The present section addresses the design of the
-order constellation with
fixed (under the assumption that the WL MMSE equalizer is used) and it is organized as follows. In Section 3.1, we address the optimum constellation design for the WL MMSE receiver by extending the results of [2] to the case of additive rotationally variant disturbance. In Section 3.2, we propose a suboptimum strategy based on the rhombic transformation of a given constellation. Such a strategy allows one to reduce both the computational complexity of the optimization procedure and the amount of information required at the transmitting side in order to adapt the constellation.
The results in the previous section allow one to state that, by using a real-valued constellation (
) instead of a complex-valued nonredundant (
) one, a performance gain can be achieved in terms of the MMSE at the equalizer output. On the other hand, not always an MSE gain provided by the WL equalizer leads to a SER gain [19]. In fact, for a fixed expended average energy per bit, the reduction of the minimum distance between the constellation points, due to the adoption of one-dimensional constellations rather than two-dimensional ones (e.g., when we adopt the
-PAM rather than the
-QAM) leads to a potential increase in the SER. Therefore, we address the constellation design minimizing the SER at the WL MMSE equalizer output by accounting for its rotationally variant properties.
In the literature (e.g., [2, 20]), most of the constellations employed by the transmission stage are circularly symmetric (
), while statistically redundant constellations are confined to the real-valued ones. Moreover, in [2], with reference to the transmission over a time nondispersive channel (
) affected by circularly symmetric noise, a procedure for constellation optimization has been proposed, showing also that, for large signal-to-noise ratios (SNR), the performance of the conventional QAM maximum-likelihood (ML) receiver is invariant with respect to rhombic transformations of the complex plane. However, it is important to point out that a rhombic transformation of a circular constellation makes it rotationally variant and, for some values of
(e.g.,
), the procedure in [2] provides a rotationally variant constellation. On the other hand, the WL equalizer is equivalent to the linear equalizer over the nondispersive channel considered in [2] and, therefore, optimizing the circularity degree of the constellation does not provide any performance advantage. On the other hand, when a time-dispersive channel is considered, the WL MMSE equalizer is sensitive to the rotationally variant properties of the transmitted signal and, therefore, we propose a transceiver structure (see Figure 1) where (i) the transmitter can switch between the available constellations of order
; (ii) the WL MMSE receiver accounts for the CSI and informs the transmitter, by means of a feedback channel, about which constellation has to be adopted to minimize the SER.
The use of a feedback channel in order to improve the bit-rate could also be exploited for choosing the constellation size rather than its circularity degree when the signal-to-noise ratio of each channel realization is not previously known. For example, the problem of the constellation choice has been addressed in [21, 22] with reference to the discrete multitone (DMT) transceiver and to multiple-input multiple-output transceiver, respectively. The two parameters of the constellations (size and circularity-degree) could also be jointly optimized by generalizing the procedures here proposed.
3.1. Constellation Optimization in the Presence of Gaussian Rotationally Variant Noise
In order to optimize over the constellation choice we need to first derive a performance analysis of the considered equalizer. Approximated evaluations of the performance of the WL receiver are available in [11] for a QAM constellation and in [3] for a PAM constellation in the presence of a PAM cochannel interference. Moreover, such performance analysis is generalized in [9] for IIR WL filters. Here, we derive an approximation of the equalizer performance suited for successive optimization over transmitter constellation.
With no loss of generality, assume that
and rewrite the output of the FIR equalizer as follows:
where
is the transmitted symbol drawn from the complex-valued constellation
with
and
, and
is the residual disturbance that includes the intersymbol interference and the noise terms after the WL equalizer filtering. The circularly symmetric model for the additive disturbance is inadequate since the output of a WL filter is, in general, rotationally variant. Therefore, we model
as rotationally variant, that is,
,
, and
. Moreover, in order to make the constellation design analytically tractable, we approximate
as Gaussian. For the sake of clarity, let us note that, if symbols
and noise are circularly symmetric (
), then the additive disturbance
and the equalizer output
will be circularly symmetric too; on the other hand, if
is rotationally variant, then
will be rotationally variant too, but nothing can be stated about the circularity properties of
also when
.
