Theory and Applications of Active Constellation Extension

Active constellation extension (ACE) was originally developed to reduce the peak-to-average-power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems with quadrature amplitude modulation (QAM). Alternatively, ACE can be a promising approach for optimizing various possible aspects of the 2-D constellation signals. However, the literature lacks a rigorous theoretical framework for ACE, and hence its applications are currently limited. This study proposes a formal mathematical framework and theory for ACE for the general case of the 2-D constellation signals. The proposed framework is used to demonstrate the roles of ACE in reducing the PAPR in the OFDM systems with QAM and improving the error performance and throughput of phase shift keying under fading channels.


I. INTRODUCTION
Active constellation extension (ACE) was first proposed in 2003 as a means of reducing the peak-to-average power ratio (PAPR) in quadrature amplitude modulated orthogonal frequency division multiplexing (OFDM) systems [1]. The principle of ACE is to move the quadrature amplitude modulated frequency-domain symbols within so-called permissive regions so as to emulate the corresponding clipped-version signal. In [1] and [2], permissive regions are described (without formal definition) as the regions within which the outer constellation points of 4-ary quadrature amplitude modulation (4-QAM) and 16-QAM signals can be moved without decreasing the minimum distance of the constellation points. It was shown in [1] and [2] that ACE achieves a significant reduction in the PAPR at the cost of a limited signal power growth in OFDM systems to less than 1 dB.
In tackling the PAPR problem, many ACE-based approaches are proposed following the concept of moving signals within permissive regions to reduce PAPR [1]- [17]. Much effort is spent in reducing the computation complexity, such as smart gradient project ACE [1], [3], least square The associate editor coordinating the review of this manuscript and approving it for publication was Jie Tang . approximation ACE [4], ACE with frame interleaving [5], clipping-based ACE [6], ACE with the bounded distortion [7], ACE with parabolic peak cancellation [8], [9], a constellation extension-based ACE [10], ACE with signals of high-order constellation [11], and neural network aided ACE [14]. Apart from OFDM, filter-bank multi-carrier (FBMC) is also considered as the signal waveform for ACE to reduce PAPR [12], [13]. 1 In addition to reducing PAPR in OFDM with QAM, ACE has also been used in DVB-T2 systems [15], in space-time block coded (STBC) systems to improve the decoding performance [16], and in encrypted transmission to ensure security [17].
Although the above-mentioned research of ACE has done much progress in various system performance [1]- [16], the approach of ACE was again given descriptively without formal definition. Due to the enhanced degree of freedom which it provides, ACE enlightens a promising opportunity for optimizing various aspects of QAM systems, including power/resource allocation, precoding, channel pre-distortion, and preamble insertion for synchronization purposes.
However, to realize this potential, a rigorous theoretical framework for ACE is required for arbitrary constellation signals. Accordingly, this study proposes a mathematical framework and the corresponding theories to perform ACE for any generalized 2-D constellation. The proposed framework is demonstrated by illustrating the roles of ACE in reducing the PAPR in the OFDM systems with QAM, especially when specific constellation size is adopted, and in improving the error performance and throughput of phase shift keying (PSK) under fading channels.
In this paper, we propose the mathematical framework of ACE for any 2D-constellation in Section II. Specific constellation is adopted following the proposed ACE mathematical framework as an example for reducing PAPR of the OFDM signal in Section III. Sections IV proposes a new pre-equalization model based on the ACE mathematical framework to fight the flat fading channel, and further extention in Section V is also proposed to take advantage of the proposed mathematical framework to seek for extra throughput. Section VI concludes the paper.

