Digital transmission system with a trellis-based, reduced-state estimation method

The invention relates to a digital transmission system comprising a receiver, in which system a trellis-based estimation method (3) with a number of states reduced as a result of feedback (5) of at least one feedback value (.xi.) forms estimates (a) for a received signal (r) by means of an estimated impulse response (h) of a transmission system (1), a feedback value (.xi.) being determined from at least one estimate (a). With reduced-state estimation methods for the digital transmission, the problem consists of the additional noise components caused by the feedback of preliminary false symbol decisions. To achieve optimum estimates (a) for the received signal (r) despite a reduced number of states due to feedback, the receiver forms the feedback value (.xi.) from at least one intermediate value (a.sub.SDF). In digital transmission systems, the transmit symbols (a) and the estimates (a) have the values -1 or 1 in the receiver. In the receiver according to the invention, intermediate values (a.sub.SDF) having a value in the range from -1 to 1 can be fed back, so that a better feedback value (.xi.) and thus better estimates (a) for the received signal (r) may be achieved.

BACKGROUND OF THE INVENTION 
The invention relates to a digital transmission system comprising a 
receiver, in which system a trellis-based estimation method with a number 
of states reduced as a result of feedback of at least one feedback value 
forms estimates for received symbols by means of an estimated impulse 
response of a transmission system, a feedback value being determined from 
at least one estimate. 
Furthermore, the invention relates to a receiver and a method of receiving 
a digital signal. 
The subject of the invention is determined for digital transmission 
systems, for example, digital mobile radio systems such as the GSM system 
or digital Continuous Phase Modulation (CPM) radio relay systems in which 
trellis-based estimation methods are used which may be used both for 
equalizing a transmission channel and for decoding other trellis-coded 
signals (such as, for example, CPM signals). 
When digital transmission takes place over dispersive channels, the 
transmit signal is distorted and disturbed by noise; in GSM, for example, 
the distortions are caused by multipath propagation and intersymbol 
interference of the modulation method. Thus, special measures for 
recovering the transmitted data from the received signal are necessary in 
the receiver i.e. an equalization method is to be used. As channel coding 
is often used for increasing the resistance to noise in digital 
transmission (for example, convolutional coding in GSM), the data 
estimated by the equalizer are still to be decoded. It is then 
advantageous when the decoder is not only supplied with estimates of the 
coded data, but also with reliability information (also termed Soft-Output 
information (SO information)) by the equalizer, which information 
indicates with what reliability the data were decided in trellis-based 
reduced-state equalization methods, which may also be used for equalizing 
long impulse responses having moderate complexity, the supplied 
soft-output information may differ considerably from the actual values, 
which has a degrading effect on the subsequent decoding. 
A Pulse Amplitude Modulation (PAM) transmission over a distorting channel 
that generates Inter-Symbol Interference (ISI), may be modeled in a 
discrete-time version in an equivalent low-pass range, as is shown in the 
left-hand part of FIG. 1. The sampled received signal r(k) occurs as a 
noise-affected convolution of the PAM transmit sequence a(k) having the 
channel impulse response h(k), whose length is referenced L: 
##EQU1## 
where n(k) represents the discrete-time noise which is assumed to be white 
noise and is a given fact prior to the sampling when a whitened matched 
filter is used as a continuous-time receiver input filter. Depending on 
the modulation method used, the amplitude coefficients and the channel 
impulse response are either real or complex. 
The optimum equalization method with minimum error probability, Maximum 
Likelihood Sequence Estimation (MLSE), is known from G.D. Forney, "Maximum 
likelihood sequence estimation of digital sequences in the presence of 
intersymbol interference", IEEE Transactions on Information Theory, vol. 
IT-18, 1972, pp. 363-378. There was more particularly shown here that MLSE 
may be implemented efficiently with the Viterbi Algorithm (VA). However, 
for long impulse responses h(k), even the VA is hard to realize, because 
the trellis diagram which is to be made with the VA has Z=ML.sup.L-1 
states per time period with M-stage amplitude coefficients, thus the 
complexity of VA exponentially increases with the length of the 
discrete-time impulse response. When the number of subsequent symbols 
affected by a transmitted symbol becomes too large, more cost-effective, 
ie reduced-state, equalization methods are to be used because of the 
limited available computing speed. 
