Low density parity check (LDPC) codes were invented by Gallager in early 1960’s [8]. The name comes from the characteristic of their parity-check matrix which contains only sparse 1’s in comparison to the amount of 0’s. Their main advantage is that they provide a performance which is very close to the capacity for a lot of different channels and linear time complex algorithms for decoding.
LDPC codes are linear codes obtained from sparse bipartite graphs. Suppose that Gis a factor graph with n upper node (bit nodes) and r lower node (check nodes). The graph gives rise to a linear code of block length n and dimension at least n− r in the following way: The n coordinates of the codeword are associated with the n bit nodes.
The codeword are those vectors (X0, ..., Xn−1) such that for all check nodes fc0, ..., fcr−1 the sum of the neighboring positions among the message nodes is zero; see Fig. 2.7 for an example.
LDP C Bit No de
LDP C Che c k N o de X0
C0
f fC1
C2
f
X1 X2 X3
X0+X3=0 X0+ X1+X2=0 X1+ X2+X3=0
Figure 2.7: The factor graph representation of a LDPC code with n = 4 and r = 3.
The parity check matrix H is a binary r × n matrix in which the entry (i, j) is 1 if and only if the ith check node is connected to the jth bit node in the graph. Then the LDPC code defined by the graph is the set of vectors X = (X0, ..., Xn−1) such that
HTX = 0.
A general class of decoding algorithms for LDPC codes is called message passing algorithms, and which is iterative decoding algorithms. The reason for their name is that at each round of the algorithms messages are passed from bit nodes to check nodes, and from check nodes back to bit nodes. The messages from bit nodes to check nodes are computed based on the observed value of the bit node and some of the messages passed from the neighboring check nodes to that bit node. An important aspect is that the message that is sent from a bit node X to a check node f must not take into account the message sent in the previous round from f to X. The same is true for messages passed from check nodes to bit nodes.
The details of the LDPC code specification are given in Appendix A.3. The codeword length used in our simulation is equal to 1248 bits, with code rate = 1/2.
Chapter 3
Joint Channel Estimation and
Decoding in SISO-OFDM Systems
Transmitting a radio signal over a multi-path fading channel, the received signal will encounter unknown amplitude and phase distortions. In order to coherently detect the received signals, accurate channel estimation is essential. In this chapter, we introduce a channel estimation method used in SISO-OFDM systems. At the beginning of the transmission, the preamble is first transmitted, and then is the data symbols transmis-sion. At the preamble, the pilot tones are inserted into all of the sub-carriers. But at the data symbols, the pilot tones are inserted into only some of the sub-carriers.
Firstly, the model based channel estimation is introduced by the combination of the LS estimation and the regression function interpolator at the preamble. In order to reduce the complexity of the channel estimation, the piecewise polynomial model based channel estimation is proposed, and the Hermite curve can be applied to smooth the disconnection caused by the piecewise scheme. Since the channel response changes rapidly, to re-estimate the channel response at the present OFDM symbol is needed.
After estimating the initial channel, the data symbols can be detected and decoded by the current channel estimate, and then, the channel can be re-estimated by the information computed from the LDPC decoder. The joint channel estimation and decoding algorithm is obtained.
3.1 Model Based Channel Estimation
As in [9], the channel response Hm is viewed as a sampled version of a continuous complex fading process. The receiver can model the true sampled fading process by a regression function
F[m] =
MO−1 k=0
akmk=qTma = Hm+ g[m], (3.1) where g[m] represents the modelling error and MO is the (polynomial) model order.
The ML estimate of the coefficients a is the solution to mina where the error between the LS estimation and the regression function is minimized, and ML is the number of frequency-domain units within our modeling range and will be referred to as the modeling block length.
Since for QPSK|Xm|2 is constant, the above optimization problem can be simplified to
Taking the derivative of the argument on the right hand side of (3.3) with respect toa, we obtain
The (regression) model-based channel estimate is given by
Hˆm=qTmˆa. (3.5)
The factor graph representation of this model-based channel estimate is shown in Fig.
3.1, with the function nodes defined by
fM B( ˆHLS) =
ˆ ,0 M odel Based C hannel Estimation
with model order Mo
C ompute the soft infromation for the LD PC decoder
NSD+NSP -1
Figure 3.1: The factor graph representation of the model-based channel estimate.
0 10 20 30 40 50 60
Figure 3.2: Performance comparison of the model-based channel estimates and the LS channel estimate.
fM B,h(ˆa) = qTma.ˆ (3.7) Fig. 3.2 plots the channel estimates obtained by various model-based estimates and the LS channel estimate. The label M B(MO, ML) denotes the channel estimate that uses an MO-order polynomial model with ML samples in the modelling block. In the preamble period, the model-based approach is used as a refinement of the initial LS channel estimate. Unlike the crude LS estimate, the model-based estimate has a much smoother channel estimate and follows the real channel response much closer. It is also observed that as the model-order MO increases, the channel estimate becomes closer to the true channel response. The data unit can apply the channel estimate obtained in the preamble unit to detect the received samples, providing the LDPC decoder a soft input.