text: 40 MARIA ANGELICA CUETO AND HANNAH MARKWIG assumptions imposed in Theorem 7.3 yield the formulas for the tropical Igusa invariants for these curves. In w(cid:101) hat remains, we discussed the remaining three types: (I), (IV) and (V). As discussed above, the initial form of A will not have a uniform value on each of these cones when char K = 3. We bypass this diﬃculty by writing A as an integer combination of four ± polynomials with coeﬃcients 1 and disjoint supports, comparing the valuation of their initial forms, and considering possible ties and cancellations. A calculation available on the Supplementary material yields A = 4A +6A +12A +120A . Any ω in the relative 4 6 12 120 interior of the Type (I), (IV) or (V) cells gives − in (A ) = α2α2α α , in (A ) = α2α α α α , (7.13) ω 4 6 5 4 3 ω 12 − 6 5 4 3 2 in (A ) = α2α2α2 , in (A ) = α α α α α α . ω 6 6 5 4 ω 120 6 5 4 3 2 1 Since ω < ω < . . . < ω in Type (I), we conclude that in (A ) > in (A ), in (A ) 1 2 6 ω 6 ω 12 ω 120 so we need only compare the weights of A and 6A . There are two cases to analyze: 4 6 − ≤ − − Case 1: If 1 ω ω , our genericity assumptions ensure that val(6α 4α ) = − 4 3 4 3 1 ω . Thus val(A) is the one predicted in Lemma 7.5. The formulas for the 4 tropical Igusa invariants described in Theorem 7.3 remain valid in this setting. − − − Case 2: If 1 ω > ω , we conclude that in (A) = 4 in (A ), and so val(A) = 4 3 ω ω 4 2ω + 2ω + ω + ω . The expressions for val(B), val(C), and val(D) obtained 6 5 4 3 from Lemma 7.5 and arithmetic manipulations as in the proof of Theorem 7.3 (cid:3) yield the desired expressions for the Igusa invariants on the Type (I) cone. − − − Remark 7.8. If 1 ω = ω and in(6α ) = in( 4α ) the above methods do not allow us − 4 3 4 3 to compute val(A). We bypass this diﬃculty by using three Laurent monomials in the 3 Igusa invariants where the polynomial A is cancelled out. Lemma 7.5 yields: − − 5jtrop(Trop X) 3jtrop(Trop X) = 2 val(D) 5 val(B) = 2L + 4L + 2L , 2 1 1 0 2 − − 3jtrop(Trop X) 2jtrop(Trop X) = 3 val(D) 5 val(C) = 3L + 16L + 3L , 3 1 1 0 2 − − 3jtrop(Trop X) 2jtrop(Trop X) = val(D) + 2 val(B) 3 val(C) = L + 8L + L . 3 2 1 0 2 The matrix describing the three integer linear combination of jtrop, jtrop and jtrop has 1 2 3 rank two, so we can only express the last two invariants in terms of jtrop. 1 8. A new Igusa invariant As was shown in Remark 7.4, the fact that the tropical Igusa functions do not yield coordinates on Mtrop raises a natural question: can we replace j , j , j by an alternative 2 1 2 3 set of three algebraic invariants better suited for tropicalization? Given the expressions in Theorem 7.3 we propose to replace j with a linear expression in j , j , j whose initial 3 1 2 3 form corresponding to a weight vector of Type (I) or (II) appears as a result of a cancella- tion in the initial forms of the j ’s. In other words, we aim to compute a Khovanskii basis i of the ring of invariants of M . 2 The computation of val(A), val(B) and val(C) on Types (I) and (II) in expressions (7.8) and (7.11) gives the linear relation − − − val(A) val(B) = val(C). Therefore, a cancellation might be produced among leading terms via the expressions ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: assumptions

text: imposed

text: in

text: Theorem

text: yield

text: the

text: formulas

text: for

text: the

text: tropical

text: Igusa

text: invariants

text: for

text: these

text: curves.

text: In

text: hat

text: remains,

text: we

text: discussed

text: the

text: remaining

text: three

text: types:

text: (I),

text: (IV)

text: and

text: (V).

text: As

text: discussed

text: above,

text: the

text: initial

text: form

text: of

text: A

text: will

text: not

text: have

text: a

text: uniform

text: value

text: on

text: each

text: of

text: these

text: cones

text: when

text: char

text: K

text: We

text: bypass

text: this

text: diﬃculty

text: by

text: writing

text: A

text: as

text: an

text: integer

text: combination

text: of

text: four

text: polynomials

text: with

text: coeﬃcients

text: and

text: disjoint

text: supports,

text: comparing

text: the

text: valuation

text: of

text: their

text: initial

text: forms,

text: and

text: considering

text: possible

text: ties

text: and

text: cancellations.

text: A

text: calculation

text: available

text: on

text: the

text: Supplementary

text: material

text: yields

text: A

text: Any

text: ω

text: in

text: the

text: relative

text: interior

text: of

text: the

text: Type

text: (I),

text: (IV)

text: or

text: (V)

text: cells

text: gives

text: in

text: (A

text: α2α2α

text: α

text: in

text: (A

text: α2α

text: α

text: α

text: α

text: ω

text: ω

text: in

text: (A

text: α2α2α2

text: in

text: (A

text: α

text: α

text: α

text: α

text: α

text: α

text: ω

text: ω

text: Since

text: ω

text: ω

text: ω

text: in

text: Type

text: (I),

text: we

text: conclude

text: that

text: in

text: (A

text: in

text: (A

text: in

text: (A

text: ω

text: ω

text: ω

text: so

text: we

text: need

text: only

text: compare

text: the

text: weights

text: of

text: A

text: and

text: There

text: are

text: two

text: cases

text: to

text: analyze:

text: Case

text: If

text: ω

text: ω

text: our

text: genericity

text: assumptions

text: ensure

text: that

text: 4α

text: ω

text: Thus

text: val(A)

text: is

text: the

text: one

text: predicted

text: in

text: Lemma

text: The

text: formulas

text: for

text: the

text: tropical

text: Igusa

text: invariants

text: described

text: in

text: Theorem

text: remain

text: valid

text: in

text: this

text: setting.

text: Case

text: If

text: ω

text: ω

text: we

text: conclude

text: that

text: in

text: (A)

text: in

text: (A

text: and

text: so

text: val(A)

text: ω

text: ω

text: 2ω

text: 2ω

text: ω

text: ω

text: The

text: expressions

text: for

text: val(B),

text: val(C),

text: and

text: val(D)

text: obtained

text: from

text: Lemma

text: and

text: arithmetic

text: manipulations

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: yield

text: the

text: desired

text: expressions

text: for

text: the

text: Igusa

text: invariants

text: on

text: the

text: Type

text: (I)

text: cone.

text: Remark

text: If

text: ω

text: ω

text: and

text: in(

text: 4α

text: the

text: above

text: methods

text: do

text: not

text: allow

text: us

text: to

text: compute

text: val(A).

text: We

text: bypass

text: this

text: diﬃculty

text: by

text: using

text: three

text: Laurent

text: monomials

text: in

text: the

text: Igusa

text: invariants

text: where

text: the

text: polynomial

text: A

text: is

text: cancelled

text: out.

text: Lemma

text: yields:

text: X)

text: X)

text: val(D)

text: val(B)

text: X)

text: X)

text: val(D)

text: val(C)

text: X)

text: X)

text: val(D)

text: val(B)

text: val(C)

text: L

text: L

text: The

text: matrix

text: describing

text: the

text: three

text: integer

text: linear

text: combination

text: of

text: jtrop,

text: jtrop

text: and

text: jtrop

text: has

text: rank

text: two,

text: so

text: we

text: can

text: only

text: express

text: the

text: last

text: two

text: invariants

text: in

text: terms

text: of

text: jtrop.

text: A

text: new

text: Igusa

text: invariant

text: As

text: was

text: shown

text: in

text: Remark

text: the

text: fact

text: that

text: the

text: tropical

text: Igusa

text: functions

text: do

text: not

text: yield

text: coordinates

text: on

text: Mtrop

text: raises

text: a

text: natural

text: question:

text: can

text: we

text: replace

text: j

text: j

text: j

text: by

text: an

text: alternative

text: set

text: of

text: three

text: algebraic

text: invariants

text: better

text: suited

text: for

text: tropicalization?

text: Given

text: the

text: expressions

text: in

text: Theorem

text: we

text: propose

text: to

text: replace

text: j

text: with

text: a

text: linear

text: expression

text: in

text: j

text: j

text: j

text: whose

text: initial

text: form

text: corresponding

text: to

text: a

text: weight

text: vector

text: of

text: Type

text: (I)

text: or

text: (II)

text: appears

text: as

text: a

text: result

text: of

text: a

text: cancella-

text: tion

text: in

text: the

text: initial

text: forms

text: of

text: the

text: j

text: ’s.

text: In

text: other

text: words,

text: we

text: aim

text: to

text: compute

text: a

text: Khovanskii

text: basis

text: i

text: of

text: the

text: ring

text: of

text: invariants

text: of

text: M

text: The

text: computation

text: of

text: val(A),

text: val(B)

text: and

text: val(C)

text: on

text: Types

text: (I)

text: and

text: (II)

text: in

text: expressions

text: and

text: gives

text: the

text: linear

text: relation

text: val(A)

text: val(B)

text: val(C).

text: Therefore,

text: a

text: cancellation

text: might

text: be

text: produced

text: among

text: leading

text: terms

text: via

text: the

text: expressions

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 46 MARIA ANGELICA CUETO AND HANNAH MARKWIG

label: paragraph

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG


text: TROPICAL GEOMETRY OF GENUS TWO CURVES 7 with Γ [30, 29]. Indeed, to each vertex v in Γ we assign the sum of the genera of all → semistable vertices of Σ(X) mapping to v under trop: Σ(X) Trop X. The semistable vertices correspond exactly to the components of the central ﬁber X [4], so we weigh them 0 with the genus of the associated component. For planar tropicalizations, where Γ is the dual complex of the Newton subdivision of the corresponding curve, we assign to each vertex v of Γ the number of interior lattice points of its dual polygon. This quantity is the genus of the initial degeneration of the curve induced by the vertex v minus the number of nodes (assuming it is nodal). → Z After the choice of a weight function g : V (Γ) ≥ , the forgetful map extracts the 0 combinatorial type and edge lengths of the graph Γ, contracts all legs and, iteratively, all bounded edges adjacent to a 1-valent rational vertex, in analogy with the procedure for abstract tropical curves. The result is an abstract weighted graph. In particular, for tropicalizations of planar hyperelliptic curves induced by (1.1), the forgetful map yields one of two possibilities: • a vertex of genus two, • − two vertices of genus one, connected by an edge of length (val(α ) val(α ))/2 ≥ ≥ 2 4 when val(α ) . . . val(α ) satisfy the Type (V) conditions in Table 1. 1 6 In both cases, we obtain abstract tropical curves in lower-dimensional cells of Mtrop, 2 namely Types (V) and (VII). In the algebraic setting, the forgetful map sending plane genus two smooth hyperelliptic curves to points in M (K) is surjective if we allow the curves to be deﬁned over valued | 2 ﬁeld extensions L K. As we observed above, the forgetful map on the associated tropical curves is far from being surjective: its image lies in two lower–dimensional cells of Mtrop. 2 Therefore, the left square in Figure 1 would never commute if we restrict ourselves to na¨ıve planar tropicalizations induced by hyperelliptic equations. A similar behavior in genus 3 and 4 was encoutered by Brodsky-Joswig-Morrison-Sturmfels [10, Theorems 5.1 and 7.1]. The solution to this problem will be provided by a faithful tropicalization in a suit- G able torus n . Section and Table 2 give explicit eﬀective methods for producing such m embeddings for smooth plane genus two curves. The main technique involved is tropical R modiﬁcations of n along tropical divisors [12, 33, 41], which we now recall. Deﬁnition 3.1. Fix a tropical polynomial F deﬁning a piecewise linear function R → T R ∪ {−∞} { } T F : n = F (X) = max C + β X + . . . + β X in [X , . . . , X ]. ∈Z β 1 1 n n 1 n β n≥ 0 − The graph of F is(cid:101) a rational polyhedral complex of pure dimension n 1. Unless F is linear, the bend locus of F has codimension 1. At each break codimension-1 cell σ, we − − attach a new cell σ spanned by σ and e := (0, . . . , 0, 1). The result is a pure rational n R R polyhedral complex in n. We call it the tropical modiﬁcation of n along F . (cid:88) It will often be useful to consider polynomial lifts of F , namely ∈ { (cid:54) −∞} (3.1) f(x) = c xβ K[x , . . . , x ] where supp(F ) := β : C = β 1 n β ∈ β supp(F) {− } satisﬁes trop(f)(X) := max val(c ) + β X + . . . + β X = F (X) as functions on R β β 1 1 n n n. By the Structure Theorem [39, Proposition 3.1.6], any polynomial lift f of F will allow R us to turn the tropical modiﬁcation of n along F into a weighted balanced complex, since it will be supported on the tropical hypersurface Trop V (f). In turn, any tropical R hypersurface Trop V (g) in n can be modiﬁed along F in a similar fashion and the attached cells can be endowed with suitable multiplicities to turn the resulting complex into a balanced one. For the precise formulas, we refer to [2, Construction 3.3]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TROPICAL

