Optimum Soft Decision Decoding of Linear Block Codes

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1 Optimum Soft Decision Decoding of Linear Block Codes {m i } Channel encoder C=(C n-1,,c 0 ) BPSK S(t) (n,k,d) linear modulator block code Optimal receiver AWGN Assume that [n,k,d] linear block code C is binary. The channel encoder maps the K information bits m 0,,m k-1 into a codeword C =(C n-1,,c 0 ) The transmitted waveform s(t) is 2εc s( t) = (2Cj 1) cos 2 π fct, t [( n 1 j) Tb, ( n j) Tb ], j = 0, n 1 T The vector representation of s(t) b s = ε [(2C 1),,(2C 1)]. c n 1 0 The receiver is optimal with respect to the coded signals

2 Optimum Soft Decision Decoding of Linear Block Codes Assume that the transmitted information are equally likely. The optimal decision rule is the minimum distance decision rule. Let s i be the vector representation of the coded signal waveform s i (t) corresponding to the codeword C i =(C i(n-1),,c i0 ) s ε C C K i = c [(2 i( n 1) 1),,(2 i0 1)], 0 i 2 1 Let us compute the block error prob. P e : Since C is linear, P e (error C i ) is the same for all codewords C i. Therefore, P e 2 1 k = 2 Pe (error C i ) i= 0 P e = P e (error C 0 ) K

3 Optimum Soft Decision Decoding of Linear Block Codes Applying the union bound K K s 2 ( ) i s0 εcwtc i Pe Q = Q i= 1 2N 0 i= 1 2N 0 K εcwtc ( i) = Q i= 1 N 0 K 2 1 nεc ε c 2E b As Eb = =,we have Pe Q RwtC c ( i) K R c i= 1 N 0 The above upper bound depends on the weight distribution of C A simpler bound is as follows K 2E b Pe 2 Q Rd c N 0 d = the minimum distance of C

4 Hard Decision Decoding of Linear Block Codes In soft decision, we need to compute M distance metric. When M is large, the computation complexity is very high. To reduce the computation burden, we can quantize the analog signals, and the decoding is performed digitally. {m i } (n,k,d) linear C=(C n-1,,c 0 ) BPSK S(t) block codec modulator Cˆ or mˆ, mˆ K 0 1 Channel decoder Hard decision decoding diagram AWGN BPSK demodulator

5 Hard Decision Decoding of Linear Block Codes Equivalent diagram {m i } (n,k,d) linear C=(C n-1,,c 0 ) S(t) block codec BSC Channel decoder ˆ C The error prob. of the BSC is 2ε c p = Q N 0 Assume that p<1/2. Assume that the transmitted information are equally likely. We want to design an optimal decoder and analyze its performance.

6 Hard Decision Decoding of Linear Block Codes Because of noise, the received vector y =(y n-1,, y 0 ) may be different from the transmitted codeword C. The difference vector e = y C = (e n-1,, e 0 ) is called an error vector. e j = 0 with probability 1-p, and e j = 1 with probability p. Since there are 2 K possible error vectors, the decoder can not decide which codeword was actually transmitted. Since the transmitted information is equally likely, the MAP rule is equivalent to the ML decision rule. Thus the optimal decoder decode y as c=c ˆ iff p( y C ) = max p( y C ) j 0 i 2 1 wt( e ) n wt ( e ) wt( e ) n wt( e ) p (1 p) = max p (1 p) j j i i 0 i 2 1 wt( e ) = min wt( e ) d( y,c ) = min d( y,c ) j i i j k j k i j i

7 Minimum Hamming Distance Decoding Rule The optimal decision rule now becomes the minimum Hamming distance decision rule. The optimal decoder decodes y as the nearest codeword d( y,c ) = min d( y,c ) j i i In other words, the decoder picks the error vector that has the least weight and then forms an estimate Cˆ = y eˆ ê Ĉ

8 Error Detection Capability Let C be a block code with block length n and minimum Hamming distance d. Suppose C is used only for error detection. That is, the receiver just tests if the received vector is codeword or not. If it is not, the receiver detects that an error has occurred, and ask for a retransmission of the codeword. The code C can detect up to d-1 errors. On the other hand, when more than d-1 errors occur, the receiver may be fooled since it is possible for an error vector and weight d to transform one codeword into another codeword. Therefore C is capable of detecting up to d-1 errors.

