mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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168 ADVANCED TOPICS<br />
to achieve spatial multiplexing (MIMO). Because subcarriers are orthogonal, we<br />
can use the simple MIMO model introduced in Chapter 1, in which the decision<br />
variables for a particular subcarrier collected over multiple receive antennas can be<br />
modeled as<br />
r f= Hs + n. (7.53)<br />
Unlike Chapter 1, we will assume complex-valued quantities.<br />
Let's consider MAPSD assuming no prior information about the two symbols.<br />
For symbol si, we need to form the MAP symbol metrics specified in (7.20). For<br />
the case of only two M-ary symbols drawn from a set S — {Sj\j = 1... M}, this<br />
becomes<br />
M<br />
P{ 8l =Sj\r} = ^Pr{r|e 1 =S j) e 2 = Sfc} (7.54)<br />
Assuming n is zero-mean Gaussian with covariance matrix C n , then<br />
fc=l<br />
Pr{r| Sl = S jt s 2 = S fc } = N) exp {(r - Hs^C^r - H Sj -. fc )} , (7.55)<br />
where Sjj. — [Sj Sk\ T ■<br />
To compute soft information for a particular bit í>¿ included in s\, we divide<br />
the set of possible symbol values S into two subsets: the set B+ corresponding to<br />
bi = +1 and the set B¿~ corresponding to 6» = — 1. The log-likelihood ratio for 6¿<br />
can then be determined using<br />
LLRi = log I 53 P{ Sl = Sjlr} j - log j ^ P{ Sl = Sj\r) J . (7.56)<br />
7.6 THE LITERATURE<br />
7.6.1 MAP symbol detection<br />
The relation between MAPSD and minimizing symbol error rate is pointed out in<br />
[Abe68j. A structure for MAPSD is given in [Gon68]. Fixed-lag (finite decision<br />
depth) MAPSD forms are developed in [Gon68, Abe70), and a parallel architecture<br />
is given in [Erf94]. Early work on MAPSD is surveyed in [Bel79]. Like MLSD,<br />
MAPSD can be formulated using an MF front end [Col05].<br />
The BCJR algorithm is described in [Bah74]. It can be viewed as a special<br />
case of Pearl's belief-propagation algorithm [Mce98]. Like the Viterbi algorithm,<br />
reduced-complexity versions of the BCJR, algorithm exist. The log-MAX approximate<br />
form that uses a forward and backward Viterbi algorithm is developed in<br />
[Vit98]. Reduced-state approximations have been proposed [ColOl, Vit07] as well<br />
as only considering a subset of symbols, treating the others as colored noise [P0088].<br />
7.6.2 Soft information<br />
In wireless communications, typical FEC codes are convolutional codes (proposed<br />
in [Eli55] according to [Vit67]), turbo codes [Ber96], and, more recently, low-density