Outline Transmitters (Chapters 3 and 4, Source Coding and Modulation) (week 1 and 2) Receivers (Chapter 5) (week 3 and 4) Received Signal Synchronization (Chapter 6) (week 5) Channel Capacity (Chapter 7) (week 6) Error Correction Codes (Chapter 8) (week 10 and 11) Equalization (Bandwidth Constrained Channels) (Chapter 10) Adaptive Equalization (Chapter 11) Spread Spectrum (Chapter 13) Fading and multi path (Chapter 14)

Error Correction Codes (Chapter 8) Trellis/Convolution Codes Trellis Codes and QAM Viterbi Algorithm Trellis for NRZI NRZI code with memory L=1 Convolution Code Generator QAM set partition via convolution code 8 state Trellis for rectangular QAM

Trellis/Convolution Codes Modulation with Memory – Example: NRZI Trellis diagram Maximum likelihood sequence detectors –Viterbi Algorithm Viterbi Algorithm Trellis for NRZI NRZI code with memory L=1

Modulation with Memory Example: NRZI (non return to zero invert) –Binary PAM NRZI code with memory L=1

Modulation with Memory Example: NRZI (non return to zero invert) –Encoded by: Modulo 2 addition Has one memory and 1/1 code rate (No error correction)

Trellis diagram Indicates state transitions for various inputs Trellis Diagram for NRZI

Maximum likelihood sequence detectors Consider NRZI matched filter decoder output:

Maximum likelihood sequence detectors Consider joint pdf of a matched filter decoder output sequence and a transmitted sequence : m = 1,2,…M (M symbols)

Maximum likelihood sequence detectors Now find maximum likely hood: Minimizes Euclidean distance

Maximum likelihood sequence detectors Viterbi algorithm –Eliminates sequences as data is collected 1.compute distances as you go 2.Keep only the smallest distances for each trellis state at each symbol time 3.After some number of symbols K decide what the first symbol was by: a) Consensus or b) Minimum D path 4.Do this for each time step

Maximum likelihood sequence detectors Viterbi algorithm for NRZI The example shows how the Viterbi algorithm by trying all possible states and digital inputs can correctly estimate the digital inputs (green)

Maximum likelihood sequence detectors Viterbi algorithm for NRZI Try it again with different initial state and data. The trellis to the right matches the Excel spreadsheet

Trellis/Convolution Codes The general Convolution Code K=number of stages k=number of input bits per stage n=number of output bits The code rate Represents how much faster data must be sent

Trellis/Convolution Codes The general Convolution Code –The code is described by the “generators” i.e., the weights on each mod 2 adder –e.g., for K = 3, k = 1, n = 3,

Trellis/Convolution Codes The general Convolution Code –There are better codes –K = 3, k = 1, n = 3, –With generator is optimal

Trellis/Convolution Codes The general Convolution Code –Trellis and Transfer function Tell us d free = minimum hamming distance between paths through the Trellis –Coding gain:

Maximum likelihood sequence detector Viterbi algorithm for K = 3, k = 1, n = 3,with generator 5 7 7

Maximum likelihood sequence detector Viterbi algorithm for K = 3, k = 1, n = 3,with generator Example of error correction? sigma = 0.5 Decision at step 11 corrects bit error at step 2

Trellis Codes and QAM Bandwidth Constrained Channels –Use set partitioning to combine code with M-ary bandwidth constrained signaling Ungerboeck (1982) –Three rules: Use all subsets with equal frequency in trellis Use Transitions that join use maximum distance subsets Parallel transitions are assigned max Euclidean distance (unencoded bit transitions) QAM set partition via convolution code 8 state Trellis for rectangular QAM

Set Partitioning Chop up the Constellations to increase Euclidean distance between points In rectangular QAM case each partition increases d by

s m1 s m2 64 QAM Set Partitioning Each 4 x 4 block of 64 QAM constellation is replaced by 16 QAM partition

64 QAM Set Partitioning Examples: Decide between these in Real decoder

Trellis Codes and QAM Eight state trellis for QAM Contains no overlapping locations

Trellis Codes and QAM The code generator

Trellis Codes and QAM The code generator –Alternate = essentially a kind of parity bit

Trellis Codes and QAM The code generator –Alternate = essentially a kind of parity bit –The weights = “parity matrix” not generator matrix –Ungerboeck IEEE Communications Magazine February 1987-Vol. 25. No. 2

Trellis Codes and QAM The code generator

Trellis Codes and QAM The code generator Memory = 3, thus 8 states

Trellis Codes and QAM The code generator

Trellis Codes and 64 QAM Send 64 QAM but only 5 bits per symbol –6 bits = 3 data bits + 3 encoded bits –The 3 encoded bits come from 2 data bits One of eight subsets – 3 bits Select the levels for each dimension One of eight symbols – 3 bits 4 levels per dimension 2 bits in, 3 bits out

Trellis Codes and QAM Trellis seems not the same? –Maybe books trellis is parity realization