Introduction to Channel Coding

DIGITAL COMMUNICATIONS

Block 3

Channel Coding

Francisco J. Escribano, 2014-15

Fundamentals of error control

● Error control:

– error detection (ARQ schemes)

– correction (FEC schemes)

● Channel model:

– discrete inputs,

– discrete (hard, rn) or continuous (soft, λ

n) outputs,

– memoryless.

Channelencoder

ChannelChannel

errorcorrector

Channelerror

detectorRTx

● Enabling detection/correction:

– Adding redundancy to the information: for every k bits, transmit n, n>k.

● Shannon's theorem (1948):

1) If , for ε>0, there is n, R=k/n const., so that Pb<ε.

2) If Pb is acceptable, rates R<R(P

b)=C/(1-H

b)) are achievable.

3) For any Pb, rates greater than R(P

b) are not achievable.

● Problem: Shannon's theorem is not constructive.

R<C=maxpX

I ( X ;Y )

● Added redundancy is structured redundancy.

● This relies on sound algebraic & geometrical basis.

● Our initial approach:

– Algebra over the Galois Field of order 2, GF(2)={0,1}.

– GF(2) is a proper field, GF(2)m is a vector field of dim. m.

– Dot product · :logical AND. Sum +(-) : logical XOR.

– Scalar product: b, d ∈ GF(2)m

b·dT=b1·d

1+...+b

– Product by scalars: a ∈ GF(2), b ∈ GF(2)m

a·b=(a·b1..a·b

– It is also possible to define a matrix algebra over GF(2).

● Given a vector b ∈ GF(2)m, its binary weight is w(b)=number of 1's in b.

● It is possible to define a distance over vector field GF(2)m, called Hamming distance:

dH(b,d)=w(b+d); b, d ∈ GF(2)m

● Hamming distance is a proper distance and accounts for the number of differing positions between vectors.

● Geometrical view:

(0110)

(1011)

(1110) (1010)

● A given encoder produces n bit outputs for each k bit inputs:

– R=k/n<1 is the rate of the code.

● The information rate decreases by R by the use of a code.

R'b=R·R

b (bit/s)

● If used jointly with a modulation with spectral efficiency η=Rb/B

(bit/s/Hz), the efficiency decreases by R

η'=R·η < η (bit/s/Hz)● In terms of P

b, the achievable E

0 region in AWGN is lower

bounded by:

N 0(dB)⩾10⋅log10 (

1η '⋅(2

η'−1) )

Source: http://www.comtechefdata.com/technologies/fec/ldpc

● How a channel code can improve Pb (BER in statistical terms).

● Cost: loss in resources (spectral efficiency, power).

Binary Linear block codes

● An (n,k) linear block code (LBC) is a subspace C(n,k) < GF(2)n

with dim(C(n,k))=k.

● C(n,k) contains 2k vectors c=(c1…c

● R=k/n is the rate of the LBC.

● n-k is the redundancy of the LBC

– we would only need vectors with k components to specify the same amount of information.

● Recall vector theory:

– A basis for C(n,k) has k vectors over GF(2)n

– C(n,k) is orthogonal to an n-k dimensional subspace over GF(2)n (its null subspace).

● c ∈ C(n,k) can be both specified as:

– c=b1·g

1+...+b

k, where {g

j=1,...,k is the basis, and

(b1...b

k) are its coordinates over it.

– c such that the scalar products c·hiT are null, when

i=1,...,n-k is a basis of the null subspace.

● Arranging in matrix form, an LBC C(n,k) can be specified by

– G={gij}

i=1,...,k, j=1,...,n, c=b·G, b ∈ GF(2)k.

– H={hij}

i=1,...,n-k, j=1,...,n, c·HT=0.

● G is a k×n generator matrix of the LBC C(n,k).

● H is a (n-k)×n parity-check matrix of the LBC C(n,k).

– In other approach, it can be shown that the rows in H stand for linearly independent parity-check equations.

– The row rank of H for an LBC should be n-k.

● Note that gj ∈ C(n,k), and so G·HT=0.

● The encoder is given by G

– Note that a number of different G generate the same LBC

● For any input information block with length k, it yields a codeword with length n.

● An encoder is systematic if b is contained in c=(b1...b

k+1...c

so that ck+1

...cn are the n-k parity bits.

– Systematicity is a property of the encoder, not of the LBC C(n,k) itself.

– GS=[I

k | P] is a systematic generator matrix.

LBCencoder

b=(b1...b

k) c=(c

n)=b·G

● How to obtain G from H or H from G.

– G rows are k vectors linearly independent over GF(2)n

– H rows are n-k vectors linearly independent over GF(2)n

– They are related through G·HT=0 (a)

● (a) does not yield a sufficient set of equations, given H or G.

– A number of vector sets comply with it (basis sets are not unique).

● Given G, put it in systematic form by combining rows (the code will be the same, but the encoding does change).

– If GS=[I

k | P], then H

S=[PT | I

n-k] complies with (a).

● Conversely, given H, put it in systematic form by combining rows.

– If HS=[I

n-k | P], then G

S=[PT | I

k] complies with (a).

● Parity check submatrix P can be on the left or on the right side (but on opposite sides of H and G simultaneously for a given LBC).

● Note that, by taking 2k vectors out of 2n, we are getting apart the binary words.

● Minimum Hamming distance between input words is

● Recall that we have added n-k redundancy bits, so that

dmin (GF (2)k )=minbi≠b j

{d H (bi ,b j ) |bi ,b j∈GF (2)k }=1

d min (C (n , k ))=minc i≠c j

{d H (c i ,c j ) | ci ,c j∈C (n , k )}>1

d min (C (n , k ))⩽n−k+1 (Singleton bound)

● The channel model corresponds to a BSC (binary symmetric channel)

● p=P(ci≠r

i ) is the bit error probability of the modulation in AWGN.

Modulator

ChannelBSC(p)

AWGNchannel

r=(r1...r

n)c=(c

Harddemodulator

● The received word is r=c+e, where P(ei=1)=p.

– e is the error vector introduced by the noisy channel

– w(e) is the number of errors in r wrt original word c

– P(w(e)=t)=pt·(1-p)n-t, because the channel is memoryless.

● At the receiver side, we can compute the syndrome s=(s1...s

n-k) as

s=r·HT=(c+e)·HT=c·HT+e·HT=e·HT.

● r ∈ C(n,k) ⇔ s=0.

Channeldecoder

s=(s1...s

n-k)=r·HTr=(r

Two possibilities at the receiver side:

● a) Error detection (ARQ schemes):

– If s≠0, there are errors, so ask for retransmission.

● b) Error correction (FEC schemes):

– Decode an estimated ĉ ∈ C(n,k), so that dH(ĉ,r) is the minimum

over all codewords in C(n,k) (closest neighbor decoding).

– ĉ is the most probable word under the assumption that p is small (otherwise, the decoding fails).

(0110)

(1011)

(1110) (1010)

● Detection and correction capabilities (worst case) of an LBC with d

min(C(n,k)).

