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:
where is an input sequence, is a sequence from output and is an impulse response for output .
A convolutional encoder is a discrete 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:
Transfer functions for the second (recursive) encoder are:
Define by
where, for any rational function ,
- .
Then is the maximum of the polynomial degrees of the, and the constraint length is defined as . For instance, in the first example the constraint length is 3, and in the second the constraint length is 4.
Read more about this topic: Convolutional Code
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