text: 5 where bold-faced variables indicate vectors, e.g., v = trajectory, such that φ follows a uniform distribution. j (v , v , v )T , v is the velocity of the colliding particle We also assume that the velocity of the buﬀer gas follows x y z b of buﬀer gas, m˜ = m /m is the buﬀer gas-to-ion mass Maxwell-Boltzmann statistics and is characterised by a b i ratio, and R is a rotation matrix determined by the scat- ﬁxed temperature T . Both the density and the tem- b tering angles. As with the position, the velocity of the perature of the buﬀer gas are taken to remain constant, ion is given by the sum of the intrinsic and forced mo- i.e., the heating of the buﬀer gas due to the collisions tion, v (τ) = v (A , φ , τ) + v (τ), where the forced is assumed to be negligible. With these assumptions, j h,j j j f,j term is independent of A , φ and so is unchanged by the the averaging can be performed over φ , v , τ and the j j j b,k collision. We therefore obtain, elements of the isotropic random rotation matrix by in- tegrating the coeﬃcients over the distributions of each of these variables in turn, see the supplemental material for (cid:48) 1 m˜ − m˜ − − v = v + (v v )+ R[v (v v )]. details. The coeﬃcients of the terms in Eq. (20) where h 1 + m˜ h 1 + m˜ b f 1 + m˜ h b f (cid:54) k = l average to zero, as do the c , c , c coeﬃcients, (19) 1 3 5 signiﬁcantly simplifying the expression. The remaining Taken together, Eq. (17) and Eq. (19) indicate that terms are given b(cid:88)y, (cid:2) (cid:3) the problem is equivalent to that of an ion with no forced − motion colliding with a particle of velocity v v . This (cid:104) (cid:48)| (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) b f E E = η E + a v2 + κ k T , is similar to the frame transformation used in Ref. [23] in j x,y,z jk k 4,jk f,k j B b ∈ which the intrinisic micromotion is assigned to the buﬀer k (x,y,z) (21) gas, but in the present case is performed only for the forced motion and is valid for all q . j where the coeﬃcients are deﬁned as, Using the procedure detailed in Ref. [14], we obtain a (cid:2) (cid:3) (cid:48) set of coupled equations for A 2 and hence the secular j energies aft(cid:88)er a co(cid:18)llision(cid:112), (cid:112) (cid:112) (cid:104) (cid:105) δ m˜ κ (3δ + 1)M η = jk + j jk ce˙ (τ)2 + se˙ (τ)2 , jk m˜ + 1 6β2c2 j k k k 0,k (cid:48) (cid:112) (22) E = η E E + a E v j jkl k l 1,jkl k b,l ∈ (k,l) (x,y,z) (cid:19) and, (20) (cid:2) (cid:3) + a v v + a E v 2,jkl b,k b,l 3,jkl k f,l + a4,jklvf,kvf,l + a5,jklvf,kvb,l , (cid:104)a v2 (cid:105) = m˜ miΩ2κj (3δ + 1)M v (τ)2 . (23) 4,jk f,k 24 jk j f,k where the coeﬃcients η and a describe the trans- jkl i,jkl In the above expressions, δ is the Kronecker delta, fer of energy between the motion along the three coor- jk κ is deﬁned by, dinate axes and between the diﬀerent components of the j ion’s velocity and the velocity of the buﬀer gas. The co- eﬃcients of this expression depend on the elements of the m˜ β2c2 random rotation matrix R, the set of phases φ , and the κ = j 0,j , (24) j j (1 + m˜ )2 W2 time of collision τ. As supplementary information to this j article, we provide a Mathematica notebook containing M details of this procedure and the full form of Eq. (20). and the operator j(cid:90)is deﬁned as, (cid:2) (cid:3) To gain a better understanding of the collision pro- cess, it is useful to average over the collision parameters toof pobret-acionlltishieonmeenaenrgpieoss,t-(cid:104)cEolj(cid:48)l|iEsioxn, Eeyn,eErgzy(cid:105). foTroadgoivseon, sweet Mj [h(τ)] = Ll→im∞ 21L −LL h(τ) cej(τ)2 + sej(τ)2 dτ. must introduce some further assumptions. The Langevin (25) model of collisions has been shown to be accurate for the classical trajectories considered here [9, 10] and so In principle, the above procedure may be adapted to we adopt this. This results in two useful simpliﬁcations. arbitrary distributions for the velocity of the buﬀer gas by Firstly, the rotation matrix R is isotropic in this model averaging over these in place of the Maxwell-Boltzmann and so is uncorrelated from the velocity of the ion and distribution. This would allow for an investigation of the neutral particle. Secondly, collisions occur at a uniform results when, e.g., Fermi-Dirac or Bose-Einstein statistics rate which is independent of the energy of the ion, and so are required to correctly characterize the buﬀer gas. In τ can be assumed to follow a uniform distribution. We practice, however, such statistics become relevant at col- assume that the density of the buﬀer gas is low and uni- lision energies low enough so that a classical description formally distributed in space, which results in collisions of the motion may no longer be valid. Moreover, at such occuring with equal probability at all points in the ion’s low energies the long-range nature and ﬁnite duration of ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: bold-faced

text: variables

text: indicate

text: vectors,

text: e.g.,

text: v

text: trajectory,

text: such

text: that

text: φ

text: follows

text: a

text: uniform

text: distribution.

text: j

text: (v

text: v

text: v

text: )T

text: v

text: is

text: the

text: velocity

text: of

text: the

text: colliding

text: particle

text: We

text: also

text: assume

text: that

text: the

text: velocity

text: of

text: the

text: buﬀer

text: gas

text: follows

text: x

text: y

text: z

text: b

text: of

text: buﬀer

text: gas,

text: m˜

text: m

text: /m

text: is

text: the

text: buﬀer

text: gas-to-ion

text: mass

text: Maxwell-Boltzmann

text: statistics

text: and

text: is

text: characterised

text: by

text: a

text: b

text: i

text: ratio,

text: and

text: R

text: is

text: a

text: rotation

text: matrix

text: determined

text: by

text: the

text: scat-

text: ﬁxed

text: temperature

text: T

text: Both

text: the

text: density

text: and

text: the

text: tem-

text: b

text: tering

text: angles.

text: As

text: with

text: the

text: position,

text: the

text: velocity

text: of

text: the

text: perature

text: of

text: the

text: buﬀer

text: gas

text: are

text: taken

text: to

text: remain

text: constant,

text: ion

text: is

text: given

text: by

text: the

text: sum

text: of

text: the

text: intrinsic

text: and

text: forced

text: mo-

text: i.e.,

text: the

text: heating

text: of

text: the

text: buﬀer

text: gas

text: due

text: to

text: the

text: collisions

text: tion,

text: v

text: (τ)

text: v

text: (A

text: φ

text: τ)

text: v

text: (τ),

text: where

text: the

text: forced

text: is

text: assumed

text: to

text: be

text: negligible.

text: With

text: these

text: assumptions,

text: j

text: h,j

text: j

text: j

text: f,j

text: term

text: is

text: independent

text: of

text: A

text: φ

text: and

text: so

text: is

text: unchanged

text: by

text: the

text: the

text: averaging

text: can

text: be

text: performed

text: over

text: φ

text: v

text: τ

text: and

text: the

text: j

text: j

text: j

text: b,k

text: collision.

text: We

text: therefore

text: obtain,

text: elements

text: of

text: the

text: isotropic

text: random

text: rotation

text: matrix

text: by

text: in-

text: tegrating

text: the

text: coeﬃcients

text: over

text: the

text: distributions

text: of

text: each

text: of

text: these

text: variables

text: in

text: turn,

text: see

text: the

text: supplemental

text: material

text: for

text: m˜

text: m˜

text: v

text: v

text: (v

text: v

text: R[v

text: (v

text: v

text: details.

text: The

text: coeﬃcients

text: of

text: the

text: terms

text: in

text: Eq.

