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Maximum Likelihood Sequence Detection & the Viterbi Algorithm EE-242 Digital Communications & Coding Miguel Ángel Galicia Ismail AlQerm EE-242 Digital Communications & Coding

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Why using MLSD? EE-242 Digital Communications & Coding

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Outline Viterbi Algorithm MLSD EE-242 Digital Communications & Coding

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Viterbi Algorithm What is Viterbi Algorithm? Viterbi Algorithm (VA) is an algorithm implemented using dynamic programming to detect and estimate sequence of symbols in digital communication and signal processing. EE-242 Digital Communications & Coding

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Viterbi Algorithm in Sequence Detection Basic Tools Needed: State diagram to represent transmitted signals. EE-242 Digital Communications & Coding

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Tools Needed The Trellis: EE-242 Digital Communications & Coding

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Viterbi Algorithm Description Main principle: Finding a noiseless output sequence with minimum distance from the detected noisy sequence of symbols. It consists of certain number of Recursion (Stages) EE-242 Digital Communications & Coding

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At each recursion, there are three main steps that must be done to achieve MLSD using VA: Branch Metric Generation: B i,j,n = (X n – C i,j ) 2 Survivor path and path metrics update. The path metric is the sum of all branch metrics at a given state. The survivor path at recursion n: M j,n = min[M i,n-1 + B i,j,n ] Most likely path traced back. EE-242 Digital Communications & Coding

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VA Example (1/4) Assume that the received noisy sequence is Xn= (0.05, 2.05, -1.05, -2.00, -0.05) and we have the following trellis diagram: First: Branch Metrics at stage 1: – B 1,1,1 = (X 1 – C 1,1 ) 2 = (0.05 -2) 2 = 3.8025 – B -1,-1,1 = (X 1 -C -1,-1 ) 2 =(0.05+2) 2 =4.2025 – B 1,-1,1 = B -1,1,1 =(0.05) 2 =0.0025 EE-242 Digital Communications & Coding

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VA Example (2/4) Second: Finding the survivor path for each state: i= 2 states. M1,0 = M-1,0 = 0 M1,1= min[Mi,0+Bi,1] from state -1 to 1 ([M 1,0 + B 1,1 ] = 3.8025) > ([M -1,0 + B -1,1 ]=0+0.0025) For state -1 similarly : M-1,1= min[Mi,0+Bi,1] from state 1 to -1 EE-242 Digital Communications & Coding

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VA Example (3/4) Stage 2: B1,1,2= (X1 – C1,1)^2= (2.05 -2)^2= 0.0025 B-1,-1,2 =(X1 -C-l,-l)^2 =(2.05+2)^2 =16.402 B1,-1,2 =B-1,1,2 =(2.05)^2 =4.202 Survivor path selection: For state1 M1,1 = M-1,1 = 0.0025 from state 1 to 1 M1,2= min[Mi,0+Bi,2]= ([M1,1+B1,2]=0.005<[M-1,1+B-1,2]= 4.2045) For state -1: M-1,2= min[Mi,0+Bi,2]= ([M-1,1+B1,2]=4.2045<[M-1,1+B-1,2]= 16.4045) From state 1 to -1 EE-242 Digital Communications & Coding

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VA Example (4/4) After 5 recursions: The path trace back as shown with the most likely inputs an outputs EE-242 Digital Communications & Coding

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MLSD Algorithm EE-242 Digital Communications & Coding Searches the minimum Euclidean distance path through the trellis memory of transmitted signal of k symbols and r(t) Optimal Detection Rule becomes:

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MLSD – NRZI Example (1/4) NRZI transmitted signal (binary modulation - PAM) – corresponding to the points S1 = -S2 = – Reduce the number of 2 k sequences in the trellis using the Viterbi algorithm – State Diagram representation: EE-242 Digital Communications & Coding (0,(0,

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MLSD – NRZI Example (2/4) – Start @ t=0: S 0 @ t=T: @t=2T: – Entering paths at S 0 at t=2T: * bits (0,0) & (1,1) * signal points: – Entering paths at S 1 at t=2T: * bits (0,1) & (1,0) * signal points: * EE-242 Digital Communications & Coding

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MLSD – NRZI Example (3/4) Euclidean distance @ S 0 Outputs r 1 and r 2 from the demodulator: Euclidean distance @ S0 Outputs r 1 and r 2 : EE-242 Digital Communications & Coding * 2 survivors: D 0 (1,1) & D 1 (1,0)

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MLSD – NRZI Example (4/4) EE-242 Digital Communications & Coding Path metrics @ t=3T: S 0 : S 1 : … and goes on… * 2 survivors again: D 0 (1,1,0) & D 1 (1,0,0)

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MLSD – M-ary Extension EE-242 Digital Communications & Coding M=4 signals – Four-state trellis: – 2 signal paths enter & 2 leave each node – 4 survivors @ each stage – 1 / 2 signal paths is eliminated @ each stage – Minimize the number of trellis’ paths

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EE-242 Digital Communications & Coding References: Tor M. Aulin, Breadth-First Maximum Likelihood Sequence Detection: Basics, IEEE Transactions on Communications, Vol. 47, No. 2, February 1999. Hui-Ling Lou, Implementing the Viterbi Algorithm: Fundamental and real-time issues for processor designers, IEEE Signal Processing Magazine, September 1995. X. Zhu and J. M. Kahn, Markov Chain Model in Maximum Likelihood Sequence Detection for Free-Space Optical Communication Through Atmospheric Turbulence Channels, IEEE Transactions on Communications, Vol. 51, No. 3, March 2003. J. R. Barry, E. A. Lee and David G. Messershmitt, Digital Communication, 3 rd ed., Ed. NY, USA: Springer-Verlag, 2004. J. G. Proakis and M. Salehi, Digital Communications, 5 th ed., Mc-Graw Hill, 2008.

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EE-242 Digital Communications & Coding Thank you! Q & A

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