International Journal of Computer Networks & Communications (IJCNC)


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 [1] and [4] 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 [6], Costa showed that the capacity for this channel doesn’t change when the interference is non-causally known at the transmitter, and later in [9], they generalize this result for an arbitrary interference distribution if the noise is Gaussian that is,

In [4], the authors followed [1] 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 [5] for the classic non fading channel and in [3] 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.


Lattices are firstly used in [5] to achieve the capacity of dirty paper channel. Let be a k-


 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 [4] 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



Figure 1. Bounds for dirty paper channel.

Again for only magnitude fading, the bounds coincide and are achieved by time sharing and so the channel capacity.

Figure 2. Bounds for dirty paper channel.
Figure 3. Bounds for dirty paper channel.


The faded dirty paper channels with arbitrary coefficients have been studied. The results in [1] and [4] 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.


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