Cryptography Reference
In-Depth Information
It is particularly instructive to study the limiting form taken by the equalizer
in the classical case where the transmitted symbols are assumed to be equiprob-
able (which corresponds to the 1 st iteration of the turbo equalizer). In this case,
σ x =0 and the equalizer coecients can be written:
f = σ x HH H + σ w I 1
h Δ σ x
(11.43)
Here we can recognize the form of a classical linear MMSE equalizer with finite
length. Inversely, under the hypothesis of perfect a priori information on the
transmitted symbols, we have σ x = σ x . The equalizer then takes the following
form:
L
k =0 |
1
σ x
2 = h Δ h Δ =
2
h Δ
f =
with
h
h k |
(11.44)
2 + σ w
σ x
h
and the equalized signal z i can be written:
2
σ x
x i− Δ + h Δ w i
h
z i =
(11.45)
2 + σ w
σ x
h
We recognize here the output of a classical MMSE interference canceller, fed
by a perfect estimation of the transmitted data. The equalized signal can be
decomposed as the sum of the useful signal x i− Δ , up to a scale factor that is char-
acteristic of the MMSE criterion, and a coloured noise term. In other words, the
equalizer suppresses all the ISI without raising the noise level and thus reaches
the theoretical matched-filter bound corresponding to ISI-free transmission.
To summarize, we see that the SISO MMSE linear equalizer adapts the
equalization strategy according to the reliability of the estimated data, measured
here by parameter σ x .
To conclude this description of the equalizer, we point out that the interfer-
ence cancellation operation defined formally by Equation (11.33) has no physical
reality in the sense that it cannot be performed directly in this way using trans-
verse linear filters. In practice, we prefer to use one of the two architectures
presented in Figure 11.15, strictly equivalent from a theoretical point of view.
The coecient g Δ appearing in implementation (1) is the central coecient
g Δ = f T h Δ of the global filter formed by the cascade of the channel with filter
f . In the case of implementation (2), we again find the classical structure of an
interference canceller type equalizer, operating here on the estimated signal x .
Filter g = f T H is given by the convolution of filter f with the impulse response
of the channel, the central coecient g Δ then being forced to zero.
SISO demapping
The role of this module is to convert the equalized data z i into extrinsic LLRs
on the interleaved coded bits, which will be then transmitted to the channel
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