Golden Angle Modulation
Abstract
Quadrature amplitude modulation (QAM) exhibits a shapingloss of , ( dB) compared to the AWGN Shannon capacity. With inspiration gained from special (leaf, flower petal, and seed) packing arrangements (spiral phyllotaxis) found among plants, a novel, shapeversatile, circular symmetric, modulation scheme, the Golden Angle Modulation (GAM) is introduced. Discshaped, and complex Gaussian approximating bellshaped, GAMsignal constellations are considered. For bellGAM, a highrate approximation, and a mutual information optimization formulation, are developed. BellGAM overcomes the asymptotic shapingloss seen in QAM, and offers Shannon capacity approaching performance. Transmitter resource limited links, such as space probetoearth, and mobiletobasestation, are cases where GAM could be particularly valuable.
I Introduction
Modulation formats, of great number and variety, have been developed and analyzed in the literature. Examples are PAM, square/rectangularQAM, phase shift Keying (PSK), StarQAM [1], and amplitudePSK (APSK) [2]. SquareQAM, hereon referred to as QAM, is the defactostandard in existing wireless communication systems. However, at high signaltonoiseratio (SNR), QAM is known to asymptotically exhibit an 1.53 dB SNRgap (a shapingloss) to the additive white Gaussian noise (AWGN) Shannon capacity [3]. This is attributed to the square shape, and the uniform discrete distribution, of the QAMsignal constellation points. Geometric and probabilistic shaping techniques have been proposed to mitigate the shapinggap [3]. An early work on geometric shaping is nonuniformQAM in [4]. More recent works in this direction are, e.g., [5, 6, 7, 8]. Existing work on modulation schemes have, in our view, not completely solved the shapingloss problem, nor offered a modulation format practically wellsuited for the task. This leads us to examine new modulation formats.
Inspired by the beautiful, and equally captivating, cylindricalsymmetric packing of scales on a cycad cone, the sphericalsymmetric packing of seeds on a thistle seed head, or the circularsymmetric packing of sunflower seeds, we have recognized that this shapeversatile spiralphyllotaxis packing principle, found among plants, is applicable to modulation signal constellation design. Based on the spiral phyllotaxis packing, the key contribution of this letter is proposing a novel, shapeversatile, highperformance modulation framework  the Golden angle modulation (GAM). We consider discrete modulation, with equiprobable constellation points, that approximate a r.v. with continuous complex Gaussian (bell)shaped distribution, as well as a baseline case, with a discshaped distribution. We find, as expected, that the MIasymptote of geometricbell GAM (GBGAM), for increasing number of constellation points, coincide with the AWGN Shannon capacity. This also supports the complex Gaussian communication signal assumption, used in many performance analysis works, for the lowtohigh SNRrange.
Ii Golden Angle Modulation
The core design of GAM is given below.
Definition II.1
(Golden angle modulation) The th constellation point of GAM has the probability of excitation , and the complex amplitude
(1) 
where is the radius of constellation point , denotes the golden angle in rads, and .
We will assume that for an increasing spiral winding. For the probability, it may be equiprobable, , or dependent on index . The later, where , corresponds to GAM with probabilistic shaping and is explored extensively in [9]. Hence, a constellation point, is located turns () relative to the previous constellation point. Replacing , with , the golden ratio, or its fractional part, gives an equivalent spiral winding, but, in the opposite direction. Note that phase rotation value deviating with approximately 1% from the golden angle (ratio) destroys the locally uniform packing. The mathematical design of the phase rotation in Def. II.1 is inspired from the work by Vogel, [10], who described an idealized growth pattern for the sunflower seeds, (in our notation). Vogel did however not consider modulation. More importantly, a key insight here, enabling the approximation of a complex Gaussian pdf, is to not restrict the radial function to as in [10]. Allowing for arbitrary radial growth of , gives the geometric shaping capability, and allowing for arbitrary probabilities , gives the dimension of probabilisticshaping.
GAM features the following advantages:

Natural constellation point indexing: In contrast to QAM, APSK, and other, without any natural index order, GAM enables a unique indexing based on signal phase, , or magnitude, , alone.

Practically nearideal circular design: A circular design can offer enhanced MI, distance, symbol error rate and PAPR performance over a squareQAM design.

Shapeflexibility in radially distribution of constellation points while retaining an evenly distributed packing: We recognize this as a central feature of GAM which allows approximation of (practically) any radialsymmetric pdfs.

Naturally lends itself for circularsymmetric probabilistic shaping: We recognize this as a central feature of GAM which allows approximation of (practically) any radialsymmetric pdfs.