Furthermore, it is known that first a pre-equalization, ie shortening, of 
the channel impulse response is to be made by means of decision feedback 
equalization (DFE). Then either the DFE itself is to make preliminary 
threshold decisions, or preliminary hard-decision symbols (a.sub.HDF) of 
the next Viterbi equalizer are fed to the DFE, which in both cases leads 
to a noticeable degradation due to error propagation. 
Furthermore, the impulse response for each state of the equalizer may be 
shortened by a state-dependent (private) DFE, instead of a 
state-independent (common) DFE for all the states. The trellis-based 
algorithm then provides that only the first part of the impulse response 
having length R+1, with 1.ltoreq.R+1.ltoreq.L, is equalized. Thus the 
branch metrics are computed for state transitions in the time interval k 
from a (reduced) state S.sup.(r) (k)=(a(k-1)a(k-2) . . . a(k-R)) to the 
(reduced) state S.sup.(r) (k+1)=(a(k)a(k-1) . . . a(k-R+1)) (the 
coefficients a(k-.mu.) indicate the data assumed in the specific state) 
with the aid of contents of path registers of the respective states. The 
registers contain hard-decision estimates a.sub.HDF (k-.mu.,S.sup.(r) 
(k)), R+1.ltoreq..mu..ltoreq.L-1 of the previous data symbols in the path 
leading to the state S.sup.(r) (k), which estimates are to be updated in 
each time period. Thus the branch metric becomes 
##EQU2## 
where h(.mu.), O.ltoreq..mu..ltoreq.L-1 which indicate the channel impulse 
response estimates available to the receiver. 
This reduced-state estimation method will be referenced Decision-Feedback 
Sequence Estimation (DFSE) hereinafter. The states and transitions are no 
longer assigned unambiguously to a special combination of symbols in the 
channel memory, but there are henceforth ambiguities, as generally occurs 
in reduced-state methods, while particularly with DFSE the oldest symbols 
in the channel memory are taken into account only for preliminary-decision 
symbol values. 
The efficiency of DFSE may be further improved by an upstream all-pass 
filter which transforms the impulse response into its minimum-phase 
equivalent. This transformation provides a concentration of the energy of 
the impulse response in the front part, while the discrete-time noise 
remains white noise as before. This property of the minimum phase total 
impulse response motivates an additional reduction of the complexity of 
the DFSE, in that only h(R+1), h(R+2), . . . , h(R'), R&lt;R'&lt;L-1 are taken 
into account by private DFEs a.sub.HDF (k-.mu.,S.sup.(r) (k))), but the 
remaining part of the impulse response h(R'+1), . . . H(L-1) only by a 
single common DFE a.sub.HDF (k-.mu.,k)). If the last part of the impulse 
response contains only a small part of the total energy, the omission of 
the individual state relation (of the private DFEs) hardly reduces the 
efficiency of the estimation. 
These methods produce only hard-decision estimates for the received 
symbols, without further information about with which certainty the 
individual decisions were made, ie how likely they are true. This 
soft-output (SO) information, referenced .sub.m (k) here (the vector 
.sub.m (k) has a degree of probability for all the transmit symbols a(k)), 
however, is necessary in many transmission systems for an additionally 
available channel coding to enhance the resistance to noise, as, in 
consequence, the results of the decoding can be considerably improved by 
this SO information after the equalization. 
For determining the symbol probabilities .sub.m (k) in block-oriented 
transmission, particularly an algorithm with a bidirectional recursion 
rule can be used. First a forward recursion is used for computing the 
probabilities .alpha.(k,S(k)) for the states S(k).di-elect cons.{1,2, . . 
. ,Z} at step k, while the received signals considered thus far up to 
instant k-1 are taken into account. Then, a backward recursion is used for 
computing probabilities .beta.(k,S(k)) for the received signals considered 
from the block end back to the step k with a presupposed state S(k) in the 
real step k. The state probabilities .psi.(k,S(k)) for the states S(k) at 
step k are then the result of 
EQU .psi.(k,S(k))=.alpha.(k,S(k)).multidot..beta.(k,S(k)). (3) 
while the total received sequence is taken into account. 
