Abstract
Constellation design has been previously addressed by assuming that there is a linear equalizer at the receiver side. However, the widely linear equalizer is well known to outperform the linear one with no significant complexity increase; we derive optimum and suboptimum techniques for constellation design in presence of such an equalizer. The proposed techniques adapt the circularity properties of the transmitted signals to the specific channel to be equalized; their performance analysis shows that also the simplest suboptimum procedure provides significant improvements over a fixed-constellation scheme.
information bits into 
points of a two-dimensional constellation was proposed [
is the imaginary unit, the superscripts 
, 
, and 
denote the complex-conjugate, the transpose and the Hermitian transpose, respectively, 
is the statistical expectation, 
is the Kronecker delta, 
is the identity matrix of size 
, 
is the vector/matrix with all zero entries (the size is omitted for brevity), 
denotes the 
th entry of the vector 
, 
denotes the 
entry of the matrix 
, 
denotes the 
th column of 
, 
and 
are the real and the imaginary part, respectively, 
denotes the 
-norm with 
, and, finally, 
is the 
-transform of 
.
are independent identically distributed (i.i.d.) zero-mean random variables drawn from the complex-valued constellation 
whose (finite) order 
determines the bit rate (
bits per symbol) of the uncoded system part. With no loss of generality, we assume that 
and 
, that is, the transmitted available power is unit, and that 
exhibits a possibly nonnull pseudocorrelation 
, such that 
(if 
, then the correlation matrix of the 
random vector 
will be positive semidefinite); note that the noncircularity of 
consists in the difference between the power of the in-phase component and the quadrature one and in the correlation between them. Such assumption allows one to consider both the conventional circularly symmetric constellations (
), such as 
-PSK and square 
-QAM with 
, and the rotationally variant constellations, such as the well-known PAM (
) and its rotated version (for which it exists 
such that 
is real-valued and, consequently, 
), non-square QAM (with 
since a different power is allocated to the in-phase and quadrature components). The time-invariant FIR channel impulse response 
of memory 
is assumed to be known at the receiver side. Finally, the additive noise 
, whose power 
is assumed known at the receiver, is modeled as zero-mean complex-valued wide-sense stationary time-uncorrelated and independent of the useful signal. The additive disturbance 
is not assumed circularly symmetric because it may include the effects of one-dimensional cochannel interferences.
received samples 
which, in a matrix notation, can be written as follows:
is the (possibly) nonnull noise pseudocorrelation (if 
, then the noise is circularly symmetric).
and consequently the received one 
in (1) can be rotationally variant, we adopt a widely linear receiver in order to exploit the statistical redundancy exhibited by the received signal. Note that such a choice improves the performance since the linear equalizers are a subset of the WL equalizers; their performances coincide only in the presence of circularly symmetric signals [
and 
that process the received vector 
and its complex conjugate version 
, respectively. The optimum filters 
and 
minimizing the mean square error 
are given by [
denotes the 
th column of 
and the processing delay 
has to be chosen in order to optimize the performance. For notational simplicity, in (4) and (5) we have omitted the dependence of 
and 
on 
. Let us point out that when 
, that is, the transmitted symbols are drawn from a circularly symmetric constellation, 
= 0 and, therefore, the WL MMSE equalizer degenerates into the conventional linear MMSE equalizer. Another special case is represented by the scenario where a real-valued constellation is adopted. In fact, since 
, 
and the WL MMSE equalizer becomes 
, that is, it is implemented by extracting the in-phase component of the linear equalizer 
, which does not coincide, however, with the linear MMSE equalizer.
of the transmitted signal, let us analyze the dependence on 
of the MMSE. To this end, denote with 
the error measured at the output of the WL MMSE equalizer for given values of 
and 
. It can be easily shown that
is the MMSE at the outputs of both the WL MMSE equalizer and the linear MMSE equalizer in the presence of a circularly symmetric constellation, 
represents the MMSE gain achieved by properly choosing the pseudocorrelation 
of the transmitted constellation. When 
, that is, the noise is circularly symmetric, 
depends on 
instead of 
and its (first) derivative with respect to 
can be written as
and 
are positive semidefinite, one has 
and, hence, increasing the degree of noncircularity of the transmitted signal improves the MMSE. For such a reason, the use of a real-valued transmitted sequence together with a WL MMSE equalizer corresponds to the optimum choice as far as the MMSE is adopted as the performance measure. On the other hand, when 
, the variations of 
with respect to 
depend on the specific values of the channel impulse response and the noise statistics.