II. FORMULATION OF ACTIVE CONSTELLATION EXTENSION
Consider a set of M 2-D constellation points {X 0 , X 1 , . . . , X M −1 }, which can be regarded as the modulated symbols mapped from the input information. Specifically, the decision region of X i is denoted as R(X i ) and represented by a Voronoi cell [18]- [19] as (1) where C represents the set of all complex numbers and |x| denotes the Euclidean norm of the complex number x.
Notably that each Voronoi cell is simply connected, i.e., every closed path in R(X i ) encloses only points belonging to R(X i ), and the boundary of the cell comprises piecewise straight line segments [20]. For convenience, in classifying the constellation points in the Voronoi cells, let a bounded set first be defined as follows: where P is a finite real number. Definition 2: When all the boundary segments of a Voronoi cell are bounded, the cell is said to be closed and the corresponding constellation point is referred to as an inner point. By contrast, when certain boundary segments of the cell are unbounded, the cell is said to be open and the corresponding constellation point is referred to as an outer point.  Proof: If there exists a straight line which connects all the constellation points in the provided constellation, every Voronoi cell within the constellation is a strip-shaped unbounded region [20] (also shown in Fig. 2(a)). In such a case, the two constellation points located at the edges have only one unbounded boundary segment in their respective open cells, whereas the other constellation points have two unbounded boundary segments (also shown in Fig. 2(a)). For the case where no such straight line exists, the boundary of a certain open cell must consist of just two unbounded boundary segments (since an unbounded 2-D region for such a case is also simply connected [20], and can thus have only two outmost boundary segments reaching to infinity no matter how the bounded boundary segments are connected).
Proposition 2: The shared boundary segment of two neighboring Voronoi cells coincides with the perpendicular bisector of the line connecting the constellation points of the cells.
Proof: When the equality in (1) holds, the solution of the equality forms the whole or partial perpendicular bisector of points X i and X j . The case of a whole perpendicular bisector applies to a constellation with more than two signal points, which can be connected by a straight line. By contrast, the case of a partial perpendicular bisector applies to a constellation with more than two signal points which cannot be connected by a straight line. Notably, if the boundary segment of two neighboring cells consists of just one point, then this point is the midpoint of the two neighboring constellation points (partial perpendicular bisector). An example is given in Fig. 2(b) where the partial perpendicular bisector is the midpoint for the neighboring points X o 0 and X o 2 . Consider a certain outer point X o i . Let the permissive region of X o i be denoted as R p (X o i ) and be defined as follows: is an unbounded region and has two permissive boundary segments which start from point X o i and run parallel to the one or two unbounded boundary segment(s) of R(X o i ). Fig. 1(b) presents an illustrative example of a specific 2-D constellation with M = 5, in which the permissive regions are indicated by the shaded regions. Using Definition 3, the permissive regions for any 2-D constellation can be easily obtained. One of the most fundamental properties of a permissive region is given in the following.
Proposition 3: The permissive region of a certain outer point , forms an open region containing points providing larger Euclidean distances to X j than X o i to X j , i.e., Proof: Since the shared boundary of two neighboring Voronoi cells coincides with the whole or partial perpendicular bisector of the associated constellation points (see Proposition 2), the two unbounded boundary segments of R p (X o i ), i.e., when the equality holds in (3), are parallel to the unbounded boundary segments of R(X o i ) and are thus perpendicular to the line connecting the two corresponding neighboring constellation points X j and X k , respectively. Consequently, for x ∈ R p (X o i ), angles xX o i X j and xX o i X k are obtuse angles and thus are opposite to the longest sides of the triangles xX o i X j and xX o i X k , respectively. As a result, Note that for an inner point, the corresponding permissive region is an empty set, i.e., R p (X 0 ) ∈ φ (empty set), as given in the following proposition.
Proposition 4: A closed cell has no permissive region. Proof: For any point x in the Voronoi cell of an inner point X i , a neighboring constellation point X j can always be found such that In other words, no point in the Voronoi cell of X i satisfies (3).  It is seen that the magnitude of the points in the permissive regions increases with an increasing constellation size of QAM by comparing Figs. 1 and 3, which results in fewer choices of points with smaller magnitude.

