Cosine-Modulated Filter Bank Design for Multicarrier VDSL Modems

Transcription

1 Cosine-Modulated Filter Bank Design for Multicarrier VDSL Modems Ari Viholainen, Tapio Saramäki, and Markku Renfors Telecommunications Laboratory, Tampere University of Technology P.O. Box 553, FIN-3311 Tampere, Finland Telefax: Int , mr@cs.tut.fi ABSTRACT: In this paper we apply the idea of cosine-modulated perfect reconstruction filter banks to develop efficient transmultiplexers for multicarrier VDSL transmission systems of the Discrete Wavelet Multitone type. The effects of the design parameters (number of channels, roll-off, filter lengths on the implementation complexity are evaluated through design examples. Based on a simplified interference model, criteria are developed for evaluating the reduction in data transmission rate due to nonideal filter bank design, i.e, finite number of channels and nonzero roll-off. The cosine-modulated transmultiplexer is also analyzed from the data modulation point-of-view and similarity of the critically sampled filter bank approach to other approaches known in the literature is discussed. 1 Introduction A multicarrier technique, DMT, has been adopted for the Asymmetric Digital Subscriber Line (ADSL standards [1], and similar techniques have been proposed also for the Very High-Speed Digital Subscriber Line (VDSL systems. As a further development of DMT, the so-called Discrete Wavelet MultiTone (DWMT technique has been proposed for VDSL standardization [2]. The main advantage of multicarrier modulation in high-speed digital subscriber line modems is that subchannels which contain interfering signals utilize lower-order modulation, or in case of strong interference, are not utilized at all. Subchannels with high S/N-ratio are utilizing high-order modulation. In this way the transmission capacity of the transmission medium, twisted pair, can be utilized efficiently. Both the DMT and DWMT can be considered as filter bank based transmultiplexer (TMUX systems, as shown in Fig. 1. In spectrally efficient systems, the subchannels are partly overlapping in the frequency domain, while orthogonality of the subcarriers is maintained. This means that, in ideal conditions, there is no crosstalk between the subchannels and each subchannel is free of Intersymbol Interference (ISI. Different approaches for spectrally efficient TMUX design using partially overlapping subchannels have been considered in the literature since 196'ies. In case of DMT, the filter banks needed in the transmitter and receiver ends are implemented through Inverse Discrete Fourier Transform (IDFT and Discrete Fourier Transform (DFT, respectively. However, the selectivity of DFT as a filter bank is rather limited. This means that a strong narrowband interference (like a near-by SW broadcasting or radio amateur station makes several subchannels unusable for data transmission. In DWMT, the idea is to make the filter bank more selective to limit the effect of the interference to the subchannels which are in the frequency range of the interference. In this paper we present efficient solutions for VDSL systems based on the idea of cosine-modulated filter bank (CMFB. Section 2 introduces the CMFBs in the VDSL TMUX application. In Section 3 we analyse the filter bank based TMUX system from the data modulation point of view. Section presents an analysis of how the nonideality of the filter banks, the finite number of subchannels and non-zero transition bandwidths, effect on the transmission capacity of the TMUX system. A figure of merit for filter optimization is proposed. Section 5 gives results of several designs with different parameters of the filter bank. The effect of the design parameters on the implementation complexity is evaluated. 2 M-Channel Filter Banks 2.1 Transmultiplexer System We consider the maximally decimated M -channel TMUX shown in Fig.1. This kind of TMUXs have been analyzed in [3] and []. For this system, the output signals at the channel sampling frequency are given by Xˆ 1 z M ( Xˆ M z M ( where [ H] ij = H j ( zw ( i 1 F T H T F M Xˆ 1 ( z M T = M Xˆ M ( z, [ ] ij F j ( zw ( i 1, = W = e j2π M.,(1 The crosstalk between the subchannels of the TMUX are cancelled if the following matrix is diagonal: [ H] T F = diag T 1 ( z M T M ( z M (2 When T i ( z are just delays, we have perfect reconstruction (PR, i.e., each subchannel is ISI-free. This condition is similar to the condition for achieving PR in analysis-synthesis filter banks. This means that the results developed for designing PR analysissynthesis filter banks can be directly applied to the TMUX design [3].