– a) It can detect error events e with binary weight up to

w(e)|max,det

=d=dmin

(C(n,k))-1

– b) It can correct error events e with binary weight up to

w(e)|max,corr

=t=⎣(dmin

(C(n,k))-1)/2⎦

● It is possible to implement a joint strategy:

– A dmin

(C(n,k))=4 code can simultaneously correct all error

patterns with w(e)=1, and detect all error patterns with w(e)=2.

● The minimum distance dmin

(C(n,k)) is a property of the set of

codewords in C(n,k), independent from the encoding (G).

● As the code is linear, dH(c

– ci, c

j, 0 ∈ C(n,k)

● dmin

(C(n,k))=min{w(c) | c ∈ C(n,k), c≠0}

– i.e., corresponds to the minimum word weight over all codewords different from null.

● dmin

(C(n,k)) can be calculated from H:

– It is the minimum number of different columns of H adding to 0.

– It is the column rank of H + 1.

● Detection limits: probability of undetected errors?

– Note that an LBC contains 2k codewords, and the received word corresponds to any of the 2n possibilities in GF(2)n.

– An LBC detects up to 2n-2k error patterns.

● An undetected error occurs if r=c+e with e≠0 ∈ C(n,k)

– In this case, r·HT=0.

– Ai is the number of codewords in C(n,k) with weight i: it is

called the weight spectrum of the LBC.

Pu (E)=∑i=dmin

Ai⋅p i⋅(1− p)n−i

● On correction, an LBC considers syndrome s=r·HT.

– Assume correction capabilities up to w(e)=t, and EC to be

the set of correctable error patterns.

– A syndrome table associates a unique si over the 2n-k

possibilities to a unique error pattern ei ∈ E

C with w(e

i)≤t.

– If si=r·HT, decode ĉ=r+e

– Given an encoder G, estimate information vector b such that b·G=ĉ.

● If the number of correctable errors #(EC)<2n-k, there are 2n-k-

#(EC) syndromes usable in detection, but not in correction.

– At most, an LBC can correct 2n-k error patterns.

● A w(e)≤t error correcting LBC has a probability of correcting erroneously bounded by

– This is an upper bound, since not all the codewords are separated by the minimum distance of the code.

● Calculating the resulting P'b of an LBC is not an easy task,

and it depends heavily on how the encoding is made through G.

● LBC codes are mainly used in detection tasks (ARQ).

P (E )⩽∑i=t+1

(ni )⋅pi⋅(1− p)n−i

● Observe that both coding & decoding can be performed with low complexity hardware (combinational logic: gates).

● Examples of LBC

– Repetition codes

– Single parity check codes

– Hamming codes

– Cyclic redundancy codes

– Reed-Muller codes

– Golay codes

– Product codes

– Interleaved codes

● Some of them will be examined in the lab.

● An example of performance: RHam

=4/7, RGol

Multiple-error correction block codes

Multiple-error-correction block codes ● There are a number of more powerful block codes, based on higher

order Galois fields.

– They use symbols over GF(r), with r>2.

– Now the operations are defined as mod r.

– An (n,k) linear block code with symbols from GF(r) is again a k-dimensional subspace of the vector space GF(r)n.

● They are used mainly for correction, and have applications in channels or systems where error bursts are frequent, i.e.

– Storage systems (erasure channel).

– Communication channels with deep fades.

● They are frequently used in concatenation with other codes that fail in short bursts when correcting (i.e. convolutional codes).

● We are going to study two important instances of such broad class of codes: Reed-Solomon, and BCH codes.

Multiple-error-correction block codes

● An (n,k,t)q Reed-Solomon (R-S) code is defined as a mapping

from GF(q)k↔GF(q)n, with the following parameters:

– Block length n=q-1 symbols, for input block length of k<n symbols.

– Alphabet size q=pr, where p is a prime number.

● Minimum distance of the code dmin

=n-k+1 (it achieves the

Singleton bound for linear block codes).

– It can correct up to t symbol errors, dmin

=2t+1.

● This is a whole class of codes.

– For any q=pr, n=q-1 and k=q-1-2t, there exists a Reed-Solomon code meeting all these criteria.

Multiple-error-correction block codes ● The R-S code is built with the help of polynomial algebra. ● The message

is mapped to a polynomial with given coefficients

● To get the corresponding codeword, the encoding function works evaluating the polynomial at n distinct given points

Note that all the operations are performed over GF(q).

s=( s1 ... sk ) ∈ GF (q ) k

p (a )=∑i=1

zi⋅ai−1 ; a , z i ∈ GF ( q )

c=(p(a1) ...p(an)) ∈ GF(q)n

Multiple-error-correction block codes ● We can rewrite the encoding in a more familiar form

where A is the transpose of a Vandermonde matrix with structure

c=z⋅A , z=( z1 ... zk )

A=(1 1 ⋯ 1a1 a2 ⋯ an

a12 a2

2 ⋯ an2

⋮ ⋮ ⋱ ⋮

a1(k−1) a2

(k−1) ⋯ an(k−1)

● The number of polynomials in GF(q) of degree less than k is clearly qk, exactly the possible number of messages.

– This guarantees that each information word can be mapped to a unique codeword by choosing a convenient mapping s↔z.

● It is only required that a1,..,a

n are distinct points in GF(q),

(these are the points where the polynomial p(a) is evaluated to build the codeword)

– The points can be chosen to meet certain properties.

– Either the polynomial can be chosen in a given way.

● One way to build the framework for R-S codes consists in choosing a

1,..,a

n as n distinct points in GF(q) and build p(a)

by forcing the condition

● The polynomial is characterized as the only polynomial of degree less than k that meets the above mentioned condition, and can be found by using known algebraic methods (Lagrange interpolation).

● The codeword is given as

This is an instance of R-S systematic encoding.

p (ai )=si ∀ i=1,. .. k

cs=(s1 ... sk , p (ak+1 ) ...p (an ))

● In other possible construction, the polynomial is given by the mapping z=s

● And the points in GF(q) are chosen to meet certain convenient properties.

– Let α be a primitive root of GF(q). This means that, for any b ∊ GF(q), except 0, there exists an integer u / αu=b (mod q).

– ai=αi, i=1,...,q-1 (this spans GF(q), except 0).

p (a )=∑i=1

si⋅ai−1 , a ∈ GF ( q )

Multiple-error-correction block codes ● Now we can rewrite

where this time A is the transpose of a Vandermonde matrix with structure

c=s⋅A , s=( s1 ... sk )

A=(1 1 ⋯ 1α α2 ⋯ αn

2⋯ (α

⋮ ⋮ ⋱ ⋮

(α)(k−1 ) (α2)(k−1) ⋯ (αn)(k−1))

● In this last case, it can be demonstrated that the parity-check matrix of the resulting R-S code is

– It can correct t or fewer random symbol errors over a span of n=q-1 symbols.

