# Dirty paper coding using sign-bit shaping and LDPC codes

TL;DR: A lattice-based DPC scheme that provides good shaping and coding gains with moderate complexity at both the encoder and the decoder and a design for superposition coding that provides rates better than time-sharing over a Gaussian broadcast channel.

Abstract: Dirty paper coding (DPC) refers to methods for pre-subtraction of known interference at the transmitter of a multiuser communication system. There are numerous applications for DPC, including coding for broadcast channels. Recently, lattice-based coding techniques have provided several designs for DPC. In lattice-based DPC, there are two codes - a convolutional code that defines a lattice used for shaping and an error correction code used for channel coding. Several specific designs have been reported in the recent literature using convolutional and graph-based codes for capacity-approaching shaping and coding gains. In most of the reported designs, either the encoder works on a joint trellis of shaping and channel codes or the decoder requires iterations between the shaping and channel decoders. This results in high complexity of implementation. In this work, we present a lattice-based DPC scheme that provides good shaping and coding gains with moderate complexity at both the encoder and the decoder. We use a convolutional code for sign-bit shaping, and a low-density parity check (LDPC) code for channel coding. The crucial idea is the introduction of a one-codeword delay and careful parsing of the bits at the transmitter, which enables an LDPC decoder to be run first at the receiver. This provides gains without the need for iterations between the shaping and channel decoders. Simulation results confirm that at high rates the proposed DPC method performs close to capacity with moderate complexity. As an application of the proposed DPC method, we show a design for superposition coding that provides rates better than time-sharing over a Gaussian broadcast channel.

## Summary (2 min read)

### Introduction

- Situations where interference is known non-causally at the transmitter but not at the receiver model several useful multiuser communication scenarios.
- In [5], multilevel coding is used, and there are different codes for different bits of the symbols.
- In [8] and [9], shaping follows channel coding and the receiver performs iterations between the shaping and channel decoders.
- The rest of the paper is organized as follows.
- After a brief review of the lattice-based DPC coding method in Section II, the authors present the proposed DPC method in Section III.

### II. LATTICE DIRTY PAPER CODES

- The transmit power is assumed to be upper-bounded by 1nE[|X|2] ≤ PX per symbol, and the interference power is denoted 1nE[|S|2] = PS per symbol.
- The noise variance per symbol is denoted PN .
- In [1], Costa shows that the capacity of the dirty paper channel is 12 log ( 1 + PXPN ) i.e. known interference can be canceled perfectly at the transmitter.
- The lattice transmission approach of [3] [4] is as follows.
- Note that the dither is assumed to be known at the transmitter and receiver (say, through the use of a common seed in a random number generator).

### III. PROPOSED SCHEME

- The proposed scheme uses a convolutional code for signbit shaping [10] and low density parity check (LDPC) codes for channel coding.
- For M = 16, the constellation and mapping are shown in Fig.
- The mapping in Fig. 1 is suited for signbit shaping, since a flip of the most significant bit results in a significant change in symbol value for all possible 4-bit inputs.
- Also, the mapping is mostly Gray except for a few symbol transitions.
- As expected, larger values of M will result in larger shaping gains in their design, and the authors stick to the 16-PAM shown in Fig. 1 for illustration and simulation.

### A. Encoder Structure

- The encoder structure for the proposed scheme is as shown in Fig.
- Note that the authors need the rate of the convolutional code to be 1−k′/s.
- Let the coset chosen by m′ be denoted C(m′).
- The sign-bit vector z is chosen from C(m′) so as to minimize the squared sum of the vector (v − αS) mod M , where α = PXPX+PN is the MMSE factor and S is the interference vector.
- In summary, the encoder structure achieves DPC shaping and LDPC coding with bit-interleaved modulation.

### B. Decoder Structure

- The decoder for the proposed scheme is as shown in Fig.3.
- The decoded shaped bits are passed through a syndrome former to get message bits used for shaping.
- The LDPC decoder outputs k−k′ message bits and s bits of the sign bit vector of the previous block.
- The demapper function at the receiver has to calculate LLRs taking into account the modulo M operation at the encoder [4].
- Therefore, the received constellation AR is a replicated version of the M -PAM constellation A used at the transmitter (assuming that scaling factors have been corrected at the receiver).

### IV. SIMULATION RESULTS

- A non-systematic convolutional code is used to avoid error propagation problems.
- The authors considered several candidate distributions originally designed for BPSK over AWGN, and chose the one that provided the best performance.
- The plot with interference did not change appreciably for all power levels of interference, and the authors have provided one plot for illustration.
- This shows that the authors are 1.46 dB away from ideal dirty paper channel capacity.
- The authors observed that trellis shaping with larger number of states results in a decrease in shaping loss in other simulations.

