Golden Angle Modulation,” IEEE Wireless Communications 1971

arXiv:1802.10022v1 [cs.IT] 27 Feb 2018 hpn,mta nomto,pa-oaeaepwrratio power peak-to-average information, mutual shaping, h oe/nryefiinyi xetdt improve. to expected capacity is channel Gaussian efficiency the power/energy approach the can design th effi GAM spectral the ampli design, target As quadrature modest classical relatively proposed with for to even relative the compared modulation, decreased and With be simpler, i can MI schemes. [1], SNR a performance to modulation in The shaping, respect classical suggested. introduced with geometric evaluated also schemes proposed numerically is GAM (S the design ratio key by SNR-dependent, signal-to-noise two Inspired full to [2]. the peak-t relative the over and range ratio, (MI), power distribution. information input mutual (GAM) average Gaussian the modulation truncated improves angle (double) design golden a a on for propose based design we systems, shaping fiber geometric optical or microwave-links, lar, cee,i scoet h diiewieGusa noise Gaussian white additive the to capacity, close (Shannon) (AWGN) as is schemes, M)v.tesga-onierto(SNR), ratio inform signal-to-noise mutual the average vs. the that (MI) desirable sch is peak/average modulation It important. the reduced is respect, this and/or In power/energy. rates, transmit communication hanced oselto onsi ahrn eed nteshaping. the on of depends number ring each the in that points is achieving constellation APSK capacity geometrically-shaped with the Ye issue symmetry symmetric. since circular circular is preferred can distribution with Gaussian) often shaping (complex constellation is geometric a APSK) While QAM, (like [15]. to [14], applied [13], be in APSK [12 Maxwel on in shaping on probabilistic based [11], distribution in Boltzmann constellations sphere/cube dimensional ok npoaiitcsaigaea uvyi 1] on [10], in survey and some a Similarly, as [8], are [9]. in nonuniform- shaping in AWGN probabilistic on on optimization for works nonuniform-PAM capacity [6], PSK/PAM on in on [7], constellations in asymmetric QAM on geometric-shapi t are on works proposed Some been shaping-loss. this QAM have overcome Hence, probabilistic-shaping [5]. and shaping-loss) capacity Geometric- (a AWGN the SNR-gap designs, and waste dB constellation MI 1.53 all the between of asymptotic deployed an and QAM, exhibits studied [4]. (APSK) most PSK amplitude the keying are and shift [3], constellations phase star-QAM (PSK), (QAM), signal modulation modulation amplitude quadrature known Well possible. ne Terms Index Abstract uuerdootclcmuiain ytm eur en- require systems communications radio/optical Future ≈ 30 I hswr,treig .. uuegnrto cellu- generation future e.g., targeting, work, this —In oetasi oe hnnee thg rates. high at needed than power transmit more % Mdlto,sga oselto,geometric constellation, signal —Modulation, ee Larsson Peter .I I. prahn h WNCapacity AWGN the Approaching NTRODUCTION odnAgeModulation: Angle Golden C log = tdn ebr IEEE Member, Student 2 ialSkoglund, Mikael S 1+ (1 ftemodulation the of , S ) bH/]as [b/Hz/s] ciencies. . ,and ], ,one t, ation tude eme NR) new The N ng asK Rasmussen K. Lars , o- l- o e s - , eirMme,IEEE Member, Senior P sdsrbe ..frAG aaiyaheig n/ra and/or achieving, capacity AWGN uniform GAM for by near-ideal e.g. offered with as desirable, shaping forms, characteristic, is radial magnitude Such and distribution. of phase phase notion decoupling the not has defy sinc spiral i.e. spiral importantly, Archimedean more design, Archimedean but, the by before, the because times to, several just used itself not been is limit framework This nor design. GAM adopt, The not consid does mapping. also [23], spiral-based [22], area wit Archimedean [21], related design coding, the a a source-channel In joint examined proposed spirals. Archimedean latter former intertwinned the four the design, Whereas