H=(1 α α

2⋯ α

(q−2)

1 α2 (α2)2 ⋯ (α2)(q−2)

1 α3 (α3)2 ⋯ (α3)(q−2)

⋮ ⋮ ⋮ ⋱ ⋮

1 α2 t

(α2 t)

2⋯ (α

2 t)(q−2)

● The weight distribution (spectrum) of R-S codes has closed form

● We can derive bounds for the probability of undetected errors for a symmetric DMC with q-ary input/output alphabets, and probability of correct reception 1-ε.

Ai=(q−1i )q−2t

{(q−1 )i+∑

(−1 )i+ j( ij)(q

2 t−q i )}

2 t+1⩽i⩽q−1

Pu ( E )<q−2 t before any correction step

Pu ( E ,λ )<q−2 t∑h=0

(q−1h ) (q−1 )

h if used first to correct λ errors

0⩽ϵ⩽q

q−1 0<λ<t

● Other kind of linear block code defined over higher-order Galois fields is the class of BCH codes.

– Named after Raj Bose and D. K. Ray-Chaudhuri.

● An (n,k,t)q BCH code is again defined over GF(q), where

q=pr, and p is prime.

● We have m, n, q, d=2t+1, l, so that

– 2≤d≤n. l will be considered later.

– gcd(n,q)=1 (“gcd” → greatest common divisor)

– m is the multiplicative order of q modulo n; m is thus the smallest integer meeting qm=1 (mod n).

– t is the number of errors that may be corrected.

● Let α be a primitive n-th root of 1 in GF(qm). This means αn=1 (mod qm).

● Let mi(x) the minimal polynomial for GF(q) of αi, ∀ i.

– This is the monic polynomial of least degree having αi as a root.

– Monic → the coefficient of the highest power of x is 1.

● Then, a BCH code is defined by a so-called generator polynomial

where “lcm” stands for least common multiple. Its degree is at most (d-1)m=2mt.

g ( x )=lcm (ml ( x ) ...ml+d−2 ( x ) )

● The encoding with a generator polynomial is done by building a polynomial containing the information symbols

● Then

● We may do systematic encoding as

s (x )=∑i=1

si⋅x i−1s=( s1... sk ) ∈ GF (q ) k →

c ( x )=∑i=1

c i⋅xi−1=s ( x )⋅g ( x ) , c i ∈ GF (q) → c=(c1 ... cn )

cs (x )= xn−k⋅s ( x )⏟

systematic symbols

+xn−k⋅s (x ) mod g ( x )⏟

redundancy symbols

● The case with l=1, and n=qm-1 is called a primitive BCH code.

– The number of parity check symbols is n-k≤(d-1)m=2mt.

– The minimum distance is dmin

≥d=2t+1.

● If m=1, then we have a Reed-Solomon code (of the “primitive root” kind)!

– R-S codes can be seen as a subclass of BCH codes.

– In this case, it can be verified that the R-S code may be defined by means of a generator polynomial, in the form

g ( x )=( x−α ) (x−α2 ) ... (x−α2 t )==g1+g2 x+g3 x2

+...+g2 t−2 x2 t−1+x2 t

● The parity check matrix for a primitive BCH code over GF(qm) is

– This code can correct t or fewer random symbol errors, d=2t+1, over a span of n=qm-1 symbol positions.

H=(1 α α

2⋯ α

(n−1)

2⋯ (α

2)(n−1)

1 α3 (α3)2 ⋯ (α3)(n−1)

⋮ ⋮ ⋮ ⋱ ⋮

1 α(d−1)

(α(d−1)

)2⋯ (α

(d−1))(n−1)

● Why these codes can correct error bursts in the channel?

BCH/R-Sencoder

BSC(p)

BCH/R-Sdecoder

Mapbits

to q-arysymbols

Mapq-ary

symbolsto bits

Mapq-ary

symbolsto bits

Mapbits

to q-arysymbols

s=(s1...s

r=(r1...r

b=(b1...b

k·log2(q))

b'=(b'1...b'

k·log2(q))

s'=(s'1...s'

c=(c1...c

BCH/R-Sencoder

BSC(p)

BCH/R-Sdecoder

Mapbits

to q-arysymbols

Mapq-ary

symbolsto bits

Mapq-ary

symbolsto bits

Mapbits

to q-arysymbols

s=(s1...s

r=(r1...r

b=(b1...b

k·log2(q))

b'=(b'1...b'

k·log2(q))

s'=(s'1...s'

c=(c1...c

Other channel encoder/decoder,modulator/demodulator,

medium access technique

BCH/R-Sencoder

BSC(p)

BCH/R-Sdecoder

Mapbits

to q-arysymbols

Mapq-ary

symbolsto bits

Mapq-ary

symbolsto bits

Mapbits

to q-arysymbols e

b=(...1111...)

s=(s1...s

r=(r1...r

b=(b1...b

k·log2(q))

b'=(b'1...b'

k·log2(q))

s'=(s'1...s'

c=(c1...c

BCH/R-Sencoder

BSC(p)

BCH/R-Sdecoder

Mapbits

to q-arysymbols

Mapq-ary

symbolsto bits

Mapq-ary

symbolsto bits

Mapbits

to q-arysymbols e

b=(...1111...)

s=(s1...s

r=(r1...r

b=(b1...b

k·log2(q))

b'=(b'1...b'

k·log2(q))

s'=(s'1...s'

c=(c1...c

If bit error burst falls within a single symbol ri in GF(q), or at most spans over t

symbols in GF(q) within word r, it can be corrected!

● Note that algebra is now far more involved and more complex than with binary LBC.

– This is part of the price to pay to get better data integrity protection.

– Other logical price to pay is the reduction in data rate, R. k and n are measured in symbols, but, as the mapping and demapping is performed from GF(2) to GF(q), and viceversa, the end-to-end effective data rate is again

● But there is still something to do to get the best from R-S and BCH codes: decoding is substantially more complex!

Rb'=Rb⋅

● We are going to see a simple instance of decoding for these families of nonbinary LBC.

● We address the general BCH case, as R-S can be seen as an instance of the former.

● Correction can be performed by identifying the pairs

● For this, we can resort to the syndrome

r ( x )=c ( x )+e ( x ) is the received codeword

e ( x )=e j1x j1+...+e jν

x j ν 0⩽ j1⩽...⩽ j ν⩽n−1

(x ji , e ji) ∀ i=1,... ,ν

(S1... S2 t ) , where S i=r (αi )=e (α

i) ∈ GF (qm)

S l=δ1β1l+...δνβν

l , i=l , ..2t

δi=e ji, βi=α

● We can build a set of equations

● Based on this, a BCH or an R-S may be decoded on 4 steps:

1. Compute the syndrome vector.

2. Determine the so-called error-location polynomial.

3. Determine the so-called error-value evaluator.

4. Evaluate error-location numbers ( ) and error values ( ), and perform correction.

● The error-location polynomial is defined as

ji e ji

σ ( x )=(1−β1 x ) ... (1−βν x )=∑l=0

σ l xl , where σ0=1

● We can find the error-location polynomial with the help of the Berlekamp's algorithm, using the syndrome vector.

– It works iteratively, in 2t steps.

– Details can be found in the references.

● Once determined, its roots can be found by substituting the elements of GF(qm) cyclically in σ(x).

