 Fig. : Expansion coefficients after optimization. The coefficients not shown here are all close to zero. the interval ω < ω < ω, 9.3% of the signal energy of the optimum is concentrated and the spectrum is nearly flat, while outside it decreases more rapidly, resulting in less spectral overlap between the modulated subcarriers. In order to compare our solution with other known shapes, we define the interference power which includes ISI and ICI and corresponds to the squared error with respect to the extended Nyquist criterion: LN e c l with c l rkχ [, ], l k --- N + () χ l where k l Div --- N and χ l od N --- Table compares the interference power and the in-bandenergy for several known s and for the new optimized. Shown are the values for N 8 subcarriers and for an length of 4 symbol intervals, i.e. L Numerical Solution The numerical optimization process results in the expansion coefficients as shown in Fig.. Except for the first few coefficients, all have values close to zero. This is intuitively clear because the first dpss v 0 [ n] has the most concentrated spectrum, the next dpss v [ n] is the sequence with the most concentrated spectrum that is orthogonal to v 0 [ n], etc. The resulting elementary s r νµ, [ n] in Fig. 3 clearly fulfill the extended Nyquist criterion: at the sampling instants except for ν µ, n 0 all samples are zero. Fig. 4 shows the absolute value of the spectrum Ge jωt A ( ) of the optimized versus the spectrum of the conventional rectangular used by OFD with DFT. In r 0,0, r, r 0,0 Re{r,0 } Im{r,0 } 3 0 n/ 3 4 Fig. 3: Elementary s for the optimized pulse shape. 9. Dortmunder Fernsehseminar; Sept. 00; Elektronische edien: Technologien, Systeme, Anwendungen; ITG-Fachbericht 67; VDE Verlag 00; S

5 Table : Interference powers and signal energy concentrations for different pulse shapes. All s have normalized signal energy. E η according to (). optimized (for η ) interference power e E E db 9.30 % % sinc (truncated) db % % Ge jωt A ( ) 30 db 0 db 0 db 0 db 0 db optimum rectangular truncated square root raised ine (srrc) From the values in Table we can draw the following conclusions: The optimized achieves by far the smallest interference power and a much better energy concentration than the rectangular or the srrc-. Of course, the energy concentration of the truncated sinc cannot be reached as this would not be in accordance with the orthogonality condition. The untruncated sinc- concentrates all its signal energy within ω ω, but the truncation leads to an increase of the interference power. As we can observe from the last column, the truncated square-root-raised-ine has practically all its signal energy in the range ω < ω. Therefore, and because of the relatively low interference power, for η (but not for η ) this would be already a feasible solution. The gaussian gt () exp( πt ( T s ) ) has some interesting mathematical properties like its invariance with respect to the Fourier transform and the very rapid decrease of its spectrum. However, even the untruncated does not satisfy the Nyquist criterion. Thus, even for a flat channel, powerful equalization has to be used [9]. 5. Conclusion -38. db 8.7 % % gaussian db 9.37 % % rectangular db a 77.% 90.8 % a. The guard interval was not considered for this calculation. With guard interval the interference power will be much lower. A procedure for designing new s for a digital orthogonal C-OQA system has been presented. The s obtained maximize the signal energy inside the frequency band around the subcarrier, resulting in minimal spectral overlap between the subcarriers. The new s provide an almost flat spectrum in the pass-band around the subcarriers and the out-of-band spectral parts are more than 0 db below the values of a conventional OFD with 0 db 30 db ω ω Fig. 4: agnitude of the spectrum Ge ( jωt A) of the optimized in comparison with the rectangular. IDFT. Also, the interference power due to ICI and ISI is much lower with the new s. These properties are of great advantage, if frequency-selective noise is present on the channel or for multi-user applications, where spectral overlap between different users has to be minimized. The C-OQA requires no guard interval, so the spectral efficiency compared to the conventional OFD is increased. 6. References [] A. Vahlin, N. Holte: "Optimal Finite Duration Pulses for OFD." IEEE Trans. Comm., vol 44, no., pp 0-4, Jan. 996 [ B. Le Floch,. Alard, C. Berrou: "Coded Orthogonal Frequency Division ultiplex." Proc. IEEE, vol. 83, no. 6, pp , June 995 [3] G. Cherubini, E. Eleftheriou, S. Ölçer, J. Cioffi: "Filter Bank odulation Techniques for Very High-Speed Digital Subscriber Lines." IEEE Comm. ag.,pp , ay 000 [4] N. J. Fliege: "Orthogonal ultiple Carrier Data Transmission." European Transactions on Telecommunications, vol. 3, no. 3, pp , ay-june 99 [5] K. D. Kammeyer, U. Tuisel, H. Schulze, H. Bochmann: "Digital ulticarrier-transmission of Audio Signals Over obile Radio Channels." European Transactions on Telecommunications, vol. 3, no. 3, pp , ay-june 99 [6] D. Slepian: "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty V: The Discrete Case." Bell Syst. Tech. J., vol. 57, no. 5, pp , ay-june 978 [7] P. H. Halpern: "Optimum Finite Duration Nyquist Signals." IEEE Trans. Comm., vol. CO-7, no. 6, pp , June 979 [8] T. Coleman,. A. Branch, A. Grace: Optimization Toolbox for Use with atlab, Version. Natick: The athworks, 999 [9] K. atheus: Generalized Coherent ulticarrier Systems for obile Communications. Dr.-Ing. Dissertation, Aachen: Shaker, Dortmunder Fernsehseminar; Sept. 00; Elektronische edien: Technologien, Systeme, Anwendungen; ITG-Fachbericht 67; VDE Verlag 00; S