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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 ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 1 / 31

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 2 / 31

Introduction

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 3 / 31

Introduction

“Constellation Shaping”

• Stephane Y. Le Goff

2007 IEEE , T. Wireless “Quantizer Optimization”

• Jaspreet Singh

2008 ISIT Transmitter Side Optimization Receiver Side Optimization Our Contribution

• Joint optimization

CISS 2011

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 4 / 31

Introduction

Mutual Information( MI ) and Channel Capacity

MI between two random variables, X and Y is given by, I(X; Y ) = E

• log

p(Y |X) 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 ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 5 / 31

Introduction

The mutual information between output Y and input X is I(X; Y ) =

M−1

• j=0

p(xj)

• p(y|xj) log2

p(y|xj) p(y) dy. M - number of input symbols. This can be solved using Gauss - Hermite Quadratures.

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• 5

5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4

SNR(dB) INFORMATION RATE Information rate results of 16 PAM under continuous output and uniform constellation Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 6 / 31

Constellation Shaping

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization 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.

• 3
• 2
• 1

1 2 3

• 1

1

Low Energy High Energy High Energy

(a) Probability of picking low-energy subconstellation = 0.5

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• 2
• 1

1 2 3

• 1
• 0.5

0.5 1

Low Energy Symbols High Energy Symbols High Energy Symbols

(b) Probability of picking low-energy subconstellation = 0.9

( Es = ΣM−1

i=0 p(xi)Ei = 1)

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 If p0, p1 represents the probability of the shaping encoder giving out a zero and one respectively, then from the table, p0= 31

40 and p1= 9 40

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 9 / 31

Constellation Shaping

16-PAM Constellation and its symbol-labeling map

1111 1011 1101 0110 1000 0010 0100 0111 1001 0011 0101 0000 0001 1100 1010 1110 Each symbol is selected with probability 8 p

Each symbol is selected with probability 8 1 p

Shaping Operation

Channel Encoder 101011101101 100101100110000100 Splitter 1 1 1 1 1 1 E N C 1 1 1 1 1 1 1 1 1 1

rd

3

bit

th

bit

st

1 bit

nd

2 bit

rd

3

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 10 / 31

Constellation Shaping

16-PAM results with continuous output optimized over p0

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5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4

SNR(dB) INFORMATION RATE

shaping uniform

0.77 dB

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 11 / 31

Quantization

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization 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.

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• 2

2 6 y



Quantizer Spacing

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 13 / 31

Quantization

Importance of Quantizer Spacing

0.5 1 1.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 QUANTIZER SPACING INFORMATION RATE

(c) Information Variation with quan- tizer spacing at SNR = 10dB, uniformly-distributed inputs and 16 quantization levels

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5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 SNR(dB) INFORMATION RATE

• ptimum quantizer

non-optimum quantizer

(d) Variation of information rate with SNR under quantizer spacing = 0.1, 16 quantization levels and uniformly- distributed inputs

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 14 / 31

Discrete Memoryless Channel

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization 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.

x y

1

x

2

x

1 M

x

2

y

3

y

1 N

y

1

y

p(yi|xj) = bi+1

bi

1 √ 2πσ exp −(y − xj)2 2σ2

• dy

where bi, bi+1 are the boundaries of the quantization region associated with level yi.

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 16 / 31

Discrete Memoryless Channel

Information Rate Evaluation

For the DMC and one-dimensional modulation, I(X; Y ) =

M−1

• j=0

N−1

• i=0

p(xj)p(yi|xj) log2 p(yi|xj) p(yi)

• where,

p(yi) is the probability of observing output yi. For finding p(yi), we use, p(yi) =

M−1

• j=0

p(yi|xj)p(xj)

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 17 / 31

Optimization Results

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 18 / 31

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 p0, given the SNR and number of quantization bits(ℓ = log2(N)). The algorithm used is,

1 Fix the SNR and number of quantization bits(ℓ). 2 Vary p0 from 0.5 to 0.99 in increments of 0.005. 3 For each value of p0, 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 p0 and δ that will produce it.