text: GEOMETRY

text: OF

text: GENUS

text: TWO

text: CURVES

text: with

text: Γ

text: Indeed,

text: to

text: each

text: vertex

text: v

text: in

text: Γ

text: we

text: assign

text: the

text: sum

text: of

text: the

text: genera

text: of

text: all

text: semistable

text: vertices

text: of

text: Σ(X)

text: mapping

text: to

text: v

text: under

text: trop:

text: Σ(X)

text: Trop

text: X.

text: The

text: semistable

text: vertices

text: correspond

text: exactly

text: to

text: the

text: components

text: of

text: the

text: central

text: ﬁber

text: X

text: so

text: we

text: weigh

text: them

text: with

text: the

text: genus

text: of

text: the

text: associated

text: component.

text: For

text: planar

text: tropicalizations,

text: where

text: Γ

text: is

text: the

text: dual

text: complex

text: of

text: the

text: Newton

text: subdivision

text: of

text: the

text: corresponding

text: curve,

text: we

text: assign

text: to

text: each

text: vertex

text: v

text: of

text: Γ

text: the

text: number

text: of

text: interior

text: lattice

text: points

text: of

text: its

text: dual

text: polygon.

text: This

text: quantity

text: is

text: the

text: genus

text: of

text: the

text: initial

text: degeneration

text: of

text: the

text: curve

text: induced

text: by

text: the

text: vertex

text: v

text: minus

text: the

text: number

text: of

text: nodes

text: (assuming

text: it

text: is

text: nodal).

text: Z

text: After

text: the

text: choice

text: of

text: a

text: weight

text: function

text: g

text: V

text: (Γ)

text: the

text: forgetful

text: map

text: extracts

text: the

text: combinatorial

text: type

text: and

text: edge

text: lengths

text: of

text: the

text: graph

text: Γ,

text: contracts

text: all

text: legs

text: and,

text: iteratively,

text: all

text: bounded

text: edges

text: adjacent

text: to

text: a

text: rational

text: vertex,

text: in

text: analogy

text: with

text: the

text: procedure

text: for

text: abstract

text: tropical

text: curves.

text: The

text: result

text: is

text: an

text: abstract

text: weighted

text: graph.

text: In

text: particular,

text: for

text: tropicalizations

text: of

text: planar

text: hyperelliptic

text: curves

text: induced

text: by

text: the

text: forgetful

text: map

text: yields

text: one

text: of

text: two

text: possibilities:

text: a

text: vertex

text: of

text: genus

text: two,

text: two

text: vertices

text: of

text: genus

text: one,

text: connected

text: by

text: an

text: edge

text: of

text: length

text: (val(α

text: val(α

text: when

text: val(α

text: val(α

text: satisfy

text: the

text: Type

text: (V)

text: conditions

text: in

text: Table

text: In

text: both

text: cases,

text: we

text: obtain

text: abstract

text: tropical

text: curves

text: in

text: lower-dimensional

text: cells

text: of

text: Mtrop,

text: namely

text: Types

text: (V)

text: and

text: (VII).

text: In

text: the

text: algebraic

text: setting,

text: the

text: forgetful

text: map

text: sending

text: plane

text: genus

text: two

text: smooth

text: hyperelliptic

text: curves

text: to

text: points

text: in

text: M

text: (K)

text: is

text: surjective

text: if

text: we

text: allow

text: the

text: curves

text: to

text: be

text: deﬁned

text: over

text: valued

text: ﬁeld

text: extensions

text: L

text: K.

text: As

text: we

text: observed

text: above,

text: the

text: forgetful

text: map

text: on

text: the

text: associated

text: tropical

text: curves

text: is

text: far

text: from

text: being

text: surjective:

text: its

text: image

text: lies

text: in

text: two

text: lower–dimensional

text: cells

text: of

text: Mtrop.

text: Therefore,

text: the

text: left

text: square

text: in

text: Figure

text: would

text: never

text: commute

text: if

text: we

text: restrict

text: ourselves

text: to

text: na¨ıve

text: planar

text: tropicalizations

text: induced

text: by

text: hyperelliptic

text: equations.

text: A

text: similar

text: behavior

text: in

text: genus

text: and

text: was

text: encoutered

text: by

text: Brodsky-Joswig-Morrison-Sturmfels

text: Theorems

text: and

text: The

text: solution

text: to

text: this

text: problem

text: will

text: be

text: provided

text: by

text: a

text: faithful

text: tropicalization

text: in

text: a

text: suit-

text: G

text: able

text: torus

text: n

text: Section

text: and

text: Table

text: give

text: explicit

text: eﬀective

text: methods

text: for

text: producing

text: such

text: m

text: embeddings

text: for

text: smooth

text: plane

text: genus

text: two

text: curves.

text: The

text: main

text: technique

text: involved

text: is

text: tropical

text: R

text: modiﬁcations

text: of

text: n

text: along

text: tropical

text: divisors

text: which

text: we

text: now

text: recall.

text: Deﬁnition

text: Fix

text: a

text: tropical

text: polynomial

text: F

text: deﬁning

text: a

text: piecewise

text: linear

text: function

text: R

text: T

text: R

text: T

text: F

text: n

text: F

text: (X)

text: max

text: C

text: β

text: X

text: β

text: X

text: in

text: [X

text: X

text: ∈Z

text: β

text: n

text: n

text: n

text: β

text: n≥

text: The

text: graph

text: of

text: F

text: a

text: rational

text: polyhedral

text: complex

text: of

text: pure

text: dimension

text: n

text: Unless

text: F

text: is

text: linear,

text: the

text: bend

text: locus

text: of

text: F

text: has

text: codimension

text: At

text: each

text: break

text: cell

text: σ,

text: we

text: attach

text: a

text: new

text: cell

text: σ

text: spanned

text: by

text: σ

text: and

text: e

text: The

text: result

text: is

text: a

text: pure

text: rational

text: n

text: R

text: R

text: polyhedral

text: complex

text: in

text: n.

text: We

text: call

text: it

text: the

text: tropical

text: modiﬁcation

text: of

text: n

text: along

text: F

text: It

text: will

text: often

text: be

text: useful

text: to

text: consider

text: polynomial

text: lifts

text: of

text: F

text: namely

text: f(x)

text: c

text: xβ

text: K[x

text: x

text: where

text: supp(F

text: β

text: C

text: β

text: n

text: β

text: β

text: supp(F)

text: satisﬁes

text: trop(f)(X)

text: max

text: val(c

text: β

text: X

text: β

text: X

text: F

text: (X)

text: as

text: functions

text: on

text: R

text: β

text: β

text: n

text: n

text: n.

text: By

text: the

text: Structure

text: Theorem

text: Proposition

text: any

text: polynomial

text: lift

text: f

text: of

text: F

text: will

text: allow

text: R

text: us

text: to

text: turn

text: the

text: tropical

text: modiﬁcation

text: of

text: n

text: along

text: F

text: into

text: a

text: weighted

text: balanced

text: complex,

text: since

text: it

text: will

text: be

text: supported

text: on

text: the

text: tropical

text: hypersurface

text: Trop

text: V

text: (f).

text: In

text: turn,

text: any

text: tropical

text: R

text: hypersurface

text: Trop

text: V

text: (g)

text: in

text: n

text: can

text: be

text: modiﬁed

text: along

text: F

text: in

text: a

text: similar

text: fashion

text: and

text: the

text: attached

text: cells

text: can

text: be

text: endowed

text: with

text: suitable

text: multiplicities

text: to

text: turn

text: the

text: resulting

text: complex

text: into

text: a

text: balanced

text: one.