text: where

text: h

text: m˜

text: h

text: m˜

text: b

text: f

text: m˜

text: h

text: b

text: f

text: k

text: l

text: average

text: to

text: zero,

text: as

text: do

text: the

text: c

text: c

text: c

text: coeﬃcients,

text: signiﬁcantly

text: simplifying

text: the

text: expression.

text: The

text: remaining

text: Taken

text: together,

text: Eq.

text: and

text: Eq.

text: indicate

text: that

text: terms

text: are

text: given

text: the

text: problem

text: is

text: equivalent

text: to

text: that

text: of

text: an

text: ion

text: with

text: no

text: forced

text: motion

text: colliding

text: with

text: a

text: particle

text: of

text: velocity

text: v

text: v

text: This

text: b

text: f

text: E

text: E

text: η

text: E

text: a

text: κ

text: k

text: T

text: is

text: similar

text: to

text: the

text: frame

text: transformation

text: used

text: in

text: Ref.

text: in

text: j

text: x,y,z

text: jk

text: k

text: f,k

text: j

text: B

text: b

text: which

text: the

text: intrinisic

text: micromotion

text: is

text: assigned

text: to

text: the

text: buﬀer

text: k

text: (x,y,z)

text: gas,

text: but

text: in

text: the

text: present

text: case

text: is

text: performed

text: only

text: for

text: the

text: forced

text: motion

text: and

text: is

text: valid

text: for

text: all

text: q

text: j

text: where

text: the

text: coeﬃcients

text: are

text: deﬁned

text: as,

text: Using

text: the

text: procedure

text: detailed

text: in

text: Ref.

text: we

text: obtain

text: a

text: set

text: of

text: coupled

text: equations

text: for

text: A

text: and

text: hence

text: the

text: secular

text: j

text: energies

text: a

text: δ

text: m˜

text: κ

text: η

text: jk

text: j

text: jk

text: ce˙

text: se˙

text: jk

text: m˜

text: j

text: k

text: k

text: k

text: E

text: η

text: E

text: E

text: a

text: E

text: v

text: j

text: jkl

text: k

text: l

text: k

text: b,l

text: (k,l)

text: (x,y,z)

text: and,

text: a

text: v

text: v

text: a

text: E

text: v

text: b,k

text: b,l

text: k

text: f,l

text: m˜

text: v

text: f,k

text: jk

text: j

text: f,k

text: where

text: the

text: coeﬃcients

text: η

text: and

text: a

text: describe

text: the

text: trans-

text: jkl

text: i,jkl

text: In

text: the

text: above

text: expressions,

text: δ

text: is

text: the

text: Kronecker

text: delta,

text: fer

text: of

text: energy

text: between

text: the

text: motion

text: along

text: the

text: three

text: coor-

text: jk

text: κ

text: is

text: deﬁned

text: by,

text: dinate

text: axes

text: and

text: between

text: the

text: diﬀerent

text: components

text: of

text: the

text: j

text: ion’s

text: velocity

text: and

text: the

text: velocity

text: of

text: the

text: buﬀer

text: gas.

text: The

text: co-

text: eﬃcients

text: of

text: this

text: expression

text: depend

text: on

text: the

text: elements

text: of

text: the

text: m˜

text: random

text: rotation

text: matrix

text: R,

text: the

text: set

text: of

text: phases

text: φ

text: and

text: the

text: κ

text: j

text: j

text: j

text: m˜

text: time

text: of

text: collision

text: τ.

text: As

text: supplementary

text: information

text: to

text: this

text: j

text: article,

text: we

text: provide

text: a

text: Mathematica

text: notebook

text: containing

text: M

text: details

text: of

text: this

text: procedure

text: and

text: the

text: full

text: form

text: of

text: Eq.

text: and

text: the

text: operator

text: deﬁned

text: as,

text: To

text: gain

text: a

text: better

text: understanding

text: of

text: the

text: collision

text: pro-

text: cess,

text: it

text: is

text: useful

text: to

text: average

text: over

text: the

text: collision

text: parameters

text: toof

text: foTroadgoivseon,

text: sweet

text: Mj

text: [h(τ)]

text: Ll→im∞

text: −LL

text: h(τ)

text: dτ.

text: must

text: introduce

text: some

text: further

text: assumptions.

text: The

text: Langevin

text: model

text: of

text: collisions

text: has

text: been

text: shown

text: to

text: be

text: accurate

text: for

text: the

text: classical

text: trajectories

text: considered

text: here

text: and

text: so

text: In

text: principle,

text: the

text: above

text: procedure

text: may

text: be

text: adapted

text: to

text: we

text: adopt

text: this.

text: This

text: results

text: in

text: two

text: useful

text: simpliﬁcations.

text: arbitrary

text: distributions

text: for

text: the

text: velocity

text: of

text: the

text: buﬀer

text: gas

text: by

text: Firstly,

text: the

text: rotation

text: matrix

text: R

text: is

text: isotropic

text: in

text: this

text: model

text: averaging

text: over

text: these

text: in

text: place

text: of

text: the

text: Maxwell-Boltzmann

text: and

text: so

text: is

text: uncorrelated

text: from

text: the

text: velocity

text: of

text: the

text: ion

text: and

text: distribution.

text: This

text: would

text: allow

text: for

text: an

text: investigation

text: of

text: the

text: neutral

text: particle.

text: Secondly,

text: collisions

text: occur

text: at

text: a

text: uniform

text: results

text: when,

text: e.g.,

text: Fermi-Dirac

text: or

text: Bose-Einstein

text: statistics

text: rate

text: which

text: is

text: independent

text: of

text: the

text: energy

text: of

text: the

text: ion,

text: and

text: so

text: are

text: required

text: to

text: correctly

text: characterize

text: the

text: buﬀer

text: gas.

text: In

text: τ

text: can

text: be

text: assumed

text: to

text: follow

text: a

text: uniform

text: distribution.

text: We

text: practice,

text: however,

text: such

text: statistics

text: become

text: relevant

text: at

text: col-

text: assume

text: that

text: the

text: density

text: of

text: the

text: buﬀer

text: gas

text: is

text: low

text: and

text: uni-

text: lision

text: energies

text: low

text: enough

text: so

text: that

text: a

text: classical

text: description

text: formally

text: distributed

text: in

text: space,

text: which

text: results

text: in

text: collisions

text: of

text: the

text: motion

text: may

text: no

text: longer

text: be

text: valid.