9 Error Detection and Error Correction Capability 1 ( 1) 2 d When C is used only for error correction, C can correct up to errors. t C t 1 C 2 t C3 C i t We associate with each codeword C i a ball of radius t and center C i, where 1 t= ( d 1) 2 All these balls are non-overlapping. If C i is transmitted and t or fewer errors occur, then the received vector y is inside the ball centered at C i, and is closer to C i than any other codeword. Thus the nearest neighbor decoding will correct these errors. On the other hand, if more than t errors occur, the received vector y may be closer to some other codeword. If this is the case, then the decoder will be fooled.

10 Error Detection and Error Correction The block code can also be used for both error correction and error detection. Suppose d=7. Then C can correct/detect 3 errors. We may take a different decoding strategy to increase the error detection capability at the expense of the error correction capability. For example, C can be used to correct up to 2 errors and at the same time detect 4 possible errors. t c d t c C i C j t d In general, a block code C with the minimum distance d can simultaneously correct t c errors and detect td errors as long as t c +t d d-1. For any two disjoint codewords C i and C j, the ball of radius t d and centre C i and the ball of radius t c and centre C j are disjoint.

11 Syndrome Decoding A brute force method for performing the nearest neighbor decoding will involve 2 k possible comparisons. More efficient method is syndrome decoding: Given the received vector y, one first computes the vector s = Hy Where H is a fixed parity check matrix for C. The (n-k)-dimensional vector s=(s n-k-1, s n-k-2,, s 0 ) is called the syndrome of y. The syndrome provides some information about the possible error vector e. There is one to one correspondence between syndromes and sets of all possible error vectors.

12 Syndrome Decoding Given the received vector y, the set of all possible error vectors is { y C : C C} y+ C i i The set y + C is called a coset of C. Note that y C. Thus A coset containing y = the set of all possible error vectors with respect to y. Property 1: Two cosets are either disjoint and coincide. Property 2: Two vectors y 1 and y 2 have the same syndrome iff y 1 and y 2 are in the same coset. = ( )' = 0 ' ' Hy Hy H y1 y2 1 2 ( y y ) C y y + C

13 Syndrome Decoding There is a one to one correspondence between syndrome and cosets. Each coset contains 2 K vectors. This implies that there are 2 n-k cosets and hence 2 n-k syndromes. In view of the nearest neighbor decoding rule, we get the following decoding algorithm: Step 1: Given a received vector y, compute the syndrome s of y. Step 2: Find the least weight vector in the coset given by the syndrome s. Step 3: Decode y as ˆ c= y eˆ The least weight vector in a coset is called the coset leader of the coset. ê

14 Syndrome Decoding Standard array for syndrome decoding ( n -k is small ) Coset leaders Syndrome Codewords 0 C 1 C 2 k-1 s 0 Coset e 1 C 1 +e 1 C 2 k-1 + e 1 s 1 Coset e 2 n-k-1 C 1 + e 2n-k-1 C 2k-1 + e 2n-k-1 s 2 n-k -1 When y is received, its position in the standard array is located. The decoder the decides that the error vector is the left-most vector in the row containing y, and y is decoded as the codeword at the top of the column containing y.