text: For

text: the

text: precise

text: formulas,

text: we

text: refer

text: to

text: Construction

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 (cid:78) { } out that, deploying distributed antennas has been shown a agonal elements a , a , . . . , a . The expectation operator 1 2 N promising technique to increase spectral efﬁciency and expand is denoted by E()˙. [A] gives the (i, j)th entry of matrix A. ij coverage of wireless communication networks [19]– [23]. As A B is the Kronecker product of A and B. We write a such, it is of great interest to consider distributed antenna de- function a(x) of x as o(x) if lim → a(x)/x = 0. Finally, CN x 0 ployment in the context of mmWave massive MIMO systems. (0, 1) denotes a circularly symmetric complex Gaussian The diversity-multiplexing tradeoff (DMT) is a compact and random variable with zero mean and unit variance. convenient framework to compare different MIMO systems II. SYSTEM MODEL in terms of the two main and related system indicators: data Consider a single-user mmWave massive MIMO system as rate and error performance [24]–[27]. This tradeoff was origi- shown in Fig. 1. The transmitter is equipped with a distributed nally characterized in [24] for MIMO communication systems antenna array to send N data streams to a receiver, which operating over independent and identically distributed (i.i.d.) s is also equipped with a distributed antenna array. Here, a Rayleigh fading channels. The framework has then ignited a distributed antenna array means an array consisting of several lot of interests in analyzing various communication systems remote antenna units (RAUs) (i.e., antenna subarrays) that are and under different channel models. For a mmWave massive distributively located, as depicted in Fig. 2. Speciﬁcally, the MIMO system, how to quantify the diversity performance and antenna array at the transmitter consists of K RAUs, each of characterize its DMT is a fundamental and open research t which has N antennas and is connected to a baseband pro- problem. In particular, to the best of our knowledge, until t cessing unit (BPU) by ﬁber. Likewise, the distributed antenna now there is no uniﬁed diversity gain analysis for mmWave array at the receiver consists of K RAUs, each having N massive MIMO systems that is applicable to both co-located r r antennas and also being connected to a BPU by ﬁbers. Such and distributed antenna array architectures. a MIMO system shall be referred to as a (K , N , K , N ) To ﬁll this gap, this paper investigates the diversity per- t t r r distributed MIMO (D-MIMO) system. When K = K = 1, formance of mmWave massive MIMO systems with the t r the system reduces to a conventional co-located MIMO (C- proposed distributed subarray architecture (the multiplexing MIMO) system. performance will be investigated in another paper). The focus The transmitter accepts as its input N data streams and is is on the asymptotical diversity gain analysis in order to ﬁnd s ≤ ≤ equipped with N(rf) RF chains, where N N(rf) N K . out the potential diversity advantage provided by multiple t s t t t Given N(rf) transmit RF chains, the transmitter can apply a distributed antenna arrays. The obtained analysis can be used t × conveniently to compare various mmWave massive MIMO low-dimension Nt(rf) Ns b×aseband precoder, Wt, followed systems with different distributed antenna array structures. by a high-dimension KtNt Nt(rf) RF precoder, Ft. Note that amplitude and phase modiﬁcations are feasible for the The main contributions of this paper are summarized as follows: First, for a single-user system employing the proposed baseband precoder Wt, while only phase changes can be made distributed subarray architecture, a diversity gain expression by the RF precoder Ft through the use of variable phase shifters and combiners. The transmitted signal vector can be is obtained when the number of antennas at each subarray written as increases without bound. This expression clearly indicates that one can obtain a large diversity gain and/or multiplexing gain x = FtWts, (1) by employing the proposed distributed subarray architecture. × E where s is the N 1 symbol vector such that [ssH] = Second, the diversity gain analysis is extended to the multiuser s P I . Thus P represents the average total input power. scenario with downlink and uplink transmission, as well as CNosnsNisdering a narrowband block fading channel, the K N ×1 the single-user system employing the conventional partially- r r received signal vector is connected RF structure based on the distributed subarrays. Simulation results are provided to corroborate the analysis y = HF W s + n (2) t t results and show that the distributed subarray architecture × × yields signiﬁcantly better diversity performance than the co- where H is KrNr KtNtCcNhannel matrix and n is a KrNr 1 located single-array architecture. vector consisting of i.i.d. (0, 1) noise samples. Throughout The remainder of this paper is organized as follows. Section this paper, H is assumed known to both the transmitter and ≤ ≤ II describes the massive MIMO system model and hybrid pro- receiver. Given that Nr(rf) RF chains (where Ns Nr(rf) cessing with the distributed subarray architecture in mmWave NrKr) are used at the receiver to detect the Ns data streams, the processed signal is given by fading channels. Section III provides the asymptotical diversity analysis for the single-user mmWave system. In Sections IV z = WHFHHF W s + WHFHn (3) and V, the diversity gain analysis is extended to the multiuser r r t t r r × scenario and the scenario with the conventional partially- where F is the K N N(rf) RF combining matrix, and W r × r r r r connected RF architecture, respectively. Numerical results are is the N(rf) N baseband combining matrix. r s presented in Section VI. Section VII concludes the paper. Furthermore, according to the architecture of RAUs at the Throughout this paper, the following notations are used. transmitting and receiving ends, H can be written as  √ √  Boldface upper and lower case letters denote matrices and  · · ·  · · cstoalnudmnfovrecttroarnss,proessepecatnivdelyc.onTjhuegastuep-terrasncsrpipotsse,( )rTespaencdtiv(e)lHy. H = √g...11H11 ... √ g1...KtH1Kt . (4) { } · · · diag a1, a2, . . . , aN stands for a diagonal matrix with di- gKr1HKr1 gKrKtHKrKt ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: out

text: that,

text: deploying

text: distributed

text: antennas

text: has

text: been

text: shown

text: a

text: agonal

text: elements

text: a

text: a

text: a

text: The

text: expectation

text: operator

text: N

text: promising

text: technique

text: to

text: increase

text: spectral

text: efﬁciency

text: and

text: expand

text: is

text: denoted

text: by

text: E()˙.

text: [A]

text: gives

text: the

text: (i,

text: j)th

text: entry

text: of

text: matrix

text: A.

text: ij

text: coverage

text: of

text: wireless

text: communication

text: networks

text: As

text: A

text: B

text: is

text: the

text: Kronecker

text: product

text: of

text: A

text: and

text: B.

text: We

text: write

text: a

text: such,

text: it

text: is

text: of

text: great

text: interest

text: to

text: consider

text: distributed

text: antenna

text: de-

text: function

text: a(x)

text: of

text: x

text: as

text: o(x)

text: if

text: lim

text: a(x)/x

text: Finally,

text: CN

text: x

text: ployment

text: in

text: the

text: context

text: of

text: mmWave

text: massive

text: MIMO

text: systems.

text: denotes

text: a

text: circularly

text: symmetric

text: complex

text: Gaussian

text: The

text: diversity-multiplexing

text: tradeoff

text: (DMT)

text: is

text: a

text: compact

text: and

text: random

text: variable

text: with

text: zero

text: mean

text: and

text: unit

text: variance.

text: convenient

text: framework

text: to

text: compare

text: different

text: MIMO

text: systems

text: II.

text: SYSTEM

text: MODEL

text: in

text: terms

text: of

text: the

text: two

text: main

text: and

text: related

text: system

text: indicators:

text: data

text: Consider

text: a

text: single-user

text: mmWave

text: massive

text: MIMO

text: system

text: as

text: rate

text: and

text: error

text: performance

text: This

text: tradeoff

text: was

text: origi-

text: shown

text: in

text: Fig.

text: The

text: transmitter

text: is

text: equipped

text: with

text: a

text: distributed

text: nally

text: characterized

text: in

text: for

text: MIMO

text: communication

text: systems

text: antenna

text: array

text: to

text: send

text: N

text: data

text: streams

text: to

text: a

text: receiver,

text: which

text: operating

text: over

text: independent

text: and

text: identically

text: distributed

text: (i.i.d.)

text: s

text: is

text: also

text: equipped

text: with

text: a

text: distributed

text: antenna

text: array.

text: Here,

text: a

text: Rayleigh

text: fading

text: channels.

text: The

text: framework

text: has

text: then

text: ignited

text: a

text: distributed

text: antenna

text: array

text: means

text: an

text: array

text: consisting

text: of

text: several

text: lot

text: of

text: interests

text: in

text: analyzing

text: various

text: communication

text: systems

text: remote

text: antenna

text: units

text: (RAUs)

text: (i.e.,

text: antenna

text: subarrays)

text: that

text: are

text: and

text: under

text: different

text: channel

text: models.

text: For

text: a

text: mmWave

text: massive

text: distributively

text: located,

text: as

text: depicted

text: in

text: Fig.

text: Speciﬁcally,

text: the

text: MIMO

text: system,

text: how

text: to

text: quantify

text: the

text: diversity

text: performance

text: and

text: antenna

text: array

text: at

text: the

text: transmitter

text: consists

text: of

text: K

text: RAUs,

text: each

text: of

text: characterize

text: its

text: DMT

text: is

text: a

text: fundamental

text: and

text: open

text: research

text: t

text: which

text: has

text: N

text: antennas

text: and

text: is

text: connected

text: to

text: a

text: baseband

text: pro-

text: problem.

text: In

text: particular,

text: to

text: the

text: best

text: of

text: our

text: knowledge,

text: until

text: t

text: cessing

text: unit

text: (BPU)

text: by

text: ﬁber.

text: Likewise,

text: the

text: distributed

text: antenna

text: now

text: there

text: is

text: no

text: uniﬁed

text: diversity

text: gain

text: analysis

text: for

text: mmWave

text: array

text: at

text: the

text: receiver

text: consists

text: of

text: K

text: RAUs,

text: each

text: having

text: N

text: massive

text: MIMO

text: systems

text: that

text: is

text: applicable

text: to

text: both

text: co-located

text: r

text: r

text: antennas

text: and

text: also

text: being

text: connected

text: to

text: a

text: BPU

text: by

text: ﬁbers.

text: Such

text: and

text: distributed

text: antenna

text: array

text: architectures.

text: a

text: MIMO

text: system

text: shall

text: be

text: referred

text: to

text: as

text: a

text: (K

text: N

text: K

text: N

text: To

text: ﬁll

text: this

text: gap,

text: this

text: paper

text: investigates

text: the

text: diversity

text: per-

text: t

text: t

text: r

text: r

text: distributed

text: MIMO

text: (D-MIMO)

text: system.

text: When

text: K

text: K

text: formance

text: of

text: mmWave

text: massive

text: MIMO

text: systems

text: with

text: the

text: t

text: r

text: the

text: system

text: reduces

text: to

text: a

text: conventional

text: co-located

text: MIMO

text: (C-

text: proposed

text: distributed

text: subarray

text: architecture

text: (the

text: multiplexing

text: MIMO)

text: system.

text: performance

text: will

text: be

text: investigated

text: in

text: another

text: paper).

text: The

text: focus

text: The

text: transmitter

text: accepts

text: as

text: its

text: input

text: N

text: data

text: streams

text: and

text: is

text: is

text: on

text: the

text: asymptotical

text: diversity

text: gain

text: analysis

text: in

text: order

text: to

text: ﬁnd

text: s

text: equipped

text: with

text: N(rf)

text: RF

text: chains,

text: where

text: N

text: N(rf)

text: N

text: K

text: out

text: the

text: potential

text: diversity

text: advantage

text: provided

text: by

text: multiple

text: t

text: s

text: t

text: t

text: t

text: Given

text: N(rf)

text: transmit

text: RF

text: chains,

text: the

text: transmitter

text: can

text: apply

text: a

text: distributed

text: antenna

text: arrays.

text: The

text: obtained

text: analysis

text: can

text: be

text: used

text: t

text: conveniently

text: to

text: compare

text: various

text: mmWave

text: massive

text: MIMO

text: low-dimension

text: Nt(rf)

text: Ns

text: b×aseband

text: precoder,

text: Wt,

text: followed

text: systems

text: with

text: different

text: distributed

text: antenna

text: array

text: structures.

text: by

text: a

text: high-dimension

text: KtNt

text: Nt(rf)

text: RF

text: precoder,

text: Ft.

text: Note

text: that

text: amplitude

text: and

text: phase

text: modiﬁcations

text: are

text: feasible

text: for

text: the

text: The

text: main

text: contributions

text: of

text: this

text: paper

text: are

text: summarized

text: as

text: follows:

text: First,

text: for

text: a

text: single-user

text: system

text: employing

text: the

text: proposed

text: baseband

text: precoder

text: Wt,

text: while

text: only

text: phase

text: changes

text: can

text: be

text: made

text: distributed

text: subarray

text: architecture,

text: a

text: diversity

text: gain

text: expression

text: by

text: the

text: RF

text: precoder

text: Ft

text: through

text: the

text: use

text: of

text: variable

text: phase

text: shifters

text: and

text: combiners.

text: The

text: transmitted

text: signal

text: vector

text: can

text: be

text: is

text: obtained

text: when

text: the

text: number

text: of

text: antennas

text: at

text: each

text: subarray

text: written

text: as

text: increases

text: without

text: bound.

text: This

text: expression

text: clearly

text: indicates

text: that

text: one

text: can

text: obtain

text: a

text: large

text: diversity

text: gain

text: and/or

text: multiplexing

text: gain

text: x

text: FtWts,

text: by

text: employing

text: the

text: proposed

text: distributed

text: subarray

text: architecture.

text: E

text: where

text: s

text: is

text: the

text: N

text: symbol

text: vector

text: such

text: that

text: [ssH]

text: Second,

text: the

text: diversity

text: gain

text: analysis

text: is

text: extended

text: to

text: the

text: multiuser

text: s

text: P

text: I

text: Thus

text: P

text: represents

text: the

text: average

text: total

text: input

text: power.