The sample
is the input of the decision device which performs the symbol-by-symbol ML detection of the transmitted symbol. By defining the following eigenvalue decomposition (the dependence on
at the right-hand-side is omitted for simplicity):
with
being the eigenvector matrix and
having on the diagonal the eigenvalues, it can be verified that the pair-wise error probability
[20], that is, the probability of transmitting
and deciding (at the receiver) in favor of
when the transmission system uses only
and
, is given by
where
denotes
+
where
and
, and, for
and
,
When
,
and analogously for
and
. By utilizing (11), assuming that the symbols
are equally probable, and resorting to both the union bound and Chernoff bound techniques, the SER
is upper-bounded as follows:
and, therefore, the optimum constellation can be approximated with the solution
of the following problem:
Unfortunately, it is difficult to find the closed-form expression of the solution of such an optimization problem. For such a reason, we propose to find a local solution by means of numerical algorithms (e.g., a projected gradient method). To this aim, we can exploit the gradient of
with respect to
, while we resort to numerical approximation of the gradient with respect to
since it is difficult to obtain its analytical expression.
Before proceeding, let us discuss the property of the locally optimum constellation for a fixed
. The
th component of the gradient of
is given by
By zeroing the gradient of the Lagrangian
one has that the locally optimum
satisfies the following equation:
with
Condition (17) generalizes the result of [2] to the case of
rotationally variant (i.e.,
or
) and with a constrained pseudocorrelation. (
.) In fact, (17) with
(i.e., no constraint is imposed on the pseudocorrelation) requires that
is proportional to the weighted sum (with weights
) of
,
, as found in [2]. For the sake of clarity, let us note that the procedure proposed in [2] does not allow one to exploit the potential advantage of a rotationally variant constellation when the WL MMSE receiver is employed. For example, when a linear MMSE equalizer is employed for
in high signal-to-noise ratio, the minimum of the SER is equivalently achieved [2] by both the conventional 4-QAM constellation and the rhombic constellations with the same perimeter, that is, the perimeter of the largest convex polygon consisting of the lines
(see [1] for further details). On the other hand, when a WL MMSE equalizer is employed, a rhombic constellation, which is rotationally variant, is not equivalent to the conventional 4-QAM since the achieved MMSE is dependent on
as shown in (8).
3.2. A Suboptimum Procedure Based on Rhombic Transformations
In this section, we propose a suboptimum constellation-design procedure for the WL MMSE equalizer. The method is based on the exploitation of a rhombic transformation that operates on a circularly symmetric constellation making it rotationally variant. Such a transformation depends on two parameters and allows one to control the pseudocorrelation
of the obtained constellation; consequently, the optimization procedure is simplified since the SER in (13) is a function of only two parameters, instead of
parameters.
Assume that
is a unit-power circularly-symmetric complex-valued constellation and define the complex-valued constellation
as follows:
or, more compactly (the compact expression is introduced for notation simplicity whereas the matrix form is utilized to understand the physical meaning),
with
and
. When
(
),
is stretched along the in-phase (quadrature) component and it becomes one-dimensional for
; when
, a correlation between
and
is introduced and for
, even if it is two-dimensional,
can be reduced to a one-dimensional constellation by a simple rotation. For symmetry, in the following we consider only the positive values of
and
. It is easily verified that, if
is drawn from
, then
The method proposed here assumes that the information-bearing symbol sequence, say
, is drawn from a fixed constellation
(e.g., the optimum constellation provided by [2]) whereas the possibly rotationally variant channel input
is obtained by resorting to the zero-memory precoding defined by the rhombic transformation (19). Clearly, such a strategy is suboptimum since it assumes that the channel input can be drawn from only those constellations
resulting from a rhombic transformation of the chosen
. However, the main advantages of such a method in comparison with the optimum one are
-
(1)
the huge reduction of the computational complexity of the constellation optimization procedure when
; in fact, the SER becomes a function of only two variables (
and
), regardless of the constellation order
;
-
(2)
the reduced implementation complexity of the transmitter stage; in fact, the symbol-mapping is implemented by means of the linear transformation (19);
-
(3)
the decrease of the information amount to be transmitted on the feedback channel; in fact, only the values of two parameters (instead of
) have to be sent to the transmitter.