spiral logarithmic proposed. were schemes 0 2,tokyGMdsgsweepooe.Tefis one, first The magnitudes had proposed. (GB-GAM), where GAM [1 bell-shaped GAM-designs in geometric key specifically More two shaping. [2], radial distribution, geometric phase c for uniform ideal allows ideal near near constellation a a enough design, offers symmetric large design For spiral-related, radians. discrete, this in angle golden the h atrtosuidAcieensiaswt amplitudes with spirals intercon- Archimedean wherea uses studied radius, f work two increasing first latter of The the segments was [18]. semicircular [17], modulation nected [16], spiral-based in Analog proposed designs. spiral-based where where where h A inlcntlainpit a eepesdas expressed be can formally points More constellation index. signal point GAM the constellation for the different on phase have the dependency to and points radial constellation the of allowing distribution is i second consecu- angle The separating golden domain. is the phase with first points The constellation indexed GAM. tively in features design key modulation ¯ n[8.I 1] 2] iceesia-ae modulation spiral-based discrete [20], [19], In [18]. in , ( h eodoe icsae A ds-A) had (disc-GAM), GAM disc-shaped one, second The . x ent htrltvl e ok,peeigGM consider GAM, preceding works, few relatively that note We eetyi 1,adetne n[] the [2], in extended and [1], in Recently r = ) m c φ c = disc gb x m e c ix gb , 2 = , GM rmwr a nrdcd hr r two are There introduced. was framework (GAM) x , s x r p m p m ln πϕm ≥ 2 / P = = ¯ P/ ¯ eirMme,IEEE Member, Senior 0 c r (ln M , disc , m ( n[7,and [17], in M ϕ e 1 + M √ iφ M , 1) + ,m m, m − m , − (3 ln( m . − = { ∈ M √ m , { !) f 5) 1 1 /M ( , , / x 2 2 { ∈ 2 = ) M , . . . , M , . . . , ) ≈ n vrg power average and , 1 , g , 0 2 ( . 381 x M , . . . , )e } } odnangle Golden , , ig and , ( x ) g , } , sand ms 2 ircular ( πϕ x the n ered ) and it e (3) (2) (1) of ≥ is s, ], h s , , PAPR-controlled, designs. In addition, none of the prior works The optimization problem can now be formulated as have used, nor benefited from, the golden angle-based phase arrangement of constellation points. maximize f(u, v) ln (f(u, v)) du dv, f(u,v) − A In [1], [2], it was seen that the MI of GB-GAM was close ZZ to the channel capacity for, say, 0 MI H/2, where subject to (u2 + v2)f(u, v) du dv =1, (5) ≤ ≈ A H = log2(M) is the signal constellation entropy. Disc-GAM, ZZ on the other hand, performed best when MI / H. Hence, f(u, v) du dv =1. there is a wide SNR-range, where neither GB-GAM, nor disc- ZZA GAM, perform optimally. An effort to handle this in [1], [2], This problem can, due to convexity of the entropy, be was by optimizing the magnitudes rm to maximize the MI for solved by classical Lagrangian optimization. The solution is every SNR. This problem could not be solved analytically, and straightforward and is simply a truncated 2-dim Gaussian a numerical optimization solving approach could only handle pdf. Writing it on the standard bi-variate Gaussian form, M / 16 constellation points. Hence, it is of interest to find an with an undetermined normalization constant c , we have 2 2 2 1 c1 −(u +v )/σ analytical expression for the magnitudes rm that improves the f(u, v) = 2 e , where (u, v) . Since ρi and πσ ∈ A MI for the full SNR-range, S (0, ), relative to GB-GAM, ρo can be tuned to any value, without loss of generality, we let ∈ ∞ 2 disc-GAM, and classical signal constellations. It is also of σ =1. As the discrete GAM magnitudes rm are sought after, interest to parameterize the analytical rm expression in one, it useful to transform f(u, v) to a marginal density f(ρ) with or two, variables that allows for numerical optimization of respect to a magnitude ρ. Via Euclidean-to-polar coordinate larger constellations sizes. variable substitution, u = ρ sin(φ) and v = ρ cos(φ), and integrating over a uniform distribution in phase, we get