text: scenario

text: with

text: downlink

text: and

text: uplink

text: transmission,

text: as

text: well

text: as

text: CNosnsNisdering

text: a

text: narrowband

text: block

text: fading

text: channel,

text: the

text: K

text: N

text: ×1

text: the

text: single-user

text: system

text: employing

text: the

text: conventional

text: partially-

text: r

text: r

text: received

text: signal

text: vector

text: is

text: connected

text: RF

text: structure

text: based

text: on

text: the

text: distributed

text: subarrays.

text: Simulation

text: results

text: are

text: provided

text: to

text: corroborate

text: the

text: analysis

text: y

text: HF

text: W

text: s

text: n

text: t

text: t

text: results

text: and

text: show

text: that

text: the

text: distributed

text: subarray

text: architecture

text: yields

text: signiﬁcantly

text: better

text: diversity

text: performance

text: than

text: the

text: co-

text: where

text: H

text: is

text: KrNr

text: KtNtCcNhannel

text: matrix

text: and

text: n

text: is

text: a

text: KrNr

text: located

text: single-array

text: architecture.

text: vector

text: consisting

text: of

text: i.i.d.

text: noise

text: samples.

text: Throughout

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: this

text: paper,

text: H

text: is

text: assumed

text: known

text: to

text: both

text: the

text: transmitter

text: and

text: II

text: describes

text: the

text: massive

text: MIMO

text: system

text: model

text: and

text: hybrid

text: pro-

text: receiver.

text: Given

text: that

text: Nr(rf)

text: RF

text: chains

text: (where

text: Ns

text: Nr(rf)

text: cessing

text: with

text: the

text: distributed

text: subarray

text: architecture

text: in

text: mmWave

text: NrKr)

text: are

text: used

text: at

text: the

text: receiver

text: to

text: detect

text: the

text: Ns

text: data

text: streams,

text: the

text: processed

text: signal

text: is

text: given

text: by

text: fading

text: channels.

text: Section

text: III

text: provides

text: the

text: asymptotical

text: diversity

text: analysis

text: for

text: the

text: single-user

text: mmWave

text: system.

text: In

text: Sections

text: IV

text: z

text: WHFHHF

text: W

text: s

text: WHFHn

text: and

text: V,

text: the

text: diversity

text: gain

text: analysis

text: is

text: extended

text: to

text: the

text: multiuser

text: r

text: r

text: t

text: t

text: r

text: r

text: scenario

text: and

text: the

text: scenario

text: with

text: the

text: conventional

text: partially-

text: where

text: F

text: is

text: the

text: K

text: N

text: N(rf)

text: RF

text: combining

text: matrix,

text: and

text: W

text: r

text: r

text: r

text: r

text: r

text: connected

text: RF

text: architecture,

text: respectively.

text: Numerical

text: results

text: are

text: is

text: the

text: N(rf)

text: N

text: baseband

text: combining

text: matrix.

text: r

text: s

text: presented

text: in

text: Section

text: VI.

text: Section

text: VII

text: concludes

text: the

text: paper.

text: Furthermore,

text: according

text: to

text: the

text: architecture

text: of

text: RAUs

text: at

text: the

text: Throughout

text: this

text: paper,

text: the

text: following

text: notations

text: are

text: used.

text: transmitting

text: and

text: receiving

text: ends,

text: H

text: can

text: be

text: written

text: as

text: Boldface

text: upper

text: and

text: lower

text: case

text: letters

text: denote

text: matrices

text: and

text: )rTespaencdtiv(e)lHy.

text: H

text: diag

text: aN

text: stands

text: for

text: a

text: diagonal

text: matrix

text: with

text: di-

text: gKrKtHKrKt

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 analysis, i.e., analyzing performance at the high signal-to-noise PDF of the instantaneous receive SNR at the receiver, denoted (SNR) region, as done in [29]. This is a reasonable approach γ , is given by l since the performance of practical interest is in the high SNR L! (cid:18) (cid:19) (cid:32)−(cid:18) −(cid:19) (cid:33) − region and in such a region, good approximation can be made f (γ ) = − − [F(γ )]L l[1 F(γ )]l 1f(γ ) l:L l (L l)!(l 1)! l l l on the exact analysis. − − In the high-SNR region, the average BER function can be = − L! − 1 γl L l + o γl L l . (14) approximated in most cases as [29] (L l)!(l 1)! γ¯ γ¯ γ¯ ≈ · − Applying the above PDF in Proposition 1 in [29] leads to the BER (Gc γ¯) Gd (10) desired result. (cid:3) Lemma 1 can be extended to the case of independent but where G and G are referred to as the diversity and coding d c not identically distributed (i.n.i.d.) Rayleigh fading channels gains, respectively, and γ¯ is the average receive SNR. The and the result is stated in the next lemma. diversity gain determines the slope of the BER curve versus γ¯ at high SNR in a log-log scale, whereas the coding gain Lemma 2: Suppose that the GSC system with L receive antennas operates over the i.n.i.d. Rayleigh fading channels. determines how the curve is shifted along the horizontal axis with respect to a benchmark BER curve γ¯−Gd. This yields a If it selects the path with the lth highest SNR for detection, then it can achieve diversity gain simple parameterized average BER characterization for high SNR, which can provide meaningful insights on the system G = L − l + 1. (15) d performance behavior. In this section, the diversity gain is ﬁrst examined for Proof: Let γ¯min and γ¯max denote the maximum and min- a generalized selection combining. The main result is then imum values of the average receive SNRs of all these L A B diversity paths, respectively. Furthermore, let and denote invoked in the diversity analysis of the distributed mmWave massive MIMO system studied in this paper. two GSC systems, each equipped with L receive antennas and operating over i.i.d. Rayleigh fading channels such that the A. Diversity Gain of Generalized Selection Combining average receive SNRs equal to γ¯ and γ¯ , respectively. max min Selection combining (SC) is the most popular low- It is known from Lemma 1 that the diversity gains of these − complexity combining scheme. With selection combining, the two systems are the same and equal to L l + 1 if both receiver estimates the SNRs of all available diversity branches systems select the branch with the lth highest instantaneous and then select the one with the highest SNR for detection. SNR for detection. Furthermore, since the GSC system under For generalized selection combining (GSC) considered here, consideration cannot have better diversity performance than A the receiver also estimates the SNRs of all available diversity System and cannot have worse diversity performance than B branches. However, instead of selecting the branch with the System , it can then be concluded that the i.n.i.d. system − (cid:3) highest SNR, it selects a branch with the lth highest SNR for must also achieve the diversity gain of L l + 1. detection. It is pointed out that, while such a GSC scheme has no practical interest in its own right, its diversity analysis B. Diversity Gain Analysis of the Distributed mmWave Mas- can be used in performance analysis of the mmWave massive sive MIMO System (cid:80) (cid:80) MIMO system considered in this paper. From the structure and deﬁnition of the channel matrix Lemma 1: Consider a GSC system with L receive antennas operating over i.i.d. Rayleigh fading channels. If the receiver H in Section II, there is a total of Ls = Ki=r1 Kj=t1 Lij selects the branch with the lth highest SNR for detection then propagation paths. Naturally, H can be decomposed into a the system achieves diversity gain sum of Ls rank-one matrices, each corresponding to one propagation p(cid:88)ath.(cid:88)Spe(cid:88)ciﬁcally, H can be rewritten as − Gd = L l + 1. (11) Kr Kt Lij H = α˜l a˜ (φrl, θrl)a˜H(φtl, θtl), (16) (cid:115)ij r ij ij t ij ij Proof: Let F(γ) and f(γ) be the probability density func- i=1 j=1 l=1 tion (PDF) and cumulative distribution function (CDF) of the where instantaneous SNRs in all branches, respectively. Let γ¯ denote N N the average receive SNR of each branch. With Rayleigh fading, α˜l = g t r αl , (17) (cid:18) (cid:19) ij ij L ij it follows from [29] that F(γ) and f(γ) can be written as ij × a˜ (φrl, θrl) is a K(cid:26)N 1 vector whose bth entry is deﬁned − − γ γ r ij ij r r F(γ) = 1 e γγ¯ = + o (12) as γ¯ (cid:18)γ¯ (cid:19) ∈ [a (φrl, θrl)] − − , b Qr [a˜ (φrl, θrl)] = r ij ij b (i 1)Nr ∈ i (18) and r ij ij b 0, b / Qr i 1 − 1 − γ γ − × f(γ) = γ¯ e γγ¯ = γ¯ γ¯2 + o γ¯ . (13) wvehcetorer Qwriho=se((bith e(cid:26)1n)tNryr,isiNderﬁ].nAedndasa˜t(φtijl, θitjl) is a KtNt 1 ∈ If the receiver selects the branch with the lth highest SNR for [a (φtl, θtl)] − − , b Qt detection, then based on the theory of order statistics [30], the [a˜t(φtijl, θitjl)]b = 0,t ij ij b (j 1)Nt b ∈/ Qjt. (19) j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: analysis,

text: i.e.,

text: analyzing

text: performance

text: at

text: the

text: high

text: signal-to-noise

text: PDF

text: of

text: the

text: instantaneous

text: receive

text: SNR

text: at

text: the

text: receiver,

text: denoted

text: (SNR)

text: region,

text: as

text: done

text: in

text: This

text: is

text: a

text: reasonable

text: approach

text: γ

text: is

text: given

text: by

text: l

text: since

text: the

text: performance

text: of

text: practical

text: interest

text: is

text: in

text: the

text: high

text: SNR

text: L!

text: region

text: and

text: in

text: such

text: a

text: region,

text: good

text: approximation

text: can

text: be

text: made

text: f

text: (γ

text: [F(γ

text: )]L

text: F(γ

text: )]l

text: l:L

text: l

text: (L

text: l)!(l

text: l

text: l

text: l

text: on

text: the

text: exact

text: analysis.

text: In

text: the

text: high-SNR

text: region,

text: the

text: average

text: BER

text: function

text: can

text: be

text: L!

text: γl

text: L

text: l

text: o

text: γl

text: L

text: l

text: approximated

text: in

text: most

text: cases

text: as

text: (L

text: l)!(l

text: γ¯

text: γ¯

text: γ¯

text: Applying

text: the

text: above

text: PDF

text: in

text: Proposition

text: in

text: leads

text: to

text: the

text: BER

text: (Gc

text: γ¯)

text: Gd

text: desired

text: result.

text: Lemma

text: can

text: be

text: extended

text: to

text: the

text: case

text: of

text: independent

text: but

text: where

text: G

text: and

text: G

text: are

text: referred

text: to

text: as

text: the

text: diversity

text: and

text: coding

text: d

text: c

text: not

text: identically

text: distributed

text: (i.n.i.d.)

text: Rayleigh

text: fading

text: channels

text: gains,

text: respectively,

text: and

text: γ¯

text: is

text: the

text: average

text: receive

text: SNR.

text: The

text: and

text: the

text: result

text: is

text: stated

text: in

text: the

text: next

text: lemma.

text: diversity

text: gain

text: determines

text: the

text: slope

text: of

text: the

text: BER

text: curve

text: versus

text: γ¯

text: at

text: high

text: SNR

text: in

text: a

text: log-log

text: scale,

text: whereas

text: the

text: coding

text: gain

text: Lemma

text: Suppose

text: that

text: the

text: GSC

text: system

text: with

text: L

text: receive

text: antennas

text: operates

text: over

text: the

text: i.n.i.d.

