22-03-2012, 01:09 PM
Channel Codes Classical and Modern
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Channel-Coding Overview
The large number of coding techniques for error prevention may be partitioned
into the set of automatic request-for-repeat (ARQ) schemes and the set of forwarderror-
correction (FEC) schemes. In ARQ schemes, the role of the code is simply
to reliably detect whether or not the received word (e.g., received packet) contains
one or more errors. In the event a received word does contain one or more errors,
a request for retransmission of the same word is sent out from the receiver back
to the transmitter. The codes in this case are said to be error-detection codes. In
FEC schemes, the code is endowed with characteristics that permit error correction
through an appropriately devised decoding algorithm. The codes for this approach
are said to be error-correction codes, or sometimes error-control codes. There also
exist hybrid FEC/ARQ schemes in which a request for retransmission occurs if
the decoder fails to correct the errors incurred over the channel and detects this
fact. Note that this is a natural approach for data-storage systems: if the FEC
decoder fails, an attempt to re-read the data is made. The codes in this case are
said to be error-detection-and-correction codes.
Coding Limits for Channels with Memory
A channel with memory is one whose output depends not only on the input, but
also on previous inputs. These previous inputs can typically be encapsulated by
a channel state, so that the channel output depends on the input and the state.
Such a channel with memory is called a finite-state channel. There exists a number
of important finite-state channel models and we introduce capacity results for
two of these, the Gilbert–Elliott channel and the constrained-input intersymbolinterference
(ISI) channel. A commonly discussed capacity result is the important
water-filling capacity result for unconstrained-input ISI channels [3, 4], but we do
not discuss this here.
Gilbert–Elliott Channels
We define the Gilbert–Elliott (GE) channel model in its most general form as
follows. A GE channel is a multi-state channel for which a different channel model
exists in each state. A common example is a two-state channel that possesses both
a “good” state G and a “bad” state B, where a low-error-rate BSC resides in the
good state and a high-error-rate BSC resides in the bad state.