– If , is an error-location number.

– The errors are thus located at such positions.

● On the other hand, the error-value evaluator is defined as

σ (αi )=0 α−i=α

qm−i−1

Z0 ( x )=∑l=1

δlβl ∏i=1, i≠l

(1−βi x )

Multiple-error-correction block codes ● It can be shown that the error-value evaluator can be calculated

as a function of known quantities

● After some algebra, the error values are determined as

– Where the denominator is the derivative of σ(x).

● With the error values and the error locations, the error vector is estimated and correction may be performed

Z0 ( x )=S1+(S2+σ1 S1 )x+(S3+σ1 S2+σ2 S1 )x2+

+...+(Sν+σ1 Sν−1+...+σν−1 S1 )xν

δk=−Z0 (βk

−1)σ ' (βk

−1 )

e (x )=∑i=1

δi xji → c (x )=r ( x )− e ( x )

● BCH and R-S codes may be very powerful, but the amount of algebra required is very high.

– The supporting theory is very complex, and designing and analyzing these codes require mastering algebra and geometry over finite-size fields.

– All the operations are to be understood in GF(q) or GF(qm), when corresponding.

● Nonbinary LBC of the kind described are usually employed in sophisticated FEC strategies for specific channels.

– Binary LBC are more usual in ARQ strategies.

● Example BPSK+AWGN: RHam

=246/255, RGol

=1/2, RR-S

=155/255.

Convolutional codes

● A binary convolutional code (CC) is another kind of linear channel code class.

● The encoding can be described in terms of a finite state machine (FSM).

– A CC can eventually produce sequences of infinite length.

– A CC encoder has memory. General structure:

MEMORY: ml bits for l-th input

Forward logic (coded bits)

Backward logic (feedback)

k input streams

n output streams

notmandatory

Systematic output

Convolutional codes

● The memory is organized as a shift register.

– Number of positions for input l: memory ml.

– ml=ν

l is the constraint length of the l-th input/register.

– The register effects step by step delays on the input: recall discrete LTI systems theory.

● A CC encoder produces sequences, not just blocks of data.

– Sequence-based properties vs. block-based properties.

1 2 3 4 ml

l-th input streaminput at instant i

to backwardlogic

to forwardlogic

d i( l ) d i−1

( l ) d i−2(l ) d i−3

(l ) d i−4(l ) d i−ml

Convolutional codes

● Both forward and backward logic is boolean logic.

– Very easy: each operation adds up (XOR) a number of memory positions, from each of the k inputs.

● , is 1 when the p-th register position for the l-th input is added to get the j-th output.

c i( j)=∑

∑q=i

i−ml

g l , q−i( j ) ⋅d q

(l ) j-th output at instant i

inputs from all the k registers at instant i

g l , p( j ) , p=0,. .. , ml

Same structure forbackward logic

Convolutional codes

● Parameters of a CC so far:

– k input streams

– n output streams

– k shift registers with length ml each, l=1,...,k

– νl=m

l is the constraint length of the l-th register

– m=maxl{ν

l} is the memory order of the code

– ν=ν1+...+ν

k is the overall constraint length of the code

● A CC is denoted as (n,k,ν).

– As usual, its rate is R=k/n, where k and n take normally small values.

Convolutional codes

● The backward / forward logic may be specified in the form of generator sequences.

– Theses sequences are the impulse responses of each output j wrt each input l.

● Observe that:

– connects the l-th input directly to the j-th output

– just delays the l-th input to the j-th output q time steps.

g l( j )=(g l , 0

( j ) , ... , g l ,ml

( j) )

g l( j )=(1,0, ... ,0)

g l( j )=(0,. .. ,1(q th) , ... ,0 )

Convolutional codes

● Given the presence of the shift register, the generator sequences are better denoted as generator polynomials

● We can thus write, for example

g l( j)=(g l ,0

( j) ,... , g l , ml

( j ) )≡ g l( j)(D)=∑

g l , q( j)⋅D q

gl( j )=(1,0, ... ,0) ≡ g l

( j )(D)=1

g l( j )=(0,. .. ,1(qth) , ... ,0 )≡ gl

( j)(D)=Dq

gl( j )=(1,1,0, ... ,0) ≡ gl

( j )(D)=1+D

Convolutional codes

● As all operations involved are linear, a binary CC is linear and the sequences produced constitute CC codewords.

● A feedforward CC (without backward logic - feedback) can be denoted in matrix from as

G(D)=(g1(1)(D) g1

(2)(D) ⋯ g1

(n )(D)

g2(1)(D) g2

(2)(D) ⋯ g2

(n )(D)

⋮ ⋮ ⋱ ⋮

gk(1)(D) gk

(2)(D) ⋯ gk(n )(D)

Convolutional codes

● If each input has a feedback logic given as

the code is denoted as

g l(0)(D)=∑q=0

ml g l , q(0)⋅Dq

G(D)=(g1(1 )(D)

g1(0)(D)

g1(2)(D)

g1(0)(D)

⋯g0(n)(D)

g1(0)(D)

g2(1 )(D)

g2(0)(D)

g2(2)(D)

g2(0)(D)

⋯g2(n)(D)

g2(0)(D)

⋮ ⋮ ⋱ ⋮

gk(1 )(D)

gk(0)(D)

gk(2)(D)

gk(0)(D)

⋯gk(n)(D)

gk(0)(D)

Convolutional codes

● We can generalize the concept of parity-check matrix H(D).

– An (n,k,ν) CC is fully specified by G(D) or H(D).

● Based on the matrix description, there are a good deal linear tools for design, analysis and evaluation of a given CC.

● A regular CC can be described as a (canonical) all-feedforward CC and through an equivalent feedback (recursive) CC.

– Note that a recursive CC is related to an IIR filter.

● Even though k and n could be very small, a CC has a very rich algebraic structure.

– This is closely related to the constraint length of the CC.

– Each output bit is related to the present and past inputs via powerful algebraic methods.

Convolutional codes

● Given G(D), a CC can be classified as:

– Systematic and feedforward (NSC).

– Systematic and recursive (RSC).

– Non-systematic and feedforward.

– Non-systematic and recursive.

● RSC is a popular class of CC, because it provides an infinite output for a finite-weight input (IIR behavior).

● Each NSC can be converted straightforwardly to a RSC with similar error correcting properties.

● CC encoders are easy to implement with standard hardware: shift registers + combinational logic.

Convolutional codes● As with the case of nonbinary LBC, we may use the

polynomial representation to perform coding & decoding.

– But now we have encoding with memory, spanning over theoretically infinite length sequences → not practical.

c ( D )=b ( D )⋅G ( D )

b ( D )=(b(1) ( D ) ... b(k) ( D ) ) ; b(i)( D )=∑j=0

b j(i) D j

b j( i) is the i−th input bit stream

c ( D )=(c(1) ( D ) ...c(n) ( D ) ) ; c(l) ( D )=∑h=0

ch(l) Dh

ch(l) is the l−th output bit stream

Convolutional codes

● We do not need to look into the algebraic details of G(D) and H(D) to study:

– Coding

– Decoding

– Error correcting capabilities

● A CC encoder is a FSM!