text: Rayleigh

text: fading

text: channels.

text: determines

text: how

text: the

text: curve

text: is

text: shifted

text: along

text: the

text: horizontal

text: axis

text: with

text: respect

text: to

text: a

text: benchmark

text: BER

text: curve

text: γ¯−Gd.

text: This

text: yields

text: a

text: If

text: it

text: selects

text: the

text: path

text: with

text: the

text: lth

text: highest

text: SNR

text: for

text: detection,

text: then

text: it

text: can

text: achieve

text: diversity

text: gain

text: simple

text: parameterized

text: average

text: BER

text: characterization

text: for

text: high

text: SNR,

text: which

text: can

text: provide

text: meaningful

text: insights

text: on

text: the

text: system

text: G

text: L

text: l

text: d

text: performance

text: behavior.

text: In

text: this

text: section,

text: the

text: diversity

text: gain

text: is

text: ﬁrst

text: examined

text: for

text: Proof:

text: Let

text: γ¯min

text: and

text: γ¯max

text: denote

text: the

text: maximum

text: and

text: min-

text: a

text: generalized

text: selection

text: combining.

text: The

text: main

text: result

text: is

text: then

text: imum

text: values

text: of

text: the

text: average

text: receive

text: SNRs

text: of

text: all

text: these

text: L

text: A

text: B

text: diversity

text: paths,

text: respectively.

text: Furthermore,

text: let

text: and

text: denote

text: invoked

text: in

text: the

text: diversity

text: analysis

text: of

text: the

text: distributed

text: mmWave

text: massive

text: MIMO

text: system

text: studied

text: in

text: this

text: paper.

text: two

text: GSC

text: systems,

text: each

text: equipped

text: with

text: L

text: receive

text: antennas

text: and

text: operating

text: over

text: i.i.d.

text: Rayleigh

text: fading

text: channels

text: such

text: that

text: the

text: A.

text: Diversity

text: Gain

text: of

text: Generalized

text: Selection

text: Combining

text: average

text: receive

text: SNRs

text: equal

text: to

text: γ¯

text: and

text: γ¯

text: respectively.

text: max

text: min

text: Selection

text: combining

text: (SC)

text: is

text: the

text: most

text: popular

text: low-

text: It

text: is

text: known

text: from

text: Lemma

text: that

text: the

text: diversity

text: gains

text: of

text: these

text: complexity

text: combining

text: scheme.

text: With

text: selection

text: combining,

text: the

text: two

text: systems

text: are

text: the

text: same

text: and

text: equal

text: to

text: L

text: l

text: if

text: both

text: receiver

text: estimates

text: the

text: SNRs

text: of

text: all

text: available

text: diversity

text: branches

text: systems

text: select

text: the

text: branch

text: with

text: the

text: lth

text: highest

text: instantaneous

text: and

text: then

text: select

text: the

text: one

text: with

text: the

text: highest

text: SNR

text: for

text: detection.

text: SNR

text: for

text: detection.

text: Furthermore,

text: since

text: the

text: GSC

text: system

text: under

text: For

text: generalized

text: selection

text: combining

text: (GSC)

text: considered

text: here,

text: consideration

text: cannot

text: have

text: better

text: diversity

text: performance

text: than

text: A

text: the

text: receiver

text: also

text: estimates

text: the

text: SNRs

text: of

text: all

text: available

text: diversity

text: System

text: and

text: cannot

text: have

text: worse

text: diversity

text: performance

text: than

text: B

text: branches.

text: However,

text: instead

text: of

text: selecting

text: the

text: branch

text: with

text: the

text: System

text: it

text: can

text: then

text: be

text: concluded

text: that

text: the

text: i.n.i.d.

text: system

text: highest

text: SNR,

text: it

text: selects

text: a

text: branch

text: with

text: the

text: lth

text: highest

text: SNR

text: for

text: must

text: also

text: achieve

text: the

text: diversity

text: gain

text: of

text: L

text: l

text: detection.

text: It

text: is

text: pointed

text: out

text: that,

text: while

text: such

text: a

text: GSC

text: scheme

text: has

text: no

text: practical

text: interest

text: in

text: its

text: own

text: right,

text: its

text: diversity

text: analysis

text: B.

text: Diversity

text: Gain

text: Analysis

text: of

text: the

text: Distributed

text: mmWave

text: Mas-

text: can

text: be

text: used

text: in

text: performance

text: analysis

text: of

text: the

text: mmWave

text: massive

text: sive

text: MIMO

text: System

text: MIMO

text: system

text: considered

text: in

text: this

text: paper.

text: From

text: the

text: structure

text: and

text: deﬁnition

text: of

text: the

text: channel

text: matrix

text: Lemma

text: Consider

text: a

text: GSC

text: system

text: with

text: L

text: receive

text: antennas

text: operating

text: over

text: i.i.d.

text: Rayleigh

text: fading

text: channels.

text: If

text: the

text: receiver

text: H

text: in

text: Section

text: II,

text: there

text: is

text: a

text: total

text: of

text: Ls

text: Lij

text: selects

text: the

text: branch

text: with

text: the

text: lth

text: highest

text: SNR

text: for

text: detection

text: then

text: propagation

text: paths.

text: Naturally,

text: H

text: can

text: be

text: decomposed

text: into

text: a

text: the

text: system

text: achieves

text: diversity

text: gain

text: sum

text: of

text: Ls

text: rank-one

text: matrices,

text: each

text: corresponding

text: to

text: one

text: propagation

text: H

text: can

text: be

text: rewritten

text: as

text: Gd

text: L

text: l

text: Kr

text: Kt

text: Lij

text: H

text: α˜l

text: a˜

text: (φrl,

text: θrl)a˜H(φtl,

text: θtl),

text: r

text: ij

text: ij

text: t

text: ij

text: ij

text: Proof:

text: Let

text: F(γ)

text: and

text: f(γ)

text: be

text: the

text: probability

text: density

text: func-

text: tion

text: (PDF)

text: and

text: cumulative

text: distribution

text: function

text: (CDF)

text: of

text: the

text: where

text: instantaneous

text: SNRs

text: in

text: all

text: branches,

text: respectively.

text: Let

text: γ¯

text: denote

text: N

text: N

text: the

text: average

text: receive

text: SNR

text: of

text: each

text: branch.

text: With

text: Rayleigh

text: fading,

text: α˜l

text: g

text: t

text: r

text: αl

text: ij

text: ij

text: L

text: ij

text: it

text: follows

text: from

text: that

text: F(γ)

text: and

text: f(γ)

text: can

text: be

text: written

text: as

text: ij

text: a˜

text: (φrl,

text: θrl)

text: is

text: a

text: vector

text: whose

text: bth

text: entry

text: is

text: deﬁned

text: γ

text: γ

text: r

text: ij

text: ij

text: r

text: r

text: F(γ)

text: e

text: γγ¯

text: o

text: as

text: γ¯

text: [a

text: (φrl,

text: θrl)]

text: b

text: Qr

text: [a˜

text: (φrl,

text: θrl)]

text: r

text: ij

text: ij

text: b

text: (i

text: i

text: and

text: r

text: ij

text: ij

text: b

text: b

text: Qr

text: i

text: γ

text: γ

text: f(γ)

text: γ¯

text: e

text: γγ¯

text: γ¯

text: γ¯2

text: o

text: γ¯

text: wvehcetorer

text: Qwriho=se((bith

text: θitjl)

text: is

text: a

text: KtNt

text: If

text: the

text: receiver

text: selects

text: the

text: branch

text: with

text: the

text: lth

text: highest

text: SNR

text: for

text: [a

text: (φtl,

text: θtl)]

text: b

text: Qt

text: detection,

text: then

text: based

text: on

text: the

text: theory

text: of

text: order

text: statistics

text: the

text: [a˜t(φtijl,

text: θitjl)]b

text: ij

text: ij

text: b

text: (j

text: b

text: Qjt.

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 Often, it is more practical to search for approximate sparse (cid:107) (cid:62) (cid:107) linear combinations rather exact ones as in (14). Thus one may ζ := (X(−11)) x(11) ∞, (7) approximately express point x(1) by solving the Lasso problem (cid:107) (cid:62) (cid:107) 1 ζ¯ := (X¯ (−1)) x¯(1) ∞, (8) [24] 1 1 ζ¯˙ := (cid:107)(X¯˙ (−1))(cid:62)x¯˙ (1)(cid:107)∞, (ζ¯ = ζ¯˙) (9) 1 1 (cid:107) (cid:107) λ (cid:107) (cid:107) min c + e 2 s.t. x(1) = X− c + e, (15) 14) Other quantities: c 1 2 2 1 1 where e represents the reconstruction error. We have the following (cid:107) (cid:107) known guarantee4: η¯ := x¯(11) 2, (10) Theorem 1 (SSC with uncorrupted data, deterministic [33]). (cid:107) (cid:107) η¯˙ := x¯˙ (1) , (η¯ = η¯˙) (11) Recall the notation of Deﬁnition 1, and suppose that 1 2 |(cid:104) (cid:105)| γ¯ := max x¯(i), P S⊥x˜(1) (12) ∈ k j i>j1∈,k[N[1N]i], 1 µλ < r and 1/ζ < λ. (16) |(cid:104) (cid:105)| γ¯˙ := ma∈x x¯˙ (ki), P S˙⊥x˜˙ (j1) . (13) Then every optimal solution to the Lasso SSC problem (15) is i>1∈,k [Ni], 1 j [N1] non-zero and subspace preserving. Theorem 1 can be interpreted as follows: If all data points from S other than x(1) are well distributed (large r), the data points 1 1 S 2 REVIEW OF SPARSE SUBSPACE CLUSTERING from other subspaces are sufﬁciently far from as measured by 1 their inner product with the dual direction vˆ (small µ ), and We begin by reviewing Sparse Subspace Clustering (SSC) for 1,λ λ § the reconstruction error is penalized sufﬁciently enough (large λ), data with no corruptions ( ), as well as the two fundamental § then the Lasso problem (15) is guaranteed to furnish non-zero and approaches to SSC for incomplete data ( ), which this paper is subspace preserving solutions. devoted to analyzing. Employing Lemmas 3 and 4 in the Appendix, Theorem 1 can 2.1 SSC With Uncorrupted Data be converted to a probabilistic statement under a simpliﬁed fully In the absence of data corruptions (noise, missing entries, ∈(cid:83)R × random model. outliers, etc.) we consider a data matrix X N D as in Deﬁni- ∈ tion 1, whose columns are unit-(cid:96) points3 that lie in an unknown Deﬁnition 2 (Random model). For each i [n], let the ith union of low-dimensional linear 2subspaces n S ⊂ RD, with subspace be chosen uniformly at random from the Grassman- d := dim(S ). Thus X = [X(1) · · · Xi(=n1)]Γi, where each nian manifold of d-dimensional subspaces of RD. Moreover, let5 Xsuib(si)pa:=ce [Sx(1,ia)n·id· ·Γxi(Nsi)ia]n∈unRkNnoi×wDn pceornmsiusttastioofnN, iindpiocianttisngsptahnant itnhge Nint/enrse=ct:ioρndo+f e1acphoisnutbssbpeaccehoasnedntuhneifuonrimt lsyphaetrreaSndDo−m1.frFoimnatlhlye, i deﬁne the quantities clustering of the points is unknown. (cid:115) (cid:115) Among a variety of methods [31] for retrieving the clusters { } X(i) , one may apply Sparse Subspace Clustering (SSC) [9], [10], whose main principle is to express each point in X as α := log(ρ), β := 6 log(N). (17) a sparse linear combination of other points in X. Speciﬁcally, 16d D we seek an expression, say, of point x(1) as a sparse linear 1\ { } combination of all other points X− := X x(1) by means of 1 1 Theorem 2 (SSC with uncorrupted data, probabilistic [33], [22]). the basis pursuit problem [7] Consider the random model of Deﬁnition 2. If ρ is larger than a (cid:107) (cid:107) min c s.t. x(1) = X− c, (14) constant, λ > 1/α, and ∈R − 1 1 1 c N1 1 and then form an afﬁnity graph in which we connect x(1) to 1 α > β, (18) those points of X− that correspond to the support (non-zero 1 coefﬁcients) of the computed optimal solution of (14). Clearly, then any optimal solution to the Lasso SSC problem (15) is non- we want these points to lie in t\he{ sam}e subspace as x(11), i.e., zero a−n√d subspace preserving, with probability at least 1−2/N2− to be points of X(−1) := X(1) x(1) , in which case we say exp( ρd). 1 1 that the solution is subspace preserving. When this is true for the Condition (18) agrees with intuition, since it effectively says that expression of each and every point in X, then the corresponding the subspace preserving property is easier to achieve for small afﬁnity graph contains no connections between points in different subspaces, i.e., it is a subspace preserving graph. Assuming that relative subspace dimen§sions d/D, fewer subspaces, and more points per subspace. In we will give analogues of Theorems 1 points within each subspace are sufﬁciently well connected, the and 2 for two fundamental variants of SSC for incomplete data, afﬁnity graph will have precisely n connected components, and described next. spectral clustering is guaranteed to furnish the correct clusters. 3. This is not an essential assumption, rather it is convenient for the purpose 4. For ease of notation, we state all results in terms of expressing x(1). 1 of theoretical analysis. 5. For simplicity, we assume that n divides N. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Often,

