Editing Convolutional code (section)

== Impulse response, transfer function, and constraint length ==
A convolutional encoder is called so because it performs a ''[[convolution]]'' of the input stream with the encoder's ''impulse responses'':

:<math>y_i^j=\sum_{k=0}^{\infty} h^j_k x_{i-k} = (x * h^j)[i],</math>

where {{mvar|x}} is an input sequence, {{mvar|y<sup>j</sup>}} is a sequence from output {{mvar|j}}, {{mvar|h<sup>j</sup>}} is an impulse response for output {{mvar|j}} and <math>{*}</math> denotes convolution.

A convolutional encoder is a discrete [[LTI system|linear time-invariant system]]. Every output of an encoder can be described by its own [[transfer function]], which is closely related to the generator polynomial. An impulse response is connected with a transfer function through [[Z-transform]].

Transfer functions for the first (non-recursive) encoder are:
* <math>H_1(z)=1+z^{-1}+z^{-2},\,</math>
* <math>H_2(z)=z^{-1}+z^{-2},\,</math>
* <math>H_3(z)=1+z^{-2}.\,</math>

Transfer functions for the second (recursive) encoder are:
* <math>H_1(z)=\frac{1+z^{-1}+z^{-3}}{1-z^{-2}-z^{-3}},\,</math>
* <math>H_2(z)=1.\,</math>

Define {{mvar|m}} by
: <math> m = \max_i \operatorname{polydeg} (H_i(1/z)) \,</math>

where, for any [[rational function]] <math>f(z) = P(z)/Q(z) \,</math>,
: <math> \operatorname{polydeg}(f) = \max (\deg(P), \deg(Q)) \,</math>.

Then {{mvar|m}} is the maximum of the [[degree of a polynomial|polynomial degrees]] of the

<math> H_i(1/z) \,</math>, and the ''constraint length'' is defined as <math> K = m + 1  \,</math>. For instance, in the first example the constraint length is 3, and in the second the constraint length is 4.