### V. APPLICATION TO GAUSSIAN BROADCAST CHANNEL

- Here, User 2 is coded using DPC considering User 1 as interference.
- User 1 is shaped using sign-bit shaping and coded using an LDPC code over M -PAM.
- The demapper at Receiver 1 calculates LLR for the i-th bit in the j-th receiver symbol Y1j using the following formula.
- Comparison with the SNR needed for a single user capacity of 3 bits per channel use (which is 17.99 dB) shows that the total loss for both the users is about 2.4642dB.
- The authors see that the (3,3) rate point is clearly outside the time-sharing region.

### VI. CONCLUSIONS

- The authors have proposed a method for designing lattice-based schemes for dirty paper coding using sign-bit shaping and LDPC codes.
- Simulation results show that the proposed design performs 1.46dB away from the dirty paper capacity for a block length of n = 40000 at the rate of 3 bits/channel use.
- This performance is comparable to other results in the literature.
- As discussed in this article, a novel method for combining shaping and coding results in good gains at lesser complexity in their design, when compared to other lattice-based strategies.
- As an application, the authors have designed a superposition coding scheme for Gaussian broadcast channels that is shown to perform better than timesharing through simulations.

Did you find this useful? Give us your feedback

...read more

##### Citations

22 citations

### Cites background from "Dirty paper coding using sign-bit s..."

..., having only two users), dirty paper coding within each group is practically feasible [8]–[10]....

[...]

8 citations

### Cites background or methods from "Dirty paper coding using sign-bit s..."

...Practical code designs for DPC have been studied in [3]–[7]....

[...]

...In this work, we design codes for the MIMO-BC using this idea from [7]....

[...]

...This is an extension of the design for the single antenna Gaussian broadcast channel in [7]....

[...]

^{1}, Princeton University

^{2}, Samsung

^{3}, Columbia University

^{4}, Texas A&M University

^{5}

5 citations

### Cites background from "Dirty paper coding using sign-bit s..."

...A low-complexity DPC scheme with low-density paritycheck (LDPC) codes and sign-bit shaping was proposed in [15] which performed 1....

[...]

5 citations

### Cites background from "Dirty paper coding using sign-bit s..."

...With small groups (having only 2 or 3 users), dirty paper coding within each group is practically feasible and is equivalent to dirty paper coding for a MISO broadcast channel with small number of users [6], [7]....

[...]

3 citations

### Cites methods from "Dirty paper coding using sign-bit s..."

...These DPC techniques typically use a combination of a vector quantizer and a capacity-approaching channel code like low density parity check (LDPC) code [14], [18], [19]....

[...]

##### References

4,068 citations

2,098 citations

489 citations

### "Dirty paper coding using sign-bit s..." refers background or methods in this paper

...In [3], a dirty paper coding (DPC) scheme based on lattice strategies was proposed and shown to achieve the capacity of the dirty paper channel....

[...]

...We follow [4] for a brief review of the transmitter and receiver structure in the lattice DPC method [3]....

[...]

...The lattice transmission approach of [3] [4] is as follows....

[...]

...Lattice-based ideas for DPC were suggested and shown to be capacity-approaching in [2], [3]....

[...]

[...]

390 citations

### "Dirty paper coding using sign-bit s..." refers methods in this paper

...A part of the message m′ = [m1 m2 · · ·mk′ ] with k′ < k bits is mapped to a coset leader of the convolutional code using an inverse syndrome former [10]....

[...]

...The proposed scheme uses a convolutional code for signbit shaping [10] and low density parity check (LDPC) codes for channel coding....

[...]

...The minimization in (6) is implemented using the Viterbi algorithm [10]....

[...]

325 citations

### "Dirty paper coding using sign-bit s..." refers background or methods in this paper

...Recently, many designs of lattice-based DPC schemes have been proposed in [4]–[8]....

[...]

...Lower bounds on achievable rates for the above equivalent channel is shown in [4] to be equal to I (V;Y′) ≥ 1 2 log2 (1 + SNR) − 1 2 log2 (2πeG (Λ)) ....

[...]

...The demapper function at the receiver has to calculate LLRs taking into account the modulo M operation at the encoder [4]....

[...]

...The lattice transmission approach of [3] [4] is as follows....

[...]