text: it

text: is

text: more

text: practical

text: to

text: search

text: for

text: approximate

text: sparse

label: equation

text: linear

text: combinations

text: rather

text: exact

text: ones

text: as

text: in

text: Thus

text: one

text: may

text: ζ

text: approximately

text: express

text: point

text: by

text: solving

text: the

text: Lasso

text: problem

text: ζ¯

text: (X¯

text: ζ¯˙

text: (ζ¯

text: ζ¯˙)

text: λ

text: min

text: c

text: e

text: s.t.

text: X−

text: c

text: e,

label: list

text: Other

text: quantities:

text: c

text: where

text: e

text: represents

text: the

text: reconstruction

text: error.

text: We

text: have

text: the

text: following

text: known

text: η¯

text: Theorem

text: (SSC

text: with

text: uncorrupted

text: data,

text: deterministic

text: η¯˙

text: x¯˙

text: (η¯

text: η¯˙)

text: Recall

text: the

text: notation

text: of

text: Deﬁnition

text: and

text: suppose

text: that

text: γ¯

text: max

text: x¯(i),

text: P

text: k

text: j

text: µλ

text: r

text: and

text: λ.

text: γ¯˙

text: ma∈x

text: x¯˙

text: (ki),

text: P

text: S˙⊥x˜˙

text: Then

text: every

text: optimal

text: solution

text: to

text: the

text: Lasso

text: SSC

text: problem

text: is

text: [Ni],

text: j

text: non-zero

text: and

text: subspace

text: preserving.

text: Theorem

text: can

text: be

text: interpreted

text: as

text: follows:

text: If

text: all

text: data

text: points

text: from

text: S

text: other

text: than

text: are

text: well

text: distributed

text: (large

text: r),

text: the

text: data

text: points

text: S

text: REVIEW

text: OF

text: SPARSE

text: SUBSPACE

text: CLUSTERING

text: from

text: other

text: subspaces

text: are

text: sufﬁciently

text: far

text: from

text: as

text: measured

text: by

text: their

text: inner

text: product

text: with

text: the

text: dual

text: direction

text: vˆ

text: (small

text: µ

text: and

text: We

text: begin

text: by

text: reviewing

text: Sparse

text: Subspace

text: Clustering

text: (SSC)

text: for

text: λ

text: the

text: reconstruction

text: error

text: is

text: penalized

text: sufﬁciently

text: enough

text: (large

text: λ),

text: data

text: with

text: no

text: corruptions

text: as

text: well

text: as

text: the

text: two

text: fundamental

text: then

text: the

text: Lasso

text: problem

text: is

text: guaranteed

text: to

text: furnish

text: non-zero

text: and

text: approaches

text: to

text: SSC

text: for

text: incomplete

text: data

text: which

text: this

text: paper

text: is

text: subspace

text: preserving

text: solutions.

text: devoted

text: to

text: analyzing.

text: Employing

text: Lemmas

text: and

text: in

text: the

text: Appendix,

text: Theorem

text: can

text: SSC

text: With

text: Uncorrupted

text: Data

text: be

text: converted

text: to

text: a

text: probabilistic

text: statement

text: under

text: a

text: simpliﬁed

text: fully

text: In

text: the

text: absence

text: of

text: data

text: corruptions

text: (noise,

text: missing

text: entries,

text: random

text: model.

text: outliers,

text: etc.)

text: we

text: consider

text: a

text: data

text: matrix

text: X

text: N

text: D

text: as

text: in

text: Deﬁni-

text: tion

text: whose

text: columns

text: are

text: that

text: lie

text: in

text: an

text: unknown

text: Deﬁnition

text: (Random

text: model).

text: For

text: each

text: i

text: [n],

text: let

text: the

text: ith

text: union

text: of

text: low-dimensional

text: linear

text: n

text: S

text: RD,

text: with

text: subspace

text: be

text: chosen

text: uniformly

text: at

text: random

text: from

text: the

text: Grassman-

text: d

text: dim(S

text: Thus

text: X

text: where

text: each

text: nian

text: manifold

text: of

text: d-dimensional

text: subspaces

text: of

text: RD.

text: Moreover,

text: Xsuib(si)pa:=ce

text: ·Γxi(Nsi)ia]n∈unRkNnoi×wDn

text: pceornmsiusttastioofnN,

text: iindpiocianttisngsptahnant

text: itnhge

text: Nint/enrse=ct:ioρndo+f

text: i

text: deﬁne

text: the

text: quantities

text: clustering

text: of

text: the

text: points

text: is

text: unknown.

text: Among

text: a

text: variety

text: of

text: methods

text: for

text: retrieving

text: the

text: clusters

text: X(i)

text: one

text: may

text: apply

text: Sparse

text: Subspace

text: Clustering

text: (SSC)

text: whose

text: main

text: principle

text: is

text: to

text: express

text: each

text: point

text: in

text: X

text: as

text: α

text: log(ρ),

text: β

text: log(N).

text: a

text: sparse

text: linear

text: combination

text: of

text: other

text: points

text: in

text: X.

text: Speciﬁcally,

text: D

text: we

text: seek

text: an

text: expression,

text: say,

text: of

text: point

text: as

text: a

text: sparse

text: linear

text: combination

text: of

text: all

text: other

text: points

text: X−

text: X

text: by

text: means

text: of

text: Theorem

text: (SSC

text: with

text: uncorrupted

text: data,

text: probabilistic

text: the

text: basis

text: pursuit

text: problem

text: Consider

text: the

text: random

text: model

text: of

text: Deﬁnition

text: If

text: ρ

text: is

text: larger

text: than

text: a

text: min

text: c

text: s.t.

text: X−

text: c,

text: constant,

text: λ

text: and

text: ∈R

text: c

text: and

text: then

text: form

text: an

text: afﬁnity

text: graph

text: in

text: which

text: we

text: connect

text: to

text: α

text: β,

text: those

text: points

text: of

text: X−

text: that

text: correspond

text: to

text: the

text: support

text: (non-zero

text: coefﬁcients)

text: of

text: the

text: computed

text: optimal

text: solution

text: of

text: Clearly,

text: then

text: any

text: optimal

text: solution

text: to

text: the

text: Lasso

text: SSC

text: problem

text: is

text: non-

text: we

text: want

text: these

text: points

text: to

text: lie

text: in

text: t\he{

text: sam}e

text: subspace

text: as

text: i.e.,

text: zero

text: a−n√d

text: subspace

text: preserving,

text: with

text: probability

text: at

text: least

text: to

text: be

text: points

text: of

text: in

text: which

text: case

text: we

text: say

text: exp(

text: ρd).

text: that

text: the

text: solution

text: is

text: subspace

text: preserving.

text: When

text: this

text: is

text: true

text: for

text: the

text: Condition

text: agrees

text: with

text: intuition,

text: since

text: it

text: effectively

text: says

text: that

text: expression

text: of

text: each

text: and

text: every

text: point

text: in

text: X,

text: then

text: the

text: corresponding

text: the

text: subspace

text: preserving

text: property

text: is

text: easier

text: to

text: achieve

text: for

text: small

text: afﬁnity

text: graph

text: contains

text: no

text: connections

text: between

text: points

text: in

text: different

text: subspaces,

text: i.e.,

text: it

text: is

text: a

text: subspace

text: preserving

text: graph.

text: Assuming

text: that

text: relative

text: subspace

text: dimen§sions

text: d/D,

text: fewer

text: subspaces,

text: and

text: more

text: points

text: per

text: subspace.

text: In

text: we

text: will

text: give

text: analogues

text: of

text: Theorems

text: points

text: within

text: each

text: subspace

text: are

text: sufﬁciently

text: well

text: connected,

text: the

text: and

text: for

text: two

text: fundamental

text: variants

text: of

text: SSC

text: for

text: incomplete

text: data,

text: afﬁnity

text: graph

text: will

text: have

text: precisely

text: n

text: connected

text: components,

text: and

text: described

text: next.

text: spectral

text: clustering

text: is

text: guaranteed

text: to

text: furnish

text: the

text: correct

text: clusters.

text: This

text: is

text: not

text: an

text: essential

text: assumption,

text: rather

text: it

text: is

text: convenient

text: for

text: the

text: purpose

text: For

text: ease

text: of

text: notation,

text: we

text: state

text: all

text: results

text: in

text: terms

text: of

text: expressing

text: of

text: theoretical

text: analysis.