Any number of constellation points, while retaining the overall circular shape: This gives full flexibility, e.g., in alphabet size of a channel coder, or a probabilistic shaper.

Rotation (and gain) invariant: The uniquely identifiable gain and rotation of signal constellation could, e.g., allow for blind channel estimation.
Some comments. First, the index range is not necessarily limited to . Second, GAM has a complex valued DC component. Possible remedies, if a problem, is to subtract the DC component, or negate every second symbol. Third, while hexagonal packing is the densest 2Dpacking (as desirable in the high SNRrange), it does not share the above listed features of GAM.
Iia DiscGAM
We first introduce discGAM, i.e. GAM without shaping, below. Besides its own merits, this also serves as a baseline to shaping. If the expression for is altered (fixed), but is fixed (altered), geometric (probabilistic) shaping result.
Definition II.2
(DiscGAM) Let , be the number of constellation points, and be the average power constraint. Then, the complex amplitude of the th constellation point is
(2)  
(3) 
In the above, (3) is found from the power normalization condition, .
We illustrate the discGAM constellation in Fig. 2, where its approximate discshape is clearly depicted.
A few remarks about discGAM. The entropy is simply . The PAPR is when . From PAPRpointofview, this makes discGAM favorable over QAM, since dB. (PSK has dB, but with very poor MIperformance for ). Letting , QAM asymptotically requires ( dB) higher average power than discGAM for the same average constellation point distances . Also when , QAM asymptotically requires ( dB) higher peak power than discGAM for the same average distance between (uniformly packed) constellation points.
IiB Geometric bellGAM
Next, two geometric shaped GAM schemes are introduced.
IiB1 Highrate approach
This first GBGAM design builds on the inverse sampling method (a highrate (HR) approximation), and approximates a complex Gaussian distributed r.v.
Theorem II.1
(GeometricbellGAM (HR)) Let , be the number of constellation points, and be the average power constraint. Then, the complex amplitude of the th constellation point is
(4) 
Proof:
The proof is given in Appendix A. \qed
We illustrate the geometricbellGAM signal constellation in Fig. III, and note that it is densest at its center, i.e. where the pdf for the complex Gaussian r.v. peaks.
We note the following characteristics. When , since , we get . The entropy is . The PAPR is , which tends to infinity with . This is expected as the PAPR of a complex Gaussian r.v. is infinite. Note here that the index range in (4) can not be , but is chosen as , as otherwise.
IiB2 MIoptimization of GBGAM: FormulationG1
In this method, we let , and vary in order to maximize the MI, for a desired SNR . The optimized signal constellation points are . More formally, allowing for a complex valued output r.v. , and a complex valued (discrete modulation) input r.v. , the optimization problem is
(5)  
subject to  
Remark II.1
For some applications, the PAPR is of interest. A PAPRinequality constraint, , being the target PAPR, can be amended to the optimization problem. Other constraints may also be of interest.
When optimizing GAM in AWGN, the MI is , where , , integrating over the complex domain, with .
Due to the nonlinearities in the MI, the optimization problem in formulationG1 is hard to solve analytically. Hence, a numerical optimization solver is used in Section III.
Iii Numerical Results
Here, we now examine the MIperformance of GAM (with its irregular cellshapes and varying cellsizes) by using a MonteCarlo simulation approach. For the optimization in formulationG1, MATLAB’s fmincon function is used together with numerical integration for the MI.
In Fig. 3, we illustrate the MI performance for GBGAM (HR) together with the Shannon capacity. As expected, for larger constellation size , a greater overlap with the Shannon capacity is seen. The MI approximation is good up to about . Naturally, the MI is limited by the entropy of the signal constellation. We observe an intermediate SNR region, a region where the MI does neither reach the channel capacity, nor the entropy of the signal constellation. It is in this SNRregion that further constellation optimization, i.e. formulationG1 , is of interest. The MI for GBGAM (G1) is also shown, but due to optimization complexity only, for .
In Tab. I, the MI of GBGAM with the HR, and G1, formulations are given. As expected, the optimized scheme, G1, perform better than HR. While the MIimprovements are modest, the optimization formulation is substantiated. For G1, when the MI is as large as the constellation entropy, we have observed that the signal constellation approaches the discGAM solution, whereas in the lowMI region, we have noted that the optimized signal constellation approaches the HR GBGAM solution. It is further observed that the extreme magnitudes of the highest constellation indices for GBGAM (HR), as seen in Fig. , are attenuated with the MIoptimization and leads to improved PAPR performance.
In Fig. 4, we compare discGAM, GBGAM (HR), and QAM, together with the Shannon capacity when the MI is nearly as large as the constellation entropy. We note the (expected) dB SNRgap for QAM to the Shannoncapacity. The (expected) SNRgap from QAM to discGAM is dB. When the MI is approximately as large as the constellation entropy, discGAM, and QAM, perform slightly better than GBGAM (HR) scheme. This observation prompted us to develop the MIoptimization formulationG1.
SNR  AWGN capacity  HR  G1 

