Optimal Intra-Symbol Transmit Windowing for Multicarrier Modulation

Transcription

1 Optimal Intra-Symbol Transmit Windowing for Multicarrier Modulation Thomas Magesacher Department of Information Technology, Lund University, Sweden ISCCSP 2006, Marrakech This work has been supported by the EU (IST-MUSE) and VINNOVA-Celtic (BANITS).

2 Transmit windowing: what, how, and why? power spectral density [db] normalised frequency Poor spectral containment of basis functions yields poor exploitation of the mask The transmit power and thus the information rate are low Windowing of the transmit blocks allows better exploitation of the PSD limits Consequently, windowing increases the transmit power and thus can increase the throughput

3 Multicarrier modulation with intra-symbol windowing x n diag {p} W H t U Z W G V 1 add H Z rem transmitter channel receiver ˆx Block-wise multiplication of the transmit data by the window U in transmitter Equivalently, we can multiply by V before the IFFT, where V[n,k]= 1 (W H u)[((k n) modn) + 1], n,k = 1,...,N N The receiver restores orthogonality by multiplication with V 1 after equalization Implementation: U (O(N)) and V 1 (O(N 2 ))

4 Standard-design: maximum mainlobe-energy window u (e) u (e) =arg max u 2π/(N+L) 0 e(e jω )Z add u 2 dω, e(e jω )= [ 1 e jω...e jω(n+l 1)] amplitude, linear u (e) sample No. State-of-the-art design [1][2] Does not require CSI

5 Objective function: information rate R(p,u) ( N { } BHA diag p Cx diag { } p (BHA) H ) [k,k] R(p,u)= log 2 1+ ( k=1 BC n B H) [k,k] A =Z add W H V: transmit processing B =V 1 GWZ rem : receive processing C x = E { xx H} C N N : covariance matrix of the transmit data C n = E { nn H} C (N+L) (N+L) : covariance matrix of the noise

6 Optimal power loading p ( ) = arg max p N k=1 1 + ( BHA diag {p} (BHA H) [k,k] ( BC n B H) [k,k] { subject to Q vec A diag {p}a H} m m R S 1 + : PSD mask Q C S (N+L)2 : PSD transform matrix Finding the optimal power values p ( ) is a semidefinite program assuming perfect CSI (H and C n are known) x N (0,I N )

7 Optimal transmit windowing ( {u (r),p (r) N BHA diag {p} (BHA) H ) [k,k] } = arg max u,p 1+ ( k=1 BC n B H) [k,k] { subject to Q vec A diag {p}a H} m Dependence of V 1 (and thus of B) on u renders the problem non-convex Numerical solutions using dedicated software [3]-[5]

8 Window shapes amplitude, linear maximum-rate window, magnitude: u (r) maximum-rate window, real: Re {u (r)} maximum-rate window, imag.: Im {u (r)} maximum mainlobe-energy window u (e) rectangular window sample No. N = 32, L = 2, SNR = 10dB, flat mask

9 Information rate 1.0 information rate / Rmax maximum-rate window u (r) maximum mainlobe-energy window u (e) rectangular window N/L N = 32, L {2,4,6,...,16}, SNR = 10dB, flat mask

10 Information-rate gain information rate gain over rectangular window maximum-rate window u (r) 0.9 maximum mainlobe-energy window u (e) SNR = 0dB SNR = 10dB SNR = 20dB SNR = 30dB N/L N = 32, L {2,4,6,...,16}, SNR = 10dB, flat mask

11 Out-of-band PSD 0 N = 32, L = 2 magnitude (db) maximum-rate window u (r) max. mainlobe-energy win. u (e) rectangular window mask frequency, normalised p (e) [k] = p (r) [k] = p (rect) [k] = 0,k {1,...,8,25,...,32}

12 Out-of-band PSD 0 N = 32, L = 8 magnitude (db) maximum-rate window u (r) max. mainlobe-energy win. u (e) rectangular window mask frequency, normalised p (e) [k] = p (r) [k] = p (rect) [k] = 0,k {1,...,8,25,...,32}

13 Out-of-band PSD 0 N = 32, L = 16 magnitude (db) maximum-rate window u (r) max. mainlobe-energy win. u (e) rectangular window mask frequency, normalised p (e) [k] = p (r) [k] = p (rect) [k] = 0,k {1,...,8,25,...,32}

14 Conclusions 1 Maximum-rate window outperforms the maximum mainlobe-energy window in terms of information rate for the all-flat case 2 Gain in rate is more pronounced for low SNRs and for short cyclic extensions 3 Superior out-of-band PSD characteristics of the maximum mainlobe-energy window gradually deteriorates with decreasing N/L

15 References [1] G. Cuypers and K. Vanbleu and G. Ysebaert and M. Moonen, Egress reduction by intrasymbol windowing in DMT based transmitters, in Proc. ICASSP03, vol. 3, pp [2] Y.-P. Lin and S.-M. Phoong, Window designs for DFT-based multicarrier systems, IEEE Trans. on Signal Processing, vol. 53, pp , Mar [3] J. Löfberg, YALMIP : A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004, Available from [4] B. Borchers, CSDP, a C library for semidefinite programming, Optimization Methods & Software, vol. 11-2, pp , 1999, Available from borchers/csdp.html [5] Mathworks, Matlab optimization toolbox (online documentation), 2005,