-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 [
) 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 [
-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.
), while statistically redundant constellations are confined to the real-valued ones. Moreover, in [
) 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 [
; (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.
and rewrite the output of the FIR equalizer as follows:
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 
.
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):
being the eigenvector matrix and 
having on the diagonal the eigenvalues, it can be verified that the pair-wise error probability 
[
and deciding (at the receiver) in favor of 
when the transmission system uses only 
and 
, is given by
denotes 
+ 
where 
and 
, and, for 
and 
,
, 
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:
of the following problem:
with respect to 
, while we resort to numerical approximation of the gradient with respect to 
since it is difficult to obtain its analytical expression.
. The 
th component of the gradient of 
is given by
satisfies the following equation:
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 [
in high signal-to-noise ratio, the minimum of the SER is equivalently achieved [
(see [
as shown in (8).
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.
is a unit-power circularly-symmetric complex-valued constellation and define the complex-valued constellation 
as follows:
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
, is drawn from a fixed constellation 
(e.g., the optimum constellation provided by [
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
; in fact, the SER becomes a function of only two variables (
and 
), regardless of the constellation order 
;
) have to be sent to the transmitter.
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.
. By means of some tedious but simple algebra operations, it can be shown that 
is lower bounded by
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:
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 
, 
.
-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 
-PAM reported in Figure 
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 
.
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
, 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 
.
) the noise sequence at the output of the channel is zero-mean white Gaussian complex-valued circularly symmetric with variance 
, that is, 
; (
) the decision delay 
is optimized; (
) the SER has been estimated by stopping the simulation after 100 errors occur; (
) each sample at the output of the WL filter is the input of the decision device that performs the symbol-by-symbol ML detection of the transmitted symbol.
and 
: note that in our search we consider 
, so we consider a finite number of points. On the other hand, we resort to the constrained gradient-based algorithm for solving (14). Since the cost function (13) exhibits local minima, 1000 starting points have been randomly generated according to a uniform distribution. Due to the amount of time required by the computer simulations to determine the solution of (14), we consider, as in [
affected by an additive circularly symmetric white Gaussian noise with variance 
. In our experiments, we have addressed the optimization of the constellation when 
and 
for different values of 
, 
, and 
.
-PAM constellation obtained by applying to it the rhombic transformation. As in [
the locally optimum constellation set includes the conventional 
-QAM (
) and 
-PAM (
), as well as the 
-QAM subject to a rhombic transformation (
); note that such constellations can be obtained by means of a rhombic transformation of the conventional 
-QAM (as also shown in Section 3.2), which has been utilized to implement our suboptimum strategy when 
. For 
, the optimum constellation set includes the noncircular 
-QAM found by Foschini et al. (
), one of the conventional 
-QAM scheme (
) called "1-7" 8-QAM [
-PAM (
) and the noncircular 
-QAM scheme that we call noncircular 8-QAM. In the following, in order to implement the rhombic-transformation-based constellation-optimization strategy, we resort to the rectangular 
-QAM; we remember that, unlike 
-QAM, such a scheme cannot be transformed into the conventional uniform 
-PAM, but in the nonoptimum nonuniform 
-PAM (the optimality of uniform PAM over additive white Gaussian noise has been shown in [
, we have set 
and we have plotted the SERs achieved by both the suboptimum strategy (26) and the optimum strategy (14) versus 
, for 
and for different values of 
(
); moreover, for each point of Figure 
-PAM when 
and outperform the conventional nonadaptive transceiver employing the QPSK modulation jointly with the linear MMSE receiver. Note also that as 
, the chosen value of 
does not affect the performance.
