aa r X i v : . [ c s . I T ] A ug N.B. A USB memory stick, storing my draft PhD thesis - comprising material partly used in this work, ”disappeared” frommy hotel room safe in July 2017. For this reason, I’m unnecessarily forced to post this preprint prior publication acceptance.
Golden Angle Modulation
Peter Larsson
Student Member, IEEE
Abstract —Quadrature amplitude modulation (QAM) exhibitsa shaping-loss of π e / , ( ≈ . dB) compared to the AWGNShannon 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) isintroduced. Disc-shaped, and complex Gaussian approximatingbell-shaped, GAM-signal constellations are considered. For bell-GAM, a high-rate approximation, and a mutual informationoptimization formulation, are developed. Bell-GAM overcomesthe asymptotic shaping-loss seen in QAM, and offers Shannoncapacity approaching performance. Transmitter resource limitedlinks, such as space probe-to-earth, and mobile-to-basestation,are cases where GAM could be particularly valuable. Index Terms —Modulation, golden angle, golden ratio, shaping,inverse sampling, Shannon capacity, optimization.
I. I
NTRODUCTION M ODULATION formats, of great number and variety,have been developed and analyzed in the literature.Examples are PAM, square/rectangular-QAM, phase shiftKeying (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 knownto asymptotically exhibit an 1.53 dB SNR-gap (a shaping-loss) to the additive white Gaussian noise (AWGN) Shannoncapacity [3]. This is attributed to the square shape, and theuniform discrete distribution, of the QAM-signal constellationpoints. Geometric and probabilistic shaping techniques havebeen proposed to mitigate the shaping-gap [3]. An early workon geometric shaping is nonuniform-QAM in [4]. More recentworks in this direction are, e.g., [5], [6], [7], [8]. Existingwork on modulation schemes have, in our view, not completelysolved the shaping-loss problem, nor offered a modulationformat practically well-suited for the task. This leads us toexamine new modulation formats.Inspired by the beautiful, and equally captivating,cylindrical-symmetric packing of scales on a cycad cone, thespherical-symmetric packing of seeds on a thistle seed head,or the circular-symmetric packing of sunflower seeds, we haverecognized that this shape-versatile spiral-phyllotaxis packingprinciple, found among plants, is applicable to modulationsignal constellation design. Based on the spiral phyllotaxispacking, the key contribution of this letter is proposing anovel, shape-versatile, high-performance modulation frame-work - the
Golden angle modulation (GAM). We considerdiscrete modulation, with equiprobable constellation points,that approximate a r.v. with continuous complex Gaussian
The author is with the ACCESS Linnaeus Center and the School ofElectrical Engineering at KTH Royal Institute of Technology, SE-100 44Stockholm, Sweden. E-mail: [email protected].
Re(x) -2 -1 0 1 2 I m ( x ) -2.5-2-1.5-1-0.500.511.522.5 Figure 1: Geometric-bell-GAM signal constellation, N = 2 .(bell)-shaped distribution, as well as a baseline case, with adisc-shaped distribution. We find, as expected, that the MI-asymptote of geometric-bell GAM (GB-GAM), for increasingnumber of constellation points, coincide with the AWGNShannon capacity. This also supports the complex Gaussiancommunication signal assumption, used in many performanceanalysis works, for the low-to-high SNR-range.II. G OLDEN A NGLE M ODULATION
The core design of GAM is given below.
Definition 2.1: (Golden angle modulation) The n th constel-lation point of GAM has the probability of excitation p n , andthe complex amplitude x n = r n e i πϕn , n ∈ { , , . . . , N } , (1)where r n is the radius of constellation point n , πϕ denotesthe golden angle in rads, and ϕ = 1 − ( √ − / .We will assume that r n +1 > r n for an increasing spiral wind-ing. For the probability, it may be equiprobable, p n = 1 /N ,or dependent on index n . The later, where p n +1 ≤ p n , corre-sponds to GAM with probabilistic shaping and is exploredextensively in [9]. Hence, a constellation point, is located ϕ ≈ . turns ( . ◦ ) relative to the previous constellationpoint. Replacing ϕ , with (1 + √ / ≈ . , the goldenratio, or its fractional part, gives an equivalent spiral winding,but, in the opposite direction. Note that phase rotation valuedeviating with approximately 1% from the golden angle (ratio)destroys the locally uniform packing. The mathematical designof the phase rotation in Def. 2.1 is inspired from the workby Vogel, [10], who described an idealized growth patternfor the sunflower seeds, x n = √ n e i πϕn (in our notation).Vogel did however not consider modulation. More importantly,This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version