2 A. Truncated Geometric Bell-shaped GAM c1 e−ρ 2πρ, ρ ρ ρ , f(ρ)= π i o (6) 0, Otherwise≤ ≤ . It can be recognized that GB-GAM and disc-GAM can be Integrating the pdf, ρo f(r) dr = 1, yields the constant seen as resulting from two extremes of a truncated Gaussian ρi − 2 − 2 input distribution. GB-GAM would then approximate a (non- c =1/(e ρi e ρo ). The corresponding cdf is then 1 − R truncated) Gaussian pdf, and disc-GAM would approximate a 2 2 −ρ −ρ truncated Gaussian pdf when the radius approaches zero. We e i e F (ρ)= 2 , ρ ρ ρ . (7) −ρ − − 2 i o can approach this line of reasoning more formally. The overall e i e ρo ≤ ≤ − procedure is to first find a continuous input distribution that Similar to this work, in [24], [25], [26], a peak-power (closely) approximates the MI-maximizing distribution, and constrained input distribution was considered. Specifically, the then using this distribution together with the GAM frame- complex AWGN channel was studied in [25]. It was found that work and inverse sampling to determine constellation point the optimal input distribution is discrete in the radial domain, magnitudes rm. First, the MI between a continuous input but uniformly continuous in phase. As a sub-optimal input signal X, and a continuous output signal Y , can be lower distribution, they derived (based on the same line of reasoning bounded as follows, I(Y ; X) = h(Y ) h(Y X) = h(Y ) as above) a peak-power truncated complex Gaussian distribu- − | − h(W ) h(X) h(W ), where the fact that conditioning tion. However, no actual modulation scheme was proposed. ≥ − reduces entropy, h(Y ) h(Y W )= h(X), is used in the last Moreover, the result on continuous uniformly distributed phase ≤ | step. Hence, instead of finding an input distribution for X that is incompatible with the idea of discrete signal constellation maximize the entropy h(Y ), the entropy h(X) is maximized, points. Even if a modulation scheme had been proposed, say thereby giving a lower bound to I(Y ; X). APSK-based, it is not obvious how to combine APSK with the Since GAM is near circular symmetric, i.e. for a sufficiently truncated complex Gaussian input distribution assumption in a large M, we target a continuous circular symmetric input simple and well-performing manner. The designer of an APSK distribution f(xre, xim) that maximizes h(X) for an average scheme must carefully split the total number of constellation power constraint. For notation simplicity and clearness below, points among the rings, and arrange the distance among rings. we use the notation f(u, v) instead of f(xre, xim). Moreover, The constellations must be designed, and optimized, for every for generality, and control over the PAPR, we let the circular distinct M. Certain values of M, e.g. prime numbered M, symmetric density f(u, v) be nonzero for magnitudes less than give APSK constellations with asymmetries. GAM, on the an outer radius, ρo, but also greater than an inner radius, ρi. other hand, together with the method of inverse sampling, is More precisely, the pdf is assumed to fulfill versatile enough to seamlessly adapt to a desired truncated complex Gaussian input distribution and any integer M. To more flexibly control PAPR, as said, we also let 0 ρi <ρo. f(u, v) 0, (u, v) , ≤ ≥ ∈ A (4) To design a discrete constellation, we consider the method f(u, v)=0, Otherwise, of inverse sampling, together with the GAM design, and the (double) truncated bi-variate Gaussian target density. In where u, v : ρ2 u2 + v2 ρ2 . inverse sampling, one equates the cdf F (ρ) with a continuous A∈{ i ≤ ≤ o } 1.5 2 1.5