text: Moreover,

text: at

text: such

text: occuring

text: with

text: equal

text: probability

text: at

text: all

text: points

text: in

text: the

text: ion’s

text: low

text: energies

text: the

text: long-range

text: nature

text: and

text: ﬁnite

text: duration

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 6 the ion-neutral interaction introduces an additional heat- Eq. (19), and so the most signiﬁcant eﬀect is the oscilla- ∝ ing eﬀect from dislocating the ion from its position in the tions described by v (τ) sin(2τ). Hence, the present f,j rf ﬁeld. At higher collision energies, this heating eﬀect results can also be adapted to the case of excess micro- becomes less signiﬁcant [24]. To simplify both the ana- motion due to an rf phase oﬀset, which also results in lytical model and the numerical calculations, we proceed forced motion with the same form of the velocity [15]. by assuming that the energy of the ion is large enough In Fig. 2(a), the predicted mean secular energies for the so that these eﬀects can be neglected. case of excess micromotion along the x-axis and T = 0 K b The mean energy after a large number of collisions are shown and compared to the results of numerical sim- can be calculated from Eq. (21) as follows. Averaging ulations (see Appendix for details), while the results in over the pre-collision energies, corresponding to setting the absence of EMM but with a nonzero buﬀer-gas tem- (cid:104) (cid:48)| (cid:105) → (cid:104) (cid:48)(cid:105) → (cid:104) (cid:105) E E , E , E E(cid:88)and E E , produces perature (T = 100 µK) are shown in Fig. 2(b) for com- j x y z j j j b parison. At low mass ratio, there is a clear diﬀerence in (cid:104) (cid:48)(cid:105) (cid:104) (cid:105)(cid:104) (cid:105) (cid:104) (cid:105) the qualitative behaviour of the mean energies obtained E = η E + (cid:15) , (26) j jk k j for the two cases. In this regime, forced motion leads k to one high-energy component (E ) and two components x → (cid:104) (cid:105) with lower energy (E , E ), and in the limit m˜ 0 these where (cid:15) is the sum of the energy-independent terms y z j all converge to 0. In contrast, a nonzero value of T re- arising in the averaging procedure. Depending on the b (cid:104) (cid:105) sults in two high-energy components and one low-energy values of the η , the mean energy will either increase → jk component which converge to non-zero values as m˜ 0. with every collision, or tend towards a ﬁnite value for (cid:104) (cid:48)(cid:105) (cid:104) (cid:105) Note that the radial and axial components have diﬀerent which E = E . In the latter case, substituting this j j mean values even in this limit. As the mass ratio in- equality into Eq. (26) and solving for the mean energies creases, the transfer of energy between the motion along produces,      the x and y axes becomes more eﬃcient such that in        both cases there are two high-energy and one low-energy (cid:104) (cid:105)  (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)−1 (cid:104) (cid:105) component. We may therefore predict that the diﬀer- E η η η (cid:15) (cid:104) x(cid:105) − (cid:104) xx(cid:105) (cid:104) xy(cid:105) (cid:104) xz(cid:105) · (cid:104) x(cid:105) ences between the two cases will be most signiﬁcant at E = I η η η (cid:15) , (cid:104) y(cid:105) 3 (cid:104) yx(cid:105) (cid:104) yy(cid:105) (cid:104) yz(cid:105) (cid:104) y(cid:105) low mass ratio. In both cases, all energies diverge at the ≈ E η η η (cid:15) z zx zy zz z same mass ratio 1.2, conﬁrming that the critical mass (27) ratio is unaﬀected by the presence of forced motion. × III. ENERGY DISTRIBUTIONS where I is the 3 3 identity matrix. The mean to- 3 (cid:104) (cid:105) tal kinetic energy, E , of the ion including the con- j,K It has previously been established that the distribution tributions from the micromotion and the forced motion (cid:104) (cid:105) of the ion’s energy does not, in general, follow a thermal can then be evaluated from the values of E (see Ap- j distribution regardless of whether or not forced motion pendix ), (cid:88) is present, and that it closely follows Tsallis statistics (cid:104) (cid:105) in both cases [8–10, 12, 14]. Before proceeding further, (cid:104)E (cid:105) = Ej c2 (β + 2m)2 + 1m Ω2 v2 , (28) we must ﬁrst conﬁrm that the Tsallis distributions ob- j,K 2β2c2 2m 2 i 4 f,j tained through numerical simulations can be successfully j 0,j m predicted from our analytical model for the collision pro- where v2 is the mean-square velocity of the forced mo- cess. To simplify matters, we investigate the distribution f,j of the total secular energy E = E + E + E by making tion. For simplicity, however, we will continue to use the x y z the change of basis, secular energy, since for low mass ratios and low values of q this is approximately equal for each axis, whereas j the time-averaged energy is signiﬁcantly larger for axes (cid:54) E = E sin2 θ cos2 φ , with q = 0 compared to axes with q = 0 [10]. It is x ρ ρ j j possible that the matrix inversion in Eq. (27) cannot be E = E sin2 θ sin2 φ , (29) y ρ ρ performed or results in a negative energy, correspond- (cid:104) (cid:48)(cid:105) (cid:104) (cid:105) E = E cos2 θ , ing to a breakdown of the assumption that E = E z ρ j j and implying that the mean energy does not converge to ∈ a ﬁxed value. The mass ratio at which this occurs for where θρ, φρ [0, π/2) describe the relative distribu- a given set of trapping parameters is referred to as the tion of the total energy E between the three axes. The √ √ critical mass ratio [10], and since it is independent of the advantage of this basis is that it allows E to be factored (cid:15)j, it is unchanged when forced motion is present. out of ex(cid:112)press(cid:112)ions involving Ek El in Eq. (20), e.g., We now focus on the case of in-phase EMM in an ideal linear quadrupole trap deﬁned by q and a , taking − −r z q = q = q , q = 0, and a = a = a /2. The con- η E E = Eη sin2 θ sin φ cos φ . (30) x y r z x y z jxy x y jxy ρ ρ ρ stant oﬀset in the ion’s position caused by the spatially- independent force does not appear in either Eq. (17) or ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: the

text: ion-neutral

text: interaction

text: introduces

text: an

text: additional

text: heat-

text: Eq.

text: and

text: so

text: the

text: most

text: signiﬁcant

text: eﬀect

text: is

text: the

text: oscilla-

text: ing

text: eﬀect

text: from

text: dislocating

text: the

text: ion

text: from

text: its

text: position

text: in

text: the

text: tions

text: described

text: by

text: v

text: (τ)

text: Hence,

text: the

text: present

text: f,j

text: rf

text: ﬁeld.

text: At

text: higher

text: collision

text: energies,

text: this

text: heating

text: eﬀect

text: results

text: can

text: also

text: be

text: adapted

text: to

text: the

text: case

text: of

text: excess

text: micro-

text: becomes

text: less

text: signiﬁcant

text: To

text: simplify

text: both

text: the

text: ana-

text: motion

text: due

text: to

text: an

text: rf

text: phase

text: oﬀset,

text: which

text: also

text: results

text: in

text: lytical

text: model

text: and

text: the

text: numerical

text: calculations,

text: we

text: proceed

text: forced

text: motion

text: with

text: the

text: same

text: form

text: of

text: the

text: velocity

text: by

text: assuming

text: that

text: the

text: energy

text: of

text: the

text: ion

text: is

text: large

text: enough

text: In

text: Fig.

text: the

text: predicted

text: mean

text: secular

text: energies

text: for

text: the

text: so

text: that

text: these

text: eﬀects

text: can

text: be

text: neglected.

text: case

text: of

text: excess

text: micromotion

text: along

text: the

text: x-axis

text: and

text: T

text: K

text: b

text: The

text: mean

text: energy

text: after

text: a

text: large

text: number

text: of

text: collisions

text: are

text: shown

text: and

text: compared

text: to

text: the

text: results

text: of

text: numerical

text: sim-

text: can

text: be

text: calculated

text: from

text: Eq.

text: as

text: follows.

text: Averaging

text: ulations

text: (see

text: Appendix

text: for

text: details),

text: while

text: the

text: results

text: in

text: over

text: the

text: pre-collision

text: energies,

text: corresponding

text: to

text: setting

text: the

text: absence

text: of

text: EMM

text: but

text: with

text: a

text: nonzero

text: buﬀer-gas

text: tem-

text: E

text: E

text: E

text: E

text: E

text: E

text: produces

text: perature

text: (T

text: µK)

text: are

text: shown

text: in

text: Fig.

text: for

text: com-

text: j

text: x

text: y

text: z

text: j

text: j

text: j

text: b

text: parison.

text: At

text: low

text: mass

text: ratio,

text: there

text: is

text: a

text: clear

text: diﬀerence

text: in

text: the

text: qualitative

text: behaviour

text: of

text: the

text: mean

text: energies

text: obtained

text: E

text: η

text: E

text: j

text: jk

text: k

text: j

text: for

text: the

text: two

text: cases.

text: In

text: this

text: regime,

text: forced

text: motion

text: leads

text: k

text: to

text: one

text: high-energy

text: component

text: (E

text: and

text: two

text: components

text: x

text: with

text: lower

text: energy

text: (E

text: E

text: and

text: in

text: the

text: limit

text: m˜

text: these

text: where

text: is

text: the

text: sum

text: of

text: the

text: energy-independent

text: terms

text: y

text: z

text: j

text: all

text: converge

text: to

text: In

text: contrast,

text: a

text: nonzero

text: value

text: of

text: T

text: re-

text: arising

text: in

text: the

text: averaging

text: procedure.

