Constellation Shaping for Communication Channels with Quantized Outputs Chandana Nannapaneni , Dr. Matthew C. Valenti and Xingyu Xiang Lane Department of Computer Science and Electrical Engineering West Virginia University CISS - March 24, 2011 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 1 / 31
Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 2 / 31
Introduction Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 3 / 31
Introduction Transmitter Side Optimization Receiver Side Optimization “Constellation Shaping” “ Quantizer Optimization” - - Stephane Y. Le Goff Jaspreet Singh 2007 IEEE , T. Wireless 2008 ISIT Our Contribution - Joint optimization CISS 2011 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 4 / 31
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 ) Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 5 / 31
Introduction The mutual information between output Y and input X is M − 1 p ( y | x j ) � � I ( X ; Y ) = p ( x j ) p ( y | x j ) log 2 p ( y ) dy. j =0 M - number of input symbols. This can be solved using Gauss - Hermite Quadratures. Information rate results of 16 PAM under continuous output and uniform constellation 4 3.5 3 INFORMATION RATE 2.5 2 1.5 1 0.5 0 -10 -5 0 5 10 15 20 25 30 SNR(dB) Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 6 / 31
Constellation Shaping Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 7 / 31
Constellation Shaping Constellation Shaping for PAM Our strategy is from S. LeGoff, IEEE T. Wireless, 2007. 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. 1 1 0.5 Low Energy Symbols High Energy Symbols High Energy Low Energy High Energy High Energy Symbols 0 0 -0.5 -1 -1 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (a) Probability of picking low-energy (b) Probability of picking low-energy subconstellation = 0.5 subconstellation = 0.9 ( E s = Σ M − 1 i =0 p ( x i ) E i = 1 ) Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 8 / 31
Constellation Shaping Shaping Encoder 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 If p 0 , p 1 represents the probability of the shaping encoder giving out a zero and one respectively, then from the table, p 0 = 31 40 and p 1 = 9 40 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 9 / 31
Constellation Shaping 16-PAM Constellation and its symbol-labeling map 1111 1011 0111 0011 0000 0100 1000 1100 1101 0101 0001 0010 0110 1010 1110 1001 p Each symbol is selected with probability 0 8 1 p Each symbol is selected with probability 0 8 Shaping Operation 101011101101 100101100110000100 Channel Splitter Encoder 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 E 0 1 0 N th rd C 0 bit 3 1 0 0 1 0 0 1 st 1 1 0 1 1 1 bit 0 0 0 1 0 0 0 0 nd bit 2 0 1 0 0 0 rd bit 3 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 10 / 31
Constellation Shaping 16-PAM results with continuous output optimized over p 0 4 3.5 3 INFORMATION RATE 0.77 dB 2.5 2 1.5 1 shaping 0.5 uniform 0 -10 -5 0 5 10 15 20 25 30 SNR(dB) Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 11 / 31
Quantization Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 12 / 31
Quantization Quantization Basics 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 2008 paper by Jaspreet Singh. Quantizer Spacing y -6 -2 2 6 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 13 / 31
Quantization Importance of Quantizer Spacing 4 1.05 1 3.5 0.95 3 INFORMATION RATE INFORMATION RATE 0.9 2.5 0.85 0.8 2 0.75 1.5 0.7 1 0.65 optimum quantizer 0.5 0.6 non-optimum quantizer 0.55 0 0 0.5 1 1.5 -10 -5 0 5 10 15 20 25 30 QUANTIZER SPACING SNR(dB) (c) Information Variation with quan- (d) Variation of information rate with tizer spacing at SNR = 10dB, SNR under quantizer spacing = 0.1, uniformly-distributed inputs and 16 16 quantization levels and uniformly- quantization levels distributed inputs Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 14 / 31
Discrete Memoryless Channel Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 15 / 31
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 0 x 0 y 1 x 1 y 2 x y 2 3 x M 1 y N 1 � b i +1 � − ( y − x j ) 2 1 � p ( y i | x j ) = √ 2 πσ exp dy 2 σ 2 b i where b i , b i +1 are the boundaries of the quantization region associated with level y i . Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 16 / 31
Discrete Memoryless Channel Information Rate Evaluation For the DMC and one-dimensional modulation, M − 1 N − 1 � p ( y i | x j ) � � � I ( X ; Y ) = p ( x j ) p ( y i | x j ) log 2 p ( y i ) j =0 i =0 where, p ( y i ) is the probability of observing output y i . For finding p ( y i ) , we use, M − 1 � p ( y i ) = p ( y i | x j ) p ( x j ) j =0 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 17 / 31
Optimization Results Outline Introduction 1 Constellation Shaping 2 Quantization 3 Discrete Memoryless Channel 4 Optimization Results 5 Implementation 6 Conclusion 7 Chandana Nannapaneni Constellation Shaping with Quantization ( Lane Department of Computer Science and Electrical Engineering West Virginia CISS - March 24, 2011 18 / 31
Recommend
More recommend