...We follow [4] for a brief review of the transmitter and receiver structure in the lattice DPC method [3]....

[...]

##### Related Papers (5)

##### Frequently Asked Questions (18)

###### Q2. What have the authors stated for future works in "Dirty paper coding using sign-bit shaping and ldpc codes" ?

Optimizing the LDPC code using genetic algorithms and asymmetric density evolution [ 12 ] along with joint optimization of shaping code and LDPC code using EXIT charts [ 8 ] are topics for future work.

###### Q3. What is the SNR for Receiver 1?

The SNR for Receiver 1 is computed as 10 log10 ( PX1PX2+PN1) = 19.1791 dB. SinceDPC is done for User 2, the effective SNR at Receiver 2 is computed as 10 log10 ( PX2 PN2 ) = 19.4574 dB.

###### Q4. What is the coding strategy for dirty paper?

In a Gaussian dirty paper channel, the received symbolvector Y = [Y1 Y2 · · ·Yn] is modeled as Y = X + S + N,where X = [X1 X2 · · ·Xn] denotes the transmitted vector, S = [S1 S2 · · ·Sn] denotes the interfering vector assumed to be known non-causally at the transmitter and N denotes the additive Gaussian noise vector.

###### Q5. What is the sign-bit vector for i-th?

The bits that map to the i-th symbol are denoted zia2ia3i · · · ali; the sign-bit vector is denoted z = [z1 z2 · · · zs], and the authors define vectors aj = [aj1 aj2 · · · ajs] for 2 ≤ j ≤ l.

###### Q6. What is the snr of the proposed design?

Simulation results show that the proposed design performs 1.46dB away from the dirty paper capacity for a block length of n = 40000 at the rate of 3 bits/channel use.

###### Q7. What is the granular gain of the channel?

The granular gain G(Λ) = 22R/6Sx is computed from the simulations to be 1.282dB [4], where R = 3.5 is the rate before channel coding, and Sx is the transmit power (obtained through simulations).

###### Q8. What is the sign-bit vector for the i-th block?

Let m′′ = [z mk′+1 · · ·mk] be input to the systematic LDPC encoder to obtain the codeword E(m′′) = [m′′ pT ], where pT is the parity-bit vector for the T -th block.

###### Q9. What is the meaning of dirty paper coding?

The interfering vector S is used as an input in the encoding process and plays an important role to determine a suitable transmit vector X. A coding strategy for choosing X needs to overcome the imminent addition of S and protect the transmitted information from the addition of the noise N. Such coding strategies are called dirty paper coding (DPC) methods.

###### Q10. Why is the LDPC decoder shown in Fig.3?

Because of the one-codeword delay, parity bits of the T + 1-th block and message plus shaped bits of the T -th block form a valid LDPC codeword.

###### Q11. What is the mapping in Fig. 1?

1. The mapping in Fig. 1 is suited for signbit shaping, since a flip of the most significant bit results in a significant change in symbol value for all possible 4-bit inputs.

###### Q12. What is the noise variance per symbol?

The transmit power is assumed to be upper-bounded by 1nE[|X|2] ≤ PX per symbol, and the interference power is denoted 1nE[|S|2] = PS per symbol.

###### Q13. What is the capacity of the dirty paper channel?

In [1], Costa shows that the capacity of the dirty paper channel is 12 log ( 1 + PXPN ) i.e. known interference can be canceled perfectly at the transmitter.

###### Q14. What is the SNR for the receiver?

The total transmit power, power for User 1 and power for User 2 required for a bit error rate of 10−5 (at both receivers) are estimated from the simulation and denoted P , Px1 and PX2 , respectively.

###### Q15. What is the a priori probability of the distribution of M -PAM symbols?

At the receiver, the authors approximate pi usinga Gaussian distribution with variance PS assuming that the distribution of M -PAM symbols is approximately Gaussian.

###### Q16. What is the proposed scheme for superposition coding in a two-user Gaussian?

The authors use the proposed scheme for superposition coding in a two-user Gaussian broadcast channel Y1 = X + N1 and Y2 = X + N2 with PN1 > PN2 .

###### Q17. What is the average power of the symbol a?

The number of replications r is chosen so that the average power of AR is approximately equal to the total average power PX + PS , and the bit mapping of the symbol a + jM (a ∈ A, 1 ≤ j ≤ r) is the same as that for a.

###### Q18. What is the LLR of the sign bits after a delay?

The s = nlog2M LLRs of the sign bits after a delay on one time step, and then−s output LLRs of the de-interleaver are given as the input to the LDPC decoder.