2 2.2 Cosine-Modulated Filter Banks We consider here CMFBs with FIR filters for this VDSL TMUX problem, because this is the most effective technique to construct the desired analysis and synthesis filters from both designing and implementation points of view []-[8]. The basic idea of CMFB is that we can derive subchannel filters with real coefficients from a single prototype filter by using cosine modulation. Those subchannel filters are uniformly shifted versions of the prototype filter. Such a filter bank can be implemented using one prototype filter and a cosine-modulation block. The main interest here is on PR filter banks. However, near perfect reconstruction (NPR filter banks can also be an interesting choice. Sometimes it is judicious to relax the PR conditions by allowing small amplitude and aliasing errors. This results in improved stopband attenuations Prototype Filter In our case the prototype filter for CMFB is a symmetric linear-phase FIR lowpass filter, whose bandwidth is π ( 2M. The transfer function of this prototype filter is of the following form N H p ( z = h p ( nz n, h p ( N n = n = h p ( n (3 The length of the prototype filter is L = 2KM, where K is integer and M is the number of channels. The prototype is a spectral factor of a 2M -th band (Nyquist filter, i.e., hn ( = h p ( n*h p ( n is an 2M -th band filter. h has regular zero-crossings with the spacing of 2M, except that the center tap value is 1 ( 2M. At ω = π ( 2M, the prototype filter approximately achieves the value of 1 2. The stopband edge for the prototype filter is usually given as ( 1 + ρπ ω s = ( 2M where ρ is the roll-off factor determining how much the adjacent subchannels overlap Channel Filters The subchannel filters (both the M analysis and synthesis filters, see Fig. 2 are derived from the prototype filter by complex modulation. One alternative is to use impulse responses of the following form [] h k ( n 2h p ( n ( 2k + 1 π N = cos n ( 1 kπ -- f k ( n 2h p ( n ( 2k + 1 π N = cos n ( 1 kπ -- (5 (6 It follows from the above equations that the synthesis filters are time-reversed versions of the analysis filters, that is, f k ( n = h k ( N n or equivalently F k ( z = z N H k ( z. The design routine takes into account the frequency response specifications for the prototype filter as well as the conditions for perfect reconstruction. There are two alternative ways to design the prototype filter: the least-mean-square (L2 and minimax (MM criteria. The synthesis methods are studied in [6]-[8]. The synthesis techniques proposed in [8] are the most powerful and flexible among the existing design methods. We have used the Dutta-Vidyasagar algorithm [9] to solve both problems Fig. 3 shows the amplitude responses of the analysis and synthesis filters, for K = 5, M =, and ρ = 1,, as well as impulse responses of the analysis bank and overall impulse responses for some of the subchannels. In the worst case the channel filters have approximately a 3 db lower stopband attenuation than the prototype filter. We have noticed that for minimax designs the channel filters have almost always 3 db lower attenuation, but for L2 designs the difference may vary between and 3 db. 3 Data Modulation When CMFBs are used for data transmission in the TMUX configuration, each subchannel in the transmitter end takes f s M real symbols per second resulting in the total symbol rate of f s. In the modulation domain, each subchannel has a bandwidth of ( 1 + ρf s ( 2M and the subchannel spacing is f s ( 2M. Consequently, as a modulation technique, the transmultiplexer based on real signals has as good spectral efficiency as any multicarrier system utilizing I/Q-modulation for the subcarriers. However, it is important to develop better understanding of the modulation induced to the baseband input signals by the filter bank. This is needed in order to be able to analyse the performance of the overall system and to make optimal design for the receiver signal processing, considering functions like channel equalization and detection. Independently of which interpretation we adopt for the data modulation, the critically sampled PR filter bank approach to TMUX design provides readily zero ISI and zero cross-talk (ICI in ideal conditions. The NPR design approach provides well-controlled ISI and ICI performance with somewhat reduced complexity for the filter banks. 3.1 Different Interpretations of Data Modulation in Filter Banks VSB Interpretation Fig. 3 indicates that subchannel 1 can be considered as a baseband channel. In this case h 1 ( n and f 1 ( n together produce the ISI-free overall impulse response satisfying the well-known Nyquist criteria. This is also one of the constraints included in the design. Now we can conclude that if the subchannel input signals are real L -level data streams, then the signal in subchannel 1 is real L -level baseband PAM signal. Also it is quite straightforward to interpret the other subchannel signals to be obtained through VSB