The ν memory positionsstore a content (among 2ν possible ones) at instant i-1

Coder is said to be atstate s(i-1)

The ν memory positionsstore a new content

at instant iCoder is said to be at

state s(i)

k input bits determinethe shifting ofthe registers

And we get n relatedoutput bits

Convolutional codes

● The finite-state behavior of the CC can be captured by the concept of trellis.

– For any starting state, we have 2k possible edges leading to a corresponding set of ending states.

ss=s(i-1)

s=1,...,2ν

se=s(i)

e=1,...,2ν

input bi=(b

i,1...b

output ci=(c

i,1...c

Convolutional codes

● The trellis illustrates the encoding process in 2 axis:

– X-axis: time / Y-axis: states

● Example for a (2,1,3) CC:

– For a finite-size input data sequence, a CC can be forced to finish at a known state (often 0) by adding terminating (dummy) bits.

– Note that one section (e.g. i-1 → i) fully specifies the CC.

output 00

output 01s2

i-1 i i+1

input 0 input 1

Convolutional codes

● The trellis illustrates the encoding process in 2 axis:

– X-axis: time / Y-axis: states

● Example for a (2,1,3) CC:

– For a finite-size input data sequence, a CC can be forced to finish at a known state (often 0) by adding terminating (dummy) bits.

– Note that one section (e.g. i-1 → i) fully specifies the CC.

output 00

output 01s2

i-1 i i+1

input 0 input 1

Memory:Memory:same input,same input,

different outputsdifferent outputs

Convolutional codes

● The trellis description allows us

– To build the encoder

– To build the decoder

– To get the properties of the code

● The encoder:

H(D)↔G(D)

RegistersCombinational

logick n

ss=s(i-1)

s=1,...,2νs

e=s(i)

e=1,...,2νinput bi=(b

i,1...b

output ci=(c

i,1...c

Convolutional codes

● The decoder is far more complicated

– Long sequences

– Memory: dependence with past states

● In fact, CC were already well known before there existed a practical good method to decode them: the Viterbi algorithm.

– It is a Maximum Likelihood Sequence Estimation (MLSE) algorithm with many applications.

● Problem: for a length N>>n sequence at the receiver side

– There are 2ν·2N·k/n paths through the trellis to match with the received data.

– Even if the coder starting state is known (often 0), there are still 2N·k/n paths to walk through in a brute force approach.

Convolutional codes

● Viterbi algorithm setup.

input bi → output c

i(s(i-1),b

start s(i-1) → end s(i)(s(i-1),bi)

received data ri

Key facts:● The encoding corresponds to a Markov

chain model: P(s(i))=P(s(i)|s(i-1))·P(s(i-1)).

● Total likelihood P(r|b) can be factorized as a product of probabilities.

● Given , P(ri|s(i),s(i-1)) depends

only on the channel kind (AWGN, BSC...).

● Transition from s(i-1) to s(i) (linked in the trellis) depends on the probability of b

P(s(i)|s(i-1))=2-k if the source is iid.

● P(s(i)|s(i-1))=0 if they are not linked in the trellis (finite state machine: deterministic).

s(i−1)→bi

Convolutional codes

● The total likelihood can be recursively calculated as:

● In the BSC(p), the observation (branch) metric would be related to:

● Maximum likelihood (ML) criterion:

P (r i|s(i) ,s(i−1))=P (r i|c i )→w (r i+ci )=d H (ri ,ci )

P (r∣b)=∏i=1

P (r i∣s(i) ,s(i−1))⋅P (s(i)∣s(i−1))⋅P (s(i−1))

b=arg {maxb

[P (r∣b ) ]}

Convolutional codes

● We know that the brute force approach to ML criterion is at least O(2N·k/n).

● The Viterbi algorithm works recursively from 1 to N/n on the basis that

– Many paths can be pruned out (transition probability=0).

– During forward recursion, we only keep the paths with highest probability: the path probability goes easily to 0 from the moment a term metric ⨯ transition probability is very small.

– When recursion reaches i=N/n, the surviving path guarantees the ML criterion (optimal for ML sequence estimation!).

● The Viterbi algorithm complexity goes down to O(N·22ν).

Convolutional codes

● The algorithm recursive rule is

● {Vj

(i)} stores the most probable state sequence wrt observation r

V j(0)=P (s(0)=s j ) ;

V j(i)=P (r i∣s

(i)=s j ,s(i−1)

=smax )⋅ maxs(i−1)

= smax

{P (s(i)=s j∣s(i−1)

=sl )⋅V l( i−1)}

i-1 i i+1

Convolutional codes

● {Vj

V j(0)=P (s(0)=s j ) ;

V j(i)=P (r i∣s

(i)=s j ,s(i−1)

= smax

{P (s(i)=s j∣s(i−1)

=sl )⋅V l( i−1)}

Probability of themost probable state sequence

corresponding to the i-1previous observations

i-1 i i+1

Convolutional codes

● {Vj

V j(0)=P (s(0)=s j ) ;

V j(i)=P (r i∣s

(i)=s j ,s(i−1)

= smax

{P (s(i)=s j∣s(i−1)

=sl )⋅V l( i−1)}

Probability of themost probable state sequence

corresponding to the i-1previous observations

i-1 i i+1

Note that we maybetter work with logs:products ↔ additions

Criterion remains the same

Convolutional codes

● Note that we have considered the algorithm when the demodulator yields hard outputs

– ri is a vector of n estimated bits (BSC(p) equivalent channel).

● In AWGN, we can do better to decode a CC

– We can provide soft (probabilistic) estimations for the observation metric.

– For an iid source, we can easily get an observation transition metric based on the probability of each b

i,l=0,1, l=1,...,k,

associated to a possible transition.

– There is a gain of around 2 dB in Eb/N

– LBC decoders can also accept soft inputs (non syndrome-based decoders).

– We will examine an example of soft decoding of CC in the lab.

Convolutional codes

● We are now familiar with the encoder and the decoder

– Encoder: FSM (registers, combinational logic).

– Decoder: Viterbi algorithm (for practical reasons, suboptimal adaptations are usually employed).

● But what about performance?

● First...

– CC are mainly intended for FEC, not for ARQ schemes.

– In a long sequence (=CC codeword), the probability of having at least one error is very high...

– And... are we going to retransmit the whole sequence?

Convolutional codes

● Given that we truncate the sequence to N bits and CC is linear

– We may analyze the system as an equivalent (N,N·k/n) LBC.

– But... equivalent matrices G and H would not be practical.

● Remember FSM: we can locate error loops in the trellis.

i i+1 i+2 i+3

Convolutional codes

● The same error loop may occur irrespective of s(i-1) and b.

i i+1 i+2 i+3

Convolutional codes

● Examining the minimal length loops and taking into account this uniform error property we can get d

min of a CC.

– For a CC forced to end at 0 state for a finite input data sequence, d

min is called d

● We can draw a lot of information by building an encoder state diagram: error loops, codeword weight spectrum...

