Chapter 1 - Coding for Reliable Digital Transmission and Storage

Block Codes

• Input split up into blocks of length k, each block denoted u
• Outputs codewords of length n, each codeword entry called a symbol, each codeword denoted v
• Block length and codeword length combo called a (n,k) block code
• Code rate = R=k/n = number of information bits per transmitted symbol
• Each message is coded independantly
• Block coder can be made as a combinational logic circuit, has no memory

Convolutional Codes

• k bit blocks, u and v represent sequences of blocks instead of a single block
• Each block depends on m previous message blocks
• Has memory
• Must be implemented with a sequential logic circuit

Modulation

How bits are represented being sent through the channel

BPSK

• Binary Phase Shift Keying
• The phase of the sin wave is shifted between 0 and π to represent a 0 and 1
• 

MPSK

• M-ary Phase Shift Keying
• has M different phased sin waves representing combinations of bits
• for 4psk (also called QPSK), each phase represents 2 bits
• for 8psk, each phase represents 3 bits
• etc... 2^{# bits}=Mpsk

• 

Discrete Memoryless Channel (DMC)

• Doesn't depend on any previous messages (memoryless)
• M-ary input modulator
• Physical channel
• Q-ary output modulator
• Uses transition probabilities, probability of j output given i input

• 

Transmission Rates and Bandwidth

• If one encoded symbol is transmitted every T seconds,
• Baud Rate = symbol transmission rate = 1/T
• Data rate = information transmission rate = R/T bits per second (bps) = (k/n)/T
• Bandwidth = Difference between upper and lower frequencies representing the range of frequencies the channel is capable of
• Bandwidth should be at least T/2 hertz

Maximum Likelihood Decoding

• u = encoder input
• v = encoder output
• r = decoder input
• u^{^} = decoder output
• Chooses output with maximum probability of being correct
• Equivalent to maximizing the log-likelyhood function
  • logP(r|v) = logP(r₀|v₀) + logP(r₁|v₁) + logP(r₂|v₂) ...
  • MLD for a DMC chooses v^{^} which maximizes the above, which is then decoded to get u^{^}

Hamming Distance

• The number of bits or symbols that differ between the encoder output v and received decoder input r
• denoted d(r,v)
• For a modulation where M=2 and Q=2 (modulator and demodulator both produce only a 0 or 1) (binary symmetric channel, BSC), MLD chooses v^{^} with smallest hamming distance
• Also called minimum distance decoder

Forward Error Correction

• Errors are only corrected at the receiver end since transmission is one way only

Error Rates

• Error probability: probability there is still an error after decoding
• Word Error Rate (WER): probability a decoded block/word has an error
• Block Error Rate (BLER): same as WER
• Bit Error Rate (BER): probability a bit in the decoded output is an error
• Signal to Noise Ratio (SNR): Energy-per-information-bit vs. noise power spectral density at receiver input, in decibels