arash saber tehrani alex dimakis mike neely
DESCRIPTION
SigSag : Iterative Detection Through Soft Message Passing. Arash Saber Tehrani Alex Dimakis Mike Neely . University of Southern California. Model. We consider a multiple access problem N users, each has a packet for an access point (base station). Each user retransmits until ack. - PowerPoint PPT PresentationTRANSCRIPT
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Arash Saber TehraniAlex Dimakis
Mike Neely
University of Southern California
SigSag:Iterative Detection Through Soft
Message Passing
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Model
• We consider a multiple access problem• N users, each has a packet for an access point
(base station).– Each user retransmits until ack.
• Flat fading where cth transmission of user i is affected by fading h(c)
i.
• Worst case: Assume they always collide. • All users transmit packets for N rounds.
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Model• Example:• 2 users transmit their packets twice.
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1 2 3
2 3 1 2 3
1 2 3
2 3 4 1 2 3U2
Xy
U1
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Model• Example:• 2 users transmit their packets twice.
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Model• Example:• 2 users transmit their packets twice.
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multiuser detection
• BS gets a bunch of noisy linear equations in the unknown bits xi, yi .
• If there was no noise, optimal detection is solving linear equations.
• Now that there is noise would ideally like to compute the likelihood for each xi ={ ±1 }B, yi ={ ±1 }B
• Integer least squares problem that is NP-hard in general.
[Boutros & Caire],[Verdu],[Reynolds,Wang,Poor] and references therein
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multiuser detection for wifi
• 802.11 BS gets noisy linear equations formed by repetition only.
• If we are at high SNR we can make hard decisions for the bits (ignore the noise) and solve linear equations.
• Gaussian elimination= • Bring in triangular form + Back-substitution.
• If the equations are in triangular form already, we only need back-substitution
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ZigZag decoding
11 1
1 11
1 11 1
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Forward ZigZag is back-substitution.
[Gollakota, Katabi], [ Zhang, Shin],[Erran, Liu, Tan, Viswanathan, Yang]
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Shortcomings of ZigZag• ZigZag can fail to decode when back-substitution is not
possible.
• Noise is accumulated as the ZigZag decoder advances through the packet.
• Previous hard decisions can push other bits to be incorrect.
• How to solve the problems? One heuristic solution is the forward-backward Zigzag.
• Here we view this as a graphical model (factor graph).
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Optimal decoding as inference
Given observations of u’s Obtain the most likely xi, yi
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fun facts about inference in graphical models
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• Optimal graphical model inference is NP-hard.
• Well-known approximation algorithm: belief propagation (aka message passing, sum-product / max-product )
• BP is optimal if the factor graph has no cycles.
• Loopy BP typically is a very good approximation algorithm.
[Pearl in AI, Gallager for LDPC codes in information theory]
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fun facts about inference in graphical models
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• Observation: ZigZag decoding is BP • (special case that deals with noise by hard
thresholding)
• (also equivalent to Luby erasure decoding and the algorithm you use to solve Sudoku)
• Heuristic approach to deal with error-accumulation, run ZigZag twice (forward-backward) and average results.
• Here we propose a systematic way to keep track of soft information about the bits: SigSag decoding
• Natural generalization of ZigZag that maintains soft information (probability distributions).
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SigSag = BP that maintains probabilities• SigSag is belief propagation on the collision graph.
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results: two users • Factor graph is cycle free for two users:
• Theorem 1 for two users transmitting without permuting their bits, the resulting factor graph is cycle-free with probability at least 1-1/w
• SigSag is performing maximum likelihood decoding whp for two users with jitter.
(max-product minimizes block error probability Sum-product minimizes bit error probability)
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results: multiple usersMultiple users
• Theorem 2 The left-regular bipartite factor graph G resulting from N packets of length B sent by N users with symbol permutation is locally tree-like with high probability.
• Specifically,
where e = (v,c) is a randomly chosen edge of the graph, 2ℓ is a fixed depth, and s is a suitable constant that depends on ℓ, N, but not on B.
•SigSag is near ML for large packets B.
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Proof ideasStart from a variable
and grow the random factor graph. Write a recursion for the probability of forming a loop at step ℓ
(similar to LDPC high girth arguments, Richardson, Urbanke, Luby, Mitzenmacher, et al.)
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importance of Jitter• Theorem 3 if the users transmit packets consecutively
without any jitter delay (W = 0), the matrix A is rank deficient (even with bit permutations)
• Without jitter ML detector fails even if there is no noise.
• Random Delay of one Symbol Suffices to make the matrix A full rank whp. (equivalent to a classic result on random matrices by Kolmos 1967)
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Experimental Results
N = 2 users packet length B = 100.
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Experimental ResultsN = 3 users packet length B = 100.
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conclusions
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• Showed how ZigZag decoding is an instance of Belief propagation
• Used this connection to develop a new decoding algorithm that maintains probabilistic information about symbols.
Showed that SigSag is optimal for two users whp and near optimal for multiple users.
Empirical performance shows significant gains.
Performance gains increase for more users.
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fin
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igSag to the rescue!!!