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A convolutional encoder is a finite state machine. An encoder with n binary cells will have 2 n states. Imagine that the encoder (shown on Img.1, above) has '1' in the left memory cell (m 0), and '0' in the right one (m −1). (m 1 is not really a memory cell because it represents a current value). We will designate such a state as "10".
There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of Reed & Solomon (1960), every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than .
A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
This is like convolution used in LTI systems to find the output of a system, when you know the input and impulse response. So we generally find the output of the system convolutional encoder, which is the convolution of the input bit, against the states of the convolution encoder, registers.
Under this definition codes such as turbo codes, terminated convolutional codes and other iteratively decodable codes (turbo-like codes) would also be considered block codes. A non-terminated convolutional encoder would be an example of a non-block (unframed) code, which has memory and is instead classified as a tree code.
Fig. 1. SCCC Encoder. The example encoder is composed of a 16-state outer convolutional code and a 2-state inner convolutional code linked by an interleaver. The natural code rate of the configuration shown is 1/4, however, the inner and/or outer codes may be punctured to achieve higher code rates as needed.
These constituent encoders are recursive convolutional codes (RSC) of moderate depth (8 or 16 states) that are separated by a code interleaver which interleaves one copy of the frame. The LDPC code, in contrast, uses many low depth constituent codes (accumulators) in parallel, each of which encode only a small portion of the input frame.
That is, the encoding function is defined via = (()) {,}. The fact that the codeword C ( x ) {\displaystyle C(x)} suffices to uniquely reconstruct x {\displaystyle x} follows from Lagrange interpolation , which states that the coefficients of a polynomial are uniquely determined when sufficiently many evaluation points are given.