3 (Vestigial SideBand modulation from real L -level baseband PAM signals [1] Offset QAM-Modulation Interpretation Based on Eq. (5 and denoting the input signal to channel k as a k ( n, the modulated signal in subchannel k can be written as x k ( n = a k (h l k ( n lm l = 2 a k (b l ( n, l cos( α( k, l + φ k (7 l = 2 a k ( l ( 1 bnl (, cos ( β( k, l + φ k l even + 2 a k ( l ( 1 b( n, l sin( β( k, l + φ k where l odd α( k, l ( 2k + 1 π N = n lm ----, 2 β( k, l = ( 2k + 1 π, 2M n N bnl (, = h p ( n lm, and φ k = ( 1 kπ --. This can be identified as offset-qam-type of modulation, with complex input data symbols at rate f s ( 2M, and a half-symbol offset between I and Q components. Here the even samples of the input data are considered as the I-components and odd samples are considered as the Q-components of the complex input data. In addition, the sign of every other sample is changed for both components. Furthermore, there is an alternating +/- 5 degree phase shift in the carriers. 3.2 Other Approaches Bandlimited orthogonal functions for multicarrier modulation were first considered by Chang [11] and [12]. In this work real modulation for the subchannels was considered. The paper [11] formulated the conditions for zero ISI and for avoiding ICI between neighbouring subchannels as symmetry conditions for amplitude and phase responses of the subchannels. It was assumed that subchannels further away from each other are sufficiently isolated by the stopband attenuation, without any special measures for avoiding ICI. The VSB modulation interpretation is cited [1] to have been presented in an unpublished proposal by Becker. The offset-qam (staggered QAM formulation was presented by Salzberg. Similar approach has later been considered by Hirosaki [13], Li and Stette [1], and Vahlin and Holte [15]. The analysis of Vahlin and Holte considers also the general case to give criteria for zero ICI between all the subchannels. The other approaches mentioned above consider only the crosstalk between neighbouring subchannels. On the other hand, the critically sampled PR filter bank approach to the TMUX system has been analysed e.g., by Vetterli [3] and Vaidyanathan []. However, in this context the similarity to the offset-qam type of multicarrier systems introduced earlier in the works mentioned above has usually not been recognized. The subchannel filters of Eq. (7 satisfy the conditions for zero ICI as formulated in the offset-qam approach. However, since ICI between all the channels is under control, it makes it possible for us to consider designs where the roll-off is so high that not only the neighbouring channels are partially overlapping. How to Deal with Interfering Signals In practical multicarrier VDSL systems, the data transmission rate of each subchannel depends on the attenuation and interference level of the subchannel in question. For example, in subchannels with moderate attenuation and some interference, lower order modulation may be used than in better subchannels. Of course, subchannels with severe attenuation or interference can not be used for data transmission. In the following analysis we make the simplifying assumption that the interference is sharply band-limited and all subchannels having interference in the passband or transition bands of the channel frequency response are not used for data transmission. The number of subchannels to be removed depends on (i how much adjacent subchannels overlap, (ii what is the bandwidth of interfering signal, and (iii where the interference is situated. For the remaining subchannels, the interference level is comparable to the stopband attenuation of the subchannel filters. Fig. shows an example of a narrowband interference. For example, isolated