III. USE OF ACE TO REDUCE PAPR IN OFDM SYSTEMS
Consider an OFDM system with N subcarriers. The frequency-domain modulated signal can be expressed as can be obtained as and A is the clipping level. The difference between the clipped signal s and the original signal s is defined by the clipped-off portion signal, i.e., c clip s−s.
In general, a frequency-domain process over c clip provides the means to approach s = s + c clip (12) without adversely affecting the error performance of the original signal. Thus, in ACE, a discrete-time Fourier transform operation over c clip is performed to obtain C clip is then moved in the respective permissive regions to form the signal in accordance with the following constraints: is an outer point and Applying the convex optimization algorithm [21]- [23], the following optimization problem with step size ς can be introduced to find the minimum PAPR: where R + is a set of non-negative real numbers and Notice that in (15) the region R p (S[n]) provides a larger Euclidean distance than S[n] to any other constellation point. Thus, when the clipped points C clip [n] for n = 0, 1, . . . , N −1 in (13) falling within R p (S[n]), C clip [n] + S[n] for n = 0, 1, . . . , N − 1 can be used to emulate a clipped OFDM signal without destroying the orthogonality between the OFDM subcarriers, and the minimum PAPR can be obtained through (16).
Consider the OFDM signal with 256 subcarriers optimized based on the ACE approach as in (16). The PAPR performance for the ACE aided OFDM with M -QAM signals is shown in Fig. 4, where the OFDM signals are oversampled 8 times to approximate the analog PAPR and the clip level A = 6.85dB above the average power was used for 16-QAM, and A = 7.23dB was used for 64-QAM to achieve similar PAPR without ACE. Obviously, the PAPR performance decreases with the increase of the constellation size M , which is reasonable that the permissive regions R p (S[n])'s to the regions of interior points shrink in a smaller ratio for the OFDM signal with higher constellation size, resulting in less ACE flexibility and therefore less PAPR reduction [1]. Further, the nontypical 5-QAM is examined in Figs. 4 and 5, where the above-mentioned PAPR performance trend is also verified.

IV. USE OF ACE TO PRE-EQUALIZE THE FADING CHANNEL FOR M-PSK
Without loss of generality, we consider a slow and flat Rayleigh fading channel with its channel gain denoted by h. 2 We assume both the transmitter and receiver are perfectly synchronized and can perfectly estimate the channel status. The received signal r can be represented by where w is the additive Gaussian white noise (AWGN) distributed with zero mean and unit variance. In order to mitigate the impact of the fading in the channel, equalization at the receiver is a common approach to counteract the fading channel gain, however, the equalized signal suffers from noise enhancement which also seriously degrades the error performance of the system. To avoid such trade-off, ACE can be utilized at the transmitter to perform pre-equalization within permissive region in a way to invert the magnitude of the channel gain, and the equal gain eqaulization is adopted at the receiver to correct the distorted phase. Through the equalization at both transmitter and receiver, the fading gain can be removed completely without affecting the statistics of AWGN. Thus, it seems like that the proposed approach alters a fading channel to an AWGN one.
To obtain maximum benefits from the adoption of ACE, we consider the M -ary phase shift keying (M -PSK) signal, where every modulated M -PSK signal x is free to be moved within R p (x) as shown in Fig. 6 with M = 8. Note that every constellation points in M -PSK are outer points. Direct inversion of the channel gain at the transmitter is not necessary, we only amplify the transmitting signal when |h|x leaves R p (x), i.e., the permissive region of x, where |h| represents the magnitude of h. Thus, the transmitted signal s can be designed to be Qx with Meanwhile, the receiver counteracts the phase of the fading channel, the received signal r is thus equal-gain equalized to be v as where h = h/|h| represents the phasor of h, and y is the circularly symmetric AWGN. We assume the fading channel gain and AWGN are independent. The mean and variance of y, denoted by E{y} and Var{y}, respectively, are identical to those of w due to h * being uniform distributed. We can further simplify (18) as Through the maximal likelihood detection, ACE-aided M -PSK under the flat Rayleigh fading channel exhibits identical error performance to M -PSK under AWGN at the receiver. However, considering practical implementaion, a proper outage threshold should be set to prevent infinite power compenstaion. Apperantly, ACE compensates the fading channel, but at the sacrifice of the extra power ϒ derived by where µ is the outage threshold, f h (h) denotes the Rayleigh distribution, and Ei (x) is the exponential integral. With an outage threshold µ, the transmitter gives up compensating the signal when in the deep fade, i.e., |h| < µ. The error performance of the ACE-aided 8-PSK signal is provided on the flat Rayleigh fading channels in Fig. 7 when σ 2 = 1. 3 Under different outage probability, the error performance of ACE-aided PSK is improved significantly comparing to that 3 Although the channel estimation error is assumed to be perfect throughout, the persmissive region R p (x) of the transmitting M -PSK signal x permits ±π/M estimation error of the phase of the channel, which is expected to outperform the conventional pre-equalization approaches when phase estimation error existed, however, this subjects to future research. of the PSK signals in Rayleigh fading channel as shown in Fig. 7. If we consider the scenario with aggressive compensation of the fading channel (i.e., µ is set to be 0.01), around 6dB gain in SNR for the ACE-aided PSK system over the PSK one without ACE is observed at the bit error rate (BER) being 10 −2 , and when the BER is lower, much more gain in SNR can be obtained for the ACE-aided PSK system. Alternatively, if we compensate the channel with µ = 0.2, the error performance of the ACE-aided PSK system approaches to that of PSK in AWGN though at a sacrifice of the outage of signal trasmission. Here, the outage transmission probability can be obtained through When |h| is Rayleigh distributed with σ 2 = 1, µ = 0.01, 0.05, 0.1, 0.2, P out (µ) = 4.9 × 10 −5 , 1.2 × 10 −3 , 0.005, 0.0198, respectively. In conclusion, a reasonable outage (e.g., P out (0.1) = 0.005) tradeoff provides comparable error performance for the ACE-aided PSK system with fading to the PSK system in AWGN.
This alternative ACE approach can also be applied to M -QAM, however, the interior points increase along with the increase of the constellation size M when M > 4, which leads to less outer points that can be moved to convert the channel gain. Since the interior points are not allowed to be moved as provided in Proposition 3, the error performance gain for M -QAM can be improved but is not so significant comparing to that of M -PSK.