text: Depending

text: on

text: the

text: b

text: sults

text: in

text: two

text: high-energy

text: components

text: and

text: one

text: low-energy

text: values

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text: η

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text: mean

text: energy

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text: either

text: increase

text: jk

text: component

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text: converge

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text: non-zero

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text: m˜

text: with

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text: collision,

text: or

text: tend

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text: latter

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text: this

text: limit.

text: As

text: the

text: mass

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text: equality

text: into

text: Eq.

text: and

text: solving

text: for

text: the

text: mean

text: energies

text: creases,

text: the

text: transfer

text: of

text: energy

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text: motion

text: along

text: produces,

text: the

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text: eﬃcient

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text: there

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text: two

text: high-energy

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text: one

text: low-energy

text: component.

text: We

text: may

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text: that

text: the

text: diﬀer-

text: E

text: η

text: η

text: η

text: ences

text: between

text: the

text: two

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text: will

text: be

text: most

text: signiﬁcant

text: at

text: E

text: I

text: η

text: η

text: η

text: low

text: mass

text: ratio.

text: In

text: both

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text: all

text: energies

text: diverge

text: at

text: the

text: E

text: η

text: η

text: η

text: z

text: zx

text: zy

text: zz

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text: same

text: mass

text: ratio

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text: that

text: the

text: critical

text: mass

text: ratio

text: is

text: unaﬀected

text: by

text: the

text: presence

text: of

text: forced

text: motion.

text: III.

text: ENERGY

text: DISTRIBUTIONS

text: where

text: I

text: is

text: the

text: identity

text: matrix.

text: The

text: mean

text: to-

text: tal

text: kinetic

text: energy,

text: E

text: of

text: the

text: ion

text: including

text: the

text: con-

text: j,K

text: It

text: has

text: previously

text: been

text: established

text: that

text: the

text: distribution

text: tributions

text: from

text: the

text: micromotion

text: and

text: the

text: forced

text: motion

text: of

text: the

text: ion’s

text: energy

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text: general,

text: follow

text: a

text: thermal

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text: evaluated

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text: the

text: values

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text: j

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text: Tsallis

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text: j,K

text: i

text: f,j

text: tained

text: through

text: numerical

text: simulations

text: can

text: be

text: successfully

text: j

text: m

text: predicted

text: from

text: our

text: analytical

text: model

text: for

text: the

text: collision

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text: where

text: is

text: the

text: mean-square

text: velocity

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text: the

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text: cess.

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text: simplify

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text: distribution

text: f,j

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text: total

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text: E

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text: by

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text: tion.

text: For

text: simplicity,

text: however,

text: we

text: will

text: continue

text: to

text: use

text: the

text: x

text: y

text: z

text: the

text: change

text: of

text: basis,

text: secular

text: energy,

text: since

text: for

text: low

text: mass

text: ratios

text: and

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text: values

text: of

text: q

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text: is

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text: energy

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text: signiﬁcantly

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text: axes

text: E

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text: θ

text: φ

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text: q

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text: q

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text: x

text: ρ

text: ρ

text: j

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text: possible

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text: the

text: matrix

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text: Eq.

text: cannot

text: be

text: E

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text: θ

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text: ρ

text: performed

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text: negative

text: energy,

text: correspond-

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text: θ

text: ing

text: to

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text: breakdown

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text: the

text: assumption

text: that

text: E

text: E

text: z

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text: j

text: j

text: and

text: implying

text: that

text: the

text: mean

text: energy

text: does

text: not

text: converge

text: to

text: a

text: ﬁxed

text: value.

text: The

text: mass

text: ratio

text: at

text: which

text: this

text: occurs

text: for

text: where

text: θρ,

text: φρ

text: describe

text: the

text: relative

text: distribu-

text: a

text: given

text: set

text: of

text: trapping

text: parameters

text: is

text: referred

text: to

text: as

text: the

text: tion

text: of

text: the

text: total

text: energy

text: E

text: between

text: the

text: three

text: axes.

text: The

text: critical

text: mass

text: ratio

text: and

text: since

text: it

text: is

text: independent

text: of

text: the

text: advantage

text: of

text: this

text: basis

text: is

text: that

text: it

text: allows

text: E

text: to

text: be

text: factored

text: it

text: is

text: unchanged

text: when

text: forced

text: motion

text: is

text: present.

text: out

text: of

text: involving

text: Ek

text: El

text: in

text: Eq.

text: e.g.,

text: We

text: now

text: focus

text: on

text: the

text: case

text: of

text: in-phase

text: EMM

text: in

text: an

text: ideal

text: linear

text: quadrupole

text: trap

text: deﬁned

text: by

text: q

text: and

text: a

text: taking

text: −r

text: z

text: q

text: q

text: q

text: q

text: and

text: a

text: a

text: a

text: The

text: con-

text: η

text: E

text: E

text: Eη

text: θ

text: sin

text: φ

text: cos

text: φ

text: x

text: y

text: r

text: z

text: x

text: y

text: z

text: jxy

text: x

text: y

text: jxy

text: ρ

text: ρ

text: ρ

text: stant

text: oﬀset

text: in

text: the

text: ion’s

text: position

text: caused

text: by

text: the

text: spatially-

text: independent

text: force

text: does

text: not

text: appear

text: in

text: either

text: Eq.