may no longer be accessible. a key insight here, enabling the approximation of a complexGaussian pdf, is to not restrict the radial function r n to √ n as in [10]. Allowing for arbitrary radial growth of r n , givesthe geometric shaping capability, and allowing for arbitraryprobabilities p n , 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, GAMenables a unique indexing based on signal phase, πϕn ,or magnitude, r n , alone. • Practically near-ideal circular design: A circular designcan offer enhanced MI-, distance-, symbol error rate- andPAPR- performance over a square-QAM design. • Shape-flexibility in radially distribution of constellationpoints while retaining an evenly distributed packing: Werecognize this as a central feature of GAM which allowsapproximation of (practically) any radial-symmetric pdfs. • Naturally lends itself for circular-symmetric probabilisticshaping: We recognize this as a central feature of GAMwhich allows approximation of (practically) any radial-symmetric pdfs. • Any number of constellation points, while retaining theoverall circular shape: This gives full flexibility, e.g., inalphabet size of a channel coder, or a probabilistic shaper. • Rotation (and gain) invariant: The uniquely identifiablegain and rotation of signal constellation could, e.g., allowfor blind channel estimation.Some comments. First, the index range is not necessarilylimited to n ∈ { , , . . . , N } . Second, GAM has a complexvalued DC component. Possible remedies, if a problem, is tosubtract the DC component, or negate every second symbol.Third, while hexagonal packing is the densest 2D-packing (asdesirable in the high SNR-range), it does not share the abovelisted features of GAM. A. Disc-GAM
We first introduce disc-GAM, i.e. GAM without shaping,below. Besides its own merits, this also serves as a base-lineto shaping. If the expression for r n is altered (fixed), but p n is fixed (altered), geometric (probabilistic) shaping result. Definition 2.2: (Disc-GAM) Let p n = 1 /N , N be thenumber of constellation points, and ¯ P be the average powerconstraint. Then, the complex amplitude of the n th constella-tion point is r n = c disc √ n, n ∈ { , , . . . , N } , (2) c disc , s PN + 1 . (3)In the above, (3) is found from the power normalizationcondition, ¯ P = P Nn =1 p n r n = c disc P Nn =1 1 N n = c disc N +12 .We illustrate the disc-GAM constellation in Fig. 2, whereits approximate disc-shape is clearly depicted.A few remarks about disc-GAM. The entropy is simply H disc = log N . The PAPR is P AP R disc = 2 ¯
P N/ ( N +1) ¯ P ≃ ≃ dB ) when N → ∞ . From PAPR-point-of-view, this makes disc-GAM favorable over QAM, since -1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5 Figure 2: Disc-GAM signal constellation, N = 2 . P AP R
QAM = 4 . dB. (PSK has P AP R
PSK = 0 dB,but with very poor MI-performance for N → ∞ ). Letting N → ∞ , QAM asymptotically requires
10 log ( π/ ( ≈ . dB) higher average power than disc-GAM for the same averageconstellation point distances . Also when N → ∞ , QAMasymptotically requires
10 log ( π/ ( ≈ . dB) higher peakpower than disc-GAM for the same average distance between(uniformly packed) constellation points. B. Geometric bell-GAM
Next, two geometric shaped GAM schemes are introduced.
1) High-rate approach:
This first GB-GAM design buildson the inverse sampling method (a high-rate (HR) approxima-tion), and approximates a complex Gaussian distributed r.v.
Theorem 2.1: (Geometric-bell-GAM (HR)) Let p n = 1 /N , N be the number of constellation points, and ¯ P be the averagepower constraint. Then, the complex amplitude of the n thconstellation point is r n = c gb s ln (cid:18) NN − n (cid:19) , n ∈ { , , . . . , N − } ,c gb , s N ¯ PN ln N − ln( N !) . (4) Proof:
The proof is given in Appendix A.We illustrate the geometric-bell-GAM signal constellationin Fig. III, and note that it is densest at its center, i.e. wherethe pdf for the complex Gaussian r.v. peaks.We note the following characteristics. When N → ∞ ,since lim N →∞ N − ln (cid:0) N N /N ! (cid:1) = 1 , we get r n ≃ p ¯ P ln ( N/ ( N − n )) . The entropy is H gb = log N . ThePAPR is P AP R gb = c gb ln( N ) = ¯ P / (1 − ln ( N !) /N ln N ) ≃ ¯ P ln ( N ) , which tends to infinity with N . This is expected asthe PAPR of a complex Gaussian r.v. is infinite. Note here thatthe index range in (4) can not be n ∈ { , , . . . , N } , but ischosen as n ∈ { , , . . . , N − } , as r N = ∞ otherwise. SNR [dB] M I and S hannon c apa c i t y [ b / H z / s ] C= log (1+ SNR ) MI GB-GAM (IS) MI asymptote QAM N=2 N=2 N=2 N=2 Formulation-G1
Figure 3: MI of GB-GAM (HR), asymptote of QAM, andShannon capacity. MI of GB-GAM (G1) with N = 2 .
