Introduction Parallel Relay Networks (PaReNet) Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Some Systems Aspects Regarding Compressive Relaying with Wireless Infrastructure Links Erhan YILMAZ Raymond KNOPP David GESBERT {yilmaz, knopp, gesbert}@eurecom.fr Mobile Communications Department EURÉCOM Sophia-Antipolis May 15, 2008 eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Outline Introduction 1 Some Information Theory Tools Parallel Relay Networks (PaReNet) 2 Outer Bounds Achievable Rates for PaReNet Amplify-and-Forward (AF) Relaying Decode-and-Forward (DF) Relaying Block-Quantization and Random Binning (BQRB) Numerical Results Cellular Networks with Relay Nodes: An Implementation 3 Single Cell System Model Numerical Results: Single-Cell Relaying Multi-Cell Relaying Numerical Results: Multi-Cell Relaying Summary and Outlook 4 eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Outline Introduction 1 Some Information Theory Tools Parallel Relay Networks (PaReNet) 2 Outer Bounds Achievable Rates for PaReNet Amplify-and-Forward (AF) Relaying Decode-and-Forward (DF) Relaying Block-Quantization and Random Binning (BQRB) Numerical Results Cellular Networks with Relay Nodes: An Implementation 3 Single Cell System Model Numerical Results: Single-Cell Relaying Multi-Cell Relaying Numerical Results: Multi-Cell Relaying Summary and Outlook 4 eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Slepian-Wolf Encoding (1/7) X� R� X� R� 1� 1� E n c o d e r 1 � � E n c o d e r � 1 � D e c o d e r 1 � � D� e� Correlated� Correlated� c� Sources� Sources� (� ,� )� o� (� ,� )� X� Y� X� Y� d� (X, Y)� (X, Y)� e� R� R� Y� Y� 2� r� 2� E n c o d e r 2 � E n c o d e r 2 � D e c o d e r 2 � �� �� �� �� R� R� 2� 2� ��� ��� ��� ��� ��� R� H(X,Y)� H(X,Y)� H(Y)� H(Y)� R� H(Y|X)� H(Y|X)� H(X|Y)� H(X)� H(X,Y)� R� H(X|Y)� H(X)� H(X,Y)� R� 1� 1� (a)� (b)� Figure: (a) SW Decoding (Correlation exploited), (b) Regular Decoding eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Slepian-Wolf Encoding (2/7) Correlated Sources with ( X , Y ) ∼ p ( x , y ) Distributed (or separated) encoding/compression Main tool ⇒ Random Binning Choose large random index for source sequences If the index range is large enough, then with high probability, different source sequences have different indexes. Thus, we can recover the source sequence from the index. What are the sufficient Rates to encode sources X and Y ? Note that R ≥ H ( X , Y ) is sufficient if we encode the sources together (regard (X, Y) pairs as a single source!!) eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Slepian-Wolf Encoding: Definitions (3/7) A ( 2 nR 1 , 2 nR 2 , n ) distributed source code for joint source ( X , Y ) consists of encoder maps f 1 : X n { 1 , 2 , . . . , 2 nR 1 } → f 2 : Y n { 1 , 2 , . . . , 2 nR 2 } → and a decoder map X n × Y n g : { 1 , 2 , . . . , 2 nR 1 } × { 1 , 2 , . . . , 2 nR 2 } → Probability of error P ( n ) Pr { g ( f 1 ( X n ) , f 2 ( Y n )) � = ( X n , Y n ) } = e eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Slepian-Wolf Theorem (4/7) Theorem [Slepian-Wolf]: For the distributed source coding problem for the source ( X , Y ) drawn i.i.d ∼ p ( x , y ) , the achievable rate region is given by R 1 H ( X | Y ) ≥ R 2 H ( Y | X ) ≥ R 1 + R 2 H ( X , Y ) . ≥ Main idea: Show that if the rate pair in the SW region, we can use a Random Binning encoding scheme with Typical Set decoding to obtain a probability of error that tends to zero. eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Slepian-Wolf Theorem: Coding Scheme (5/7) Source X assigns every sourceword x ∈ X n randomly among 2 nR 1 bins, and source Y independently assigns every sourceword y ∈ Y n randomly