Since a finite-length impulse response channel generating ISI may always be 
interpreted as a trellis coder with a FIR structure, the a-posteriori 
probabilities of the input symbols are the direct result of the state 
probabilities. Since the bidirectional algorithm also works on the basis 
of a trellis, it may be a reduced-state one in similar manner to the 
Viterbi algorithm. For the forward recursion, for computing the 
.alpha.(k,S.sup.(r) (k)), each of the M.sup.R reduced states is assigned a 
path register for computing a metric, which path register may be updated 
in each time period. Two path metrics are stored and used once again for 
the backward recursion. 
For the computation of the symbol probabilities .sub.m (k) in a continuous 
transmission without block limits formed by trellis-terminated symbols, 
only a forward recursion, that is, unidirectional recursion, is used 
contrary to the bidirectional algorithm. Then, similarly to the 
bidirectional algorithm, state probabilities .alpha.(k,S(k)) are computed. 
Since, finally, probabilities for the symbols a(k-D) are to be determined 
while the received signal is known up to instant k, a second recursion is 
necessary for determining state-related symbol probabilities 
##EQU3## 
With the results of the two recursions, the desired a-posteriori 
probabilities 
##EQU4## 
can be determined. A state reduction may also be effected with the 
unidirectional algorithm. By forming overlapping blocks on the receiving 
side, the bidirectional algorithm may also be used for a continuous 
transmission. As a result of its smaller complexity, bidirectional 
algorithm is typically more suitable than the unidirectional algorithm. 
For all the known trellis-based reduced-state equalization methods 
together, the fact is that the quality of the soft-output information 
.sub.m (k) for a received symbol with a strong state reduction with 
hard-decision feedback values is insufficient. 
OBJECTS AND SUMMARY OF THE INVENTION 
It is an object of the invention to achieve, with a reduced number of 
states, optimum estimates for the received symbols via feedback. 
With the subject according to the invention, the object is achieved in that 
the receiver forms the feedback value from at least one intermediate 
value. Contrary to known receivers, in which discrete-value estimates are 
fed back for state reduction, according to the invention it is also 
possible to use intermediate values for forming the feedback value, which 
intermediate values may adopt other values. In digital transmission 
systems, the transmit symbols and the estimates in the receiver have the 
values -1 or 1, for example. In the receiver according to the invention, 
also intermediate values having a value in the range from -1 to 1 can be 
fed back, so that a better feedback value and thus better estimates can be 
achieved for the received symbols. 
In a further embodiment of the invention, the receiver forms soft-output 
information for at least one estimate for a received symbol. In the 
receiver, an estimation method may be used which produces soft-output 
information which indicates with what reliability the data were decided. 
In digital transmission, for example, for enhancing the resistance to 
noise, channel coding is often used (for example, convolutional coding in 
GSM), so that the next decoder is still to decode the estimated data. For 
this purpose, it is advantageous when the decoder is supplied with 
Soft-Output information in addition to estimates of the coded data, from 
which SO information the receiver forms the intermediate value in a 
preferred embodiment. 
In an advantageous further embodiment, the receiver forms the soft-output 
information for a received symbol by means of optimum single-symbol 
estimation methods with a unidirectional or bidirectional recursion rule. 
The SO information for a received symbol may further be formed by the 
receiver of the received symbol by means of simplified sub-optimum 
estimation methods. Whereas the, for example, unidirectional or 
bidirectional algorithms represent optimum methods for computing the 
symbol reliability, there are furthermore a plurality of simplified, 
sub-optimum methods which may be derived systematically from the two 
algorithms. A possible simplification is, for example, the use of additive 
path metrics which are updated by ACS operations (Add-Compare-Select), 
instead of reckoning with real probabilities for which multiplications are 
necessary. The pure VA may be modified for producing symbol reliabilities, 
which leads to SOVA (Soft-Output-VA). Sub-optimum methods may generally 
also be reduced-state as a result of decision feedback. 
In a further embodiment, the receiver divides the feedback value into a 
first, common, component and into a second, private, component. The 
principle of the feedback according to the invention with an arbitrary 
division into private and/or common DFEs may also be combined with 
hard-decision feedback strategies. As a result, the feedback terms may be 
formed by up to four different components. For example, the energy 
distribution of the estimated impulse response may be used as a dividing 
criterion. 
Furthermore, the object is achieved by a receiver and a method for 
receiving a digital signal. 