text: or

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text: ( ) = 7 a Tb 0 K, E×MM a faster timescale than E. Therefore, the correlations 5 × between E and θρ, φρ can be neglected and these angles 4 00..1105 ××× ×× ×× ×× < > aEvqe.ra(3g1ed) foovrera, sriensgulletinvagriinabtlhe.e lHinoewaervreerc,usrirnecnecethreesleataionn- 0.05 < > 3 E gles reﬂect the distribution of energy between the axes, 0.00 0.1 ∼0.2 0.3 ×× × <Ex> their mean values will diﬀer depending on the presence 2 × y 1 ××× ××× ××× ××× ××× m××× ××× ××× ×× × Ez atrnidbuftoiromn ooff fηo.rcTedhamt oist,iownh, illeeadthinegηtjok aarcehainndgeepienntdheendtios-f the form of the additive noise, η is not. Note that, how- 0 ∼ ever, for any given collision η and (cid:15) are approximately 0.0 0.2 ( )0.4= 0.6μ 0.8 1.0 1.2 independent of each other and of E. m A. The existence of the steady state b T 100 K, no EMM b 5 ×× ×× We now move on to the question of ﬁnding the en- 4 0.14 ××× × × < > ergy distribution of the ion given that it evolves in each 0.12 0.1 < > collision according to Eq. (31). Linear stochastic recur- 3 0.08 ∼ <Ex> rence relations of this form have been widely studied [25– 2 0.1 0.2 0.3 ×× Ey 29] and so we summarise the relevant results. Fi(cid:48)rstly, if m ×× × Ez Tb = gj(τ) = 0 such that (cid:15) is always zero, then E = ηE. 1 ××× ××× ××× ××× ××× ××× ××× ××× ××× × Note that in this model, E = 0 represents a ﬁxed point, i.e., an absorbing state, since for E = 0 and any value (cid:48) 0 ∼ of η the result of a collision is E = E = 0. The energy 0.0 0.2 0.4 0.6 0.8 1.0 1.2 after n collisions is given by [14], m (cid:89) FIG. 2: (color online) (a) The analytically calculated (cid:104) (cid:105) i=n value of the mean secular energy E for j = x (blue j E = E η . (32) solid line), y (red dashed line) and z (black dotted line) (n) (0) (i) as a function of the neutral-to-ion mass ratio, m˜ , due to i=1 the presence of excess micromotion (EMM) and Here, we use the notation x to indicate the value collisions with a buﬀer gas of temperature T = 0 K. (n) b of the variable x at collision number n. We assume that The points indicate the results obtained from numerical simulations (106 iterations for each value of m˜ ). The collisions are infrequent enough that there is n(cid:80)o correla- tion between them, and so the η are independent and trap parameters correspond to an ideal linear trap with (i) identically distributed. By taking the logarithm of both q = 0.2, a = 0.000625 and the excess micromotion is r z sides of Eq. (32), we ﬁnd ln E = ln E + ln η , gen≈erated by a uniform electric ﬁeld with a magnitude (n) (0) n (i) and so ln E undergoes a random walk with steps of of 0.5 V/m, corresponding to a displacement of the (n) size ln η [28]. The long-term behaviour of ln E there- ion from the centre of the trap by 100 nm along the (cid:104) (cid:105) (n) → fore depends on the sign of ln η to determine in which x-axis. The inset shows the trend as m˜ 0. Error bars (cid:104) (cid:105) direction this random walk is biased. If ln η > 0, then represent the standard error of the mean calculated → ∞ → ∞ (cid:104) (cid:105) ln E as n . Conversely, if ln η < 0, then from the numerical data and are typically smaller than (n) → −∞ ln E in this limit, and so the ion’s energy tends the size of the symbols. (b) As (a), but with a buﬀer (n) towards zero. Using the terminology of Ref. [29], we re- gas temperature of T = 100 µK and no EMM. In both (cid:104) (cid:105) (cid:81) b fer to the ln η < 0 situation as the contractive case and ﬁgures, the vertical asymptote indicates the point at (cid:104) (cid:105) ln η > 0 as the divergent case. which the mean energy diverges, see main text for For large n, the product η = Π follows a log- details. (i) η,n normal distribution and the distribution of E may be (n) found by avera(cid:90)ging over the initial conditions [30], Summing over j in Eq. (20), applying this change of ∞ basis, and neglecting terms with a mean value of zero we 1 obtain, (cid:88) (cid:34) (cid:88) (cid:35) fE(n)(E(n)) = 0 E(0) fΠη,n(E(n)/E(0))fE(0)(E(0))dE(0). (33) (cid:48) E = ηE+ a2,jvn,j + (a4,jkvd2,k) = ηE+(cid:15), (31) Evaluating this integral requires specifying the initial j k energy distribution, see Ref. [14] for the result when E(0) follows a thermal distribution. The energy distribution where η contains the η multiplied by functions of obtained through this method does not converge to a jkl θ , φ . As a consequence of the random rotation of the steady-state as the number of collisions increases which is ρ ρ trajectory during a collision, these two angles evolve on a known property of an unbounded purely multiplicative ) ( > < ) ( > < Km E Km E j j ) ( > < ) ( > < Km E Km E j j ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: a

text: Tb

text: K,

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text: Therefore,

text: the

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text: θρ,

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text: The

text: existence

text: of

text: the

text: steady

text: state

text: b

text: T

text: K,

text: no

text: EMM

text: b

text: We

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text: to

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text: question

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text: collision

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text: Linear

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text: <Ex>

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text: always

text: zero,

text: then

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text: Note

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text: i.e.,

text: an

text: absorbing

text: state,

text: since

text: for

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text: η

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text: n

text: collisions

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text: The

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text: we

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text: that

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text: points

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text: numerical

text: simulations

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text: tion

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text: r

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text: random

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text: with

text: steps

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text: V/m,

text: corresponding

text: to

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text: displacement

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text: (n)

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text: long-term

text: behaviour

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text: trap

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text: The

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text: if

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text: numerical

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text: limit,

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text: ion’s

text: energy

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text: size

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text: the

text: symbols.

text: (b)

text: As

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text: but

text: with

text: a

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text: (n)

text: towards

text: zero.

text: Using

text: the

text: terminology

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text: gas

text: temperature

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text: µK

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text: contractive

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text: the

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text: see

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text: details.

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text: η,n

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text: distribution

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text: (n)

text: found

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text: initial

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text: and

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text: obtain,

text: fE(n)(E(n))

text: E

text: ηE+

text: Evaluating

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text: initial

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text: The

text: energy

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text: steady-state

text: as

text: the

text: number

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text: collisions

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text: is

text: ρ

text: ρ

text: trajectory

text: during

text: a

text: collision,

text: these

text: two

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text: known

text: property

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text: an

text: unbounded

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text: multiplicative

text: Km

text: E

text: Km

text: E

text: j

text: j

text: Km

text: E

text: Km

text: E

text: j

text: j

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text: 2 energy efﬁciency, which is throughput to power consumption ratio, circuit power consumption should be involved. Resource allocation with energy efﬁciency objective function is one of active area of research for wireless system design during past years. In [6]- [16] an energy efﬁcient power allocation developed. Where in [6]- [8] circuit power consumption is ignored and [9] assume a constant circuit power con- sumption. Another aspect, which gets a high point of importance in resource allocation is QoS. QoS constraint in addition to providing a roughly fair network leads the resource allocation in a practical way. However, in [8]- [13] QoS constraint is not considered. In [6] and [12]- [15] assume number of transmit antenna is very greater than users, where this assumption is impractical, but we consider general condition, consider any number of transmit antenna and users. Authors in [16] have an error in design, they consider imperfect channel state information (CSI), then in design utilize estimated channel and real channel together. BS to acquire channel state information use time division duplexing (TDD) protocol as shown in ﬁgure1, where ﬁrst users uplink their data then uplink a pilot sequence to BS. BS utilize the pilot, which receive from users to estimate their channel and downlink data to them. It goes without saying that to have perfect CSI, the pilots, which are sent in the same bandwidth, must be orthogonal related to them. But because of resource limitation, it is not possible for all of the users in all cells to have orthogonal pilots, therefore this pilot reused in the neighbor cell by users, that it creates pilot contamination. Pilot contamination by a bad effect on channel estimation, reduce system performance [17]. Pilot contamination in [9] and [12]- [15] is not considered too. In this paper, we investigate energy efﬁcient power allocation in a cellular network with large number of transmit antenna, that users cooperate with their BS. We formulate an energy efﬁciency maximization problem related to transmitting power with limited overall transmit power in a cell and constraint on minimum data rate should be provided for all users. Because the energy efﬁciency objective function is a fractional function, we use fractional programming technique to address it. With assume cooperation between users and their BS, we found a closed form optimal transmit power for users. The algorithm of cooperative energy efﬁcient power allocation is given in Algorithm 1. Also, this work extended in multi cell case under pilot contamination considering. Finally, we evaluate purposed algorithm with simulation, which showed convergence of our iterative algorithm, and preparing a validate for our algorithm As an answer to problem. In simulation part, pilot contamination effect on energy efﬁciency was evaluated too. The remainder of this paper is organized as follows.In Section II the system model is presented. In Section III the constrained energy efﬁcient optimization problem is formulated. The solutions for these problems with detailed algorithm designs are presented in Section IV. In section V our scenario extended

text: energy

text: efﬁciency,

text: which

text: is

text: throughput

text: to

text: power

text: consumption

text: ratio,

text: circuit

text: power

text: consumption

text: should

text: be

text: involved.

text: Resource

text: allocation

text: with

text: energy

text: efﬁciency

text: objective

text: function

text: is

text: one

text: of

text: active

text: area

text: of

text: research

text: for

text: wireless

text: system

text: design

text: during

text: past

text: years.

text: In

text: an

text: energy

text: efﬁcient

text: power

text: allocation

text: developed.

text: Where

text: in

text: circuit

text: power

text: consumption

text: is

text: ignored

text: and

text: assume

text: a

text: constant

text: circuit

text: power

text: con-

text: sumption.

text: Another

text: aspect,

text: which

text: gets

text: a

text: high

text: point

text: of

text: importance

text: in

text: resource

text: allocation

text: is

text: QoS.