amateur radio or shortwave broadcast radio transmitters cause a narrowband interference to the VDSL system, since their bandwidths ( khz are very small in relation to the practical bandwidths of the subchannels of the filter bank. The parameter ρ has an effect on how much the subchannels are spectrally overlapping. With the aid of Fig. 5, it is easy to observe that at any point on the frequency axis, the minimum number of overlapping subchannels is 1 + ρ, the maximum number is 2 + ρ, and the average number of overlapping subchannels is 1 + ρ. This means that if there is a single narrowband interference signal at a random frequency, and the probability distribution of the interference frequency can assumed to be uniform, the expected value of lost subchannels is 1 + ρ. Consequently, the reduction in data transmission rate is B = ( 1 + ρ M. (8 However, an extremely narrowband signal doesn't reduce the theoretical channel capacity of the system, since by using a very high number of subchannels with low value of roll-off, the reduction in data transmission rate could be made arbitrarily small. Therefore, in the narrowband case the value of B can be considered as a figure of merit describing the reduction in data

4 transmission rate due to nonideal filter bank design. This parameter B is proportional to the full bandwidth of the subchannels, from the lower stopband edge to the upper stopband edge. It can be shown that the value of B is a suitable figure of merit also in the case of sharply bandlimited and randomly located wideband interference, where the edges of the interference band are uniformly distributed on the frequency axis. The same figure of merit can also be applied to the case of several isolated narrowband interferences. Practical situations are more complicated, but we believe that the factor B = ( 1 + ρ M, can be used as a basis for optimizing the filter bank. Different designs with different values of ρ and M, giving the same values for B, stopband attenuation, and aliasing distortion, are considered to be equally good. Then the optimization problem is to minimize the hardware complexity needed for the required performance. 5 Optimization Results Now we are studying PR TMUXs with different values of ρ and K. All results in the following apply to all the subchannel filters of the corresponding design. At first we determined how attenuations change with different number of channels ( M. Here M is a power of two and designs with M up to 6 have been carried out. It was observed that for M = 8 or higher, the number of channels has only a minor effect on the stopband attenuation when the other parameters are fixed. In case of L2-designs, the differences in stopband attenuations are only fractions of db and in case of minimax designs, stopband attenuation is slightly increasing with increasing M. Based on this experience, it is safe to make the comparisons with M = 8 and assume that with higher values of M the results are at least as good. Table 1 shows how the stopband attenuations change with different values of K in L2 and minimax optimization.. Table 1: Attenuation of the highest stopband ripple (M=8 and ρ=1.. K=2 K=3 K= K=5 K=6 K=7 K=8 L MM Next we consider finding the optimum tradeoff between the roll-off parameter ρ and the filter length parameter K to minimize the implementation complexity. For the complexity evaluation, we estimate the multiplication and addition rates, r mult and r add to be performed at the channel sample rate in the transmitter and receiver as [6] 1 r mult = -- ( 2K + log (9 2 2 M + 3 and 1 r add = -- ( 2K + 3log. (1 2 2 M + 1 These formulas are valid when the number of channels M is a power of two. Even though this may be the best practical choice, in the following idealized analysis we consider also other choices for the number of channels and regard r mult and r add as estimates for the multiplication and addition rates. Then Table 2 shows the required number of channels and the resulting estimates for the multiplication and addition rates for db and 5 db stopband attenuations of the subchannel filters with two different values of B = ( 1 + ρ M, and with different values of K in L2- designs. In the case of db stopband attenuation, it can be seen that when K ranges from 3 to 6, ρ changes from to.722. For db and 5 db stopband attenuations, the minimum values for K were 3 and, respectively. The results of Table 2 indicate the differences in the multiplication and addition rates are not very big. Anyway, it seems that the lowest values for the multiplication and addition rates can be achieved by using the lowest value of K for which the required stopband attenuation can be achieved. Table 2: Multiplication and addition rates for L2- designs with db and 5 db stopband attenuations and with B=1/1 and B=1/1. 