Constellation Shaping for Communication Channels with Quantized Outputs

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1 Constellation Shaping for Communication Channels with Quantized Outputs, Dr. Matthew C. Valenti and Xingyu Xiang Lane Department of Computer Science and Electrical Engineering West Virginia University CISS - March 24, 2 Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West / 3 Virg

2 Outline Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 2 / 3 Virg

3 Outline Introduction Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 3 / 3 Virg

4 Introduction Transmitter Side Optimization Receiver Side Optimization Constellation Shaping - Stephane Y. Le Goff 27 IEEE, T. Wireless Quantizer Optimization - Jaspreet Singh 28 ISIT Our Contribution - Joint optimization CISS 2 Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 4 / 3 Virg

5 Introduction Mutual Information( MI ) and Channel Capacity MI between two random variables, X and Y is given by, [ ( )] p(y X) I(X; Y ) = E log p(y ) Channel Capacity is the highest rate at which information can be transmitted over the channel with low error probability. Given the channel and the receiver, capacity is defined as C = max p(x) I(X; Y ) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 5 / 3 Virg

6 Introduction The mutual information between output Y and input X is M p(y x j ) I(X; Y ) = p(x j ) p(y x j ) log 2 p(y) dy. j= M - number of input symbols. This can be solved using Gauss - Hermite Quadratures Information rate results of 6 PAM under continuous output and uniform constellation INFORMATION RATE SNR(dB) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 6 / 3 Virg

7 Outline Constellation Shaping Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 7 / 3 Virg

8 Constellation Shaping Constellation Shaping for PAM Our strategy is from S. LeGoff, IEEE T. Wireless, 27. Transmit low-energy symbols more frequently than high-energy symbols. Shaping encoder helps in achieving the desired symbol distribution. For a fixed average energy, shaping spreads out the symbols with uniform spacing maintained. High Energy Low Energy High Energy.5 High Energy Symbols Low Energy Symbols High Energy Symbols (a) Probability of picking low-energy subconstellation =.5 (b) Probability of picking low-energy subconstellation =.9 ( E s = Σ M i= p(x i)e i = ) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 8 / 3 Virg

9 Shaping Encoder Constellation Shaping We design the shaping encoder to output more zeros than ones. One example is, Table: (5,3) shaping code. 3 input bits 5 output bits If p, p represents the probability of the shaping encoder giving out a zero and one respectively, then from the table, p = 3 4 and p = 9 4 Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and Electrical - March 24, Engineering 2 West 9 / 3 Virg

10 Constellation Shaping 6-PAM Constellation and its symbol-labeling map Each symbol is selected with probability Each symbol is selected with probability p 8 p 8 Shaping Operation Channel Encoder Splitter thrd 3 bit st nd 2 bit bit E N C rd 3 bit C O D E W O R D S Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 West / 3 Virg

11 Constellation Shaping 6-PAM results with continuous output optimized over p INFORMATION RATE db.5 shaping uniform SNR(dB) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 West / 3 Virg

12 Outline Quantization Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 2 West / 3 Virg

13 Quantization Basics Quantization The output of most communications channels must be quantized prior to processing. Quantizer approximates its input to one of the predefined levels. results in loss of precision. The idea of improving information rate by optimizing the quantizer is from the ISIT 28 paper by Jaspreet Singh. Quantizer Spacing y Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 3 West / 3 Virg

14 Quantization Importance of Quantizer Spacing INFORMATION RATE QUANTIZER SPACING (c) Information Variation with quantizer spacing at SNR = db, uniformly-distributed inputs and 6 quantization levels INFORMATION RATE optimum quantizer non-optimum quantizer SNR(dB) (d) Variation of information rate with SNR under quantizer spacing =., 6 quantization levels and uniformlydistributed inputs Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 4 West / 3 Virg

15 Outline Discrete Memoryless Channel Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 5 West / 3 Virg

16 Discrete Memoryless Channel Discrete Memoryless Channel AWGN channel with discrete inputs and outputs can be modelled by a DMC. Channel described by transition or crossover probabilities. y x y x y 2 x 2 y 3 p(y i x j ) = x M bi+ b i 2πσ exp y N ( (y xj ) 2 2σ 2 ) dy where b i, b i+ are the boundaries of the quantization region associated with level y i. Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 6 West / 3 Virg

17 Discrete Memoryless Channel Information Rate Evaluation For the DMC and one-dimensional modulation, where, I(X; Y ) = M j= N i= ( ) p(yi x j ) p(x j )p(y i x j ) log 2 p(y i ) p(y i ) is the probability of observing output y i. For finding p(y i ), we use, p(y i ) = M j= p(y i x j )p(x j ) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 7 West / 3 Virg