Diagram of a (2,1,3) CC, from Lin & Costello (2004).

Convolutional codes

● With a fairly amount of algebra, related to FSM, modified encoder state diagrams and so on, it is possible to get an upper bound for optimal MLSE decoding.

● For the BSC(p), the bound can be calculated as

P b⩽∑d

B d⋅erfc(√d RE b

Bd is the total number of nonzero information bits associated with

CC codewords of weight d, divided by the number of information bits k per unit time... A lot of algebra behind...

BPSK inAWGN

Pb⩽∑d

Bd⋅(2√ p⋅(1−p ))d

Convolutional codes

● There are easier, suboptimal ways to decode a CC, and performance will vary accordingly.

● A CC may be punctured to match other rates lower than R=k/n, but the resulting equivalent CC is clearly weaker.

– Performance-rate trade-off.

– Puncturing is a very usual tool that provides flexibility to the usage of CC's in practice.

Nativeconvolutional

encoderk n

Puncturingalgorithm

(prune bits) n'<n

Convolutional codes● Examples with BPSK+AWGN using ML bounds.

Turbo codes

● Canonically, turbo codes (TC) are parallel concatenated convolutional codes (PCCC).

● Coding concatenation has been known and employed for decades, but TC added a joint efficient decoding.

– Example of concatenated coding with independent decoding is the use of ARQ + FEC hybrid strategies (CRC + CC).

k input streams n=n1+n

2 output streams

Rate R=k/(n1+n

b c=c1∪c2

Turbo codes

● Canonically, turbo codes (TC) are parallel concatenated convolutional codes (PCCC).

● Coding concatenation has been known and employed for decades, but TC added a joint efficient decoding.

– Example of concatenated coding with independent decoding is the use of ARQ + FEC hybrid strategies (CRC + CC).

2 output streams

We will see this isa key element...

Rate R=k/(n1+n

b c=c1∪c2

Turbo codes

● We have seen that standard CC decoding with Viterbi algorithm relied on MLSE criterion.

– This is optimal when binary data at CC input is iid.

● For CC, we also have decoders that provide probabilistic (soft) outputs.

– They convert a priori soft values + channel output soft estimations into updated a posteriori soft values.

– They are optimal from the maximum a posteriori (MAP) criterion point of view.

– They are called soft input-soft output (SISO) decoders.

Turbo codes

● What's in a SISO?

SISO(for a CC)

0 10 1

P (bi=b )=12

P (bi=b∣r )

● Note that the SISO works on a bit by bit basis, but produces a sequence of APP's.

Probability density function of bi

Turbo codes

SISO(for a CC)

0 10 1

P (bi=b )=12

P (bi=b∣r )

Soft demodulated valuesfrom channel

Turbo codes

SISO(for a CC)

0 10 1

P (bi=b )=12

P (bi=b∣r )

A priori probabilities(APR)

Turbo codes

SISO(for a CC)

0 10 1

P (bi=b )=12

P (bi=b∣r )

A priori probabilities(APR)

A posteriori probabilities(APP) updated withchannel information

Turbo codes

● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm.

– BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs).

– BCJR works on a trellis: recall transition metrics, transition probabilities and so on.

– Assume the block length is N: trellis starts at , ends at .

αi ( j )=P (s( i)=s j ,r1,⋯ ,r i )

βi ( j )=P (r i+1 ,⋯,rN∣s(i)=s j )

γi ( j , k )=P (r i , s(i)=s j∣s(i−1)=sk )

s(0) s(N )

Turbo codes

αi ( j )=P (s( i)=s j ,r1,⋯ ,r i )

βi ( j )=P (r i+1 ,⋯,rN∣s(i)=s j )

FORWARD term

s(0) s(N )

Turbo codes

αi ( j )=P (s( i)=s j ,r1,⋯ ,r i )

βi ( j )=P (r i+1 ,⋯,rN∣s(i)=s j )

FORWARD term

BACKWARD term

s(0) s(N )

Turbo codes

αi ( j )=P (s( i)=s j ,r1,⋯ ,r i )

βi ( j )=P (r i+1 ,⋯,rN∣s(i)=s j )

FORWARD term

BACKWARD term

TRANSITION

s(0) s(N )

Turbo codes

αi ( j )=P (s( i)=s j ,r1,⋯ ,r i )

βi ( j )=P (r i+1 ,⋯,rN∣s(i)=s j )

FORWARD term

BACKWARD term

TRANSITION

s(0) s(N )

Remember,n components

for an (n,k,ν) CC

Turbo codes

● BCJR algorithm in action:

– Forward step i=1,...,N:

– Backward step i=N-1,...,0:

– Compute the joint probability sequence i=1,...,N:

α0 ( j )=P (s(0)=s j ) ; αi ( j )=∑k=1

αi−1 ( k )⋅γi ( k , j )

βN ( j )=P (s(N )=s j ) ; βi ( j )=∑

βi+1 (k )⋅γi+1 ( j ,k )

P (s(i−1)=s j ,s(i)=sk ,r )=βi (k )⋅γi ( j ,k )⋅αi−1 ( j )

Turbo codes

● Finally, the APP's can be calculated as:

● Decision criterion based on these APP's:

P (bi=b∣r )=1

p (r )⋅ ∑

s( i−1)→

b i=bs(i )

P (s(i−1)=s j ,s(i )=sk ,r )

log( P (bi=1∣r )P (bi=0∣r ) )=log(

∑s(i−1) →

b i=1s(i )

P (s(i−1)=s j ,s(i)=sk ,r )

∑s(i−1)

→b i=0

s(i )P (s(i−1)

=s j ,s(i)=sk , r ) ) >bi=10

Turbo codes

● Finally, the APP's can be calculated as:

● Decision criterion based on these APP's:

P (bi=b∣r )=1

p (r )⋅ ∑

s( i−1)→

b i=bs(i )

P (s(i−1)=s j ,s(i )=sk ,r )

log( P (bi=1∣r )P (bi=0∣r ) )=log(

∑s(i−1) →

b i=1s(i )

P (s(i−1)=s j ,s(i)=sk ,r )

∑s(i−1)

→b i=0

s(i )P (s(i−1)

=s j ,s(i)=sk , r ) ) >bi=10

Its modulus is thereliability of the

decision

Turbo codes

● How do we get γi(j,l)?

● This probability takes into account

– The restrictions of the trellis (CC).

– The estimations from the channel.

γi ( j , l )=P (r i ,s(i )=s j∣s(i−1)

=sl )=

= p (ri∣s(i )=s j , s(i−1 )

=sl )⋅P (s(i)=s j∣s(i−1 )

Turbo codes

γi ( j , l )=P (r i ,s(i )=s j∣s(i−1)

=sl )=

= p (ri∣s(i )=s j , s(i−1 )

=sl )⋅P (s(i)=s j∣s(i−1 )

=0 if transition is not possible=1/2k if transition is possible

(binary trellis, k inputs)

Turbo codes

γi ( j , l )=P (r i ,s(i )=s j∣s(i−1)

=sl )=

= p (ri∣s(i )=s j , s(i−1 )

=sl )⋅P (s(i)=s j∣s(i−1 )

=0 if transition is not possible=1/2k if transition is possible

(binary trellis, k inputs) in AWGN for unmodulated c

(2πσ2 )n /2⋅e

−∑m=1

( ri ,m−ci ,m )2

Turbo codes

● Idea: what about feeding APP values as APR values for other decoder whose coder had the same inputs?

