mohatta2015.pdf

158 ADVANCED TOPICS
Thus, for MAPSD, we can also form an LLR by taking the log of the ratio of the
two symbol metrics.
Now for an example. Referring to Table 7.1, the signed soft value for s% would be
log(0.969975/1.4529) = -0.4043. Similarly, the signed soft value would be 0.5120
for «2·
How do we use LLRs in decoding? Because the likelihood of the symbol being
+ 1 is in the numerator, we can think of the LLR as the log of the symbol metric
for the symbol being +1. The symbol metric for the symbol being —1 is simply the
negative of the LLR. As a result, we can perform decoding by correlating the soft
values to the hypothetical message values. A correlation is obtained by forming
products of corresponding values and summing. Thus, for the MAPSD case, the
message metric for message 1 would be (+l)(-0.4043) + (-1)(0.5120) = -0.9163.
7.2.2.2 The (7,4) Hamming code There are different kinds of FEC codes. One
kind is a block code, in which the message symbols are divided up into blocks, and
each block is encoded separately. For example, consider a Hamming code in which 4
information bits (¿j, ¿2, ¿3 and ¿4) are encoded into 7 modem bits (a 7-bit codeword),
sometimes called a (7,4) Hamming code. This is done by also transmitting 3 check
bits (ci, C2 and C3), which are computed as
C\ = ¿l¿2¿4
C 2 = íi¿3¿4
C.-j = ¿2*3*4- ( 7·10)
If no errors occur during detection, then the detected values will also satisfy these
equations. Otherwise, an error will have been detected. Note that there can be
errors in the check bits as well as the information bits.
If we are sure only one error has occurred, there is a way to detect it without soft
information. This particular Hamming code has the property that it can correct
single errors using only the hard decisions. It works like this. We form a syndrome
by forming products
Si = Cl¿l¿2¿4
Si = C2¿ii:s¿4
S-Λ = C;iÍ2Í\iÍA· (7.11)
If there are no errors, then all three syndrome values should be +1. Why? Consider
Si. We know that ¿i¿2¿4 = Ci, so that ci¿i¿2¿4 should equal c\ which is always +1.
Similar arguments apply to the other syndromes.
If one or more of the syndrome elements are —1, we know there has been an
error. It turns out that the Hamming code is designed to tell us the location of
the error, assuming only one error occurred. First, we need to think of the bits as
having the positions shown in Table 7.5. Second, we need to map the +1 and —1
syndrome values to 0 and 1, i.e.,
Third, we can determine the position of the error using
Bfc = (l-S fc )/2. (7.12)
p = Bi + 2B 2 + 4B.S. (7.13)