Channel coding fundamental

1. Channel Coding
Dr. Varun Kumar
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2. Outlines
1 Introduction to Encoding
2 Channel Coding
3 Conclusion
4 References
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3. Introduction to Encoding
During conversion of analog signal into digital, following steps are taken
1 Sampling
2 Quantization
3 Encoding
Encoding is further classified into three sections.
Source Coding −→ Bandwidth suppression, Decrease in redundancy,
Removal of extra bit (Data compression−→ Lossless & Lossy
Compression).
Different Technique for Source Coding:
i. Huffman Code (Application–JPEG, MPEG, MP3)
ii. Fano Code
iii. Shannon Code
iv. Arithmetic Code
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4. Channel Coding[1],[2]
The key to achieving error free digital communication in the presence of
distortion, noise, and interference is the addition of appropriate
redundancy to the original data bits.
Channel Coding
It is used for error correction and detection of bit stream, which is sent
from the information sink.
For error correction and detection, extra/redundant bits (parity bits)
are added into bit stream.
Due to more (redundant+information) bits, bit error rate (BER) is also
more during channel encoding.
Due to more (redundant+information) bits transmission from
information sink causes loss in spectral efficiency or we enhance error
correction and detection capability at the cost of sacrifice of bandwidth.
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5. Continued–
Types of Channel Coding:
i. Linear block codes
Cyclic codes (e.g., Hamming codes)
Turbo codes
Polynomial codes (e.g., BCH codes)
ReedSolomon codes
Algebraic geometric codes
ii. Convolution codes
Low Density Parity Check (LDPC) code
Forward Error Correction (FEC):
i. Block Codes
In Linear block codes, every block of k data digits is encoded into a
longer codeword of n digits (n > k).
In block codes, k data digits are accumulated and then encoded into
n-digit codeword.
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6. Continued–
ii. Convolution codes
In convolution codes, the coded sequence of n digits not only
depends on the k data digits, but also on the previous N − 1 data
digits (N − 1). In short encoder has a memory.
Redundancy for Error Correction:
In FEC codes, a codeword is a unit of bits that can be decoded
independently.
The number of bits in codeword is known as code length.
If k data digits are transmitted by a codeword of n digit (n > k), then
number of check bit m = n − k.
Code-rate = k
n . Code is known as (n, k).
dmin is the minimum distance between t error correcting codewords
without overlapping, is
dmin = 2t + 1 (1)
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7. Continued–
n k Code Code rate
Single error correcting, t=1 or dmin=3 3 1 (3,1) 0.33
4 1 (4,1) 0.25
15 11 (15,11) 0.73
Double-error correcting, t=2 or dmin=5 10 4 (10,4) 0.4
15 8 (15,8) 0.533
Triple error correcting, t=3 or dmin=7 10 2 (10,2) 0.2
Table: Some example of error correcting codes
−→ The minimum distance between t error detecting codewords is
dmin = t + 1 (2)
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8. Linear Block Codes:
c = dG (3)
where, c = Codeword vector (1 × n)
d = Data or information vector (1 × k)
G = Generator Matrix (k × n)
Generator matrix G = [Ik P], where P −→ k × m matrix, m = n − k
c =dG
=d[Ik P]
=[d dP]
=[d cp]
(4)
Hamming distance between two codeword ca and cb is
d(ca cb) = weight of (ca ⊕ cb) (5)
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9. Continued–
Figure: Information processing across transmitter end
Line Coding −→ It is a process through which bit stream is
converted into electrical pulse.
Different Technique for Line Coding:
i. NRZ
ii. RZ
iii. Manchester
iv. AMI
v. Bipolar etc
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10. Conclusion
Channel encoding ensure that how can we detect and correct the
error in information bit stream.
Linear block code and convolution code are two types through which
we do the channel coding.
Greater the redundancy lesser be code efficiency.
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11. References
B. P. Lathi, Z. Ding et al., “Modern Digital and Analog Communication Systems /
BP Lathi, Zhi Ding.” 2010.
M. Borda, Fundamentals in information theory and coding. Springer Science &
Business Media, 2011.
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12. Thank You
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