A short and detailed presentation about two types of codes.

1. REED SOLOMON CODE
AND CONVOLUTION
CODE

2. REED SOLOMON
CODE

3. CONTENTS
 Introduction
 Properties of RS code
 RS Encoder
 RS Decoder
 Software Implementation
 Advantages
 Disadvantages
 Applications

4. INTRODUCTION
 Reed–Solomon codes are an important group of error-correcting codes introduced
by Irving S. Reed and Gustave Solomon in 1960.
 RS codes operate on the information by dividing the message stream into blocks of
data, adding redundancy per block depending only on the current inputs.
 It is capable to correct both burst errors (where a series of bits in the codeword are
received in error) and erasures.

5. PROPERTIES OF RS CODE
 RS codes are generally represented as RS (n, k), with s-bit symbols.
 Block Length: n
No. of Original Message symbols: k
Number of Parity Digits: n – k
 A Reed-Solomon decoder can correct up to t symbols that contain errors in a
codeword, where 2t = n-k.
 The relationship between the symbol size, m, and the size of the codeword n, is
given by n=2s-1

6.  The following diagram shows a typical Reed-Solomon codeword:
k 2t
n
Data Parity

7. Example:-
 RS(255,223) with 8-bit symbols.
 Each codeword contains 255 code word bytes, of which 223 bytes are data and 32
bytes are parity. For this code:
n = 255, k = 223, s = 8
2t = 32, t = 16
 The decoder can correct any 16 symbol errors in the code word: i.e. errors upto 16
bytes anywhere in the codeword can be automatically corrected.

8.  Given a symbol size s, the maximum codeword length (n) for a Reed-Solomon code
is n = 2s – 1
 For example, the maximum length of a code with 8-bit symbols (s=8) is 255 bytes.
 Reed-Solomon codes may be shortened by (conceptually) making a number of data
symbols zero at the encoder, not transmitting them, and then re-inserting them at
the decoder.

9.  Example: The (255,223) code described above can be shortened to (200,168). The
encoder takes a block of 168 data bytes, (conceptually) adds 55 zero bytes, creates
a (255,223) codeword and transmits only the 168 data bytes and 32 parity bytes.

10. ENCODER

11.  Message Polynomial-
c(x) = m(x). xn-k
 RS generator Polynomial-
g(x) = g0 + g1. x+ g2 x2+ …. + g2t-1. x2t-1 + x2t

12. DECODER

13. SOFTWARE IMPLEMENTATION
 Until recently, software implementations in real-time required too much
computational power for all but the simplest of Reed-Solomon codes (i.e. codes
with small values of t).
 The following Table gives some example benchmark figures on a 166 MHz Pentium
PC:
Code Data rate
RS(255,251) 12 Mb/s
RS(255,239) 2.7 Mb/s
RS(255,223) 1.1 Mb/s

14. ADVANTAGES
 Reed-Solomon codes are most widely used to correcting burst errors.
 Coding gain is very high.
 The Coding rate is very high for Reed Solomon code so it is suitable for many
applications including storage and transmission.

15. DISADVANTAGES
 Unlike BCH codes , RS Codes does not perform considerably well in BPSK
modulation schemes.
 Bit Error Ratio(BER) for Reed-Solomon Codes is not as good as BCH codes.

16. APPLICATIONS
 Data Storage
 Bar Code
 Satellite Broadcasting
 Spread-Spectrum System
 Ultra Wideband(UWB)

17. CONVOLUTION CODE

18. ERROR CORRECTION CODE
 There are four important error correction codes that find applications in digital
transmission. They are :
18
Block Parity
Hamming Code
Interleaved Code
Convolutional Code

19. INTRODUCTION
 Convolutional codes are introduced in 1955 by Elias.
 Convolution coding is a popular error-correcting coding method used in
digital communications. A message is convoluted, and then transmitted
into a noisy channel.
 This convolution operation encodes some redundant information into
the transmitted signal.
19

20. CONVOLUTIONAL ENCODER
 Convolutional encoding of data is accomplished using a shift register
and associated combinatorial logic that performs modulo-two addition.
 A shift register is merely a chain of flip-flops.

21. PARAMETERS OF CONVOLUTION ENCODER
 Convolutional codes are commonly specified by three parameters:
 n = number of output bits
 k = number of input bits
 K = number of shift registers
 Code Rate: The quantity k/n is called as code rate. It is a measure of the efficiency
of the code.
 Constraint Length: The quantity L(or K) is called the constraint length of the code. It
represents the number of bits in the encoder memory that affect the generation of
the n output bits.

22. CONVOLUTIONAL CODE ENCODER
+
+
Shift Register
Linear Algebraic Function Generator

23. CONVOLUTION CODE ENCODER
+
+
Constraint Length (K) = 3Code Rate = k/nNo of input bits (k)= 1state = K-1state = 2k=1 K=3 n=2
No of linear Algebraic Function
Generator(n) = 2

24. CONVOLUTION CODE ENCODER
+
0 0 0
+
11001
k=1 K=3 n=2
Input 1 0 0 1 1
State 10 01 00 10 11
Outpu 11 10 11 11 01
1 1 111100110111 1

25. REPRESENTATION OF CONVOLUTION CODES
 State Diagram
 Tree Diagram
 Trellis Diagram

26. STATE DIAGRAM
 Contents of shift registers make up "state" of code:
 Most recent input is most significant bit of state.
 Oldest input is least significant bit of state.
 (this convention is sometimes reverse)

27. TREE DIAGRAM
1101
k=1 K=3 n=2

28. TRELLIS DIAGRAM REPRESENTATION
 The trellis diagram is basically a redrawing of the state diagram. It
shows all possible state transitions at each time step. Then we connect
each state to the next state.
 There are only two choices possible at each state. These are
determined by the arrival of either a 0 or a 1 bit.
 The arrows show the input bit.
 The arrows going upwards represent a 0 bit and going downwards
represent a 1 bit.

29. TRELLIS DIAGRAM

30. DIFFERENCE BETWEEN BLOCK CODE AND CONVOLUTION
CODE
 The difference between block codes and convolution codes is the encoding
principle.
 In the block codes, the information bits are followed by the parity bits while in
convolution codes the information bits are spread along the sequence.
 The block codes can be applied only for the block of data whereas convolution
coding can be applied to a continuous data stream as well as to blocks of data.

31. ADVANTAGES
 Convolution coding is a popular error-correcting coding method used in digital
communications.
 The convolution operation encodes some redundant information into the
transmitted signal.
 It is simple and has good performance with low implementation cost.

32. FACTORS AND PROPERTIES
 The performance of a convolutional code depends on the coding rate and the
constraint length.
 Longer constraint length K
More powerful code
More coding gain
 Smaller coding rate R=k/n
More powerful code due to extra redundancy
Less bandwidth efficiency

33. THANK YOU
E047 E048 E049 E050 E056 E058CREATED