V. FURTHER EXTENSION BASED ON THE PROPOSED ACE MATHEMATICAL FRAMEWORK
A new design is proposed in this section to utilize the proposed ACE mathematical framework to seek for more advantage in signal transmission. In Section IV, the PSK signal is moved primarily based on the inverse of the fading gain |h| when |h| ≤ 1, whereas extra benefits can be further explored by properly devising the transmitting signal in the permissive region based on the proposed ACE mathematical framework. From Fig. 8, when we move the signal along the arc with the radius being the inverse of the fading gain within the considered permissive region, the error performance of the pre-equalization system (as provided in Section IV) remains the same. Thus, there are many possible signal designs can be performed on the arc within the considered permissive region. To enlight the field of research, we demonstrate an approach to gain extra information based on the developed ACE mathematical framework in the following.
To extend the signal design from the proposed ACE with M -PSK system, two possible signal points x a and x b in the considered permissive region R p (x) are chosen to be the intersection of the two boundary lines of R p (x) and the arc with radius of |h| −1 , where the boundary lines of R p (x) are derived from Propositions 1 and 2. In Fig. 8, the hollow and solid stars represent the manitude of |h| and |h| −1 , respectively, and the two red points x a and x b can convey extra information or act as pilot siganls. Here, we try to calculate the extra information by moving the signal to either x a or x b . Then, the angle ϕ in Fig. 8 is derived based on the proposed ACE mathematical framework that the boundary lines for Vonoroi and permissive regions are parallel to each other as proved in Proposition 2 as In (19), when |h| is small, ϕ tends to π M , and when |h| = 1, ϕ = 0, which implies when the fading is severe, the adoption of extra signal points benefits from enlarging the Eclidean distance between each other. With (19), the Euclidean distance between x a and x b can be immediately calculated as 2 √ γ sin(ϕ)/|h| (20) with γ being the signal-to-noise power ratio for the PSK symbol. Consider transmitting extra one bit information through x a and x b . The extra capacity denoted by C cap can be represented as where p(·) and E[·] denote probability and expectation, respectively. Under the assumption of flat Rayleigh fading, a pre-determined channel gain outage µ, and equal apriori probability for the PSK symbols, the extra capacity can be derived to be The extra capacity in (22) is evaluated in Fig. 9 for different SNR. Obviously, when the constellation size M increases, the extra capacity decreases due to the Euclidean distance being smaller for the two boundary points x a and x b . Also, when considering higher µ (outage threshold), the extra capacity decreases for lower probability enabling the boundary points x a or x b . Around 0.2 bits/s/symbol is gained by modifying the ACE pre-equalized PSK system as shown in Fig. 9, but the application of the proposed ACE mathematical framework is not limited to obtained extra information. The design of an extra bit in this section provides an example to utilize the proposed ACE mathematical framework. A joint design of PAPR and pre-equalization, pilot insertion, and coding in the permissive region is also possible to explore various applicatons to achieve better system performance based on the proposed ACE mathematical framework.

VI. CONCLUSION
We propose a mathematical framework for ACE for the case of a generalized 2 -D constellation signal. The proposed framework has been used to demonstrate the applications of ACE in improving error performance and system throughput. The proposed framework provides a useful basis for applying ACE to a variety of possible applications in modulation field with the 2-D constellation.