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Maximum Likelihood Sequence Detection & the Viterbi Algorithm

Maximum Likelihood Sequence Detection & the Viterbi Algorithm. EE-242 Digital Communications & Coding Miguel Ángel Galicia Ismail AlQerm. EE-242 Digital Communications & Coding. Why using MLSD?. EE-242 Digital Communications & Coding. Outline Viterbi Algorithm MLSD.

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Maximum Likelihood Sequence Detection & the Viterbi Algorithm

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

  2. Why using MLSD? EE-242 Digital Communications & Coding

  3. Outline Viterbi Algorithm MLSD EE-242 Digital Communications & Coding

  4. 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

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

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

  7. 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

  8. At each recursion, there are three main steps that must be done to achieve MLSD using VA: • Branch Metric Generation: Bi,j,n = (Xn – Ci,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: Mj,n = min[Mi,n-1 + Bi,j,n] Most likely path traced back. EE-242 Digital Communications & Coding

  9. 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: • B1,1,1 = (X1– C1,1)2= (0.05 -2)2= 3.8025 • B-1,-1,1 = (X1-C-1,-1)2=(0.05+2)2=4.2025 • B1,-1,1 = B-1,1,1=(0.05)2=0.0025 EE-242 Digital Communications & Coding

  10. 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 ([M1,0 + B1,1] = 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

  11. 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

  12. 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

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

  14. (0, MLSD – NRZI Example (1/4) • NRZI transmitted signal (binary modulation - PAM) • corresponding to the points S1 = -S2 = • Reduce the number of 2k sequences in the trellis • using the Viterbi algorithm • State Diagram representation: EE-242 Digital Communications & Coding

  15. MLSD – NRZI Example (2/4) • Start @ t=0: S0 @ t=T: @t=2T: • Entering paths at S0 at t=2T: * bits (0,0) & (1,1) * signal points: • Entering paths at S1at t=2T: * bits (0,1) & (1,0) * signal points: * EE-242 Digital Communications & Coding

  16. MLSD – NRZI Example (3/4) • Euclidean distance @ S0 • Outputs r1 and r2 from the demodulator: • Euclidean distance @ S0 • Outputs r1 and r2: * 2 survivors: D0(1,1) & D1(1,0) EE-242 Digital Communications & Coding

  17. MLSD – NRZI Example (4/4) • Path metrics @ t=3T: S0: S1: … and goes on… * 2 survivors again: D0(1,1,0) & D1(1,0,0) EE-242 Digital Communications & Coding

  18. MLSD – M-ary Extension • 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 EE-242 Digital Communications & Coding

  19. 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, 3rd ed., Ed. NY, USA: Springer-Verlag, 2004. • J. G. Proakis and M. Salehi, Digital Communications, 5th ed., Mc-Graw Hill, 2008. EE-242 Digital Communications & Coding

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

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