These and other aspects of the invention will be apparent from and 
elucidated with reference to the embodiments described hereinafter.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
In FIG. 1 can be recognized in the left-hand part the channel 1 that 
generates intersymbol interference and noise 2. The right-hand part is the 
diagrammatic representation of an equalizer 4 with a reduced-state 
trellis-based estimation method 3 having a private decision feedback 5. 
Common to all the known trellis-based reduced-state equalization methods 3 
is the fact that the quality of the soft-output information .sub.m (k) for 
a received symbol with a strong state reduction and hard-decision feedback 
values is insufficient. Generally, for all the paths starting from a 
certain state, the influence of the pulse followers is taken into account 
for the computation of the associated path metrics in that the 
discrete-value symbols are fed back from the path history, which 
discrete-value symbols are taken from the available symbol alphabet. Thus 
there are always hard decisions relating to the feedback symbols 
available, as a consequence of which this procedure will be denoted 
Hard-Decision Feedback HDF in the following. The principle may be 
explained in FIG. 1 by feeding back (private) hard-decision symbol values 
a.sub.HDF (k-.mu.,S.sup.(r) (k)) 5. The channel model in this figure has 
already been slightly redrawn to indicate clearly that the component h(O) 
up to h(R) of the impulse response is taken into account by the 
trellis-based reduced-state algorithm 3 having M.sup.R states, and the 
remaining components h(R+1) to h(L-1) of the impulse response (the 
so-called pulse followers) are taken into account by a decision feedback 
5, which may be different for each hyper state S.sup.(r) (k) in the most 
general case. The channel impulse response to be equalized h(k), however, 
has a total length of L just like before, despite the redrawing. 
Since there is a separate path history for each hyper state, the overall 
value of the hard-decision feedback shown in FIG. 1 is 
##EQU5## 
usually differing from one hyper state to another. 
The block circuit diagram in FIG. 1 may be redrawn like in FIG. 2, to 
feature an additional noise term n.sub.HDF (k,S.sup.(r) (k)) which evolves 
from the feedback of false decisions from the respective path register of 
the equalizer. The noise term n.sub.HDF (k,S.sup.(r) (k)) is typically 
different from one state to another because each hyper state S.sup.(r) (k) 
may have typically different values a.sub.HDF (k-.mu.,S.sup.(r) (k)) for 
the symbols along the path leading to this state. For the errors caused by 
the hard-decision feedback in the hyper state S.sup.(r) (k) it holds that 
##EQU6## 
Assuming that all the decisions relating to the feedback symbols are 
correct, it holds that (with ideal channel estimation) n.sub.HDF 
(k,S.sup.(r) (k))=0 and the known HDF methods produce high-quality SO 
values (m) However, when the channel has small signal-to-noise ratios, the 
feedback symbol values cannot be assumed to be correct, so that additional 
noise is introduced into the system by the decision feedback, which 
degrades the quality of the SO. Also with strongly reduced-state SO 
equalization methods it appears that the SO value is clearly lower than 
with the method having a moderate state reduction, ie the produced 
estimates for the symbol error probabilities and the real values differ 
significantly, because the errors in the computation of the metric become 
too large because false symbols a.sub.HDF (k-.mu.,S.sup.(r) (k)) occur too 
frequently. This leads to an increased bit error rate after channel 
decoding. 
The further embodiments describe the methods by means of their use for the 
equalization. The basic procedure, however, may also be extended to other 
trellis-coded signals. CPM signals may be interpreted as trellis-coded 
signals just like signals distorted by a channel. In CPM, impulse noise is 
intentionally inserted into the signal to compress the transmit spectrum. 
Thus, trellis-based estimation methods may be used for CPM in the same 
manner as for signals transmitted over dispersive channels, so that the 
subject of the invention may also be used for coded transmission with CPM. 
In the described embodiment, a modified feedback term is used in the 
computation of the metric. A possibility is to choose the feedback term so 
that the average weight of the feedback error is minimized. Soft-decision 
feedback values a.sub.SDF (k-.mu.,S.sup.(r) (k)) are necessary for this 
purpose, which values are computed with this method. When the weight of 
the error is minimized, the knowledge about the current state S.sup.(r) 
(k) and about the received signal up to the instant k-1 is to be used. The 
result of a mathematical formula value can be minimized at any instant and 
for each state: 
##EQU7## 
The expression for the error weight is 
##EQU8## 
This expression for the squared error is used in the formation of the 
expected value according to (6) and derived from the total expression 
after the complex feedback. For finding the minimum value, the expression 
is set to zero. 
