Constellation Design for Widely Linear Transceivers

2.1. System Model

Let us consider the following finite-impulse-response (FIR) baseband-equivalent noisy communication channel

(1)

where the transmitted symbols 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.

At the receiver side, the feedforward-based equalization is performed by processing the block of received samples which, in a matrix notation, can be written as follows:

(2)

According to the previous assumptions, the following correlation and pseudocorrelation matrices can be written as

(3)

where is the (possibly) nonnull noise pseudocorrelation (if , then the noise is circularly symmetric).

2.2. Feedforward-Based MMSE Equalizer

Since the transmitted sequence 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 [10]. Therefore, we resort to the two FIR filters and that process the received vector and its complex conjugate version , respectively. The optimum filters and minimizing the mean square error are given by [16, 18]

(4)

(5)

where 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.

Since the optimum equalizer and, hence, its performance depends on the pseudocorrelation 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

(6)

(7)

Since 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

(8)

Since 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.

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:

(9)

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):

(10)

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

(11)

where denotes + where and , and, for and ,

(12)

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:

(13)

and, therefore, the optimum constellation can be approximated with the solution of the following problem:

(14)

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

(15)

By zeroing the gradient of the Lagrangian

(16)

one has that the locally optimum satisfies the following equation:

(17)

with

(18)

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:

(19)

or, more compactly (the compact expression is introduced for notation simplicity whereas the matrix form is utilized to understand the physical meaning),

(20)

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

(21)

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. (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. (2)

   the reduced implementation complexity of the transmitter stage; in fact, the symbol-mapping is implemented by means of the linear transformation (19);

    
3. (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

(22)

with

(23)

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

(24)

where

(25)

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:

(26)

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:

(27)

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

(28)

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 .

4.1. Fixed Channel

In this section, we compare the performances of the constellation design procedures (26) and (14) in terms of SER. In our simulations, we solve (26) by means of an exhaustive search over 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 [16], the transmission over a two-tap channel 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 .

Let us first plot in Figure 2 some of the optimum constellations obtained during our simulations when solving the optimization problem (14) over the considered channel model; moreover, we plot the suboptimum constellation utilized to implement our suboptimum strategy and the -PAM constellation obtained by applying to it the rhombic transformation. As in [2], we have found many local optima, some of them were rotated version of the constellations of Figure 2 while others appeared as their rhombic transformation. For 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 [2], the -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 [23]).

In Figure 3, with reference to the case , 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 3, the constellation typically obtained by the optimum procedure is specified by the letter used to denote it in Figure 2. The results show that the two strategies have the same performance: more specifically, both strategies switch to the -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.

In the next experiments, we have addressed the constellation optimization when ; more specifically, in Figures 4 and 5 we have considered the transmission over when and , respectively. Figure 4 reports the SER achieved by both the suboptimum strategy and the optimum strategy versus 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.

Finally, we observe that, when , 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.

4.2. Random Channel

In the following simulations, we assume that 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 Figure 6, the SERs of the considered architectures are plotted versus the (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.

In the next experiment, we compare the considered architectures by evaluating their capability to guarantee the required quality of service (QoS). More specifically, in Table 1, we report the percentages of the channels over which the SNR required to achieve the target SER (assumed to be , , ) is not larger than 21 . Moreover, Figure 7 reports the probability, say , 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 .

Target SER	OPT-based	QAM-based	PAM-based	Two-choice-based