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Myagmarbayar Nergui et al.

Fig. 11.7. Performance of (15, 11) Hamming code, (15, 11) RS code, and (15, 11) BCH code on the transmission of retinal fundus images through burst error channels that are corrupted by (various SNR levels) burst noise.

11.5.1.Using Turbo Codes for Transmission of Retinal Fundus Image

After getting the simulation results of the (15, 11) Hamming code, the (15, 11) BCH code, the (15, 11) RS code, and the convolutional code, we have experimented with a turbo code. The turbo code encoder used is shown in Fig. 11.9a. This turbo encoder consists of two recursive systematic convolutional (RSC) encoders, which are connected in parallel, an interleaver, which is placed before the second RSC encoder, and puncturing block. Input bits are encoded by both RSC encoders. The RSC encoder 1 encodes the input bits in their original order, while the RSC encoder 2 encodes the input bits in interleaved order. We can change the code rate of the turbo code by using puncturing scheme.21 We have shown the performance of the original rate one-third turbo code and the punctured rate one-half turbo code.

Figure 11.9b shows an RSC encoder, which is used in the turbo encoder. uk is an input bit and ck is the coded output bit.

340

Reliable Transmission of Retinal Fundus Images

Fig. 11.8. Performance of convolutional code with a code rate 0.5 on transmission of retinal fundus images through AWGN channel.

A recursive systematic convolution encoder consists of linear finite-state shift registers and module 2 adder. The interleaver plays a main role in the turbo codes. In this chapter, a pseudo random interleaver is applied, because its performance is better in Ref. [27].

Mainly, puncturing and multiplexing is used to enhance a given code rate, while deleting bits. Figure 11.9c shows the turbo decoder block diagram. Turbo decoder consists of demultiplexer, two MAP decoders, interleavers, deinterleaver and decision maker. Turbo decoding begins with the formation of a posteriori probabilities (APPs) for each data bit. MAP (maximum a posteriori) decoder determines most likely information bit.

A logarithmic ratio of the APP of uk, conditioned on the received signal y, is defined as:

k

 

 

 

P(uk = 0/y1N )

 

L(u

)

 

log

 

P(uk = 1/y1N )

(11.1)

 

 

 

341

Myagmarbayar Nergui et al.

 

 

c1k

Input bits

RSC

c2k

uk

Encoder1

Output bits

 

 

 

Puncturing

 

 

&

Interleaver

 

multiplexing

u

 

 

 

c3k

RSC

k

 

 

Encoder2

 

 

 

 

 

 

(a)

ck

+

uk

+ D D D

+

(b)

(c)

Fig. 11.9. (a) Turbo code encoder, (b) RSC encoder, and (c) block diagram of a turbo decoder.26

342

Reliable Transmission of Retinal Fundus Images

The decoding decision of u˜ k is made based on the sign of L(uk ), i.e.

u˜ k = sign[L(uk)].

(11.2)

L(uk) is computed by three terms, L_apriori, L_channel, and Le(uk). L_apriori is a priori information based on the input bit uk at time k. The

previous decoder provides it.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

αk

1(s )

βk (s)

·

γe

(s , s)

L(u

)

= [

Le(u )

+

Lc

·

y1,s

] +

log

u+ ˜

 

 

· ˜

k

 

 

 

 

 

 

 

 

e

 

 

k

 

k

 

k

 

u˜

1

(s )

· ˜

·

 

 

 

 

 

 

 

 

 

 

 

 

k

 

 

 

 

 

 

 

 

 

 

 

 

αk

 

βk (s)

 

γ (s , s)

 

 

= L_apriori + L_channel + Le(uk),

 

 

 

 

 

 

(11.3)

where L_apriori and L_channel denote Le(uk ) and Lc · yk1,s, respectively. u+ () is the summation of all the possible transition branch pairs (sk1, sk) at time k given input uk = 1, and u() is the summation over all the possible transition branch pairs (sk1, sk ) at time k, given input uk = 0. Lc

is the channel reliable factor; its computation is given as the following,

Lc

=

4 · A · SNR_b

,

(11.4)

 

 

p

 

where A is a fading amplitude, equal to one for an AWGN channel, SNR_b

is the bit SNR ( Eb ), and p denotes 1/rc, rc is code rate of the turbo encoder.

N0

Le(uk ) is extrinsic information based on all parity and systematic information except the systematic value at time k. It can be passed to a subsequent decoder and is computed using the following equations:

 

 

 

αk

1

(s )

·

γe(s , s)

·

βk(s)

 

Le(u )

 

log

u+

˜

 

 

 

k

˜

,

 

 

 

 

(s )

 

γe(s , s)

 

 

k

 

αk

1

·

·

βk(s)

 

where

 

 

u

˜

 

 

 

k

˜

 

 

 

 

 

 

 

 

 

 

 

 

 

 

γe(s , s) = exp i

q

Lc ·

2

· yki,p · cki .

 

2

 

 

 

 

 

 

 

 

1

 

 

 

 

=

(11.5)

(11.6)

˜ k

˜ k1

(s )

can be computed recursively, with initial conditions, as

α

(s), β

described below:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

α

(s )

·

γ (s , s)

 

 

 

α

(s)

=

 

 

s ˜ k1

 

 

 

k

 

 

,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

˜ k

 

 

 

α

 

(s

)

γ (s

, s)

 

 

 

 

 

 

 

s

s ˜ k1

 

 

 

 

· k

 

(11.7)

 

 

 

α0

(s)

=

1

if s = 1 .

 

 

 

 

 

 

 

˜

 

0

otherwise

 

 

 

 

 

 

343