1.5 1 1 1 0.5 0.5 0.5

0 0 0

-0.5 -0.5 -0.5 -1 -1 -1.5 -1

-2 -1.5 -1.5 -2 -1 0 1 2 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5

(a) (b) (c)

Figure 1: Signal constellations of TGB-GAM, ρi =0, M = 256: (a) S = 10 dB; (b) S = 22.5 dB; (c) S = 35 dB.

2.5 2.5 2.5 TGB-GAM TGB-GAM TGB-GAM GB-GAM GB-GAM GB-GAM Disc-GAM Disc-GAM Disc-GAM 2 2 2

1.5 1.5 1.5

1 1 1 Magnitude of constelation point Magnitude of constelation point Magnitude of constelation point 0.5 0.5 0.5

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Normalized constellation point index n/N Normalized constellation point index n/N Normalized constellation point index n/N (a) (b) (c)

Figure 2: Magnitude distribution of TGB-GAM, ρi =0, M = 256: (a) S = 10 dB; (b) S = 22.5 dB; (c) S = 35 dB. uniformly distributed r.v., say t (0, 1), and solve for ρ. to optimizing all M values for rm. The advantage of this is that Generating t uniformly on (0, 1)∈, then gives ρ with pdf numerical optimization is not seriously limited by constellation f(ρ). With large number of constellation points, we can size. The optimization problem, including an optional PAPR approximate the uniformly distributed continuous r.v with a constraint P AP R0, can then be written uniformly distributed discrete r.v. τ with a pmf with, e.g., maximize I(Y ; X), δ(τ = m/M)=1/M,m 1, 2,...,M , or alternatively ρi,ρo with ∈ { . Thus, we} have subject to δ(τ = (m 1/2)/M)=1/M F (ρm)= ρi 0, (11) , from which− we solve for and get ≥ m/M ρm ρo ρi, 2 ≥ 2 2 −ρ m −ρ −ρ2 rM P AP R0, ρm = ln e i e i e o . (8) ≤ − − M − r where the input distribution for X is now given by xm in It is convenient to normalize the magnitudes ρm, to an (1), rm in (10), and where each constellation point have the ¯ average power constraint, P , by letting rm = ctgbρm, and probability pm =1/M. After closer scrutiny, it is seen that the where the constant ctgb is determined as below optimization problem in (11) can not easily be solved analytically. It is straightforward to further simplify the expressions M ¯ 1 2 MP (and the optimization) above by letting . r = P¯ ctgb = . (9) ρi =0 M m ⇒ M 2 m=1 s m=1 ρm To simplify the expression for rm further, we may chose X a form that converges to GB- and disc-GAM for S = 0 and Henceforth, we let P¯ = 1, and the AWGN noise variance is P S = , respectively. First let ρ = 0. Inspecting (10), and then σ2 = 1/S. In total, the (unit-power normalized) GAM i knowing∞ that ρ diminishes with increasing SNR, we propose constellation point magnitudes are o the simple, yet well-performing, form − 2 − 2 − 2 ρi m ρi ρo ln e M e e m rm = − 2 ′ − 2 . (10) ln 1 1 M −ρ m −ρ −ρ2 S+M v ′ ln e i e i e o − M m =1 M rm = ′ . (12) u − − v 1 M m u u ′ ln 1 t u M m =1 S+M Note that sinceP the constellation is unit-power normalized , the u − t resulting PAPR is simply P AP RdB = 20 log 10(r ) dB. When S 0, (12) (essentially)P converges to the GB-GAM’s M → As rm depends only on two parameters ρi and ρo, a (2) (by letting m = 0, 1,...,M 1 ), whereas when S , simpliﬁed optimization problem can be formulated compared (12) converges to the{ disc-GAM’s− (3).} → ∞ 10 3.08 9 3.06 8 10 10 4 3.04 M=2 M=2 M=2

7 3.02

6 3 ++ + + + ≈ 0.01 dB ≈ 2.98 0.5 dB 5 ≈ 0.76 dB ≈ 0.85 dB 2.96 4

MI and Shannon capacity [b/Hz/s] MI TGB-GAM, (12) MI and Shannon capacity [b/Hz/s] 2.94 MI TGB-GAM, (12) 3 MI QAM MI QAM C=log (1+S) C=log (1+S) 2 2.92 2 2 5 10 15 20 25 30 35 40 8.2 8.4 8.6 8.8 9 9.2 9.4 SNR [dB] SNR [dB] Figure 3: MI of TGB-GAM (12), QAM, and the AWGN Figure 4: MI of TGB-GAM (12), QAM, and the AWGN capacity, for H = 4, 6, 8, 10 . capacity, where C 3 [b/Hz/s]. { } ≈