text: QoS

text: constraint

text: in

text: addition

text: to

text: providing

text: a

text: roughly

text: fair

text: network

text: leads

text: the

text: resource

text: allocation

text: in

text: a

text: practical

text: way.

text: However,

text: in

text: QoS

text: constraint

text: is

text: not

text: considered.

text: In

text: and

text: assume

text: number

text: of

text: transmit

text: antenna

text: is

text: very

text: greater

text: than

text: users,

text: where

text: this

text: assumption

text: is

text: impractical,

text: but

text: we

text: consider

text: general

text: condition,

text: consider

text: any

text: number

text: of

text: transmit

text: antenna

text: and

text: users.

text: Authors

text: in

text: have

text: an

text: error

text: in

text: design,

text: they

text: consider

text: imperfect

text: channel

text: state

text: information

text: (CSI),

text: then

text: in

text: design

text: utilize

text: estimated

text: channel

text: and

text: real

text: channel

text: together.

text: BS

text: to

text: acquire

text: channel

text: state

text: information

text: use

text: time

text: division

text: duplexing

text: (TDD)

text: protocol

text: as

text: shown

text: in

text: where

text: ﬁrst

text: users

text: uplink

text: their

text: data

text: then

text: uplink

text: a

text: pilot

text: sequence

text: to

text: BS.

text: BS

text: utilize

text: the

text: pilot,

text: which

text: receive

text: from

text: users

text: to

text: estimate

text: their

text: channel

text: and

text: downlink

text: data

text: to

text: them.

text: It

text: goes

text: without

text: saying

text: that

text: to

text: have

text: perfect

text: CSI,

text: the

text: pilots,

text: which

text: are

text: sent

text: in

text: the

text: same

text: bandwidth,

text: must

text: be

text: orthogonal

text: related

text: to

text: them.

text: But

text: because

text: of

text: resource

text: limitation,

text: it

text: is

text: not

text: possible

text: for

text: all

text: of

text: the

text: users

text: in

text: all

text: cells

text: to

text: have

text: orthogonal

text: pilots,

text: therefore

text: this

text: pilot

text: reused

text: in

text: the

text: neighbor

text: cell

text: by

text: users,

text: that

text: it

text: creates

text: pilot

text: contamination.

text: Pilot

text: contamination

text: by

text: a

text: bad

text: effect

text: on

text: channel

text: estimation,

text: reduce

text: system

text: performance

text: Pilot

text: contamination

text: in

text: and

text: is

text: not

text: considered

text: too.

text: In

text: this

text: paper,

text: we

text: investigate

text: energy

text: efﬁcient

text: power

text: allocation

text: in

text: a

text: cellular

text: network

text: with

text: large

text: number

text: of

text: transmit

text: antenna,

text: that

text: users

text: cooperate

text: with

text: their

text: BS.

text: We

text: formulate

text: an

text: energy

text: efﬁciency

text: maximization

text: problem

text: related

text: to

text: transmitting

text: power

text: with

text: limited

text: overall

text: transmit

text: power

text: in

text: a

text: cell

text: and

text: constraint

text: on

text: minimum

text: data

text: rate

text: should

text: be

text: provided

text: for

text: all

text: users.

text: Because

text: the

text: energy

text: efﬁciency

text: objective

text: function

text: is

text: a

text: fractional

text: function,

text: we

text: use

text: fractional

text: programming

text: technique

text: to

text: address

text: it.

text: With

text: assume

text: cooperation

text: between

text: users

text: and

text: their

text: BS,

text: we

text: found

text: a

text: closed

text: form

text: optimal

text: transmit

text: power

text: for

text: users.

text: The

text: algorithm

text: of

text: cooperative

text: energy

text: efﬁcient

text: power

text: allocation

text: is

text: given

text: in

text: Algorithm

text: Also,

text: this

text: work

text: extended

text: in

text: multi

text: cell

text: case

text: under

text: pilot

text: contamination

text: considering.

text: Finally,

text: we

text: evaluate

text: purposed

text: algorithm

text: with

text: simulation,

text: which

text: showed

text: convergence

text: of

text: our

text: iterative

text: algorithm,

text: and

text: preparing

text: a

text: validate

text: for

text: our

text: algorithm

text: As

text: an

text: answer

text: to

text: problem.

text: In

text: simulation

text: part,

text: pilot

text: contamination

text: effect

text: on

text: energy

text: efﬁciency

text: was

text: evaluated

text: too.

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.In

text: Section

text: II

text: the

text: system

text: model

text: is

text: presented.

text: In

text: Section

text: III

text: the

text: constrained

text: energy

text: efﬁcient

text: optimization

text: problem

text: is

text: formulated.

text: The

text: solutions

text: for

text: these

text: problems

text: with

text: detailed

text: algorithm

text: designs

text: are

text: presented

text: in

text: Section

text: IV.

text: In

text: section

text: V

text: our

text: scenario

text: extended


text: 6 Where a and b are auxiliary multiplicative and additive variables for user i, now use rˆ = a log (Γ sinr (p ))+ i i i i i i i b instead of r = log (1 + Γ sinr (p )) and apply the change of variable for transmit power as pˆ = ln p, i i i i i we arrive at the following approximated pro(cid:80)blem (cid:80) − P max K rˆ (pˆ) η (pˆ) l=1 l pˆ s.t. C1 : K epˆk ≤ P max (11) k=1 ≥ ∀ C2 : rˆ (pˆ) Rmin k k Because log-sum-exp is concave [23], thus problem (11) is a convex optimization problem. For solve (9) we solve (11) iteratively until transmit power converged, also update a and b in any iteration by i i obtained transmit power from current iteration with following equation sinr (pˆ ) a = i i (12a) i 1 + sinr (pˆ ) i i − b = log(1 + sinr (pˆ )) a log(sinr (pˆ )) (12b) i i i i i i Utilizing Lagrangian dual function to solve problem (11). Let λ and φ corresponding the two constraint, Lagrangian function of (11) can be written as [23] (cid:80) L (pˆ, λ, φ) (cid:80) − P = K rˆ (pˆ) η (pˆ) (cid:80)k=1 k (13) − − φ( K epˆk P max) k=1 − + K λ (Rmin r ) k=1 l k Based on Karush Kuhn Tucker condition we can ﬁnd optimal power allocation for user ith by following expression L ∂ (pˆ, λ, φ) = 0 (14) ∂p i According to (14) optimal power allocation for user i obtained as following ##LTLine## ##LTLine##

text: Where

text: a

text: and

text: b

text: are

text: auxiliary

text: multiplicative

text: and

text: additive

text: variables

text: for

text: user

text: i,

text: now

text: use

text: rˆ

text: a

text: log

text: (Γ

text: sinr

text: (p

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: b

text: instead

text: of

text: r

text: log

text: Γ

text: sinr

text: (p

text: and

text: apply

text: the

text: change

text: of

text: variable

text: for

text: transmit

text: power

text: as

text: pˆ

text: ln

text: p,

text: i

text: i

text: i

text: i

text: i

text: we

text: arrive

text: at

text: the

text: following

text: approximated

label: equation

text: P

text: max

text: K

text: rˆ

text: (pˆ)

text: η

text: (pˆ)

text: l

text: pˆ

text: s.t.

text: K

text: epˆk

text: P

text: max

text: rˆ

text: (pˆ)

text: Rmin

text: k

text: k

text: Because

text: log-sum-exp

text: is

text: concave

text: thus

text: problem

text: is

text: a

text: convex

text: optimization

text: problem.

