Decoding methods

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This article discusses common methods in communication theory for decoding codewords sent over a noisy channel (such as a binary symmetric channel).

Notation

Henceforth [C \subset \mathbb_2^n] shall be a (not necessarily linear) code of length [n]; [x,y] shall be elements of [\mathbb_2^n]; and [d(x,y)] shall represent the Hamming distance between [x,y].

Ideal observer decoding

Given a received codeword [x \in \mathbb_2^n], ideal observer decoding picks a codeword [y \in C] to maximise:

[\mathbb(y \mbox \mid x \mbox)]

-the codeword (or a codeword) [y \in C] that is most likely to be received as [x].

Where this decoding result is non-unique a convention must be agreed. Popular such conventions are:

# Request that the codeword be resent;

# Choose any one of the possible decodings at random.

Maximum likelihood decoding

Given a received codeword [x \in \mathbb_2^n] maximum likelihood decoding picks a codeword [y \in C] to maximise:

[\mathbb(x \mbox \mid y \mbox)]

-the codeword that was most likely to have been sent given that [x] was received. Note that if all codewords are equally likely to be sent during ordinary use, then this scheme is equivalent to ideal observer decoding:

[\begin\mathbb(x \mbox \mid y \mbox) & = & \frac(x \mbox , y \mbox) }(y \mbox )} \\&& \\& = & \frac(x \mbox , y \mbox) }(y \mbox)} \frac(y \mbox)}(x \mbox)} \\&& \\& = & \frac(x \mbox , y \mbox) }(y \mbox)} \\&& \\& = & \mathbb(y \mbox \mid x \mbox) \\\end]

As for ideal observer decoding, a convention must be agreed for non-unique decoding. Again, popular such conventions are:

# Request that the codeword be resent;

# Choose any one of the possible decodings at random.

Minimum distance decoding

Given a received codeword [x \in \mathbb_2^n], minimum distance decoding picks a codeword [y \in C] to minimise the Hamming distance :

[d(x,y) = \# \]

-the codeword (or a codeword) [y \in C] that is as close as possible to [ x \in \mathbb_2^n].

Notice that if the probability of error on a discrete memoryless channel [p] is strictly less than one half, then minimum distance decoding is equivalent to maximum likelhood decoding since if

[d(x,y) = d]

then:

[\begin\mathbb(y \mbox \mid x \mbox) & = & (1-p)^p^d \\&& \\& = & (1-p)^n \left( \frac\right)^d \\\end]

which (since [p] is less than one half) is maximised by minimising [d].

As for other decoding methods, a convention is agreed for non-unique decoding. Popular such conventions are:

# Request that the codeword be resent;

# Choose any one of the possible decodings at random.

Syndrome decoding

Syndrome decoding is a highly efficient method of decoding a linear code over a noisy channel - ie one on which errors are made. In essence, syndrome decoding is minimum distance decoding using a reduced lookup table. It is the linearity of the code which allows for the lookup table to be reduced in size.

Suppose that [C\subset \mathbb_2^n] is a linear code of length [n] and minimum distance [d] with parity check matrix [H]. Then clearly [C] is capable of correcting upto

[t=\lfloor\frac\rfloor]

errors made by the channel (since if no more than [t] errors are made then minimum distance decoding will still correctly decode the incorrectly transmitted codeword).

Now suppose that a codeword [x \in \mathbb_2^n] is sent over the channel and the error pattern [e \in \mathbb_2^n] occurs. Then [z=x+e] is received. Ordinary minimum distance decoding would lookup the vector [z] in a table of size [|C|] for the nearest match - ie an element (not necessarily unique) [c \in C] with

[d(c,z) \leq d(y,z)]

for all [y \in C]. Syndrome decoding takes advantage of the property of the parity matrix that:

[Hx = 0]

for all [x \in C]. The syndrome of the received [z=x+e] is defined to be:

[Hz = H(x+e) =Hx + He = 0 + He = He]

Under the assumption that no more than [t] errors were made during transmission the receiver looks up the value [He] in a table of size

[\begin\sum_^t \binom < |C| \\\end]

(for a binary code) against pre-computed values of [He] for all possible error patterns [e \in \mathbb_2^n]. Knowing what [e] is, it is then trivial to decode [x] as:

[x = z - e ]

Notice that this will always give a unique (but not necessarily accurate) decoding result since

[Hx = Hy]

if and only if [x=y]. This is because the parity check matrix [H] is a generator matrix for the dual code [C^\perp] and hence is of full rank.

 

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