text: For

text: simplicity,

text: we

text: assume

text: that

text: n

text: divides

text: N.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 15 (cid:88) not be polymatroid in general, for the constant gap Gaussian distribution, we can guarantee a certain monotone property of K L − − − K the Marton’s rate expression by appropriate choice of the noise T(U( ), X( )) T(U , . . . , U ) C I(U ; Y ) 1 j k k variance leading to a polymatroid rate region. k=j+1 A result in submodular optimization [23] is that for a linear − . ≺ ≺ ≺ { } I(Uj; Yj) I(Uj; Uj−1, . . . , 1) ordering i i . . . i of 1, . . . , K , an extreme point (135) P 1 2 K of (C) can be greedily computed as (R , . . . , R ) where 1 K ∈ ≥ Note that α (0, 1] due to (128) and the fact that R 0. j { } − { } Now we construct the time-sharing of two rate-tuples cor- Rij = f( i1, . . . , ij ) f( i1, . . . , ij−1 ). (127) responding to the generalized compression strategy (42) that achieves this above rate (R , . . . , R ) as follows: 1 K P − Moreover, all extreme points of (C) can be enumerated 1) For (1 α) fraction of the time, transmit messages for by considering all linear orderings. Since each ordering of users 1, . . . , j, only, i.e., set p(u , . . . , u , x , . . . , x ) to { } 1 j 1 L 1, . . . , K is analyzed in the same manner, for notational be the marginal distribution of the original distribution, sim(cid:88)plicity, we consider the natural ordering ij = j. but let Uj+1, . . . , UK be constants. Let j be the ﬁrst index for which 2) For the rest α fraction −of time, transmit messages for users 1, . . . , (j 1) only, i.e., set − K L I(Uk; Yk) + C(cid:88)T(U( ), X( )) < p(u1, . . . , uj−1, x1, . . . , xL) to be the marginal ∈K distribution of the original distribution, but let k j − Uj, . . . , UK be constants. I(U ; Y ) T(U , . . . , U ). (128) By construction, the time-sharing of these two compression k k 1 j k=1 schemes achieves the same rate tuple as the extreme point of Then, by construction, ∀k < j P(C), (130), (131), and (134). (cid:88) To calculate the fronthaul capacity consumption of this time-sharing scheme, we have k − Rk = (cid:88)I(Ui; Yi) T(U1, . . . , Uk) − L L i=1 C¯ = (1 α) (I(U , . . . , U ; X( )) + T(X( ))) − 1 j − k 1 I(Ui; Yi) − T(U1, . . . , Uk−1) (129) α (I(U1, . . . , UjL−1; X(L)) +LT(−X(L))) (136) = I(U , . . . , U ; X( )) + T(X( )) i=1 − 1 j L − L = I(Uk; Yk) I(Uk; Uk−1, . . . , U1). (130) α(I(U1, . . . , Uj; X( )) I(U1, . . . , Uj−1; X( ))) (137) L L − D doFesurtnhoetrmdoepree,ndusionng tDhe, fsaoctwthhaetnththeesesceocnodndtertmermofisf(the) = I(−U1I,.(U. . ,,U. .j;. X, U( ;)X) (+LT))(−XI((U)) , . .C. , U − ; X(L)) ∀ 1 j − 1 j 1 minimum, i.e., k > j, we have  I(Uj; Yj) I(Uj; Uj−1, . . . , U1) R = 0. (131) · T(U(K), X(L)) −T(U , . . . , U ) − C k (cid:88) 1 j  Finally,(cid:88)we express R as − K j I(U ; Y ) + C (138) k k k=j+1 − K L Rj = I(cid:88)(Uk; Yk) + C T(U( ), X( )) (≤a) L L − ∈K I(U , . . . , U ; X( )) + T(X( )) C k − 1 (cid:88)K j L − − j 1 − T(U( ), X( )) + T(U1, . . . , Uj) + C I(U ; Y ) T(U , . . . , U − ) (132) k k 1 j 1 K k(cid:88)=1 − (cid:32)(cid:88)+ I(Uk; Yk) + C (cid:33) (139) = I(U ; Y ) I(U ; U − , . . . , U ) j j j j 1 1 k=j+1 − K L + ∈K(cid:88)I(Uk; Yk) + C T(U( ), X( )) = (cid:32)I(cid:88)(Uk; Yk) + C − T(U(K), X(L)) (cid:33) k ∈K k j − I(U ; Y ) + T(U , . . . , U ) (133) − j − L k k 1 j I(U ; Y ) + C T(U , . . . , U , X( )) k k 1 j − k=1 − = (1 α) (I(U ; Y ) I(U ; U − , . . . , U )) , (134) k=1 j j j j 1 1 + C (140) (≤b) where α can be written explicitly as below C. (141) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: not

text: be

text: polymatroid

text: in

text: general,

text: for

text: the

text: constant

text: gap

text: Gaussian

text: distribution,

text: we

text: can

text: guarantee

text: a

text: certain

text: monotone

text: property

text: of

text: K

text: L

text: K

text: the

text: Marton’s

text: rate

text: expression

text: by

text: appropriate

text: choice

text: of

text: the

text: noise

text: T(U(

text: X(

text: T(U

text: U

text: C

text: I(U

text: Y

text: j

text: k

text: k

text: variance

text: leading

text: to

text: a

text: polymatroid

text: rate

text: region.

text: A

text: result

text: in

text: submodular

text: optimization

text: is

text: that

text: for

text: a

text: linear

text: I(Uj;

text: Yj)

text: I(Uj;

text: ordering

text: i

text: i

text: i

text: of

text: K

text: an

text: extreme

text: point

text: P

text: K

text: of

text: (C)

text: can

text: be

text: greedily

text: computed

text: as

text: (R

text: R

text: where

text: K

text: Note

text: that

text: α

text: due

text: to

text: and

text: the

text: fact

text: that

text: R

text: j

text: Now

text: we

text: construct

text: the

text: time-sharing

text: of

text: two

text: rate-tuples

text: cor-

text: Rij

text: f(

text: ij

text: f(

text: responding

text: to

text: the

text: generalized

text: compression

text: strategy

text: that

text: achieves

text: this

text: above

text: rate

text: (R

text: R

text: as

text: follows:

text: K

text: P

text: Moreover,

text: all

text: extreme

text: points

text: of

text: (C)

text: can

text: be

text: enumerated

text: For

text: α)

text: fraction

text: of

text: the

text: time,

text: transmit

text: messages

text: for

text: by

text: considering

text: all

text: linear

text: orderings.

text: Since

text: each

text: ordering

text: of

text: users

text: j,

text: only,

text: i.e.,

text: set

text: p(u

text: u

text: x

text: x

text: to

text: j

text: L

text: K

text: is

text: analyzed

text: in

text: the

text: same

text: manner,

text: for

text: notational

text: be

text: the

text: marginal

text: distribution

text: of

text: the

text: original

text: distribution,

text: we

text: consider

text: the

text: natural

text: ordering

text: ij

text: j.

text: but

text: let

text: UK

text: be

text: constants.

text: Let

text: j

text: be

text: the

text: ﬁrst

text: index

text: for

text: which

text: For

text: the

text: rest

text: α

text: fraction

text: −of

text: time,

text: transmit

text: messages

text: for

text: users

text: (j

text: only,

text: i.e.,

text: set

text: K

text: L

text: I(Uk;

text: Yk)

text: X(

text: xL)

text: to

text: be

text: the

text: marginal

text: ∈K

text: distribution

text: of

text: the

text: original

text: distribution,

text: but

text: let

text: k

text: j

text: Uj,

text: UK

text: be

text: constants.

text: I(U

text: Y

text: T(U

text: U

text: By

text: construction,

text: the

text: time-sharing

text: of

text: these

text: two

text: compression

text: k

text: k

text: j

text: schemes

text: achieves

text: the

text: same

text: rate

text: tuple

text: as

text: the

text: extreme

text: point

text: of

text: Then,

text: by

text: construction,

text: ∀k

text: j

text: P(C),

text: and

text: To

text: calculate

text: the

text: fronthaul

text: capacity

text: consumption

text: of

text: this

text: time-sharing

text: scheme,

text: we

text: have

text: k

text: Rk

text: Yi)

text: Uk)

text: L

text: L

text: C¯

text: α)

text: (I(U

text: U

text: X(

text: T(X(

text: j

text: k

text: I(Ui;

text: Yi)

text: α

text: X(L))

text: +LT(−X(L)))

text: I(U

text: U

text: X(

text: T(X(

text: j

text: L

text: L

text: I(Uk;

text: Yk)

text: I(Uk;

text: Uj;

text: X(

text: X(

text: L

text: L

text: D

text: doFesurtnhoetrmdoepree,ndusionng

text: tDhe,

text: ,,U.

text: .j;.

text: X,

text: U(

text: ;)X)

text: (+LT))(−XI((U))

text: .C.

text: U

text: X(L))

text: j

text: j

text: minimum,

text: i.e.,

text: k

text: j,

text: we

text: have

text: I(Uj;

text: Yj)

text: I(Uj;

text: R

text: T(U(K),

text: X(L))

text: −T(U

text: U

text: C

text: k

text: j

text: express

text: R

text: as

text: K

text: j

text: I(U

text: Y

text: C

text: k

text: k

text: K

text: L

text: Rj

text: Yk)

text: C

text: T(U(

text: X(

text: (≤a)

text: L

text: L

text: ∈K

text: I(U

text: U

text: X(

text: T(X(

text: C

text: k

text: j

text: L

text: j

text: T(U(

text: X(

text: Uj)

text: C

text: I(U

text: Y

text: T(U

text: U

text: k

text: k

text: j

text: K

text: I(Uk;

text: Yk)

text: C

text: I(U

text: Y

text: I(U

text: U

text: U

text: j

text: j

text: j

text: j

text: K

text: L

text: Yk)

text: C

text: T(U(

text: X(

text: Yk)

text: C

text: T(U(K),

text: X(L))

text: k

text: ∈K

text: k

text: j

text: I(U

text: Y

text: T(U

text: U

text: j

text: L

text: k

text: k

text: j

text: I(U

text: Y

text: C

text: T(U

text: U

text: X(

text: k

text: k

text: j

text: α)

text: (I(U

text: Y

text: I(U

text: U

text: U

text: j

text: j

text: j

text: j

text: C

text: (≤b)