text: For

text: solve

text: we

text: solve

text: iteratively

text: until

text: transmit

text: power

text: converged,

text: also

text: update

text: a

text: and

text: b

text: in

text: any

text: iteration

text: by

text: i

text: i

text: obtained

text: transmit

text: power

text: from

text: current

text: iteration

text: with

text: following

text: equation

text: sinr

text: (pˆ

text: a

text: i

text: i

text: i

text: sinr

text: (pˆ

text: i

text: i

text: b

text: sinr

text: (pˆ

text: a

text: log(sinr

text: (pˆ

text: i

text: i

text: i

text: i

text: i

text: i

text: Utilizing

text: Lagrangian

text: dual

text: function

text: to

text: solve

text: problem

text: Let

text: λ

text: and

text: φ

text: corresponding

text: the

text: two

text: constraint,

text: Lagrangian

text: function

text: of

text: can

text: be

text: written

text: as

text: L

text: (pˆ,

text: λ,

text: φ)

text: P

text: K

text: rˆ

text: (pˆ)

text: η

text: (pˆ)

text: k

text: φ(

text: K

text: epˆk

text: P

text: max)

text: K

text: λ

text: (Rmin

text: r

text: l

text: k

text: Based

text: on

text: Karush

text: Kuhn

text: Tucker

text: condition

text: we

text: can

text: ﬁnd

text: optimal

text: power

text: allocation

text: for

text: user

text: ith

text: by

text: following

text: expression

text: L

text: (pˆ,

text: λ,

text: φ)

text: ∂p

text: i

text: According

text: to

text: optimal

text: power

text: allocation

text: for

text: user

text: i

text: obtained

text: as

text: following

text: ##LTLine##

text: ##LTLine##


text: 8 requirement constraint and w is beamforming vector for user k in cell l respectively. Also, L shows lk the number of cells, φ is the Lagrangian multiplier for constraint on overall transmit power for cell j j and η is whole network energy efﬁciency. Optimization problem, Problem solving techniques in details and Lagrangian multipliers update formula for this case comes in appendix A. Algorithm 1 Energy Efﬁcient Cooperative Power Allocation 1: Initialize convergence tolerance (cid:15) and initialize arbitrary η(t ) = 0, con = 0 and Dinkelbach algorithm 1 iteration index t = 1 1 repeat 2: 3: initialize with a feasible pˆ , Set a (t ) = 1, t l 2 2 4: b (t ) = 0 and SCA iteration index t = 1 l 2 2 repeat 5: 6: initialize arbitrary λ (t ), φ(t ), γ , γ and Lagrangian multipliers iteration index t = 1 l 3 3 λ φ 3 repeat 7: 8: Compute I by user and feed back to BS l 9: Compute the optimal power transmit pˆ according to (15) t +1 2 10: Update φ and λ according to (17) and (18) l 11: t = t + 1 3 3 12: until Convergence of λ and φ l 13: Compute sinr (pˆ ) and update a (t + 1) and b (t + 1) (cid:80) l t2+1 l 2 l 2 14: t = t + 1 2 2 15: until convergence of pˆ − P ≤ 16: if K rˆ (Pˆ(t + 1)) η(t ) (pˆ ) (cid:15) then l=1 l 2 1 t2+1 17: Set con = 1 18: Return η(t ) as optimal energy efﬁciency and pˆ 1 19: as optimal transmit power else 20: 21: Set con = 0 (cid:80) 22: t = t + 1 1 1 Return K r (pˆ ) 23: η(t1 + 1) = lP=1(pˆl )t2 t2 until 24: con = 1 VI. ANALYSIS AND SIMULATION RESULTS In this section, we evaluate energy efﬁcient cooperative power allocation in Algorithm 1. Simulation parameters are given in Table I. At ﬁrst, we show the convergence of the algorithm. In Algorithm, we have three nested loop where any of them should be converged. Fig. 2 shows convergence of Algorithm 1 over external loop iterations for number of antennas M = 100 and number of users K = 5, where converged in three iterations. In ﬁg. 3 energy efﬁciency versus Number of transmitter antenna M and for K = 5 has been shown. We can see in this ﬁgure energy efﬁciency curve is concave. When the number of transmit antennas ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: requirement

text: constraint

text: and

text: w

text: is

text: beamforming

text: vector

text: for

text: user

text: k

text: in

text: cell

text: l

text: respectively.

text: Also,

text: L

text: shows

text: lk

text: the

text: number

text: of

text: cells,

text: φ

text: is

text: the

text: Lagrangian

text: multiplier

text: for

text: constraint

text: on

text: overall

text: transmit

text: power

text: for

text: cell

text: j

text: j

text: and

text: η

text: is

text: whole

text: network

text: energy

text: efﬁciency.

text: Optimization

text: problem,

text: Problem

text: solving

text: techniques

text: in

text: details

text: and

text: Lagrangian

text: multipliers

text: update

text: formula

text: for

text: this

text: case

text: comes

text: in

text: appendix

text: A.

text: Algorithm

text: Energy

label: caption

text: Efﬁcient

text: Cooperative

text: Power

text: Allocation

text: Initialize

text: convergence

text: tolerance

text: and

text: initialize

text: arbitrary

text: η(t

text: con

text: and

text: Dinkelbach

text: algorithm

text: iteration

text: index

text: t

text: repeat

text: initialize

text: with

text: a

text: feasible

text: pˆ

text: Set

text: a

text: (t

text: t

text: l

text: b

text: (t

text: and

text: SCA

text: iteration

text: index

text: t

text: l

text: repeat

text: initialize

text: arbitrary

text: λ

text: (t

text: φ(t

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text: Lagrangian

text: multipliers

text: iteration

text: index

text: t

text: l

text: λ

text: φ

text: repeat

text: Compute

text: I

text: by

text: user

text: and

text: feed

text: back

text: to

text: BS

text: l

text: Compute

text: the

text: optimal

text: power

text: transmit

text: pˆ

text: according

text: to

text: t

text: Update

text: φ

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text: λ

text: according

text: to

text: and

text: l

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text: of

text: λ

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text: φ

text: l

text: Compute

text: sinr

text: (pˆ

text: and

text: update

text: a

text: (t

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text: b

text: (t

text: l

text: l

text: l

text: t

text: t

text: until

text: convergence

text: of

text: pˆ

text: P

text: if

text: K

text: rˆ

text: (Pˆ(t

text: η(t

text: (pˆ

text: then

text: l

text: Set

text: con

text: Return

text: η(t

text: as

text: optimal

text: energy

text: efﬁciency

text: and

text: pˆ

text: as

text: optimal

text: transmit

text: power

text: else

text: Set

text: con

text: t

text: t

text: Return

text: K

text: r

text: (pˆ

text: until

text: con

text: VI.

text: ANALYSIS

text: AND

text: SIMULATION

text: RESULTS

text: In

text: this

text: section,

text: we

text: evaluate

text: energy

text: efﬁcient

text: cooperative

text: power

text: allocation

text: in

text: Algorithm

text: Simulation

text: parameters

text: are

text: given

text: in

text: Table

text: I.

text: At

text: ﬁrst,

text: we

text: show

text: the

text: convergence

text: of

text: the

text: algorithm.

text: In

text: Algorithm,

text: we

text: have

text: three

text: nested

text: loop

text: where

text: any

text: of

text: them

text: should

text: be

text: converged.

text: Fig.

text: shows

text: convergence

text: of

text: Algorithm

text: over

text: external

text: loop

text: iterations

text: for

text: number

text: of

text: antennas

text: M

text: and

text: number

text: of

text: users

text: K

text: where

text: converged

text: in

text: three

text: iterations.

text: In

text: ﬁg.

text: energy

text: efﬁciency

text: versus

text: Number

text: of

text: transmitter

text: antenna

text: M

text: and

text: for

text: K

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text: been

text: shown.

text: We

text: can

text: see

text: in

text: this

text: ﬁgure

text: energy

text: efﬁciency

text: curve

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text: concave.