##EQU9## 
This leads to 
##EQU10## 
and the value 
##EQU11## 
is obtained with the aid of the estimated (expected) channel parameters 
h(.mu.) as the optimum soft-decision feedback value for minimum error 
weight for all the branches leaving the hyper state S.sup.(r) (k). 
The actual result which is the optimum value for the respective single 
symbol that is fed back is found at 
##EQU12## 
where A.sub.m,m.di-elect cons.{0,1, . . . M-1} denote the permitted 
amplitude coefficients from the symbol alphabet A of the M-stage 
transmission. The individual feedback values are formed according to the 
invention by weighted components of the discrete-values A.sub.m from the 
number A of the transmit symbols and are thus not exclusively selected 
from A, but have an analog range of values. Thus, soft-decision feedback 
values (SDF) are to be used for minimizing the error weight. For example, 
in binary transmission with A.sub.0 =1 and A.sub.1 =-1, it holds that 
EQU -1.ltoreq.a.sub.SDF (k-.mu.,S.sup.(r) (k)).ltoreq.1. (12) 
The state-dependent symbol probabilities (.sub.m) necessary for the 
equation (11) can be computed with the respectively used trellis-based 
equalization method. 
The necessary symbol probabilities used in equation (11) are already 
available anyhow as intermediate magnitudes in a reduced-state 
unidirectional algorithm, so that the use of soft-decision feedback values 
does not imply more circuitry and cost here. 
For explaining the method of producing the soft-decision feedback values, a 
reduced-state version of the unidirectional algorithm will be outlined 
here as an embodiment. Approximate values 
##EQU13## 
for the optimum soft-output information Pr{.alpha.(k)=A.sub.m 
.vertline.&lt;r(.nu.)&gt;} for the symbols a(k) are computed by means of the 
algorithm. The approximate value is, in the first place, the result of the 
reduction of states itself and is, in the second place, caused by the fact 
that only D received values r(v) (thus only up to v=k+D) lying further in 
the future are evaluated. Thus, not the whole sequence of received values 
is used for determining the soft-output information for individual a(k). 
But there may be assumed that for sufficiently large D the approximation 
is already very exact. The algorithm thus works continuously and gives, 
after decision delay D, the soft-output information resulting from the 
equation 
##EQU14## 
Here, the decision delay in the argument of the soft-output information 
produced in the time period k was taken into account for signal element 
a(k-D) and the sequence of the received values up to instant k was assumed 
to be known. 
The computation of the desired soft-output information to be read from the 
trellis-based algorithm presupposes the total distribution rate function 
in the numerator and denominator of (14), which function may again be 
computed with 
##EQU15## 
In the last step, the contracted equation 
##EQU16## 
was used for the total probability of S.sup.(r) (k) and partial receiving 
sequence &lt;r(.nu..sub.0.sup.k-1 &gt;, for which reason this magnitude is to be 
denoted the state probability. 
It should be observed that the second probability magnitude in (15) exactly 
corresponds to the value that is necessary for determining the 
soft-decision feedback and which value was denoted state-determined 
soft-output information 
##EQU17## 
This is a type of preliminary and state-dependent symbol probability for 
the symbol a(k-.delta.) under the condition that state S.sup.(r) (k) and 
the receiving sequence are observed only up to r(k-1). 
Consequently, the unidirectional algorithm can determine the desired 
soft-output information (corresponds to the symbol probability of a(k-D)) 
from the sum of products from state probability and state-determined 
symbol probability, while one recursion can be given for each of the two 
magnitudes (by approximation in the reduced-state case). The first 
recursion for the state probability may be given as 
##EQU18## 
where V(S.sup.(r) (k+1)) represents all the states which are permitted 
predecessors of S.sup.(r) (k+1). In this equation, .gamma.(S.sup.(r) (k), 
S.sup.(r) (k+1)) represents the branch probability for the transition 
(branch) from state S.sup.(r) (k) to S.sup.(r) (k+1) and can be expressed 
in Gaussian noise by the proportionality 
##EQU19## 
(see (2)). .sigma..sup.2 is the variance of the noise process. 