II. NUMERICAL RESULTS AND DISCUSSION 10 In the following, the MI vs. SNR performance for coded 9.8 modulation (CM) is Monte Carlo simulated, whereas the opti- 9.6 mized MI performance is numerically computed. In Fig. 1(a)- (c), we plot the signal constellations of optimized TGB-GAM 9.4 9.2 (10)+(11), for ρi =0, M = 256, and S = 10, 22.5, 35 dB. { } We note that the signal constellation change in shape from 9 an approximate discrete complex Gaussian like design, over 8.8 an intermediate design, to a disc-shaped design. In Fig. 2(a)- MI TGB-GAM, (10)+(11) MI 8.6 GB-GAM, (2) (c), we plot the magnitude distributions for the TGB-GAM MI Disc-GAM (3) MI QAM 8.4 together with GB-GAM and disc-GAM for the same SNRs MI and Shannon capacity [b/Hz/s] MI NU-QAM C=log (1+S) as in Fig. 1(a)-(c). We observe how the signal constellation 8.2 2 magnitudes have nearly the same distribution as GB-GAM MI TGB-GAM, (12) 8 for low SNRs, and nearly the same distribution as disc-GAM 24 26 28 30 32 34 36 for high SNR. We also note that the truncated Gaussian SNR [dB] design overcomes the oscillatory problem seen in [2] for the Figure 5: MI of TGB-GAM (10)+(11), GB-GAM (2), disc- GB-GAM G2 scheme at low SNRs. For the G2 scheme, a GAM (3), QAM, NU-QAM, and TGB-GAM (12) with M = polynomial described the growth of rm, and its coefﬁcients 210, and the AWGN capacity. where optimized to maximize the MI. In Fig. 3, we illustrate the MI-performance for the SNR-dependent TGB-GAM (12) together with QAM and the AWGN capacity. It is observed SNR for a desired rate R is unnecessarily high. Let’s instead that the MI performance of TGB-GAM is generally better than consider the MI-performance for very large M, say M =210. QAM, and approaches the AWGN capacity as SNR decreases. Then, the SNR-gaps from 1024-QAM and 1024-TGB-GAM For M = 16, TGB-GAM and QAM performs about the same. (12) to C =3 [b/Hz/s] are, respectively, 0.5 dB and 0.01 A possible misconception could be that since 16-QAM and dB. So a designer could on one hand choose≈ 16-QAM≈ with 16-GAM have almost the same MI vs. SNR performance, there a r =3/4 code, or on another hand choose 1024-TGB-GAM is no point in favoring GAM over QAM for relatively small with a rate r =3/10 code. The latter choice reduces the SNR- constellation sizes, e.g. M = 16. Such idea would be based gap with 0.84 dB. This has implication for modem design. on a certain design paradigm. Let’s consider an example. Say Instead of implementing many different constellation sizes, that a rate R =3 [b/Hz/s] is desired. Then, the idea could be one may implement just the largest desired constellation size, to use a constellation size M = 24 and a channel code rate say M = 210, and then adapt the code rate r to achieve r = 3/4. Zooming in around C = 3 [b/Hz/s], Fig. 4 shows the desired rate R. Of course, this reasoning assumes that that the SNR-gaps for 16-QAM and 16-TGB-GAM (12) to the other costs, e.g. any additional energy consumption, are of less AWGN capacity at C = 3 [b/Hz/s] are, respectively, 0.85 concern. Another aspect speaking for GAM is that the PAPR dB and 0.76 dB. (Note that the MI can be improved≈ for performance. For example, in [1], [2], we found that QAM ≈ GAM using (10)+(11) or with optimization of rm). However, asymptotically requires 1.96 dB higher PAPR than for disc- this is not necessarily the best design approach, as the required GAM for the same average≈ constellation point distances. In 8 [4] C. Thomas, M. Weidner, and S. Durrani, “Digital amplitude-phase 10 7.5 M=2 TGB-GAM (10)+(11) keying with M-ary alphabets,” IEEE Transactions on Communications, TGB-GAM (12) vol. 22, no. 2, pp. 168–180, February 1974. 7 [5] G. D. Forney and G. Ungerboeck, “Modulation and coding for linear 8 Gaussian channels,” IEEE Transactions on Information Theory, vol. 44, 6.5 M=2 no. 6, pp. 2384–2415, Oct 1998. 6 [6] D. Divsalar, M. Simon, and J. Yuen, “Trellis coding with asymmetric modulations,” IEEE Transactions on Communications, vol. 35, no. 2, 5.5 6 pp. 130–141, February 1987. M=2 5 [7] W. Betts, A. R. Calderbank, and R. Laroia, “Performance of nonuni-

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