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Bounds on the Achievable Rates of Faded Dirty Paper Channel
Zouhair Al-qudah1, Wael Abu Shehab2
1Department of Communication Engineering, Al-Hussein Bin Talal University, Ma’an, Jordan
2Department of Electrical Engineering, Al-Hussein Bin Talal University, Ma’an, Jordan
Bounds on the achievable rate of a Gaussian channel in the case that the transmitter knows the interference signal but not its fading coefficients are given. We generalize the analysis which were studied in  and  so that their results are special cases of our analysis. We enforce our bounds by simulations in which many numerical examples are drawn and investigated under different cases.
Dirty paper coding, lattice encoding and decoding, fading channel, Viterbi Algorithm.
Dirty paper coding (DPC) is a communication model deals with Gaussian noise non-causally known at the transmitter. In this communication model, an encoding technique is used with such a noise such that its effects can be cancelled completely as Costa showed in his valued paper. The received signal, is described as
In , Costa showed that the capacity for this channel doesn’t change when the interference is non-causally known at the transmitter, and later in , they generalize this result for an arbitrary interference distribution if the noise is Gaussian that is,
In , the authors followed  and studied the compound channel formulation instead of outage probabilities. In particular, they derived lower and upper bounds on the achievable bounds when the interference phase is unknown at the transmitter. Their derivation based on an adaption of the nested lattice approach that is described in  for the classic non fading channel and in  for the case of magnitude fading.
In this paper, we generalize the results of [1, 3, and 4] where we assume that the transmitter has lack of magnitude and phase knowledge. We refer to such a situation as the faded dirty paper channel (F-DPC). In this case, the received signal can be modelled as follows
The rest of this paper is organized as follows, we review the principles of lattice dirty paper coding in Section 2. Lower bounds and upper bounds are presented in Section 3. Bounds are simulated and discussed in Section 4. Finally, we conclude our paper in Section 5.
2. LATTICE DIRTY PAPER CODING: DEFINITIONS AND PRELIMINARIES
Lattices are firstly used in  to achieve the capacity of dirty paper channel. Let be a k-
3. BOUNDS OF FADED DIRTY PAPER CHANNEL
In this section, a lower bound and an upper bound on the achievable rates are derived when the transmitter knows the interference non-causally but not the fading coefficients, that is the channel adds faded interference.
3.1. Lower Bound
We follow  in which the signal has a block of bits long. Then, each block is divided into k blocks where every block contains n/k bits. A k dimensional lattice is used to construct the transmitted signal at each time block, the lattice is defined over complex number field. The encoding and decoding procedures are as follows. In this procedure, by using modulo lattice channel encoding scheme, the phase faded AWGN noise channel is converted such that it contains only the transmitted codeword and the total noise over this channel. In this case, the transmitted signal is given by
4. SIMULATIONS AND DISCUSSIONS
Again for only magnitude fading, the bounds coincide and are achieved by time sharing and so the channel capacity.
The faded dirty paper channels with arbitrary coefficients have been studied. The results in  and  have been generalized. We show our bounds by simulations for different cases. We have studied also the performance of faded dirty paper channel and we have showed how the system performance changed when the interference is faded.
 P. Grover and A. Sahai,(2007) “What is needed to exploit knowledge of primary transmissions?,” available at arXiv:cs/0702071.
 P.Mitran,N.Devroye and V.Tarokh, (2006) “ On Compound Channels with Side Information at the Transmitter”, IEEE Trans. on Info. Theory, vol. 52, no. 4, pp. 1745-1755.
 A.Khina and U.Erez, (2010)“On the Robustness of Dirty Paper Coding”, IEEE Trans. on Comm., vol. 58, no. 5, pp. 1437-1446.
 A.Bennatan,V.Aggarwal,Y.Wu,A.R.Calderbank,J.Hoydis ,and A.Chindapol, (2007) “Bounds and Lattice-Based Transmission Strategies for the Phase-Faded Dirty-Paper Channel”, IEEE Trans. on Wireless Comm., vol. 8, no. 7, pp. 3620-3627.
 U. Erez, S. Shamai and R. Zamir, (2005) “Capacity and Lattice-Strategies for Cancelling Known Interference,” IEEE Trans. on Inform. Theory, vol. 51, no. 11, pp. 3820-3833.
 M.Costa, (1983) “Writing on Dirty Paper”,IEEE Trans. On Information theory,vol 29,no.3,pp 439-441.
 A.Cohen and A.Lapidoth, (2002) “The Gaussian watermarking game”,IEEE Trans. Info. Theory vol.48, pp.1639-1667.
 A.Cohen and A.Lapidoth, (2002) “The Gaussian watermarking game”,IEEE Trans. Info. Theory vol.48 ,pp.1639-1667
 U. Erez, S. Shamai (Shitz), and R. Zamir,(2005) “Capacity and lattice strategies for cancelling known interference,” IEEE Trans. Information. Theory, vol. 51, pp. 3820–3833.
 G.D.Forney, (1988) “Coset Codes-Part I:Introduction and Geometrical Clasifications”, IEEE Trans. On Information Theory , vol. 34, no. 5, pp. 1123-1151
 R. Zamir, (2009) “Lattices are Everywhere”, Information Theory and Applications (ITA09), University of California at San Diego, pp. 392-421.
 R.Zamir,S.Shamai and U.Erez, (2002) ”Nested Linear /Lattice codes for structured multidimensional binning”,IEEE Trans. On Information Theory, vol. 48, no. 6, pp. 1250-1276.
 R.Zamir and M.Ferder, (1996) ”On lattice quantization noise “, IEEE Trans. On Information Theory,vol 42,no.4,pp1152-1159.
 U.Erez and S.t,Brink, (2005) “A close-to-capacity Dirty Paper Coding Scheme”, IEEE Trans. on Information Theory, vol. 51, no. 10, pp. 3417-3432.
 M.Mzaaotti and M.Chiani, (2006)“A simple rate ½ co-decoding scheme for Writing on Dirty Paper”, IEEE International Conference on Communications, Istanbul, pp. 1622-1627.
 G.B.Kyung and C. Wang, (2009) “On the Design and Challenges of Practical Binary Dirty Paper Coding”, on the proceeding of 2009 IEEE Wireless Communications and Networking Conference, Budapest, pp. 1-6..
 J.Chen and M.Fossorier, (2002)“Near optimum Universal Belief propogation Based Decoding of Low-Density Parity Check Codes”, IEEE Trans. on Communications, vol. 50, no. 3, pp. 406-414.