2) MI-optimization of GB-GAM: Formulation-G1:
In thismethod, we let p n = 1 /N , and vary r n in order to maximizethe MI, for a desired SNR S . The optimized signal constella-tion points are x ∗ n = r ∗ n e i πϕn . More formally, allowing for acomplex valued output r.v. Y , and a complex valued (discretemodulation) input r.v. X , the optimization problem ismaximize r n I ( Y ; X ) , subject to r n +1 ≥ r n , n = { , , . . . , N } ,r ≥ , N X n =1 p n r n σ = S. (5) Remark 2.1:
For some applications, the PAPR is of interest.A PAPR-inequality constraint, r N / P N − n =1 p n r n ≤ P AP R , P AP R being the target PAPR, can be amended to the opti-mization problem. Other constraints may also be of interest.When optimizing GAM in AWGN, the MI is I ( Y ; X ) = h ( Y ) − h ( W ) , where h ( W ) = log ( π e σ ) , h ( Y ) = − R C f Y log ( f Y ) d y , integrating over the complex domain,with f Y = P Nn =1 p n f ( y | x n ) = πσ P Nn =1 p n e − | y − xn | σ .Due to the non-linearities in the MI, the optimization prob-lem in formulation-G1 is hard to solve analytically. Hence, anumerical optimization solver is used in Section III.III. N UMERICAL R ESULTS
Here, we now examine the MI-performance of GAM (withits irregular cell-shapes and varying cell-sizes) by using aMonte-Carlo simulation approach. For the optimization informulation-G1, MATLAB’s fmincon function is used togetherwith numerical integration for the MI.In Fig. 3, we illustrate the MI performance for GB-GAM(HR) together with the Shannon capacity. As expected, forlarger constellation size N , a greater overlap with the Shannoncapacity is seen. The MI approximation is good up to about ≈ H/ . Naturally, the MI is limited by the entropy of thesignal constellation. We observe an intermediate SNR region, SNR [dB]
22 24 26 28 30 32 34 M I and S hannon c apa c i t y [ b / H z / s ] MI GB-GAM (HR),
N=2 MI Disc-GAM,
N=2 MI QAM,
N=2 C=log (1+S) Figure 4: MI of disc-GAM, GB-GAM (HR), QAM, with N =2 , and Shannon capacity. SNR AWGN capacity HR G1 ≈ . dB 2 1.921 1.961 ≈ . dB 4 3.440 3.549 dB ≈ . N = 16 .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 N = 16 .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 aremodest, the optimization formulation is substantiated. For G1,when the MI is as large as the constellation entropy, wehave observed that the signal constellation approaches thedisc-GAM solution, whereas in the low-MI region, we havenoted that the optimized signal constellation approaches theHR GB-GAM solution. It is further observed that the extrememagnitudes of the highest constellation indices for GB-GAM(HR), as seen in Fig. , are attenuated with the MI-optimizationand leads to improved PAPR performance.In Fig. 4, we compare disc-GAM, GB-GAM (HR), andQAM, together with the Shannon capacity when the MI isnearly 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-GAMis ≈ . dB. When the MI is approximately as large asthe constellation entropy, disc-GAM, and QAM, performslightly better than GB-GAM (HR) scheme. This observationprompted us to develop the MI-optimization formulation-G1.IV. S UMMARY AND C ONCLUSIONS
In this letter, we have introduced a new modulationformat, the golden angle modulation . With geometric- (orprobabilistic-) shaping, GAM can approximate virtually anycircular-symmetric pdf. We studied geometrically-shaped
GAM to approximate the pdf of a continuous complexGaussian r.v. A high-rate solution, was developed. We alsointroduced the notion of MI-optimized GAM under an aver-age SNR-constraint, and optionally also a PAPR-inequality-constraint. The MI-performance was observed to asymptoti-cally approach the Shannon capacity as the number of signalconstellation points tended to infinity. In contrast to QAM’s1.53 dB shaping-loss, GAM exhibit no asymptotic loss. Thecomplex Gaussian communication signal model assumption,as often used for performance analysis, was substantiated froma practical modulation point-of-view.We believe that GAM may find many applications intransmitter-resource-limited links, such as space probe-to-earth, satellite-to-earth, or mobile-to-basestation communica-tion. This is so since high data-rate is desirable from thepower-, energy-, and complexity-limited transmitter side, buthigher decoding complexity is acceptable at the receiver side.Certain cases may also benefit from using the PAPR-inequalityconstrained optimization. With bell-GAM, the power reductionin the high-MI regime could be up to 30% ( − / . / ≈ . ), which is of environmental interest. Moreover, cellular-system operators, could potentially also reduce energy con-sumption (and cost) with up to 30%. It is hoped that, GAM,with its attractive characteristics and performance, could be ofinterest for most, wireless, optical, and wired, communicationsystems. A PPENDIX
A. Proof in Theorem 2.1Proof:
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