dB  2  1.921  1.961 
dB  4  3.440  3.549 
dB  3.828  3.926 
Iv Summary and Conclusions
In this letter, we have introduced a new modulation format, the golden angle modulation. With geometric (or probabilistic) shaping, GAM can approximate virtually any circularsymmetric pdf. We studied geometricallyshaped GAM to approximate the pdf of a continuous complex Gaussian r.v. A highrate solution, was developed. We also introduced the notion of MIoptimized GAM under an average SNRconstraint, and optionally also a PAPRinequalityconstraint. The MIperformance was observed to asymptotically approach the Shannon capacity as the number of signal constellation points tended to infinity. In contrast to QAM’s 1.53 dB shapingloss, GAM exhibit no asymptotic loss. The complex Gaussian communication signal model assumption, as often used for performance analysis, was substantiated from a practical modulation pointofview.
We believe that GAM may find many applications in transmitterresourcelimited links, such as space probetoearth, satellitetoearth, or mobiletobasestation communication. This is so since high datarate is desirable from the power, energy, and complexitylimited transmitter side, but higher decoding complexity is acceptable at the receiver side. Certain cases may also benefit from using the PAPRinequality constrained optimization. With bellGAM, the power reduction in the highMI regime could be up to 30% (), which is of environmental interest. Moreover, cellularsystem operators, could potentially also reduce energy consumption (and cost) with up to 30%. It is hoped that, GAM, with its attractive characteristics and performance, could be of interest for most, wireless, optical, and wired, communication systems.
a Proof in Theorem ii.1
Proof:
The distribution in phase, for the constellation points, is already given by the factor. However, the radial distribution need to be determined. A complex Gaussian r.v. with variance is initially considered. The inverse sampling method, assumes a uniform continuous pdf on . We modify the inverse sampling method and use a discrete uniform pmf at steps , where is assumed large (for the highrate approximation).
Setting , and solving for , yields
Thus, the general solution for the signal constellation has the form . The constant is given by the average power constraint as follows,
References
 [1] L. L. Hanzo, S. X. Ng, T. Keller, and W. Webb, Star QAM Schemes for Rayleigh fading channels. WileyIEEE Press, 2004, pp. 307–335.
 [2] C. Thomas, M. Weidner, and S. Durrani, “Digital amplitudephase keying with Mary alphabets,” IEEE Transactions on Communications, vol. 22, no. 2, pp. 168–180, February 1974.
 [3] G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2384–2415, Oct 1998.
 [4] W. Betts, A. R. Calderbank, and R. Laroia, “Performance of nonuniform constellations on the Gaussian channel,” IEEE Transactions on Information Theory, vol. 40, no. 5, pp. 1633–1638, Sep 1994.
 [5] L. Szczecinski, S. Aissa, C. Gonzalez, and M. Bacic, “Exact evaluation of bit and symbolerror rates for arbitrary 2D modulation and nonuniform signaling in AWGN channel,” IEEE Transactions on Communications, vol. 54, no. 6, pp. 1049–1056, June 2006.
 [6] Z. Mheich, P. Duhamel, L. Szczecinski, and M. L. A. Morel, “Constellation shaping for broadcast channels in practical situations,” in 2011 19th European Signal Processing Conference, Aug 2011, pp. 96–100.
 [7] X. Xiang and M. C. Valenti, “Closing the gap to the capacity of APSK: Constellation shaping and degree distributions,” in 2013 International Conference on Computing, Networking and Communications (ICNC), Jan 2013, pp. 691–695.
 [8] F. Buchali, W. Idler, L. Schmalen, and Q. Hu, “Flexible optical transmission close to the Shannon limit by probabilistically shaped QAM,” in Optical Fiber Communications Conference and Exhibition (OFC)’17, March 2017, pp. 1–3.
 [9] P. Larsson, “Golden angle modulation: Geometric and probabilisticshaping approaches,” To be Submitted, 2017.
 [10] H. Vogel, “A better way to construct the sunflower head,” Mathematical Biosciences, vol. 44, no. 34, pp. 179–189, Jun. 1979.