{ }

•

•

IK K ×KXT X∗ XH

X

X Y X ⊗ Y1K×K 0K×K

K × K �tr(X) CN (a,R)

a R ‖x‖0x [A(k, n)] (k, n)th

A

C

L

M

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τ

si �

[si1 . . . siτ ]T ∈ C1×τ

{|sij |2 = Pτ ,∀j ∈ τ} P

CM×1

lth

cth hlc ∼ CN (0, αlcRlc) ∈ CM×1 αlclth cth

Rlc ∈ CM×M Rlc = E[hlchHlc ]

si M×τcth

Yc =∑∀l∈Ki

hlcsTi +Nc

Ki si

cth

Yc =C∑l=1

hlcsT +Nc

Nc ∈ CM×τ

σ2 l = 1, . . . , Cs

a(ω) = [1 e−jω . . . e−j(M−1)ω]T

ω = 2πd sin θλ d

λ θ

hlc =

Q∑i=1

γia(ωi)

γi θiith Q

R = DaBσωDHa

σω = 2πdλσθ cos θ Da = [a(ω)] σθ

Bσω

ω ∈ N (0, σω)

[Bσωlc (m,n)] = e((m−n)√3δω)2/2.

ω [−δω, δω]

[Bδω (m,n)] =sin((m− n)δω)(m− n)δω .

σω =√3δω

md[k] = c[k]

M−1∑n=0

m[n] cos

(π(2n+ 1)k

2M

),

m mdmlc

m[n] =M−1∑k=0

c[k]md[k] cos

(π(2n+ 1)k

2M

),

k n

c[k]

c[k] =

⎧⎨⎩√1M , k = 0,√2M , .

Uc

UD = [UD(k, n)] = c[k] cos

((2n+ 1)kπ

2M

).

UHcE[mdm

Hd ] =

UcE[mmH ]UHc

•

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a [k, ωi] =

⎧⎪⎨⎪⎩

√1Me−jωi

M−12

sin M2ωi

sinωi

2

, k = 0,√2M

sin((ωi+πk

M)M2)

sin((ωi+πk

M) 12)ej(ωi+

πk

M)(M−1)

2+πkM +

√2M

sin((ωi−πkM )M2)

sin((ωi−πkM ) 12)ej(ωi−

πk

M)(M−1)

2−πkM , 0 ≤ k ≤M − 1.

hlc [k, ωi] =

⎧⎪⎪⎨⎪⎪⎩

∑P

i=1

γi

√1M√Pe−jωi

M−12

sin M2ωi

sinωi

2

, k = 0,

∑P

i=1

γi

√2M√P

(sin((ωi+

πk

M)M2)

sin((ωi+πk

M) 12)ej(ωi+

πk

M)(M−1)

2+πkM +

sin((ωi−πkM )M2)

sin((ωi−πkM ) 12)ej(ωi−

πk

M)(M−1)

2−πkM

), 0 ≤ k ≤M − 1.

•

M → ∞

SHyc

ω∗i = argmaxω∗i

a (ωi)RlcaH(ωi)

π2

qlc ∈ {0, 1}M×1hlc

η =‖qlc‖0M

.

K

S = s⊗IM SHS = τIM

yc = S

C∑l=1

hlc + n

n ∈ CMτ×1 = (N)

hlc = SHyc.

hlc = hlc +

C∑m�=lhmc +

nSH

τ.

ELSlc = Ehlc{(hlc −

C∑m=1

hlm

)(hlc −

C∑m=1

hlm

)H}

ELSlc = tr( C∑m=1,m �=c

αmcRmc

)

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4731

• si•

hlslc = SHyc

hlslc hlc = Uchlslc

hlc = hlc � qlchFlc = U

Hc hlc

hFlc

ξlcR12

lchlchlc

ξlcR12

lcSH

Rmc = ξlcR12

lcRmcR12

lc

ξlcR12

lchlc

Elc =tr(R

12

lc

∑Cm=1,m�=c αmcRmcR

12

lc)

ξlc.

ξlc√tr(Rlc)

• si•

glc = R12

lcSH

glc glc

hlc = R− 12lc glc

R12

lc

lth mth

cth

δclm =tr(αlcαmcRlcRmc

)tr(αlcRlc)tr(αmcRmc)

0 ≤ δclm ≤ 1 δclmδclm

mth cth

δcl =tr(αlcRlc

∑Cm=1,m �=l αmcRmc

)tr(αlcRlc)tr(

∑Cm=1,m�=l αmcRmc)

.

sl U(sl) UM

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δ(U(sl)) =∑l∈U(sl)

tr(αlcRlc

∑m∈U(sl),m�=l αmcRmc

)tr(αlcRlc

)tr(∑

m∈U(sl),m�=c αmcRmc)

•s1

• K•

U(sl)• u ∈M = U − ∪l−1n=1U(sn)• U(sl) = u• |U(sl)| ≤ K M(1) M(|M|)

u∗ =u∈M

δ(U(sl) ∪ u)M =M− u∗U(sl) = U(sl) ∪ u∗

β

0dBd = λ

2

ε = 10 log10

(∑ ‖hlc − hlc‖2∑ ‖hlc‖2).

η = 0.5 K = 3 C = 3 β = 1 P = 0dB

Number of antennas

MSE(dB)

LS-PA

DLS-PA

MDLS-PA

C = 3 K = 3 β = 1 θ1 = [10◦, 25◦] θ2 =[20◦35◦] θ3 = [30

◦45◦] P = 0dB

K = 2 C = 2β = 1 P = 0dB

η

η

η

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Number of antennas

MSE(dB)

MDLS-WPA β = 1

LS-WPA β = 1

DLS-WPA β = 1

C = 2 K = 2 β = 1 η = 0.5 P = 0

η = 0.5 β = 1 M = 20

η

MSE(dB)

DLS-WPA

MDLS-WPA

C = 2 K = 2 β = 1 P = 0

Pilot reuse factor

MSE(dB)

LS-PA

DLS-PA

MDLS-PA

M = 20 P =0dB β = 1 η = 0.5

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