text: where

text: α

text: can

text: be

text: written

text: explicitly

text: as

text: below

text: C.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Pallavi Gupta and A. Upadhyay School of Physics and Materials Science, Thapar University, Patiala - 147004, Punjab, INDIA (Dated: May 30, 2020) ∗ Using the eﬀective Lagrangian approach, we examine the recently observed charm states D (2460), ∗ ∗ ∗ J D (2560), D (2740), D (3000) and their spin partners D (2680), D (2760) and D (3000) with JP J J − J − − −J J J states 1P 2+, 2S 0 , 1D 2 , 2P 1+ and 2S 1 , 1D 3 , 2P 0+ respectively. We study their two 3 1 5 1 1 5 1 2 2 2 2 2 2 2 body strong decays, coupling constants and branching ratios with the emission of light pseudo-scalar ∗ mesons (π, η, K). We also analyze the newly observed charm state D (3000) and suggest it to be 2 either 1F(2+) or 2P(2+) state and justify one of them to be the most favorable assignment for ∗ D (3000). We study the partial and the t(cid:101)otal decay w(cid:101)idth of unobserved states D(11F ), D (11F ) 2 ∗ 3 s 3 and D (11F ) as the spin and the strange partners of the D (3000) charmed meson. The branching s 2 2 ratios and the coupling constants g , g , g , g and g calculated in this work can be TH HH Y H SH ZH confronted with the future experimental data. PACS numbers: 14.40.Lb, 13.25.Ft − I. INTRODUCTION the D0π+ and D+π mass spectra respectively. BaBar collaboration in 2010, observed D (2560)0, D (2600)0, ∗ J ∗ J The excitation spectrum of (cq) heavy-light charmed DJ(2600)+, DJ(−27→50)0, DJ(2760)+ and DJ(2760)0 in mesons have received considerable theoretical and exper- the inclusive e+e cc interaction [3]. Masses and the imental attention, as it provide opportunities to study widths of charm states predicted by BaBar and LHCb the QCD properties within the context of diﬀerent mod- are so close, that they are considered to be in the same els. Recently, LHCb collaboration have used the Dalitz JP state. Masses and widths of these charm states ob- − → plot analysis to study the resonant substructures B served by various collaborations are presented in Table I. − − D+π π decays in the pp collision at a center-of-mass energy 7 TeV. The masses and the widths of charm reso- It is very crucial to assign a proper JP to the heavy- − nances with spins 1, 2 and 3 at high D+π masses are de- light system in a given spectra, as large amount of ex- termined [1]. The study gives indication that, these res- perimental information like decay width, branching ra- onances are mainly coming from the contribution of the tios and hyperﬁne splitting are based on their JP . Vari- ∗ ∗ ∗ ∗ D (2460), D (2680), D (2760) and D (3000) charmed ous theoretical models have suggested diﬀerent JP states 2 1 3 2 mesons. The measured Breit-Wigner masses and widths to the observed charm mesons. In this paper, we ana- of these charmed mesons are lyze the available theoretical and experimental data on the excited charm states and specify their proper JP . ∗ In our analysis, we mentioned D (2460) to be the well ∗ ± ± ± 2 D (2460) : M = 2463.7 0.4 0.4 0.6MeV, established state having JP = 2+ in the charm spec- 2 ± ± ± tra [4]. The information provided by Babar (2010) and Γ = 47.0 0.8 0.9 0.3MeV, (1) ∗ ∗ LHCb (2013) for the states D (2680) and D (2760) were J J conﬁrmed in 2016 by LHCb, which had provided their ∗ ± ± ± J values as 1 and 3 respectively. Theoretical study of D (2680) : M = 2681.1 5.6 4.9 13.1MeV, − 1 ± ± ± these two states concluded their JP to be 1 for n = 2 Γ = 186.7 8.5 8.6 8.2MeV, (2) − S-wave and 3 for n = 1 D wave respectively [5–9]. States D (2560)0 and D (2740)0 being the spin part- J∗ ∗J − ∗ ± ± ± ners of D (2680)0 and D (2760)0, are assigned JP = 0 D (2760) : M = 2775.5 4.5 4.5 4.7MeV, J J − 3 ± ± ± for S-wave (n = 2) and 2 for D-wave (n = 1) re- ∗ Γ = 95.3 9.6 7.9 33.1MeV, (3) spectively. Higher charm states D (3000) and D (3000) J J were studied by various models like 3P model, heavy (cid:48) 0 ∗ ± ∓ ∓ quark eﬀective theory, but their JP s are not yet con- ∗ D2(3000) : M = 3214 29 ±33 ±36Me±V, ﬁrmed. Authors in [11] assigned DJ(3000) as the 1F5 2+ Γ = 186 38 34 63MeV (4) or 1F 4+ state and D (3000) as the 1F 3+ or 2P2 1+ 7 J 7 1 2 2 2 state, but Ref.[10] have suggested various other possi- In 2010 and 2013, a great achievement have been (cid:48) ∗ bilities for the JP s of (D (3000)), (D (3000)) and con- made by BaBar and LHCb collaboration. LHCb J J (cid:48) collaboration observed two natural parity resonances cluded 2P(0+, 1+) to be the most favorable nLJP s in ∗ ∗ D (2650)0, D (2760)0 and two unnatural parity reso- the charm spectra by studying their branching ratio. J J − nances D (2580)0 and D (2740)0 by studying the D+π , Now, the main interest of theorists is on the newly J ∗ − J ∗ D0π+ and D +π invariant mass spectra [2]. Along predicted D (3000) state, whose mass and decay width 2 ∗ with these states, LHCb has also observed D (3000)0 in is comparable with the former D (3000) state. It is ∗ − ∗ J∗ J the D +π ﬁnal state and D (3000)+ and D (3000)0 in suggested by Zhi-Gang Wang in Ref.[12], that the en- J J ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Pallavi

label: author

text: Gupta

text: and

text: A.

text: Upadhyay

text: School

text: of

text: Physics

text: and

text: Materials

text: Science,

text: Thapar

text: University,

text: Patiala

text: Punjab,

text: INDIA

text: (Dated:

text: May

label: date

label: abstract

text: Using

text: the

text: eﬀective

text: Lagrangian

text: approach,

text: we

text: examine

text: the

text: recently

text: observed

text: charm

text: states

text: D

text: J

text: D

text: D

text: D

text: and

text: their

text: spin

text: partners

text: D

text: D

text: and

text: D

text: with

text: JP

text: J

text: J

text: J

text: −J

text: J

text: J

text: states

text: and

text: respectively.

text: We

text: study

text: their

text: two

text: body

text: strong

text: decays,

text: coupling

text: constants

text: and

text: branching

text: ratios

text: with

text: the

text: emission

text: of

text: light

text: pseudo-scalar

text: mesons

text: (π,

text: η,

text: K).

text: We

text: also

text: analyze

text: the

text: newly

text: observed

text: charm

text: state

text: D

text: and

text: suggest

text: it

text: to

text: be

text: either

text: or

text: state

text: and

text: justify

text: one

text: of

text: them

text: to

text: be

text: the

text: most

text: favorable

text: assignment

text: for

text: D

text: We

text: study

text: the

text: partial

text: and

text: the

text: decay

text: of

text: unobserved

text: states

text: D

text: s

text: and

text: D

text: as

text: the

text: spin

text: and

text: the

text: strange

text: partners

text: of

text: the

text: D

text: charmed

text: meson.

text: The

text: branching

text: s

text: ratios

text: and

text: the

text: coupling

text: constants

text: g

text: g

text: g

text: g

text: and

text: g

text: calculated

text: in

text: this

text: work

text: can

text: be

text: TH

text: HH

text: Y

text: H

text: SH

text: ZH

text: confronted

text: with

text: the

text: future

text: experimental

text: data.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: the

text: and

text: D+π

text: mass

text: spectra

text: respectively.

text: BaBar

text: collaboration

text: in

text: observed

text: D

text: D

text: J

text: J

text: The

text: excitation

text: spectrum

text: of

text: (cq)

text: heavy-light

text: charmed

text: and

text: in

text: mesons

text: have

text: received

text: considerable

text: theoretical

text: and

text: exper-

text: the

text: inclusive

text: e+e

text: cc

text: interaction

text: Masses

text: and

text: the

text: imental

text: attention,

text: as

text: it

text: provide

text: opportunities

text: to

text: study

text: widths

text: of

text: charm

text: states

text: predicted

text: by

text: BaBar

text: and

text: LHCb

text: the

text: QCD

text: properties

text: within

text: the

text: context

text: of

text: diﬀerent

text: mod-

text: are

text: so

text: close,

text: that

text: they

text: are

text: considered

text: to

text: be

text: in

text: the

text: same

text: els.

text: Recently,

text: LHCb

text: collaboration

text: have

text: used

text: the

text: Dalitz

text: JP

text: state.

text: Masses

text: and

text: widths

text: of

text: these

text: charm

text: states

text: ob-

text: plot

text: analysis

text: to

text: study

text: the

text: resonant

text: substructures

text: B

text: served

text: by

text: various

text: collaborations

text: are

text: presented

text: in

text: Table

text: I.

text: D+π

text: π

text: decays

text: in

text: the

text: pp

text: collision

text: at

text: a

text: center-of-mass

text: energy

text: TeV.

text: The

text: masses

text: and

text: the

text: widths

text: of

text: charm

text: reso-

text: It

text: is

text: very

text: crucial

text: to

text: assign

text: a

text: proper

text: JP

text: to

text: the

text: heavy-

text: nances

text: with

text: spins

text: and

text: at

text: high

text: D+π

text: masses

text: are

text: de-

text: light

text: system

text: in

text: a

text: given

text: spectra,

text: as

text: large

text: amount

text: of

text: ex-

text: termined

text: The

text: study

text: gives

text: indication

text: that,

text: these

text: res-

text: perimental

text: information

text: like

text: decay

text: width,

text: branching

text: ra-

text: onances

text: are

text: mainly

text: coming

text: from

text: the

text: contribution

text: of

text: the

text: tios

text: and

text: hyperﬁne

text: splitting

text: are

text: based

text: on

text: their

text: JP

text: Vari-

text: D

text: D

text: D

text: and

text: D

text: charmed

text: ous

text: theoretical

text: models

text: have

text: suggested

text: diﬀerent

text: JP

text: states

text: mesons.

text: The

text: measured

text: Breit-Wigner

text: masses

text: and

text: widths

text: to

text: the

text: observed

text: charm

text: mesons.

text: In

text: this

text: paper,

text: we

text: ana-

text: of

text: these

text: charmed

text: mesons

text: are

text: lyze

text: the

text: available

text: theoretical

text: and

text: experimental

text: data

text: on

text: the

text: excited

text: charm

text: states

text: and

text: specify

text: their

text: proper

text: JP

text: In

text: our

text: analysis,

text: we

text: mentioned

text: D

text: to

text: be

text: the

text: well

text: D

text: M

text: established

text: state

text: having

text: JP

text: in

text: the

text: charm

text: spec-

text: tra

text: The

text: information

text: provided

text: by

text: Babar

text: and

text: Γ

text: LHCb

text: for

text: the

text: states

text: D

text: and

text: D

text: were

text: J

text: J

text: conﬁrmed

text: in

text: by

text: LHCb,

text: which

text: had

text: provided

text: their

text: J

text: values

text: as

text: and

text: respectively.

text: Theoretical

text: study

text: of

text: D

text: M

text: these

text: two

text: states

text: concluded

text: their

text: JP

text: to

text: be

text: for

text: n

text: Γ

text: S-wave

text: and

text: for

text: n

text: D

text: wave

text: respectively

text: States

text: D

text: and

text: D

text: being

text: the

text: spin

text: part-

text: J∗

text: ∗J

text: ners

text: of

text: D

text: and

text: D

text: are

text: assigned

text: JP

text: D

text: M

text: J

text: J

text: for

text: S-wave

text: (n

text: and

text: for

text: D-wave

text: (n

text: re-

text: Γ

text: spectively.

text: Higher

text: charm

text: states

text: D

text: and

text: D

text: J

text: J

text: were

text: studied

text: by

text: various

text: models

text: like

text: model,

text: heavy

text: quark

text: eﬀective

text: theory,

text: but

text: their

text: JP

text: s

text: are

text: not

text: yet

text: con-

text: M

text: ﬁrmed.

text: Authors

text: in

text: assigned

text: as

text: the

text: Γ

text: or

text: state

text: and

text: D

text: as

text: the

text: or

text: J

text: state,

text: but

text: have

text: suggested

text: various

text: other

text: possi-

text: In

text: and

text: a

text: great

text: achievement

text: have

text: been

text: bilities

text: for

text: the

text: JP

text: s

text: of

text: (D

text: (D

text: and

text: con-

text: made

text: by

text: BaBar

text: and

text: LHCb

text: collaboration.

text: LHCb

text: J

text: J

text: collaboration

text: observed

text: two

text: natural

text: parity

text: resonances

text: cluded

text: to

text: be

text: the

text: most

text: favorable

text: nLJP

text: s

text: in

text: D

text: D

text: and

text: two

text: unnatural

text: parity

text: reso-

text: the

text: charm

text: spectra

text: by

text: studying

text: their

text: branching

text: ratio.

text: J

text: J

text: nances

text: D

text: and

text: D

text: by

text: studying

text: the

text: D+π

text: Now,

text: the

text: main

text: interest

text: of

text: theorists

text: is

text: on

text: the

text: newly

text: J

text: J

text: and

text: D

text: +π

text: invariant

text: mass

text: spectra

text: Along

text: predicted

text: D

text: state,

text: whose

text: mass

text: and

text: decay

text: width

text: with

text: these

text: states,

text: LHCb

text: has

text: also

text: observed

text: D

text: in

text: is

text: comparable

text: with

text: the

text: former

text: D

text: state.

text: It

text: is

text: J∗

text: J

text: the

text: D

text: +π

text: ﬁnal

text: state

text: and

text: D

text: and

text: D

text: in

text: suggested

text: by

text: Zhi-Gang

text: Wang

text: in

text: that

text: the

text: en-

text: J

text: J

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##