15 Example G = Coset leaders Syndrome Codewords d min = How about the actual error is (10100)?

16 Syndrome Computing in the Case of Cyclic Codes Let g(x) = x n-k +g n-k-1 x n-k-1 + g 1 x+g 0 be the generator polynomial of C. Let y=(y n-1, y 0 ) be the received vector. Associate with y a polynomial y(x) = y n-1 x n-1 + +y 1 x + y 0 Assume that we compute the syndrome s=(s n-k-1, s 0 ) of y by using the systematic parity check matrix H. Associate with s a polynomial s(x) = s n-1 x n-1 + +s 1 x + s 0 One can show that s(x) is the remainder obtained by dividing y(x) by g(x) y(x) = f(x)g(x) + s(x)

17 Example The [7,4,3] binary Hamming code revisited g(x) = x 3 +x+1 G = H=? Suppose y = [ ] What is syndrome s 1. Use matrix multiplication 2. Use polynomial division

18 Shift Register Implementation y 0 y 1 y Quotient

19 Performance of Hard Decision Decoding The error probability of BSC 2ε c p= Q < 1/ 2 N 0 Block error probability K 2 1 P = P (error C ) = P(error C ) = P(error C ) e e i i M i= 0 n-k 2 1 i = 1- p (1 p) i=0 wt( e ) n wt( e ) e i s are the coset leaders, and e 0 is the zero vector. i 0

20 Let t Performance of Hard Decision Decoding 1 = ( d 1) 2 The balls of radius t and centers C i are all disjoint. Thus any vector with weight t is a coset leader. Therefore, n k 2 1 n i i p (1 p) p (1 p) l t wt ( e ) n wt ( e ) l n l i= 0 l= 0 t n n n Pe 1 p (1 p) = p (1 p) l= 0 l l= t+ 1 l l n l l n l The equality holds if the coset leaders consist of all vectors with weight t. In this case, we have t k n 2 = 2 l= 0 l The linear code is called a perfect code. n

21 Performance of Hard Decision Decoding In general, t l= 0 n l n k 2 -Hamming bound Hamming bound gives another relationship among n, k, d. An [n,k,d] linear binary code is called quasi-perfect if 2 n k t+ 1 l= 0 n l For an [n,k,d] perfect code, the balls of radius t are disjoint and together contain all vectors of length n. For an [n,k,d] quasi-perfect code, the balls of radius t+1 may overlap and together contain all vectors of length n.

22 Error Probability for Quasi-Perfect Codes The coset leaders have weight t+1, The number of the cost leaders having weight t+1 is nk 2 1 n k 2 - n t l= 0 l n n i i p (1 p) = p (1 p) + 2 p (1 p) l l t t wt ( e ) n wt ( e ) l nl n k t+ 1 n t 1 i= 0 l= 0 l= 0 n n n Pe = p (1 p) 2 p (1 p) l=+ t 1 l l= 0 l t l n l n k t+ 1 n t 1 This formula is also a lower bound to P e for any [n, k, d] linear binary block code.

23 Other Bounds Consider the communication of two equally likely n-dimensional vectors a = (a n-1,, a 0 ), b =(b n-1,, b 0 ) over the BSC. The minimum decoding error prob. depends only on the Hamming distance d(a, b) between a and b. Denote this minimum error prob. as 2u+ 1 2u+ 1 p p l= u+ 1 l p2 ( dab (, )) = 2u+ 1 2u l 2u l 1 2u u u p (1 p) + p (1- p) if dab (, ) = 2u l= u+ 1 l 2 u l 2u+ 1 l (1 ) if dab (, ) = 2u+ 1 p 2 (.) is strictly decreasing on the set of possible integers p 2 (l+1) < p 2 (l)

24 Other Bounds (cntd) In terms of union bounds i max p ( d( C, C )) P (error C ) p ( d( C, C )) 2 0 i e i i= That is, p ( d) P p ( wt( C )) 2 e 2 i= 1 K d = the minimum distance of C i K The upper bound depends on the weight distribution of C. Since p 2 (l+1) < p 2 (l), we have p 2 (d) p e (2 K -1) p 2 (d)