According to such a choice, the constellation optimization is carried out by solving the minimization problem
with
where
denotes
,
denotes
−
, and
denotes
+
−
−
−
, and where (23) follows from (13) and (19), and the dependence of the disturbance parameters on
has been replaced by the dependence on
and
. Since finding the closed-form expression of
and
is a difficult problem, here we propose to approximate
with a function, say
, whose minimization can be carried out by evaluating it only over a very limited set of points. In the sequel, such an approximation is derived for a
-QAM constellation
, though it can be analogously determined for denser constellations.
First, we approximate the cost function (23) by assuming that the components of the residual disturbance are uncorrelated, that is,
. By means of some tedious but simple algebra operations, it can be shown that
is lower bounded by
where
where
denotes
and
denotes
. Since the right-hand side of (24) is minimized by large values of
, we propose to approximate the solution of (22) with the following one:
where
is the
-curve corresponding to the maximum value of
for a fixed
(or, equivalently, to the maximum value of
for a fixed
). Of course, the restriction to
leads to a significant decrease in the computational complexity. Let us point out that, interestingly, such a restricted optimization procedure accounts for the possible transmission of the conventional 4-PAM: in fact, it can be easily verified that when
,
.
This also suggests an extreme simplification obtained by choosing just between the 4-PAM and 4-QAM constellation (two-choice procedure), that is, one can resort to an architecture that switches between the 4-QAM and the 4-PAM constellations according to the following rule:
Three remarks about the suboptimum procedure (26) follow.
Remark 1.
The results carried out here with reference to the
-QAM constellation can be easily generalized to higher-order constellations. More specifically, the SER-bound approximations (analogous to the one in (24)) can be obtained by assuming that the inner summation in (23) is restricted to those constellation points closest to the
th one. Moreover, it can be shown that the conventional square
-QAM constellations (with
) can be transformed by (19) into the conventional uniform
-PAM. Note, however, that such a property is not satisfied by the constellations of any order; for example, as also shown in Section 4, when using the rectangular
-QAM (see Figure 2(g)) the rhombic transformation allows one to obtain the nonuniform
-PAM reported in Figure 2(i).
Remark 2.
The optimum transmission strategy proposed here requires that the receiver sends on the feedback channel the whole optimum constellation. If the suboptimum procedure is used, the transmitter architecture can be simplified. In fact, a unique symbol mapper for the alphabet
is needed and the constellation is adapted by adjusting the zero-memory WL filter (19). Unfortunately, the main disadvantage in terms of the computational complexity of the receiver remains the adaptation of the decision mechanism for the constellation
.
Remark 3.
When the proposed suboptimum strategy is used, the channel input
is obtained by performing a zero-memory WL filtering of the information-bearing sequence
. For such a reason, it is reasonable to consider an alternative receiver structure that performs the WL MMSE equalization of the received signal in order to estimate
, instead of
. After some matrix manipulations, it can be verified that such WL MMSE equalizer is the cascade of the WL MMSE equalizer in (4) and (5) and the WL zero-memory filter performing the inverse of the transformation (19) (note that (19) is not invertible for every value of
and
, e.g., when a real-valued constellation is adopted (
), however, in such a case, an ad hoc inverse transformation can be easily defined). This allows one to use a unique symbol de-mapper and the standard decision mechanism for the constellation
. The MMSE achieved by such a structure is
It can be easily shown that (a) if
, then
, unless
, and (b)
since
. Such results show that the minimum-distance decision based on the WL MMSE estimation of
outperforms the (computationally simpler) minimum-distance decision based on the WL MMSE estimation of
.