SISO(for CC

0 10 1

P (bi=b∣r1 )P (bi=b∣r 2)

From CC1 SISO

Turbo codes

● Idea: what about feeding APP values as APR values for other decoder whose coder had the same inputs?

SISO(for CC

0 10 1

P (bi=b∣r1 )P (bi=b∣r 2)

From CC1 SISO

This will happenunder someconditions

Turbo codes

● APP's from first SISO used as APR's for second SISO increase updated APP's reliability iff

– APR's are uncorrelated wrt channel estimations for second decoder.

– This is achieved by permuting input data for each encoder.

2 output streams

Rate R=k/(n1+n

b c=c1∪c2

Turbo codes

● APP's from first SISO used as APR's for second SISO increase updated APP's reliability iff

– APR's are uncorrelated wrt channel estimations for second decoder.

– This is achieved by permuting input data for each encoder.

2 output streams

INTERLEAVER(permutor)

Rate R=k/(n1+n

b c=c1∪c2

Turbo codes

● The interleaver preserves the data (b), but changes its position within the second stream (d).

– Note that this compels the TC to work with blocks of N=size(Π) bits.

– The decoder has to know the specific interleaver used at the encoder.

b1 b2 b3 b4 bN

d π (2) d π (N ) d π (3 ) d π (1 ) d π (4)

d π (i )=b i

Turbo codes

● The mentioned process is applied iteratively (l=1,...).

– Iterative decoder → this may be a drawback, since it adds latency (delay).

– Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!).

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Turbo codes

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Turbo codes

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Turbo codes

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Turbo codes

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Turbo codes

SISO 1 SISO 2

Π−1

APP1(l) APR

APP2(l)

APR1(l+1)

from channel

Initial APR1(l=0)

is taken withP(b

i=b)=1/2

Turbo codes

● When the interleaver is adequately chosen and the CC's employed are RSC, the typical BER behavior is

– Note the two distinct zones: waterfall region / error floor.

Turbo codes

● The location of the waterfall region can be analyzed by the so-called density evolution method

– Based on the exchange of mutual information between SISO's.

● The error floor can be lower bounded by the minimum Hamming distance of the TC

– Contrary to CC's, TC relies on reducing multiplicities rather than just trying to increase minimum distance.

P bfloor>

wmin⋅M min

N⋅erfc(√d min R

Turbo codes

P bfloor>

wmin⋅M min

N⋅erfc(√d min R

Hamming weightof the error

with minimumdistance

Turbo codes

P bfloor>

wmin⋅M min

N⋅erfc(√d min R

Errormultiplicity(low value!!)

Turbo codes

P bfloor>

wmin⋅M min

N⋅erfc(√d min R

Errormultiplicity(low value!!)

Interleaver gain(only if recursive

CC's!!)

Turbo codes

● Examples of 3G TC. Note that TC's are intended for FEC...

Low Density Parity Check codes

● LDPC codes are just another kind of channel codes derived from less complex ones.

– While TC's were initially an extension of CC systems, LDPC codes are an extension of the concept of binary LBC, but they are not exactly our known LBC.

● Formally, an LDPC code is an LBC whose parity check matrix is large and sparse.

– Almost all matrix elements are 0!!!!!!!!!!

– Very often, the LDPC parity check matrices are randomly generated, subject to some constraints on sparsity...

– Recall that LBC relied on extreme powerful algebra related to carefully and well chosen matrix structures.

● Formally, a (ρ,γ)-regular LDPC code is defined as the null space of a parity check matrix J⨯n H that meets these constraints:

a) Each row contains ρ 1's.

b) Each column contains γ 1's.

c) λ, the number of 1's in common between any two columns, is 0 or 1.

d) ρ and γ are small compared with n and J.

● These properties give name to this class of codes: their matrices have a low density of 1's.

● The density r of H is defined as r=ρ/n=γ/J.

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

]● Example of a (4,3)-regular LPDC parity check matrix

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

15⨯20

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

This H defines a(20,7) LBC!!!15⨯20

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

This H defines a(20,7) LBC!!!

r=4/20=3/15=0.2

15⨯20

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

r=4/20=3/15=0.2

Sparse!

15⨯20

H=[11110 000 00 00 000 00 00 00 000 111100 00 000 00 00 00 000 00 001111 000 00 00 00 000 00 00 000 011110 00 00 000 00 00 000 00 00 0111110 0010 00 100 0100 00 00 001 000 100 010 000 00 100 00 0100 010 00 000 100 010 00 0010 00 00 0100 010 001 00 000 00 010 0010 00 100 0110 000 100 00 0100 00 010 001 000 010 001 000 010 00 00 0100 00 100 0010 00 001 00 0010 00 010 000 100 100 00 000 100 001 000 010 00 01

r=4/20=3/15=0.2

Sparse!

λ=0,1

15⨯20

● Note that the J rows of H are not necessarily linearly independent over GF(2).

– To determine the dimension k of the code, it is mandatory to find the row rank of H = n-k < J.

– That's the reason why in the previous example H defined a (20,7) LBC instead of a (20,5) LBC as could be expected!

● The construction of large H for LDPC with high rates and good properties is a complex subject.

– Some methods relay on smaller Hi used as building blocks,

plus random permutations or combinatorial manipulations; resulting matrices with bad properties are discarded.

– Other methods relay on finite geometries and lot of algebra.

● LDPC codes yield performances equal or even better than TC's, but without the problem of their relatively high error floor.

– Both LDPC codes and TC's are capacity approaching codes.

● As in the case of TC, their interest is in part related to the fact that

– The encoding can be easily done (even when H or G are large, the low density of 1's reduces complexity of the encoder).

– At the decoder side, there are powerful algorithms that can take full advantage of the properties of the LDPC code.

● There are several algorithms to decode LDPC codes.

– Hard decoding.

– Soft decoding.

– Mixed approaches.

● We are going to examine two important instances thereof:

– Majority-logic (MLG) decoding; hard decoding, the simplest one (lowest complexity).

– Sum-product algorithm (SPA); soft decoding, best error performance (but high complexity!).

● Key concepts: Tanner graphs & belief propagation.

● MLG decoding: hard decoding; r=c+e → received word.

– The simplest instance of MLG decoding is the decoding of a repetition code by the rule “choose 0 if 0's are dominant, 1 if otherwise”.

● Given a (ρ,γ)-regular LDPC code, for every bit position i=1,...,n, there is a set of γ rows

that have a 1 in position i, and do not have any other common 1 position among them...

Ai={h1(i) ,⋯ ,hγ

● We can form the set of syndrome equations

● Si gives a set of γ checksums orthogonal on e

● ei is decoded as 1 if the majority of the checksums give 1;

0 in the opposite case.