18 Outline Optimization Results Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 8 West / 3 Virg

19 Optimization Results Joint Optimization( Our Contribution ) The variation of information rate with quantizer spacing follows a pattern. We have two values to optimize over, δ and p, given the SNR and number of quantization bits(l = log 2 (N)). The algorithm used is, Fix the SNR and number of quantization bits(l). 2 Vary p from.5 to.99 in increments of.5. 3 For each value of p, find the optimum quantizer spacing and compute the corresponding information rate. 4 By the end of step 3, we have an array of information rate values. We then go over the array and find the highest information rate that can be achieved, and the combination of p and δ that will produce it. Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 9 West / 3 Virg

20 Capacity Results Optimization Results INFORMATION RATE Cont l=8 l=7 l=6 l=5 l=4 INFORMATION RATE Cont l=8 l=7 l=6 l=5 l= SNR(dB) (e) Uniform Distribution SNR(dB) (f) Shaped Distribution Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 2 West / 3 Virg

21 Optimization Results Shaping Gain and its evaluation.8 4 SHAPING GAIN(dB) Cont l=8 l=7 l=6 l=5 l=4 INFORMATION RATE shaped l=7 uniform l=7 shaped l=4 uniform l= INFORMATION RATE SNR(dB) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 2 West / 3 Virg

22 Optimization Results Quantization Loss and its evaluation QUANTIZATION LOSS(dB).5.5 l=4 l=5 l=6 l=7 l=8 INFORMATION RATE shaped cont shaped l=5 shaped l= INFORMATION RATE SNR(dB) (i) Shaped Distribution Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 22 West / 3 Virg

23 Quantization Loss Optimization Results.5 QUANTIZATION LOSS(dB).5 l=4 l=5 l=6 l=7 l= INFORMATION RATE Figure: Uniform Distribution Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 23 West / 3 Virg

24 Optimization Results Optimization Results.8.9 QUANTIZER SPACING l=4 l=5 l=6 l=7 l=8 OPTIMUM P Cont l=8 l=7 l=6 l=5 l= SNR(dB) (a) Optimal Quantizer Spacing SNR(dB) (b) Optimal p (probability of selecting lower-energy subconstellation) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 24 West / 3 Virg

25 Outline Implementation Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 25 West / 3 Virg

26 Implementation Implementation BICM - ID System Information sequence Turbo Encoder Bit Separator Shaping Encoder Modulator Estimated bit probabilities Turbo Decoder Shaping Decoder Bit Separator Demodulator Channel - Feedback Information Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 26 West / 3 Virg

27 BER Curves Implementation Table: Parameters used for simulation of a 6PAM system. R r c r s Quantizer Spacing l = 8 l = 7 l = 6 l = 5 l = 4 Uniform / Shaping /2479 7/ BER - -2 uniform l=4 shaped l=4 uniform l=5 shaped l=5 uniform l=6 shaped l=6 BER - -2 uniform l=7 uniform cont uniform l=8 shaped l=7 shaped cont shaped l= Eb/No(dB) Eb/No(dB) Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 27 West / 3 Virg

28 Implementation Theoretical Actual Table: SNR required by 6PAM at rate R = 3. E b /N in db Cont l = 8 l = 7 l = 6 l = 5 l = 4 Uniform Shaping Uniform Shaping The E b /N values in actual case are taken at BER = 5 Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 28 West / 3 Virg

29 Outline Conclusion Introduction 2 Constellation Shaping 3 Quantization 4 Discrete Memoryless Channel 5 Optimization Results 6 Implementation 7 Conclusion Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 29 West / 3 Virg

30 Conclusion Conclusion The simple constellation-shaping strategy considered in this paper can achieve shaping gains of over.7 db. When a finite-resolution quantizer is used, there will necessarily be a quantization loss. The loss can be minimized by using an optimal quantizer spacing. When properly optimized, a resolution of 8 bits is sufficient to provide performance that is very close to that of an unquantized system. This work can be further extended to a more-complex two-dimensional modulations like 6-APSK and to higher-dimensional modulations. Using a non-uniform vector quantizer can also be investigated for such systems. Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 3 West / 3 Virg

31 Conclusion Questions Constellation ( Lane Shaping Department with Quantization of Computer ScienceCISS and-electrical March 24, Engineering 2 3 West / 3 Virg

Constellation Shaping for Communication Channels with Quantized Outputs

Constellation Shaping for Communication Channels with Quantized Outputs Chandana Nannapaneni, Matthew C. Valenti, and Xingyu Xiang Lane Department of Computer Science and Electrical Engineering West Virginia