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: 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

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: – 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

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 ([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

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 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:

14. 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,

15. 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

16. 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)

17. 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)

18. 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

19. 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.

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