6 Conclusions A s = db, B=.1 /.1 K=6 K=5 K= K=3 ρ M 172/ /19 199/ /2222 r mult 11.2/ / /11. 8./1.1 r add 17.6/ / / /2.2 A s =5 db, B=.1 /.1 ρ M 19/ / /239 - r mult 11.3/ / / r add 17.9/ / / In this paper we have presented the CMFBs approach for highly selective critically sampled TMUX systems. We have also analysed the TMUX system from the modulation point of view. An analysis of the available transmission capacity of practical filter banks in channels with narrowband interference was also carried out. The assumptions in the analysis were

5 simplified from the practical situation. But we believe that the simple criterion of using B = ( 1 + ρ M as a measure of limited selectivity of the filter bank is fruitful. This criterion, together with the results from practical CMFB designs indicate that the lowest implementation complexity is obtained by using the smallest filter length which gives sufficient stopband attenuation for the channel filters. However, other aspects of the implementation complexity have to be considered before making the final conclusions. The performance of NPR filter bank designs in this context remains as a topic for future work. Preliminary results indicate that it may be possible to reduce the minimum possible value for K by allowing some controlled level of ISI and/or crosstalk between the subchannels. References: [1] ANSI T : "Telecommunications - Network and Customer Installation Interfaces - Asymmetric Digital Subscriber Line (ADSL Metallic Interface." [2] M. A. Tzannes, M. C. Tzannes, J. Proakis, P. N. Heller, "DMT systems, DWMT systems and digital filter banks," in Proc. ICC'9. pp [3] M. Vetterli, "Perfect Transmultiplexers," in Proc. IEEE ICASSP, Tokyo, pp , April [] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall [5] H. S. Malvar, "Modulated QMF Filter Banks with Perfect Reconstruction," Electronics Letters, vol. 26, pp , June 199. [6] H. S. Malvar, "Extended Lapped Transforms: Properties, Applications, and Fast Algorithms," IEEE Trans. SP, vol., pp , Nov [7] T. Saramäki, "Designing Prototype Filters for Perfect- Reconstruction Cosine-Modulated Filter Banks," in Proc. IEEE Int. Conf. on Circuits and Systems, San Diego, CA, pp , May [8] T. Saramäki, "Cosine-Modulated Filter Banks - A Tutorial Review," an invited tutorial paper presented at the 1996 IEEE ICECS'96, Rhodos, Greece, Oct , [9] S. R. K. Dutta, M. Vidyasagar, "New Algorithms for Constrained Minimax Optimization," Mathematical Programming, vol. 13, pp , [1] B.R. Salzberg, "Performance of an efficient parallel data transmission system," IEEE Trans. Comm., vol.15, pp , Dec [11] R.W. Chang, "Synthesis of band-limited orthogonal signals for multichannel data transmission," Bell Sys. Tech. J., vol. 5, pp , Dec [12] R.W. Chang, R.A. Gibby, "A theoretical study of performance of an orthogonal multiplexing data transmission scheme," IEEE Trans. Comm., vol. 16, pp , Aug [13] B. Hirosaki, "An analysis of automatic equalizers for orthogonally multiplexed QAM systems," IEEE Trans. Comm., vol. 28, pp , Jan [1] R. Li, G. Stette, "Time-limited orthogonal multicarrier modulation schemes, " IEEE Trans. Comm., vol. 3, pp , Feb./March/April [15] A. Vahlin, N. Holte, "A new class of OFDM systems," in proc. Nordic SP Symposium, NORSIG'9 (Ålesund, Norway, pp , June 199. X 1 ( z Xˆ 1 ( z M F 1 ( z H 1 ( z M X 2 ( z Xˆ 2 ( z M F 2 ( z H 2 ( z M + X M ( z Xˆ M ( z M F M ( z H M ( z M Fig. 1 Maximally decimated M-channel transmultiplexer. π ( 2M π ( 2M π M π M 2π M ( M 1π M π Fig. 2 Typical ideal responses for the prototype filter and channel filters h 3 h 3 * f h Fig. 3 Amplitude and impulse responses for a cosinemodulated transmultiplexer. Fig. The effect of narrowband interference to an 8-channel filter bank. Fig. 5 Idealized channel frequency responses with different values of ρ ρ =.1 ρ = 1.1 ρ = 2.1 h * f 2 H (e jω, F (e jω H 1 (e jω, F 1 (e jω H 2 (e jω, F 2 (e jω H 3 (e jω, F 3 (e jω ρ = 1.