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 19 / 31

Optimization Results

Capacity Results

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5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 SNR(dB) INFORMATION RATE

Cont l=8 l=7 l=6 l=5 l=4

16.1 16.2 16.3 2.99 3 3.01

(e) Uniform Distribution

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5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 SNR(dB) INFORMATION RATE

Cont l=8 l=7 l=6 l=5 l=4

15.35 15.4 15.45 2.995 3 3.005

(f) Shaped Distribution

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 20 / 31

Optimization Results

Shaping Gain and its evaluation

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 INFORMATION RATE SHAPING GAIN(dB)

Cont l=8 l=7 l=6 l=5 l=4

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• 5

5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 SNR(dB) INFORMATION RATE

shaped l=7 uniform l=7 shaped l=4 uniform l=4 Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 21 / 31

Optimization Results

Quantization Loss and its evaluation

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 INFORMATION RATE QUANTIZATION LOSS(dB)

l=4 l=5 l=6 l=7 l=8

(i) Shaped Distribution

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• 5

5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 SNR(dB) INFORMATION RATE

shaped cont shaped l=5 shaped l=4

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 22 / 31

Optimization Results

Quantization Loss

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5

INFORMATION RATE QUANTIZATION LOSS(dB)

l=4 l=5 l=6 l=7 l=8

Figure: Uniform Distribution

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 23 / 31

Optimization Results

Optimization Results

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5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SNR(dB) QUANTIZER SPACING

l=4 l=5 l=6 l=7 l=8

(a) Optimal Quantizer Spacing

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5 10 15 20 25 30 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 SNR(dB) OPTIMUM P0 Cont l=8 l=7 l=6 l=5 l=4

(b) Optimal p0 (probability of selecting lower-energy subconstellation)

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 24 / 31

Implementation

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 25 / 31

Implementation

Implementation

BICM - ID System

Channel Demodulator Turbo Decoder Information sequence Estimated bit probabilities

• 1

Turbo Encoder Bit Separator Shaping Encoder Modulator Bit Separator Shaping Decoder Feedback Information Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 26 / 31

Implementation

BER Curves

Table: Parameters used for simulation of a 16PAM system.

R rc rs Quantizer Spacing ℓ = 8 ℓ = 7 ℓ = 6 ℓ = 5 ℓ = 4 Uniform 2.9940 2000/2672 1 0.0139 0.0272 0.0534 0.1058 0.2165 Shaping 2.9836 2000/2479 7/10 0.0172 0.0339 0.0670 0.1270 0.2781

12 13 14 15 16 17 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Eb/No(dB) BER

uniform l=4 shaped l=4 uniform l=5 shaped l=5 uniform l=6 shaped l=6

11.5 12 12.5 13 13.5 14 14.5 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Eb/No(dB) BER

uniform l=7 uniform cont uniform l=8 shaped l=7 shaped cont shaped l=8 Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 27 / 31

Implementation

Table: SNR required by 16PAM at rate R = 3.

Eb/N0 in dB Cont ℓ = 8 ℓ = 7 ℓ = 6 ℓ = 5 ℓ = 4 Theoretical Uniform 11.387 11.393 11.410 11.476 11.751 12.74 Shaping 10.613 10.618 10.640 10.728 11.082 12.051 Actual Uniform 13.378 13.394 13.585 13.703 14.184 15.524 Shaping 12.789 12.811 12.922 13.154 13.742 15.112 The Eb/N0 values in actual case are taken at BER = 10−5

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 28 / 31

Conclusion

Outline

1

Introduction

2

Constellation Shaping

3

Quantization

4

Discrete Memoryless Channel

5

Optimization Results

6

Implementation

7

Conclusion

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 29 / 31

Conclusion

Conclusion

The simple constellation-shaping strategy considered in this paper can achieve shaping gains of over 0.7 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 16-APSK and to higher-dimensional modulations. Using a non-uniform vector quantizer can also be investigated for such systems.

Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 30 / 31

Conclusion Questions Chandana Nannapaneni ( Lane Department of Computer Science and Electrical Engineering West Virginia Constellation Shaping with Quantization CISS - March 24, 2011 31 / 31