Digital Communications I: Modulation and Coding Course Period - 2007 Catharina Logothetis Lecture 12 Last time, we talked about: How the decoding is performed for Convolutional codes? „ What is a Maximum likelihood decoder? „ What are the soft decisions and hard decisions? „ How does the Viterbi algorithm work? „ Lecture 12 Trellis of an example ½ Conv code Tail bits Input bits 1 0 00 10 11 0/00 0/00 0/00 0/11 0/11 Output bits 11 10 0/00 0/00 1/11 1/11 1/11 0/10 0/11 1/00 1/01 1/01 0/10 0/10 0/01 t1 t t 1/01 Lecture 12 0/01 t t t Block diagram of the DCS Information source Rate 1/n Conv encoder Modulator Input sequence Codeword sequence U i = u1i , ,u ji , ,u ni 144244 Channel U = G(m) = (U1 , U , U , , U i , ) 144424443 m = (m1 , m2 , , mi , ) 1442443 Branch word ( n coded bits) Information sink Rate 1/n Conv decoder ˆ = (mˆ , mˆ , , mˆ i , ) m Demodulator Z = ( Z1 , Z , Z , , Z i , ) 14442444 received sequence Zi { Demodulator outputs for Branch word i = z1i , ,z ji , ,z ni 14 4244 n outputs per Branch word Lecture 12 Soft and hard decision decoding „ In hard decision: „ „ The demodulator makes a firm or hard decision whether one or zero is transmitted and provides no other information for the decoder such that how reliable the decision is In Soft decision: „ The demodulator provides the decoder with some side information together with the decision The side information provides the decoder with a measure of confidence for the decision Lecture 12 Soft and hard decision decoding … „ ML soft-decisions decoding rule: „ „ Choose the path in the trellis with minimum Euclidean distance from the received sequence ML hard-decisions decoding rule: „ Choose the path in the trellis with minimum Hamming distance from the received sequence Lecture 12 The Viterbi algorithm „ The Viterbi algorithm performs Maximum likelihood decoding „ It finds a path through trellis with the largest metric (maximum correlation or minimum distance) „ At each step in the trellis, it compares the partial metric of all paths entering each state, and keeps only the path with the largest metric, called the survivor, together with its metric Lecture 12 Example of hard-decision Viterbi decoding ˆ = (100) m ˆ = (11 10 11 00 11) U Z = (11 10 11 10 01) m = (101 ) U = (11 10 00 10 11 ) 2 1 0 1 0 1 0 Partial metric Γ(S (ti ), ti ) 2 Branch metric t1 t2 t3 t4 Lecture 12 t5 t6 Example of soft-decision Viterbi decoding 2 −2 −2 −2 Z = (1, , , , ,1, , − 1, ,1) 3 3 3 m = (101 ) U = (11 10 00 10 11 ) -5/3 -5/3 -5/3 5/3 5/3 10/3 -1/3 1/3 ˆ = (101) m ˆ = (11 10 00 10 11) U 1/3 8/3 14/3 1/3 -1/3 1/3 4/3 -1/3 1/3 5/3 -5/3 1/3 5/3 Partial metric 13/3 Γ(S (ti ), ti ) 5/3 -4/3 1/3 5/3 -5/3 Branch metric 10/3 -5/3 t1 t2 t3 t4 Lecture 12 t5 t6 Today, we are going to talk about: „ The properties of Convolutional codes: „ „ „ „ „ Free distance Transfer function Systematic Conv codes Catastrophic Conv codes Error performance Interleaving „ Concatenated codes „ Error correction scheme in Compact disc „ Lecture 12 10 Transfer function … „ Write the state equations ( X a , , X e dummy variables) ⎧ X b = D LNX a + LNX c ⎪ ⎪ X c = DLX b + DLX d ⎨ ⎪ X d = DLNX b + DLNX d ⎪ X = D LX c ⎩ e „ Solve T ( D, L, N ) = X e / X a D L3 N T ( D , L, N ) = = D L3 N + D L4 N + D L5 N + − DL(1 + L) N One path with weight 5, length and data weight of One path with weight 6, length and data weight of One path with weight 5, length and data weight of Lecture 12 15 Systematic Convolutional codes „ A Conv Coder at rate k / n is systematic if the k-input bits appear as part of the n-bits branch word Input „ Output Systematic codes in general have smaller free distance than non-systematic codes Lecture 12 16 Catastrophic Convolutional codes „ Catastrophic error propagations in Conv code: „ „ „ A Convolutional code is catastrophic if there is a closed loop in the state diagram with zero weight Systematic codes are not catastrophic: „ „ A finite number of errors in the coded bits cause as infinite number of errors in the decoded data bits At least one branch of output word is generated by input bits Small fraction of non-systematic codes are catastrophic Lecture 12 17 Catastrophic Conv … „ Example of a catastrophic Conv code: „ „ „ „ Assume all-zero codeword is transmitted Three errors happens on the coded bits such that the decoder takes the wrong path abdd…ddce This path has ones, no matter how many times stays in the loop at node d It results in many erroneous decoded data bits 10 Input Output a 11 00 b 10 01 10 c 01 d 11 11 01 e 00 00 Lecture 12 18 Performance bounds for Conv codes „ Error performance of the Conv codes is analyzed based on the average bit error probability (not the average codeword error probability), because „ „ „ Codewords with variable sizes due to different size of input For large blocks, codeword error probability may converges to one bit the bit error probability may remain constant … Lecture 12 19 Performance bounds … „ Analysis is based on: „ „ Assuming the all-zero codeword is transmitted Evaluating the probability of an “error event” (usually using bounds such as union bound) „ An “error event” occurs at a time instant in the trellis if a non-zero path leaves the all-zero path and remerges to it at a later time Lecture 12 20 ... f Lecture 12 11 Free distance … The path diverging and remerging to all-zero path with minimum weight Hamming weight of the branch All-zero path df =5 0 2 0 2 2 1 1 1 t1 t t t Lecture 12 t t 12. .. provides the decoder with a measure of confidence for the decision Lecture 12 Soft and hard decision decoding … „ ML soft-decisions decoding rule: „ „ Choose the path in the trellis with minimum Euclidean... outputs per Branch word Lecture 12 Soft and hard decision decoding „ In hard decision: „ „ The demodulator makes a firm or hard decision whether one or zero is transmitted and provides no other