TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 173

1

2 log 2 (1 þ 1 =s 2 ) applicable in the Gaussian case. We observe, as expected, that the
capacity levels off at 1 bit per channel use for the binary input case.

3.8 BLIND TURBO EQUALIZATION

The schemes reviewed thus far all assume that the channel parameters, comprised of
the impulse response terms ( h k ) and the background noise variance s 2

¼ E ( b i ), are
known to the receiver. In practice, these quantities must be estimated at the receiver,
using either some sort of channel identification procedure, or by integrating the chan-
nel estimation step into the iterative procedure.
Channel identification can be accomplished by incorporating a training sequence
into the transmitted sequence, which training sequence is known to the receiver,
and using adaptive filtering techniques [11, 12] to identify (an approximation to)
the channel, and / or adapt the interference canceler [37, 38, 48]. Alternatively, blind
channel identification methods may be employed [49-54], provided multiple antennae
and / or oversampling are employed at the receiver.
The approach pursued in this section examines instead how the impulse response
and noise variance of the channel can be estimated as part of the iterative decoding
procedure [18], using the expectation maximization algorithm [55, 56]. This approach
has the advantage of operating concurrently with the turbo equalizer using essentially
the same operations. The training methods or blind methods referenced above, by
contrast, require the receiver to switch to a different mode of operation. They can,
however, be considered as candidate methods for deducing initial channel estimates
for the more refined identification method pursued here.
To begin, we first examine how the noise-free channel output relates to the channel
impulse response h . Since the channel is modeled as a tapped delay line, the noise-free
channel outputs may be placed in one-to-one correspondence with the configurations
of that delay line. With L denoting the channel order (or number of delay elements in
the tapped delay line), introduce an augmented state vector

2 3

d i
d i 1
.
d iL

¼ ¼

4 5 :

d i
x i 1

x i
d iL

j i 4

For the example three-tap channel of Figure 3.7 (which has L¼ 2 delay elements),
this augmented vector appears as

2 3

o x i

d i
d i 1
d i 2

j i ¼ x i 1 n

4 5

:

We observe that, as j i encompasses both x i 2 1 and x i , it encompasses the state transition
from x i 2 1 to x i . As it is comprised of Lþ 1 binary elements d i , ... , d i 2 L , it has 2 L +1