Low Density Parity Check codes

Low Density Parity Check codes • Performance similar to turbo codes • Do not require long interleaver to achieve good performance • Better block error performance • Error floor occurs at lower BER • Decoding is not trellis based • Iterative decoding • More iterations than turbo decoding • Simpler operations in each iteration step • Not covered by patents!

Introduction • A binary parity check matrix Hn-kn • C = { vGF(2)n : vHT = 0 } • (Regular) LDPC code: The null space of a matrix H which has the following properties: • Its J rows are not necessarily linearly independent •  1s in each row •  1s in each column • No pair of columns has more than one common 1 •  and  are small compared to the number of rows (J) and columns (n) in H • Density r = /n = /J • Irregular: The number of 1s in rows/columns may vary

Example • J = n = 15 •  =  = 4

Gallager’s LDPC codes • Select  and  • Form a kk matrix H consisting of  submatrices H1,...,H, each of dimensions kk • H1 is a matrix such that its ith row has its  1s confined to columns (i-1)+1 to i. • The other submatrices are column permutations of H1 • The actual column permutations define the properties of the code

Gallager’s LDPC codes: Example • Select  = 4,  = 3 • Column permutations for H2 and H3 selected so that no two columns have more than one 1 in common • (20,7,6) code

Rows and parity checks • Each row of H forms a parity check • Let Al be the set of rows {h1(l),...,h(l)} that have a one in position l, i. e. the set of parity checks on symbol l. • Since no pair of columns has more than one common 1, any code bit other than bit l is checked by at most one of the rows in Al. • Thus there are  rows orthogonal on l, and any error pattern of weight  /2 can be decoded by one-step majority logic decoding. However, (especially when  is small) this gives poor performance • Gallager proposed two iterative decoding algorithms, one hard decision based and one soft decision based

Brief overview of graph theoretic notations • An undirected graph G(V,E) • Degree of a vertex/node : Number of edges adjacent to it • Adjacent edges: Connected to the same vertex • Path in a graph: Alternating sequence of vertices and edges (starting and ending in a vertex), where all vertices are distinct except for maybe the first and the last • A tree: A graph without cycles • Connected graph: There is a path between any pair of vertices • Bipartite graph: the set of vertices can be partitioned into two disjoint parts; so that each edge goes from a vertex in one part to a vertex in the other

Graphical description of LDPC codes • Tanner graph: A bipartite graph G(V,E), where • V = V1 V2 • V1is a set of ncode-bit nodes v0,...,vn-1, each representing one of the n code bits • V2is a set of Jcheck nodes s0,...,sJ-1, each representing one of the J parity checks • There is an edge between v V1and s V2iff the code-bit corresponding to v is contained in the parity check s. • No cycles of length 4 for the Tanner graph of an LDPC code

Example

Decoding of LDPC codes • Several decoding techniques available. Listed in order of increasing complexity (and improving performance): • Majority logic decoding (MLG) • Bit flipping (BF) • Weighted bit flipping (WBF) • Iterative decoding based on belief propagation (IDBP) also known as the sum-product algorithm (SPA) • A posteriori probability decoding (APP)

Decoding notation • Codeword = channel input : v = (v0,..., vn-1), binary • Assume BPSK modulation on AWGN channel • Channel output : y = (y0,..., yn-1) • Hard decisions : z = (z0,..., zn-1), zj = 1 (0)yj > (<) 0 • Rows of H : h1,..., hJ • Syndrome : s = (s1,..., sJ) = z  HT, sj = z hj = l=0...n-1zlhj,l • Parity failure : at least one syndrome bit nonzero • Error pattern e = v + z • Syndrome : s = (s1,..., sJ) = e  HT, sj = e hj = l=0...n-1elhj,l

Majority logic decoding • One-step MLG (see Chapter 8*) • Let Al be the set of rows {h1(l),...,h(l)} that have a one in position l, i. e. the set of parity checks on symbol l. • Set of  parity checks orthogonal on code bit l: • Sl = { s= z  h = e  h = l=0...n-1elhl : h Al } • If a majority of the parity checks on bit l are satisfied; conclude that el = 0. Otherwise, assume that el = 1. • This decoding is guaranteed to result in a correct decoding if at most /2 errors occur. • Problem:  is by assumption a small number.

Bit flipping decoding • Introduced by Gallager in 1961 • Hard decision decoding: First, form the hard decisions z = (z0,..., zn-1) • Compute the parity check sumss = (s1,..., sJ) = z  HT, sj = z hj = l=0...n-1zlhj,l • For each bit l, find fl = the number of failed parity checks for bit l • Let S = the set of bits for which fl is large • Flip the value of the bits in S • Repeat from a) until all parity checks are satisfied, or a maximum number of iterations have been reached

Bit flipping decoding (continued) • Let S = the set of bits for which fl is large • For example: • S = the set of bits for which fl exceeds some threshold  that depends on the code parameters: , , minimum distance and channel SNR. If decoding fails, try again with reduced value of  • Simple alternative: S = the set of bits for which fl is maximum • Comments: • If few errors occur, decoding should commence in a few iterations • On a poor channel, the number of iterations should be (allowed to grow) large • Improvement: Adaptive thresholds

Weighted MLG and BF decoding • Use weighting to include soft decision/reliability information in the decoding decision • Hard decision decoding: First, form the hard decisions z = (z0,..., zn-1) • Weight: | yj|(l)min = min{| yi| : 0in-1, hj,i =1, h Al } • The reliability of the jth parity check on l • Weighted MLG: Hard decision on • El = s Sl (2s-1)| yj|(l)min • ...where Sl = { s= z  h = e  h = l=0...n-1elhl : h Al } • Weighted BF: • Compute the parity checks s = (s1,..., sJ) = z  HT • For each bit l, compute El . • Flip the value of the bit for which El is largest • Repeat from a) until all parity checks are satisfied, or a maximum number of iterations have been reached

The sum-product decoding algorithm • Iterative decoding based on belief propagation • Symbol-by-symbol SISO: The goal is to compute • P( vl | y ) • L(vl) = log ( P( vl = 1| y )/P( vl = 0| y ) ) • A priori values: plx = P( vl = x), x{0,1} • qj,lx,(i) = P(vl = x |check sums  Al \ {hj} at ith iteration} • hj = (hj,0, hj,1, ..., hj,n-1); support of hj = B(hj) = { l : hj,l = 1} • j,lx,(i) = P(sj=0| vl = x,{vt:tB(hj)\{l}})   tB(hj)\{l}qj,tvt,(i) • qj,lx,(i+1) = j,l(i+1)  ht  Al \ {hj}t,lx,(i) • j,l(i+1) chosen so that qj,l0,(i+1) +qj,l1,(i+1) =1 • P (i)(vl = x | y )=l(i)  hj  Alj,lx,(i-1) •  l(i) chosen so that P (i)(vl = 0| y )+ P (i)(vl = 1 | y )=1

The SPA on the Tanner graph qj,lx,(i) qj,mx,(i) qj,nx,(i) qj,lx,(i+1) j,lx,(i) a,lx,(i) b,lx,(i) qj,mx,(i+1) qj,nx,(i+1) vl sj + • qj,lx,(i) = P(vl = x |check sums  Al \ {hj} at ith iteration} • j,lx,(i) = P(sj=0| vl = x,{vt:tB(hj)\{l}})   tB(hj)\{l}qj,tvt,(i) • qj,lx,(i+1) = j,l(i+1)  ht  Al \ {hj}t,lx,(i)

Suggested exercises • 17.1-17.11

Low Density Parity Check codes