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TURBO CODES

Presented by,
K Swaraj Gowtham
G Srinivasa Rao
B Gopi Krishna

Turbo Codes

 Turbo Code Concepts

 Log Likelihood Algebra

 Interleaving & Concatenated Codes

 Encoding With R S C

Introduction

Objectives
 Studying channel coding
 Understanding channel capacity

 Ways to increase data rate

 Provide reliable communication link

Communication System
Structural modular approach with various
components

Formatting Source Channel Access
Multiplexing Modulation
Digitization Coding Coding techniques

send

receive

CHANNEL CODING

Can be categorized
Wave form signal design Structured
sequences

Better detectible signals Added redundancy
Waveform Structured sequence
 M-ary signaling  Block
 Antipodal  Convolution
 Orthogonal  Turbo
 Trellis coded modulation

Structured Redundency

Channel
Channel
Input word encoder Output word
encoder
n-bit
k-bit
codeword
Code sequence
Redundancy = (n-k)
Code rate = k/n

TURBO CODES
 A turbo code is a refinement of the
concatenated encoding structure
plus an iterative algorithm for
decoding the associated code
sequence.
 Concatenated coding scheme is a
method for achieving large coding
gains by combining two or more
relatively simple building blocks or
component codes.

TURBO CODE CONCEPTS

Likelihood Functions:
 The mathematical foundations of hypothesis
testing rests on Baye’s theorem.
 A Posteriori Probability (APP)of a decision in
terms of a continuous-valued random variable x
as

Before the experiment, there generally
exists an a priori probability P(d = i). The
experiment consists of using Equation (1)
for computing the APP, P(d = i|x), which can
be thought of as a “refinement” of the prior
knowledge about the data, brought about by
examining the received signal x.

The Two-Signal Class Case:

H1
>
P (d = + 1| x) < P(d=-1|x)
H2

 Binary logical elements 1 and 0 are represented
electronically by voltages +1 and -1 where ‘d’
represents this voltages.
 The rightmost function, p(x|d = +1), shows the
pdf of the random variable x conditioned on d = +1
being transmitted. The leftmost function, p(x|d =
-1), illustrates a similar pdf conditioned on d = -1
being transmitted.
 A line subtended from Xk an arbitrary value taken
from the full range of values of X, intercepts the
two likelihood functions, yielding two likelihood
values ℓ1 = p(xk|dk = +1) and ℓ2 = p(xk|dk = -1).

General expression for the MAP rule in terms of APPs is

The previous equation is expressed in terms of ratio,
yielding the so-called likelihood ratio test, as follows

Log-Likelihood Ratio
By logging on both sides to the MAP ruled APPs is

To simplify the notation, it is rewritten as

At the decoder it is equal
to
This equation shows the output LLR of a systematic
decoder Consists of channel measurement , a prior
knowledge of the data, and an extrinsic LLR stemming
solely from the decoder.This soft decoder output L(dˆ ) is
a real number that provides a hard decision as well as the
reliability of that decision. The sign of L(dˆ ) denotes the
hard decision; that is, for positive values of L(dˆ ) decide
that d = +1, and for negative values decide that d = -1.
The magnitude of L(dˆ ) denotes the reliability of that
decision.

Log Likelihood Algebra

For Statistically independent data d, the sum
of two log likelihood ratios are defined as

INTERLEAVING

 This is the concept that aids much
in case of channels with memory.
 A channel with memory exhibits
mutually dependent transmission
impairments.
 A channel with multipath fading is
an example for channel with
memory.

 Errors caused due to disturbances in
these types of channels – Burst
Errors.
 Interleaving only requires a
knowledge of span of the memory
channel.
 Interleaving at the Tx’r side and de-
interleaving at the Rx’r side causes
the burst errors to be corrected.

 The interleaver shuffles the code
symbols over a span of several
block lengths or constraint lengths.
 It makes the memory channel look
like memoryless one for decoder.
 Two types of interleavers:-
->Block Interleavers.
->Convolutional Interleavers.

Block Interleaving
 A block interleaver accepts the
coded symbols in blocks from the
encoder,permutes the symbols,and
then feeds the rearranged ones to
the modulator.
 The minimum end-to-end delay is
(2MN-2M+2) symbol times where
the encoded sequence is written as
M*N array format.

 It needs a memory of 2MN symbol
times.
 The choice of M is dependent on the
coding scheme used.
 The choice of N for t-error-
correcting codes must overbound
the expected burst length divided
by t.

Convolutional Interleaving
 In this type, the code symbols are
sequentially shifted into the bank of
N registers; each successive
register contains J symbols more
storage than the preceding one.
 In this case, the end-to-end delay is
M(N-1) and the memory required is
M(N-1)/2.

Concatenated Codes
 A concatenated code uses two
levels on coding : an inner code and
an outer code (higher rate).

o Popular concatenated codes :-
Convolutional codes with
Viterbi decoding as the inner code
and Reed-Solomon codes as the
outer code.

 The purpose is to reduce the
overall complexity, yet achieving
the required error performance.

 However, the concatenated system
performance is severely degraded
by correlated errors among
successive smbols.

Encoding with Recursive
systematic codes
 Turbo codes are generated by
parallel concatenation of component
convolutional codes
 Consider an encoder with data rate
½ ,constraint length K ,i/p to
encoder dk. The corresponding code
word (Uk,Vk) is

 G1 = { g1i } and G2 = { g2i } are
the code generators, and dk is
represented as a binary digit
 This encoder can be visualized as a
discrete-time finite impulse
response (FIR) linear system, giving
rise to the familiar nonsystematic
convolutional (NSC) code

An example for NSC code
 G1={111},G2={101}, K=3,
bit rate = 1/2.

 At large Eb/N0 values, the error
performance of an NSC is better
than that of a systematic code
 infinite impulse response (IIR)
convolutional codes [3] has been
proposed as building blocks for a
turbo code
 For high code rates RSC codes result
in better error performance than the
best NSC codes at any value of
Eb/N0

 an RSC code, with K = 3, where ak
is recursively calculated as

g′i is respectively equal to g1i
if uk = dk, and to g2i if vk = dk.

An ex for Recursive encoder and
its trellis diagram:6(a),6(b)

 Example: Recursive Encoders
and Their Trellis Diagrams
a) Using the RSC encoder in Figure
6(a), verify the section of the trellis
structure (diagram) shown in
Figure 6(b).
b) For the encoder in part a), start
with the input data sequence{dk}
= 1 1 1 0, and show the step-by-
step encoder procedure for finding
the output codeword.

 Validation of trellis diagram

 Encoding a bit sequence with RSC
encoder

Concatenation of RSC codes
Good turbo codes have been
constructed from component codes
having short lengths(K = 3 to 5).
There is no limit to the number of
encoders that may be concatenated.
we should avoid pairing low-weight
codewords from one encoder with
low-weight codewords from the other
encoder. Many such pairings can be
avoided by proper design of the
interleaver

Fig :parallel concatenation of RSC codes

 If the component encoders are not
recursive, the unit weight input
sequence 0 0 … 0 0 1 0 0 … 0 0 will
always generate a low-weight
codeword at the input of a second
encoder for any interleaver design.
 if the component codes are
recursive,a weight-1 input sequence
generates an infinite impulse
response

 For the case of recursive codes, the
weight-1 input sequence does not
yield the minimum-weight codeword
out of the encoder
 Turbo code performance is largely

influenced by minimum-weight
codewords

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