text: When

text: the

text: number

text: of

text: transmit

text: antennas

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Ofﬂine Object Extraction from Dynamic Occupancy Grid Map Sequences(cid:63) Daniel Stumper1, Fabian Gies1, Stefan Hoermann1, and Klaus Dietmayer1 Abstract— A dynamic occupancy grid map (DOGMa) allows a fast, robust, and complete environment representation for automated vehicles. Dynamic objects in a DOGMa, however, are commonly represented as independent cells while modeled objects with shape and pose are favorable. The evaluation of algorithms for object extraction or the training and validation of learning algorithms rely on labeled ground truth data. Man- ually annotating objects in a DOGMa to obtain ground truth data is a time consuming and expensive process. Additionally the quality of labeled data depend strongly on the variation of ﬁltered input data. The presented work introduces an automatic labeling process, where a full sequence is used to extract the best possible object pose and shape in terms of temporal consistency. A two direction temporal search is executed to trace single objects over a sequence, where the best estimate of its extent and pose is reﬁned in every time step. Furthermore, the presented Fig. 1. Overlay of the DOGMa and Google Maps in Ulm inner city with extracted objects as orange rectangles after the full sequence is processed. algorithm only uses statistical constraints of the cell clusters for the object extraction instead of ﬁxed heuristic parameters. Experimental results show a well-performing automatic labeling algorithm with real sensor data even at challenging scenarios. types. Consequently, a detection of the entire environment including all trafﬁc participants is possible. Due to the independence of cells, there is no infor- I. INTRODUCTION mation of the associated object generating these measure- ments. However, for autonomous applications, e.g. behavior For automated driving or modern driver assistant systems, planning [8], full knowledge of the single object state is a detection of the vehicle surrounding is essential. For that favorable. To achieve this, a major challenge is to extract reason, more and more sensors are mounted on the vehicle objects from the grid map by associating cells to objects and to generate dense and precise measurements of the environ- represent them with spatial and dynamic information. ment. A well-studied topic to detect and track external dy- In this work we present an ofﬂine approach to extract namic objects in the environment is using temporal ﬁltering dynamic objects from a DOGMa. Therefore, the presented al- algorithms [1]. These object-model-based representations use gorithm uses acausal information from the future and past to Bayesian ﬁltering techniques and manage to suppress clutter generate a ground truth object state to any time. Starting from and false alarms, and are able to track multiple objects at a moment where an object is clearly visible, it can be traced once [2], [3]. Despite the impressive success, object tracking forward and backward in time, while the correct shape, pose in crowded urban shared space scenarios is still an tough and trajectory is reﬁned via best ﬁt on the entire sequence. challenge. Using occupancy grid maps is a complementing Due to this algorithm, even challenging separations of objects alternative to process sensor measurements and represent the moving next to each other and precise spatial information of complete environment object-model-free [4]. Therefore, the occluded or barely visible objects are possible. An example local environment is separated in grid cells, where the state of the algorithm’s result is shown in Figure 1. Therefore, of each cell is an estimation of the probabilities for occupied an overlay of the satellite image from Google Maps and a and free. The extension to a dynamic occupancy grid map DOGMA generated with Lidar sensors at an urban area is (DOGMa) [5], [6], [7] enables the estimation of dynamics depicted. Rectangles represent the extracted objects pose and in any cell. A main advantage of grid maps is the simple shape, as width and length, after the presented two directional accumulation of sensor measurements from different sensors search. It is clearly visible that even objects with only a at different time steps. Each cell is processed independently few number of cells hold the exact estimation of the shape. without any assumptions of object shapes, movements or Additionally the algorithm is designed to require almost no heuristic parameters and uses statistical constraints instead. 1All authors are with the Institute of Measurement, Control and Microtechnology, Ulm University, Germany. The main goal of the algorithm’s result is intended to serve firstname.lastname@uni-ulm.de (cid:63)"This work has been as ground truth to evaluate object extraction algorithms, e.g. submitted to the IEEE for possible publication. Copyright may be based on clustering techniques like DBSCAN [9], or as labels transferred without notice, after which this version may no longer be accessible" for learning techniques on grid maps what gains interest in ##LTFigure## ##LTFigure## ##LTFigure##

text: Ofﬂine

label: title

text: Object

text: Extraction

text: from

text: Dynamic

text: Occupancy

text: Grid

text: Map

text: Daniel

label: author

text: Fabian

text: Stefan

text: and

text: Klaus

text: Abstract—

text: A

label: abstract

text: dynamic

text: occupancy

text: grid

text: map

text: (DOGMa)

text: allows

text: a

text: fast,

text: robust,

text: and

text: complete

text: environment

text: representation

text: for

text: automated

text: vehicles.

text: Dynamic

text: objects

text: in

text: a

text: DOGMa,

text: however,

text: are

text: commonly

text: represented

text: as

text: independent

text: cells

text: while

text: modeled

text: objects

text: with

text: shape

text: and

text: pose

text: are

text: favorable.

text: The

text: evaluation

text: of

text: algorithms

text: for

text: object

text: extraction

text: or

text: the

text: training

text: and

text: validation

text: of

text: learning

text: algorithms

text: rely

text: on

text: labeled

text: ground

text: truth

text: data.

text: Man-

text: ually

text: annotating

text: objects

text: in

text: a

text: DOGMa

text: to

text: obtain

text: ground

text: truth

text: data

text: is

text: a

text: time

text: consuming

text: and

text: expensive

text: process.

text: Additionally

text: the

text: quality

text: of

text: labeled

text: data

text: depend

text: strongly

text: on

text: the

text: variation

text: of

text: ﬁltered

text: input

text: data.

text: The

text: presented

text: work

text: introduces

text: an

text: automatic

text: labeling

text: process,

text: where

text: a

text: full

text: sequence

text: is

text: used

text: to

text: extract

text: the

text: best

text: possible

text: object

text: pose

text: and

text: shape

text: in

text: terms

text: of

text: temporal

text: consistency.

text: A

text: two

text: direction

text: temporal

text: search

text: is

text: executed

text: to

text: trace

text: single

text: objects

text: over

text: a

text: sequence,

text: where

text: the

text: best

text: estimate

text: of

text: its

text: extent

text: and

text: pose

text: is

text: reﬁned

text: in

text: every

text: time

text: step.

text: Furthermore,

text: the

text: presented

text: Fig.

text: Overlay

text: of

text: the

text: DOGMa

text: and

text: Google

text: Maps

text: in

text: Ulm

text: inner

text: city

text: with

text: extracted

text: objects

text: as

text: orange

text: rectangles

text: after

text: the

text: full

text: sequence

text: is

text: processed.

text: algorithm

text: only

text: uses

text: statistical

text: constraints

text: of

text: the

text: cell

text: clusters

text: for

text: the

text: object

text: extraction

text: instead

text: of

text: ﬁxed

text: heuristic

text: parameters.

text: Experimental

text: results

text: show

text: a

text: well-performing

text: automatic

text: labeling

text: algorithm

text: with

text: real

text: sensor

text: data

text: even

text: at

text: challenging

text: scenarios.

text: types.

text: Consequently,

text: a

text: detection

text: of

text: the

text: entire

text: environment

text: including

text: all

text: trafﬁc

text: participants

text: is

text: possible.

text: Due

text: to

text: the

text: independence

text: of

text: cells,

text: there

text: is

text: no

text: infor-

text: I.

text: INTRODUCTION

text: mation

text: of

text: the

text: associated

text: object

text: generating

text: these

text: measure-

text: ments.

text: However,

text: for

text: autonomous

text: applications,

text: e.g.

text: behavior

text: For

text: automated

text: driving

text: or

text: modern

text: driver

text: assistant

text: systems,

text: planning

text: full

text: knowledge

text: of

text: the

text: single

text: object

text: state

text: is

text: a

text: detection

text: of

text: the

text: vehicle

text: surrounding

text: is

text: essential.

text: For

text: that

text: favorable.

text: To

text: achieve

text: this,

text: a

text: major

text: challenge

text: is

text: to

text: extract

text: reason,

text: more

text: and

text: more

text: sensors

text: are

text: mounted

text: on

text: the

text: vehicle

text: objects

text: from

text: the

text: grid

text: map

text: by

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