; for each point, the letter specifies the constellation (of those in Figure 
; more specifically, in Figures 
when 
and 
, respectively. Figure 
for 
and 
. The optimum strategy provides performance gain over the non-adaptive transceiver employing the conventional rectangular 8-QAM by using the "1-7" 8-QAM and the noncircular 8-QAM for smaller values of 
, and, as 
, by using the 8-PAM. In such a case, the performance difference between the suboptimum strategy and the optimum one is important, especially for large values of 
, since the suboptimum one employs the non-uniform 8-PAM. Such a result was expected since, when 
increases, the optimum strategy can exploit a number of degrees of freedom significantly larger than the suboptimum strategy.
; for each point, the letter specifies the constellation (of those in Figure 
, an architecture switching between the 
-QAM and the 
-PAM can provide a good trade-off between performance and complexity. Instead, when 
, the transceiver should switch among the Foschini&All, the noncircular 
-QAM and the 
-PAM.
the channel has memory 
and its taps 
are randomly generated according to a complex-valued circularly-symmetric zero-mean white Gaussian process with unit variance (i.e., 
and 
); 
the WL MMSE equalizer has 
taps; 
the results have been averaged over 500 independent channel realizations. We compare the performances achieved by four architectures: (I) the OPTimum-based architecture (OPT-based) that selects 
and 
in order to minimize the symbol error rate (i.e., 
, instead of 
); (II) the QAM-based architecture adopting the conventional circularly symmetric 4-QAM constellation; (III) the PAM-based architecture utilizing the conventional rotationally variant 4-PAM (
which corresponds to the maximum WL gain); (IV) the two-choice-based architecture that switches between the 4-QAM and the 4-PAM constellations according to (27). For clarity, we point out that the solution of (26) loses about 0.3
in comparison with the OPT-based one; we consider the OPT-based architecture in order to provide a lower bound to the SER. The OPT-based and the two-choice-based architectures, unlike the QAM-based and the PAM-based ones, require the existence of a feedback channel between the receiver and the transmitter for constellation adaptation; however, the two-choice-based architecture only needs to transmit a binary information on such feedback channel.
(in dB). The OPT-based architecture outperforms all the others and provides an SNR-gain over the nonoptimized architectures of almost 
for a 
. Interestingly, the two-choice-based architecture performs well loosing only 
in comparison with the OPT-based one. Let us also note that the PAM-based architecture performs poorly for low 
, but, as the 
increases, it outperforms the QAM-based one.
, 
, 
) is not larger than 21
. Moreover, Figure 
, that each architecture loses 
in comparison with the OPT-based one for a given target SER under the condition that the SNR is not larger than 21
.
in comparison with the OPT-based architecture to compensate for the reduction of the dimension of the signal space. Note also that such an SNR loss, which is uniformly distributed between 0 
and 4 
when the target SER is 
and 
, assumes often two specific values (0 
and 3 
) for a target SER equal to 
. In practice, the PAM architecture achieves optimum performance on 50
of the channels where the linear equalizer performs unsatisfactorily and the WL processing gain, specific to rotationally variant constellations, compensates for the smaller minimum-distance of the PAM constellation.
on 37
of the channels. This is due to the circular symmetry of the constellation that does not allow one to improve by means of the WL processing the unsatisfactory performance of the linear equalizer.
) associated to the 
-order constellation exhibiting, therefore, an unaffordable computational complexity for high-order constellations. To overcome such a problem, a second design method based on a rhombic transformation of a fixed alphabet of order 
is proposed. It performs the optimization of only two parameters (instead of 
) leading to a huge reduction of the computational complexity for large 
. For low-order constellations, the simulation results show that the two techniques are practically equivalent in terms of symbol error rate; moreover, they also show that a WL MMSE transceiver with constellation adaptation is clearly superior to the same equalizer with fixed constellation. Finally, for 
, it has been shown that the method that switches between a real-valued and a complex-valued constellation exhibits a limited performance loss versus the optimum adaptation scheme, while it achieves a strong reduction of the computational complexity and it requires to feed back to the transmitter only a binary information.