LDPC — low density parity check code

5G AN uses LDPC for channel coding on the traffic channel. LDPC corrects channel errors by maintaining parity bits for a selection of the data bits. The figure below shows the relation between the data and parity bits.

LDPC parity bit mapping (Credit: Wikipedia)

Most data bits are backed by multiple parity bits. When a parity check failure is detected, information from the multiple parity bits can be used to retrieve the original data bit.

Block code basics

Consider a parity based code that operates on a block on n bits. Out of the n bits in the block, k bits carry data and r bits carry parity. Thus

n = k + r

Now consider a code with a 7-bit block length with 3 parity bits per block. This means:

n = 7, k = 4, r = 3

This code with be considered a rate 4/7 code as there are 4 bits of data for every 7 bits of transmission.

If the 7 bits of the block are labeled c1 to c7. Here c1, c2, c3 and c4 are data bits. c5, c6 and c7 are parity bits. We can define the parity constraints as shown below. The constraints are defined with ⊕ modulo 2 additions (essentially an exclusive-or operation).

c1 ⊕ c2 ⊕ c3 ⊕ c5 = 0              e1
c1 ⊕ c2 ⊕ c4 ⊕ c6 = 0 e2
c1 ⊕ c3 ⊕ c4 ⊕ c7 = 0 e3

The above equation can also be represented in a matrix form. Each equation maps to a row in the matrix. 0 terms have been introduced for columns that did not exist in the original equation.

c1  c2  c3  c4  c5  c6  c7
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1

The equations described above are typically represented as a Tanner graph. The data and parity bits are shown on the top. The equations placing constraints on the data and parity bits are represented below with connecting lines to the bits covered by each constraint.

Tanner graph

Encoding

Let’s assume that the user data to send is 1001.

c1  c2  c3  c4
1 0 0 1

We apply this to our parity equations.

1 ⊕ 0 ⊕ 0 ⊕ c5 = 0
1 ⊕ 0 ⊕ 1 ⊕ c6 = 0
1 ⊕ 0 ⊕ 1 ⊕ c7 = 0

The parity bits c5, c6 and c7 must be assigned values so that the above equations are satisfied. The equations will be satisfied when even number of bits are set in each equation.

1 ⊕ 0 ⊕ 0 ⊕ 1 = 0
1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
1 ⊕ 0 ⊕ 1 ⊕ 0 = 0

This results in the following seven bits.

c1  c2  c3  c4  c5  c6  c7
1 0 0 1 1 0 0

The Tanner graph for the data bits and the constraint nodes is shown below.

Tanner graph example

Decoding

Bit flipping algorithm

Let’s assume that c3 bit is received in error at the receiver (orange node in the top row). This error results in the two parity check equation failures (orange nodes in the bottom row).

Tanner graph with the impact of a single bit error

Here we can apply the simplest LDPC decoding algorithm:

If the node receives more negative votes than positive votes, it decides to flip itself.

Here the c3 bit is getting two negative votes from the parity check equations. There are no positive votes, so c3 flips itself. Flipping the bit results in two positive votes to c3. At this point all parity checks are passing, thus completing the decode.

Tanner graph after correcting the bits

Message passing algorithm

Bit flipping algorithm is simple but does not have great error detection properties. Message passing algorithms improve the decoder performance by iteratively exchanging estimates between the data bit nodes and the parity check nodes. The scheme is described in the following presentation.

The iterative nature of LDPC is visually illustrated by the following short video. The video shows rate 1/2 code iterations for the transmission of an image. Each iteration recovers more of the image. The iterations proceed until the full image is recovered.

Visualize iterative decoding

Belief propagation decoder

This scheme used belief propagation techniques that were developed for iterating in neural networks. Belief propagation algorithms offer the best LDPC decode performance. They are introduced in the following videos.

Introduction to belief propagation
Overview of channel coding techniques with focus on belief propagation

Rate matching and HARQ support with LDPC

5G NR has unique requirements for supporting incremental redundancy via HARQ. In addition to this, rate matching needs to be supported to dynamically adjust the coding to the allocated resources.

3GPP has standardized on QC-LDPC (Quasi Cyclic LDPC) for HARQ and rate matching support. Details about the scheme are available in the following patent application.

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