among 2 nR 2 bins Each sends the bin index corresponding to the message The receiver decodes correctly if there is exactly one jointly typical sourceword pair corresponding to the received bin indexes, otherwise it declares error. eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook SW Theorem: Error Analysis (6/7) Error events: The transmitted sourcewords are not jointly typical: ∈ A ( n ) E 0 { ( X , Y ) / ǫ } = There exists another pair of jointly typical sourcewords in the same pair of bins, i.e. one or more of the following events {∃ x ′ � = X : f 1 ( x ′ ) = f 1 ( X ) , ( x ′ , Y ) ∈ A ( n ) E 1 ǫ } = {∃ y ′ � = Y : f 2 ( y ′ ) = f 2 ( Y ) , ( X , y ′ ) ∈ A ( n ) E 2 ǫ } = ∃ ( x ′ , y ′ ) : x ′ � = X , y ′ � = Y , f 1 ( x ′ ) = f 1 ( X ) , f 2 ( y ′ ) = f 2 ( Y ) , E 12 � = � ( x ′ , y ′ ) ∈ A ( n ) ǫ Using union of events bound: P ( n ) Pr ( E 0 ∪ E 1 ∪ E 2 ∪ E 12 ) = e Pr ( E 0 ) + Pr ( E 1 ) + Pr ( E 2 ) + Pr ( E 12 ) . ≤ eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook SW Theorem: Error Analysis (7/7) Pr ( E 0 ) → 0 by the AEP, p ( x , y ) P {∃ x ′ � = x : f 1 ( x ′ ) = f 1 ( x ) , ( x ′ , y ) ∈ A ( n ) Pr ( E 1 ) � = ǫ } ( x , y ) � p ( x , y ) � P ( f 1 ( x ′ ) = f 1 ( x )) ≤ ( x , y ) x ′� = x ( x ′ , y ) ∈A ( n ) ǫ p ( x , y ) 2 − nR 1 |A ǫ ( X | y ) | � = ( x , y ) 2 − nR 1 2 n ( H ( X | Y )+ 2 ǫ ) = 2 − n ( R 1 − H ( X | Y ) − 2 ǫ ) ≤ R 1 > H ( X | Y ) 0 if → Similarly, 2 − nR 2 2 n ( H ( Y | X )+ 2 ǫ ) → 0 Pr ( E 2 ) if R 2 > H ( Y | X ) ≤ 2 − n ( R 1 + R 2 ) 2 n ( H ( X , Y )+ ǫ ) → 0 Pr ( E 12 ) if R 1 + R 2 > H ( X , Y ) ≤ eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Rate-Distortion Theory (1/7) Goal: Rate-distortion theory calculates minimum transmission bit-rate R for a given distortion D and source. Determine the minimum rate at which information about the source must be conveyed to the sink in order to achieve a prescribed fidelity. Question 1: Given a random binary variable X , how well can one encode strings of length n with less than nH ( X ) bits. Compression that tries to go below Shannon’s limit → Lossy Data Compression Question 2: Consider a continuous random variable; how well can we encode this information using bits? Digitizing of Analog signals → Quantization For both cases, the optimal trade-off between the information rate and the inevitable distortion: Rate Distortion Theory eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Rate-Distortion Theory: Geometric View (2/7) Figure: A Geometric Representation of Rate-Distortion Theory. eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Rate-Distortion Theory: Distortion (3/7) The distortion between sequences x n and ˆ x n is defined by n 1 d ( x n ;ˆ x n ) � d ( x i ;ˆ x i ) = n i = 1 In other words, it is the distortion per symbol. The analysis here is based on this average distortion measure between sequences. A distortion function or distortion measure is a mapping X → R + ∪ { 0 } d : X × ˆ from the set of source-reproduction pairs into the set of non-negative real numbers. Squared Error distortion function: d ( x , ˆ x ) = ( x − ˆ x ) 2 eY Seminar EURÉCOM 15 May 2008
Introduction Parallel Relay Networks (PaReNet) Some Information Theory Tools Cellular Networks with Relay Nodes: An Implementation Summary and Outlook Rate-Distortion Code (4/7) A ( 2 nR , n ) rate distortion code consists of an encoding function { 1 , 2 , . . . , 2 nR } , f n : X → and a decoding function g n : { 1 , 2 , . . . , 2 nR } ˆ → X , The distortion associated with this code is E [ d ( X n , g n ( f n ( X n )))] = p ( x n ) d ( x n , g n ( f n ( x n ))) . � D = x n eY Seminar EURÉCOM 15 May 2008
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