The second recursion for the state-determined soft-output information may 
be found by computing 
##EQU20## 
This computation is to be made for all the D.gtoreq..delta..gtoreq.R+1 and 
all the a(k-.delta.).di-elect cons.A, and also or each new state S.sup.(r) 
(k+1). 
For the state-determined soft-output information to be entered of the 
symbols in the time period k, which do not take the state into account 
(thus .delta.=R), the computation holds via 
##EQU21## 
and this computation may also be made for each new hyper state and all the 
a(k-.delta.).di-elect cons.A. The set 
EQU S(.alpha.(k-R)).apprch.{S.sup.(r) (k)=(.alpha.(k-1).alpha.(k-2) . . . 
.alpha.(k-R)).vertline..alpha.(k-R)=.alpha.(k-R)} (21) 
was used, containing all the states which include the defined symbol a(k-R) 
and the respective positions in time. Thus, S(a(k-R)).LAMBDA.V(S.sup.(r) 
(k+1)) is to represent the one state S.sup.(r) (k) that is the predecessor 
of S.sup.(r) (k+1) and contains the element a(k-R). The magnitude 
.gamma.(S(a(k-R)).LAMBDA.V(S.sup.(r) (k+1)), S.sup.(r) (k+1) is the 
probability for the branch that leads from this S.sup.(r) (k) to S.sup.(r) 
(k+1). 
In this manner, all the magnitudes which are necessary for determining the 
soft-output information can be computed recursively. 
In addition to the unidirectional algorithm, it is also possible to utilize 
the reduced-state bidirectional algorithm with soft-decision feedback. 
Since the forward recursion for determining the .alpha.(k,S(k)) of the 
bidirectional algorithm at the same time forms an essential part of the 
unidirectional algorithm, this requires only moderate additional circuitry 
and cost, which consists of including the additional second recursion of 
the unidirectional algorithm, which recursion is necessary for determining 
the probabilities necessary for the soft-decision feedback. The length of 
the path register to be updated for this purpose is only L-1-R, because 
the unidirectional part of this combined algorithm is only necessary for 
decision feedback, but not for computing the symbol probabilities, which 
are computed by the bidirectional part. In contrast, a clearly more 
complex second forward recursion is necessary in the purely unidirectional 
algorithm, because an estimate delay of D.apprxeq.2L is necessary in 
minimum-phase channels for substantially fully registering the statistical 
relations. 
The solution for .xi..sub.SDF (k,S.sup.(r) (k)) according to equation (10) 
with the minimum square error may only be considered one possibility from 
a large range of conceivable soft-decision feedback strategies. Especially 
with sub-optimum methods, the use of sub-optimum soft-decision feedback 
values presents itself, or is necessary, respectively. 
The principle already described of the subdivision of the decision feedback 
into private DFEs and one common DFE is also possible without further 
problems with soft-decision values with the two extreme cases of an 
exclusive use of the common DFE, or the exclusive use of private DFEs, 
respectively. The feedback term 
##EQU22## 
shown in FIG. 3, by which the first part of the pulse follower is taken 
into account in the computation of the metric, is formed individually for 
each state, while again state-dependent soft-decision symbol values are 
used. On the other hand, the feedback term 
##EQU23## 
which takes account of the last part of the pulse follower is identical in 
all cases, thus needs to be computed only once with common soft-decision 
symbols a.sub.SDF (k-.mu.,k). The principle of the soft-decision feedback 
with its arbitrary division into private and/or common DFEs may also be 
combined with hard-decision feedback strategies. In general, the feedback 
terms may be formed by up to four different portions: .xi..sub.SDF (k, 
S.sup.(r) (k)), .xi..sub.HDF (k, S.sup.(r) (k)), .xi..sub.SDF (k) and 
.xi..sub.HDF (k), where, for example, the energy distribution of the 
estimated impulse response O.ltoreq..mu..ltoreq.L-1 can be used as a 
dividing criterion. 
When the SO values of the estimation method (for example, equalization 
method) whose quality has benefitted from the invention are subsequently 
utilized in a downstream decoder which is capable of processing 
soft-decision input information, the method according to the invention 
with the feedback of soft-decision symbol values provides a clear gain in 
the power efficiency of the digital transmission. 
Furthermore, compared with the respective methods which use hard-decision 
feedback, the noise power additionally caused by decision feedback methods 
is reduced by about 1 dB due to the estimation of the minimum mean square 
error.