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- Digital Communications I: Modulation and Coding Course Period 3 - 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 2
- Trellis of an example ½ Conv. code Input bits Tail bits 1 0 1 0 0 Output bits 11 10 00 10 11 0/00 0/00 0/00 0/00 0/00 1/11 1/11 1/11 0/11 0/11 0/11 0/10 1/00 0/10 0/10 1/01 1/01 0/01 0/01 1/01 t1 t 2 t 3 t 4 t 5 t 6 Lecture 12 3
- Block diagram of the DCS Information Rate 1/n Modulator source Conv. encoder m = (m1 , m2 ,..., mi ,...) U = G(m) 144 44 2 3 = (U1 , U 2 , U 3 ,..., U i ,...) Channel Input sequence 144 2444 4 3 Codeword sequence U i = u1i ,...,u ji ,...,u ni 14 2444 3 Branch word ( n coded bits) Information Rate 1/n Demodulator sink Conv. decoder m = (m1 , m2 ,..., mi ,...) ˆ ˆ ˆ ˆ Z = ( Z1 , Z 2 , Z 3 ,..., Z i ,...) 144 2444 4 3 received sequence Zi = z1i ,...,z ji ,...,z ni { 14 2444 3 Demodulator outputs n outputs per Branch word for Branch word i Lecture 12 4
- 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 5
- 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 6
- 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 7
- Example of hard-decision Viterbi decoding m = (100) ˆ Z = (11 10 11 10 01) U = (11 10 11 00 11) ˆ m = (101 ) U = (11 10 00 10 11 ) 0 2 2 1 3 2 0 1 1 1 2 1 0 0 0 3 0 2 1 1 1 2 Partial metric 0 0 3 0 2 Γ(S (ti ), ti ) 1 2 2 2 1 3 Branch metric 1 t1 t2 t3 t4 t5 t6 Lecture 12 8
- Example of soft-decision Viterbi decoding 2 2 −2 −2 2 −2 m = (101) ˆ Z = (1, , , , ,1, , − 1, ,1) 3 3 3 3 3 3 U = (11 10 00 10 11) ˆ m = (101 ) U = (11 10 00 10 11 ) 0 -5/3 -5/3 0 -5/3 -1/3 10/3 1/3 1/3 -1/3 14/3 0 1/3 1/3 5/3 5/3 5/3 8/3 1/3 -5/3 -1/3 4/3 1/3 Partial metric 3 2 5/3 13/3 Γ(S (ti ), ti ) -4/3 5/3 -5/3 Branch metric 1/3 5/3 10/3 -5/3 t1 t2 t3 t4 t5 t6 Lecture 12 9
- 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
- Free distance of Convolutional codes Distance properties: Since a Convolutional encoder generates codewords with various sizes (as opposite to the block codes), the following approach is used to find the minimum distance between all pairs of codewords: Since the code is linear, the minimum distance of the code is the minimum distance between each of the codewords and the all-zero codeword. This is the minimum distance in the set of all arbitrary long paths along the trellis that diverge and remerge to the all-zero path. It is called the minimum free distance or the free distance of the code, denoted by d free or d f Lecture 12 11
- Free distance … The path diverging and remerging to Hamming weight All-zero path of the branch all-zero path with minimum weight df =5 0 0 0 0 0 2 2 2 2 2 2 1 0 1 1 1 1 1 1 t1 t 2 t 3 t 4 t 5 t 6 Lecture 12 12
- Transfer function of Convolutional codes Transfer function: Transfer function of generating function is a tool which provides information about the weight distribution of the codewords. The weight distribution specifies weights of different paths in the trellis (codewords) with their corresponding lengths and amount of information. T ( D , L, N ) = ∑ ∑∑ Di Lj N l i = d f j = K l =1 D, L, N : place holders i : distance of the path from the all - zero path j : number of branches that the path takes until it remerges to the all - zero path l : weight of the information bits corresponding to the path Lecture 12 13
- Transfer function … Example of transfer function for the rate ½ Convolutonal code. 1. Redraw the state diagram such that the zero state is split into two nodes, the starting and ending nodes. 2. Label each branch by the corresponding D i L j N l LN a = 00 b = 10 c = 01 e = 00 D 2 LN DL D2L DLN DL d =11 DLN Lecture 12 14
- Transfer function … Write the state equations ( X a ,..., X e dummy variables) ⎧ X b = D 2 LNX a + LNX c ⎪ ⎪ X c = DLX b + DLX d ⎨ ⎪ X d = DLNX b + DLNX d ⎪ X = D 2 LX ⎩ e c Solve T ( D, L, N ) = X e / X a D 5 L3 N T ( D , L, N ) = = D 5 L3 N + D 6 L4 N 2 + D 6 L5 N 2 + .... 1 − DL(1 + L) N One path with weight 5, length 3 and data weight of 1 One path with weight 6, length 4 and data weight of 2 One path with weight 5, length 5 and data weight of 2 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 finite number of errors in the coded bits cause as infinite number of errors in the decoded data bits. A Convolutional code is catastrophic if there is a closed loop in the state diagram with zero weight. Systematic codes are not catastrophic: 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 6 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 b 10 c 01 e 00 10 01 00 01 d 11 11 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
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