25 Other Bounds (cntd) We can also show that p 2 (l) p( x 1 + +x l l/2) [4p(1-p)] l/2 For any l 1, where x 1 x l are i.i.d. random variables and X 1 1 with prob. p = 0 with prob. 1-p 2 1 p ( d ) P p ( wt( C )) 2 e 2 i= 1 2 K 2 1 p ( d ) P [4 p(1 p)] e K i= 1 i wt ( C ) / 2 i

26 Comparison of Performance Between Hard- Decision and Soft-Decision Decoding Method 1: use the bounds developed in the last two sections to evaluate the hard decision performance and soft decision performance of specific linear block codes. Method 2: Hard decision results in the BSC channel 0 1 p p 1-p 1-p 0 1 2E p= Q b N On the other hand, soft decision gives rise to the discrete input, continuous output channel: Y =± E+ n n is a Gaussian random variable. Compute the channel capacity of these two channels and compare them. c 0

27 Comparison of Performance Between Hard- Decision and Soft-Decision Decoding Method 3: Use the random coding argument to find the hard decision and soft decision random coding rates. Then compare these two rates. All these methods reveal that soft decision performance is roughly 2dB better than hard decision performance.

28 Concatenated Block Codes Most of the codes discussed so far are designed for correcting and detecting random errors, but not suitable for correcting burst errors. Definition: An error burst of length b in an n-bit received vector is a contiguous sequence of b bits in which the first and the last bits and any number of intermediate bits are received in error. An error vector containing a single error burst of length 7 looks like this: 00 1xxxxx10 0, Where x may be 0 or 1.

29 Concatenated Block Codes Fact: Binary codes obtained from nonbinary codes, particularly from Reed-Solumn (RS) codes are particularly suited to correcting burst errors. Consider a [255, 249, 7] RS code C over GF(2 8 ). Each code symbol is an element in GF(2 8 ) and hence represent 8 bits. Replace each code symbol in every codeword in C by its binary representation. We then gets a binary code C. The resulting binary code C has parameters: n = 255 x 8 =2040, k = 249 x 8 =1992, d 7 The original nonbinary RS code C can correct 3 symbol errors. The binary code C can correct an error burst of length 17.

30 Concatenated Block Codes In general, if C is an [N, K, D] RS code over GF(2 m ) with N = 2 m -1, K = N-2t, D = 2t + 1 Then the binary code C obtained from C can correct an error burst of length m(t-1)+1. The binary code C obtained from [N, K, D] RS code C over GF(2 m ) has good burst error correction capability, but it does not help correct random errors. To improve its capability of correcting random errors, we may use a linear [n, m, d] block code to further encode the output sequence of the binary code C. The resulting overall code is called a concatenated code.

31 Concatenated Block Codes Input data Outer encoder (N,K,D] Inner Encoder [n,m,d] Modulator Channel Output data Outer Decoder Inner Decoder Demodulator A concatenated code

32 Concatenated Block Codes K x m bits N x m bits N x n bits Thus the concatenated code has parameters block length = N x n # of information bits = k x m minimum distance d x D

33 Interleavers Another effective method for dealing with error bursts is to interleave the block coded sequence so that a whole codeword is not transmitted in consecutive time intervals. As a result, error bursts are spread out among many codewords so that errors within a codeword appear to be random.

34 Interleavers Read out bits to modulator Read in coded bits from the encoder Mn-4 Mn-3 Mn-2 Mn-1 M rows C 0,C 1, C n,c n Mn n-k parity bits k inform. If the original [n, k, d] block code can correct an error burst of length b, then the combination of the original block and the block interleaver of degree m can correct an error burst of length up to mb

Lecture 12. Block Diagram

Lecture 12. Block Diagram Lecture 12 Goals Be able to encode using a linear block code Be able to decode a linear block code received over a binary symmetric channel or an additive white Gaussian channel XII-1 Block Diagram Data