Golden Angle Modulation

Golden Angle Modulation

Peter Larsson Student Member, IEEE The author is with the ACCESS Linnaeus Center and the School of Electrical Engineering at KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. E-mail: pla@kth.se.
Abstract

Quadrature amplitude modulation (QAM) exhibits a shaping-loss 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, shape-versatile, circular symmetric, modulation scheme, the Golden Angle Modulation (GAM) is introduced. Disc-shaped, and complex Gaussian approximating bell-shaped, GAM-signal constellations are considered. For bell-GAM, a high-rate approximation, and a mutual information optimization formulation, are developed. Bell-GAM overcomes the asymptotic shaping-loss seen in QAM, and offers Shannon capacity approaching performance. Transmitter resource limited links, such as space probe-to-earth, and mobile-to-basestation, are cases where GAM could be particularly valuable.

Modulation, golden angle, golden ratio, shaping, inverse sampling, Shannon capacity, optimization.

I Introduction

Modulation formats, of great number and variety, have been developed and analyzed in the literature. Examples are PAM, square/rectangular-QAM, phase shift Keying (PSK), Star-QAM [1], and amplitude-PSK (APSK) [2]. Square-QAM, hereon referred to as QAM, is the de-facto-standard in existing wireless communication systems. However, at high signal-to-noise-ratio (SNR), QAM is known to asymptotically exhibit an 1.53 dB SNR-gap (a shaping-loss) to the additive white Gaussian noise (AWGN) Shannon capacity [3]. This is attributed to the square shape, and the uniform discrete distribution, of the QAM-signal constellation points. Geometric and probabilistic shaping techniques have been proposed to mitigate the shaping-gap [3]. An early work on geometric shaping is nonuniform-QAM 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 shaping-loss problem, nor offered a modulation format practically well-suited for the task. This leads us to examine new modulation formats.

Figure 1: Geometric-bell-GAM signal constellation, .

Inspired by the beautiful, and equally captivating, cylindrical-symmetric packing of scales on a cycad cone, the spherical-symmetric packing of seeds on a thistle seed head, or the circular-symmetric packing of sunflower seeds, we have recognized that this shape-versatile spiral-phyllotaxis 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, shape-versatile, high-performance 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 disc-shaped distribution. We find, as expected, that the MI-asymptote of geometric-bell GAM (GB-GAM), 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 low-to-high SNR-range.

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 probabilistic-shaping.

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 near-ideal circular design: A circular design can offer enhanced MI-, distance-, symbol error rate- and PAPR- performance over a square-QAM design.

  • Shape-flexibility 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 radial-symmetric pdfs.

  • Naturally lends itself for circular-symmetric probabilistic shaping: We recognize this as a central feature of GAM which allows approximation of (practically) any radial-symmetric 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 2D-packing (as desirable in the high SNR-range), it does not share the above listed features of GAM.

Ii-a Disc-GAM

We first introduce disc-GAM, i.e. GAM without shaping, below. Besides its own merits, this also serves as a base-line to shaping. If the expression for is altered (fixed), but is fixed (altered), geometric (probabilistic) shaping result.

Definition II.2

(Disc-GAM) 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 disc-GAM constellation in Fig. 2, where its approximate disc-shape is clearly depicted.

Figure 2: Disc-GAM signal constellation, .

A few remarks about disc-GAM. The entropy is simply . The PAPR is when . From PAPR-point-of-view, this makes disc-GAM favorable over QAM, since dB. (PSK has dB, but with very poor MI-performance for ). Letting , QAM asymptotically requires ( dB) higher average power than disc-GAM for the same average constellation point distances . Also when , QAM asymptotically requires ( dB) higher peak power than disc-GAM for the same average distance between (uniformly packed) constellation points.

Ii-B Geometric bell-GAM

Next, two geometric shaped GAM schemes are introduced.

Ii-B1 High-rate approach

This first GB-GAM design builds on the inverse sampling method (a high-rate (HR) approximation), and approximates a complex Gaussian distributed r.v.

Theorem II.1

(Geometric-bell-GAM (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 geometric-bell-GAM 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.

Ii-B2 MI-optimization of GB-GAM: Formulation-G1

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 PAPR-inequality 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 non-linearities in the MI, the optimization problem in formulation-G1 is hard to solve analytically. Hence, a numerical optimization solver is used in Section III.

Iii Numerical Results

Here, we now examine the MI-performance of GAM (with its irregular cell-shapes and varying cell-sizes) by using a Monte-Carlo simulation approach. For the optimization in formulation-G1, MATLAB’s fmincon function is used together with numerical integration for the MI.

Figure 3: MI of GB-GAM (HR), asymptote of QAM, and Shannon capacity. MI of GB-GAM (G1) with .

In Fig. 3, we illustrate the MI performance for GB-GAM (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 SNR-region that further constellation optimization, i.e. formulation-G1 , is of interest. The MI for GB-GAM (G1) is also shown, but due to optimization complexity only, for .

In Tab. I, the MI of GB-GAM with the HR-, and G1-, formulations are given. As expected, the optimized scheme, G1, perform better than HR. While the MI-improvements 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 disc-GAM solution, whereas in the low-MI region, we have noted that the optimized signal constellation approaches the HR GB-GAM solution. It is further observed that the extreme magnitudes of the highest constellation indices for GB-GAM (HR), as seen in Fig. , are attenuated with the MI-optimization and leads to improved PAPR performance.

In Fig. 4, we compare disc-GAM, GB-GAM (HR), and QAM, together with the Shannon capacity when the MI is nearly as large as the constellation entropy. We note the (expected) dB SNR-gap for QAM to the Shannon-capacity. The (expected) SNR-gap from QAM to disc-GAM is dB. When the MI is approximately as large as the constellation entropy, disc-GAM, and QAM, perform slightly better than GB-GAM (HR) scheme. This observation prompted us to develop the MI-optimization formulation-G1.

Figure 4: MI of disc-GAM, GB-GAM (HR), QAM, with , and Shannon capacity.
SNR AWGN capacity HR G1
dB 2 1.921 1.961
dB 4 3.440 3.549
dB 3.828 3.926
Table I: MI in [b/Hz/s] for HR- and G1-schemes with .

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 circular-symmetric pdf. We studied geometrically-shaped GAM to approximate the pdf of a continuous complex Gaussian r.v. A high-rate solution, was developed. We also introduced the notion of MI-optimized GAM under an average SNR-constraint, and optionally also a PAPR-inequality-constraint. The MI-performance 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 shaping-loss, GAM exhibit no asymptotic loss. The complex Gaussian communication signal model assumption, as often used for performance analysis, was substantiated from a practical modulation point-of-view.

We believe that GAM may find many applications in transmitter-resource-limited links, such as space probe-to-earth, satellite-to-earth, or mobile-to-basestation communication. This is so since high data-rate is desirable from the power-, energy-, and complexity-limited transmitter side, but higher decoding complexity is acceptable at the receiver side. Certain cases may also benefit from using the PAPR-inequality constrained optimization. With bell-GAM, the power reduction in the high-MI regime could be up to 30% (), which is of environmental interest. Moreover, cellular-system 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 high-rate 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,

\qed

References

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  • [2] C. Thomas, M. Weidner, and S. Durrani, “Digital amplitude-phase keying with M-ary 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.
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