### 2.1. Construction of Cross-QAM

Since we deal with cross-QAMs in this paper, we define with , that is, . As mentioned in [1], by introducing "block" parameter

cross-QAM constellation can be constructed by a square block array with the 4 corner blocks deleted, each block with uniform distributed points, as shown in Figure 1.

### 2.2. Decision Boundaries for Symbols in Cross-QAM

Since the quadrature components of cross-QAM are not independent as in square QAM, the optimal decision regions are not all rectangular. As shown in Figure 2, where the dots represent signal points while the lines indicate decision boundaries, there are three types of points: interior symbols, edge symbols, and corner symbols. The decision regions of interior symbols and edge symbols are closed square and semi-infinite rectangular, respectively. But that of the corner symbols are slightly complicated and can be represented by a combination of vertical, horizontal, and lines. According to the symmetry, it is enough to consider the symbols in one quadrant of the cross-QAM constellation. The numbers of the three types of points in one quadrant of the constellation are, respectively, given as

### 2.3. System Parameters for -ary Cross-QAM

We will define some parameters in order to make our expressions simpler and compact. Let denote half of the minimum Euclidean distance between adjacent symbols in the constellation, and let denote the two-sided power spectral density of the zero-mean AWGN (i.e., its variance ). Especially, the exact SEP expressions will be written in terms of the Gaussian -function

and a well-known integral function related to the alternate representation of one- and two-dimensional joint Gaussian -functions [21–24]

In particular, [22, equation ] and [23, equation ], both for .

Note that, in addition to the advantage of having finite integration limits, the form in (4) has the argument contained in the integrand rather than in the integration limits as is the case in (3), and it also has an integrand that is exponential in the argument , so that it can be numerically evaluated with more accuracy. Moreover, the form in (4) has some interesting implications with regard to simplifying the evaluation of performance results related to communication problems, for example, as seen later to the SEP performance evaluation over fading channels, wherein the argument of the -function is dependent on random system parameters and, thus, requires averaging over the statistics of these parameters.

To simplify the mathematical expressions, in the following derivation the argument of the above two functions (3) and (4) sometimes will be expressed as a multiple of

which denotes the normalized least distance (in noise standard deviation) from a signal point to a decision boundary.

Assuming that the signal points are equally probable and according to the symmetry of the constellation, it can be easily shown that the average symbol energy for cross -QAM constellation is given by

where

Since the symbol's signal-to-noise ratio (SNR) can be written as

thus,

So, we can also leave the SEP expression in terms of as mentioned above.

### 2.4. Overview of SEP Approximations in AWGN

For nonrectangular -ary QAM signal constellations, Proakis has given an obvious upper bound in [15, page 279] as

where is the minimum Euclidean distance between signal points, and for uniform cross-QAM. This bound may be loose when is large. In such a case, Proakis suggested replacing by , the largest number of neighboring points that are at distance from any constellation point. Obviously, for cross-QAM, . So, (10) can be reproduced for cross-QAM as

Alternately, Gilbert approximation [1, equation (1)] to the SEP for any -ary QAM can also be used for cross-QAM, and it is given as

where is the average number of nearest neighbors for a symbol in the constellation, and for cross-QAM (In [1], the expression for appears erroneously as [16].) when . Note that, as increases, increases and approaches . In fact, the principle behind the above two approximate expressions is very intuitive since they can be interpreted as the sum of the probabilities that a given point is mistaken for its neighbors. At the same time, since the sum has recalculate some error regions, both the above two expressions overestimate the actual SEP as shown later.