Convolutional codes Tomashevich Victor Name Introduction Convolutional codes
- Slides: 35
Convolutional codes Tomashevich Victor Name
Introduction • Convolutional codes map information to code bits sequentially by convolving a sequence of information bits with “generator” sequences • A convolutional encoder encodes K information bits to N>K code bits at one time step • Convolutional codes can be regarded as block codes for which the encoder has a certain structure such that we can express the encoding operation as convolution Name 2
Properties of convolutional codes Name 3
Example: Consider a rate ½ convolutional code with K=1 and N=2 defined by the circuit: Name 4
• The convolutional code is linear • The encoding mapping is bijective • Code bits generated at time step i are affected by information bits up to M time steps i – 1, i – 2, …, i – M back in time. M is the maximal delay of information bits in the encoder • Code memory is the (minimal) number of registers to construct an encoding circuit for the code. • Constraint length is the overall number of information bits affecting code bits generated at time step i: =code memory + K=MK + K=(M + 1)K • A convolutional code is systematic if the N code bits generated at time step i contain the K information bits Name 5
Example: The rate ½ code defined by the circuit Name 6
Tree Name 7
Trellis Name 8
Example: Name 9
Example: Constructing a trellis section Name 10
Trellis section: Name 11
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State diagram Name 13
Description with submatrices Name 14
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Generator matrix Name 16
Example: Name 17
Description with polynomials Name 18
Example: Name 19
Punctured convolutional codes Name 20
Example: Name 21
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Decoding of convolutional codes The Viterbi algorithm Name 25
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