● Repeating this for all i, we estimate ê, and ĉ=r+ê.

– Correct decoding of ei is guaranteed if there are less than

γ/2 errors in e.

S i={si=r⋅h j(i )T=e⋅h j

(i)T , h j( i)∈Ai , i=1,⋯, γ}

● Tanner graphs. Example for a (7,3) LBC.

● It is a bipartite graph with interesting properties for decoding.

– A variable node is connected to a check node iff the corresponding code bit is checked by the corresponding parity sum equation.

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

Variable nodes orcode-bit vertices

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

Check nodes orcheck-sum vertices

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

Check nodes orcheck-sum vertices

The absence of shortloops is necessary for

iterative decoding

● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA).

● SPA is performed by belief propagation (which is an instance of a message passing algorithm).

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

“Messages” (soft values)are passed to and from

related variable andcheck nodes

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

This process, appliediteratively and under

some rules, yields

P (ci∣λ )

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

some rules, yields

P (ci∣λ )λ softvalues

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

some rules, yields

P (ci∣λ )λ softvalues

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

some rules, yields

P (ci∣λ )

N(c5): check nodes neighbors of variable node c

λ softvalues

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

some rules, yields

P (ci∣λ )

λ softvalues

+ + + ++++

c1 c2 c3 c4 c5 c6 c7

s1 s2 s3 s4 s5 s6 s7

some rules, yields

P (ci∣λ )

λ softvalues

● If we get P(ci | λ), we have an estimation of the codeword sent ĉ.

● The decoding aims at calculating this through the marginalization

● Brute-force approach for LDPC is impractical, hence the iterative solution through SPA. Messages interchanged at step l:

P (ci∣λ )= ∑c ' :c ' i=c i

P (c '∣λ )

μc i→ s j

(l )(ci=c )=αi , j

(l)⋅P (ci=c∣λ i )⋅ ∏

sk∈N (c i )sk≠s j

μ sk→c i

(l−1)(c i=c )

μ s j→c i

(l )(ci=c )= ∑

c ∖ ck∈N (s j )c k≠c i

P (s j=0∣ci=c ,c )⋅ ∏c ' k∈N ( s j )

μck→ s j

(l)(c ' k=ck )

P (ci∣λ )= ∑c ' :c ' i=c i

P (c '∣λ )

μc i→ s j

(l )(ci=c )=αi , j

(l)⋅P (ci=c∣λ i )⋅ ∏

μ sk→c i

(l−1)(c i=c )

μ s j→c i

(l )(ci=c )= ∑

P (s j=0∣ci=c ,c )⋅ ∏c ' k∈N ( s j )

μck→ s j

(l)(c ' k=ck )

From variable nodeto check node

P (ci∣λ )= ∑c ' :c ' i=c i

P (c '∣λ )

μc i→ s j

(l )(ci=c )=αi , j

(l)⋅P (ci=c∣λ i )⋅ ∏

μ sk→c i

(l−1)(c i=c )

μ s j→c i

(l )(ci=c )= ∑

P (s j=0∣ci=c ,c )⋅ ∏c ' k∈N ( s j )

μck→ s j

(l)(c ' k=ck )

From variable nodeto check node

From check nodeto variable node

● Note that:

– is a normalization constant.

– plugs into the SPA the values from the

channel → it is the APR info.

● : APP value.

● Based on the final values of P(ci | λ), a candidate ĉ is chosen

and ĉ·HT is tested. If 0, the information word is decoded.

αi , j(l )

P (ci=c∣λ i )

P(l) (c i=c∣λ )=βi(l)⋅P (c i=c∣λi )⋅∏

s j∈N (c i)

μs j→ ci

(l) (c i=c )

● Note that:

– is a normalization constant.

– plugs into the SPA the values from the

channel → it is the APR info.

● : APP value.

● Based on the final values of P(ci | λ), a candidate ĉ is chosen

and ĉ·HT is tested. If 0, the information word is decoded.

αi , j(l )

P (ci=c∣λ i )

P(l) (c i=c∣λ )=βi(l)⋅P (c i=c∣λi )⋅∏

s j∈N (c i)

μs j→ ci

(l) (c i=c )

Normalization

● LDPC BER performance examples (DVBS2 standard).

Short n=16200

Long n=64800

Coded modulations

● We have considered up to this point channel coding and decoding isolated from the modulation process.

– Codewords feed any kind of modulator.

– Symbols go through a channel (medium).

– The info recovered from received modulated symbols is fed to the suitable channel decoder

● As hard decisions.● As soft values (probabilistic estimations).

– The abstractions of BSC(p) (hard demodulation) or soft values from AWGN ( ⋉ exp[-|r

j|2/(2σ2)] ) -and the like for

other cases- are enough for such an approach.● Note that there are other important channel kinds not

considered so far.

Coded modulations

● Coded modulations are systems where channel coding and modulation are treated as a whole.

– Joint coding/modulation.

– Joint decoding/demodulation.

● This offers potential advantages (recall the improvements made when the demodulator outputs more elaborated information -soft values vs. hard decisions).

– We combine gains in BER with spectral efficiency!

● As a drawback, the systems become more complex.

– More difficult to design and analyze.

Coded modulations

● TCM (trellis coded modulation).

– Normally, it combines a CC encoder and the modulation symbol mapper.

output mk

output mj

i-1 i i+1

Coded modulations

● If the modulation symbol mapper is well matched to the CC trellis, and the decoder is accordingly designed to take advantage of it,

– TCM provides high spectral efficiency.

– TCM can be robust in AWGN channels, and against fading and multipath effects.

● In the 80's, TCM become the standard for telephone line data modems.

– No other system could provide better performance over the twisted pair cable before the introduction of DMT and ADSL.

● However, the flexibility of providing separated channel coding and modulation subsystems is still preferred nowadays.

– Under the concept of Adaptive Coding & Modulation (ACM).

Coded modulations

● Other possibility of coded modulation, evolved from TCM and from the concatenated coding & iterative decoding framework is Bit-Interleaved Coded Modulation (BICM).

– What about if we use an interleaver between the channel coder (normally a CC) and the modulation symbol mapper?

– A soft demodulator can also accept APR values and update as APP's its soft outputs in an iterative process!

Softdemapper

Channel corruptedoutputs

APR values(interleaved

from CC SISO)

APP values(to interleaverand CC SISO)

Coded modulations

● As TCM, BICM has special good behavior (even better!)

– In channels where spectral efficiency is required.

– In dispersive channels (multipath, fading).

– Iterative decoding yields a steep waterfall region.

– Being a serial concatenated system, the error floor is very low (contrary to the parallel concatenated systems).

● BICM has already found applications in standards such as DVB-T2.

● The drawback is the higher latency and complexity of the decoding.

Coded modulations

● Examples of BICM.

References

● S. Lin, D. Costello, ERROR CONTROL CODING, Prentice Hall, 2004.

● S. B. Wicker, ERROR CONTROL SYSTEMS FOR DIGITAL COMMUNICATION AND STORAGE, Prentice Hall, 1995.

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