1 / 20

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.

francesca-roberts
Download Presentation

Maximum Likelihood Sequence Detection & the Viterbi Algorithm

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related