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arXiv:2112.02331v1 [eess.SP] 4 Dec 2021

1

RIS-Aided D2D Communications Relying on

Statistical CSI with Imperfect Hardware

Zhangjie Peng, Tianshu Li, Cunhua Pan, Member, IEEE, Hong Ren, Member, IEEE,

and Jiangzhou Wang, Fellow, IEEE

Abstract—In this letter, we investigate a reconﬁgurable intelli-

gent surfaces (RIS)-aided device to device (D2D) communication

system over Rician fading channels with imperfect hardware

including both hardware impairment at the transceivers and

phase noise at the RISs. This paper has optimized the phase shift

by a genetic algorithm (GA) method to maximize the achievable

rate for the continuous phase shifts (CPSs) and discrete phase

shifts (DPSs). We also consider the two special cases of no

RIS hardware impairments (N-RIS-HWIs) and no transceiver

hardware impairments (N-T-HWIs). We present closed-form

expressions for the achievable rate of different cases and study the

impact of hardware impairments on the communication quality.

Finally, simulation results validate the analytic work.

Index Terms—Reconﬁgurable intelligent surface (RIS), hard-

ware impairment, D2D communication, intelligent reﬂecting

surface (IRS).

I. INTRO DUC TIO N

Reconﬁgurable intelligent surface (RIS) is a new trans-

mission technology that can conﬁgure the radio channel in

a desired manner by optimizing the phase shift of each

reﬂecting element [1]. Due to their appealing properties of

low power consumption and low cost, RIS-aided wireless

communications have attracted much research attentions [2]–

[5]. Some initial attempts to study RIS-aided communication

systems include RIS-aided full-duplex systems [2], RIS-aided

physical layer security [3], RIS-aided wireless power transfer

[4], RIS-aided mobile edge computing [5], and RIS-aided

multiuser transmission [6]. The cascaded channel estimation

was studied in [7].

(Corresponding author: Cunhua Pan.)

Z. Peng is with the College of Information, Mechanical, and Electrical

Engineering, Shanghai Normal University, Shanghai 200234, China, also

with the National Mobile Communications Research Laboratory, Southeast

University, Nanjing 210096, China, and also with the Shanghai Engineering

Research Center of Intelligent Education and Bigdata, Shanghai Normal

University, Shanghai 200234, China (e-mail: pengzhangjie@shnu.edu.cn).

T. Li is with the College of Information, Mechanical and Electri-

cal Engineering, Shanghai Normal University, Shanghai 200234, China

(e-mail: 1000479056@smail.shnu.edu.cn).

C. Pan is with the School of Electronic Engineering and Computer

Science at Queen Mary University of London, London E1 4NS, U.K.

(e-mail: c.pan@qmul.ac.uk).

H. Ren is with the National Mobile Communications Research Laboratory,

Southeast University, Nanjing 210096, China (e-mail: hren@seu.edu.cn).

J. Wang is with the School of Engineering and Digital Arts, University of

Kent, CT2 7NT Canterbury, U.K. (e-mail: j.z.wang@kent.ac.uk).

D2D communication has received great attention to meet

the rapidly increasing demand for data trafﬁc [8]. In D2D

communication in cellular networks, cellular and D2D users

transmit signals simultaneously using the same spectrum of

cellular users. However, there is a paucity of contributions on

RIS-aided D2D communications. In [9], the authors proposed

to deploy an RIS that can assist pairs of devices in their

communication when the direct links are blocked by high

buildings, plants and walls.

However, most of the existing contributions on D2D com-

munication are based on the assumption of perfect hardware

in the radio frequency chains [10] and the RISs [9]. Existing

studies have shown that hardware impairments adversely affect

the system performance of the multiple-input multiple-output

systems [11], [12]. In addition, an RIS has low hardware cost

[2]–[5], [9], which is prone to hardware imperfections.

Based on above, a natural question is whether we can

use non-ideal low-cost hardware to assist the D2D commu-

nications. Speciﬁcally, our contributions are summarized as

follows: 1) We consider an RIS-aided D2D communication

system over Rician fading channels, considering hardware

impairments both at the terminals and at the RISs. We assume

that the hardware impairment is coupling with the transmission

signal at the transceiver. Moreover, the random phase error

caused by the phase noise at the RIS follows the Von Mises

distribution; 2) we present the closed-form expressions for the

achievable rate in the general case of both hardware impair-

ments both at the terminals and at the RISs, which is then

maximized by a genetic algorithm (GA) method. To obtain

more design insights, we also study the two special cases of no

RIS hardware impairments (N-RIS-HWIs) and no transceiver

hardware impairments (N-T-HWIs); 3) we provide simulation

results to verify our derived expressions. In addition, the

exhaustive search method is used to show that our results can

achieve the globally optimal solution.

The rest of the letter is organized as follows. The model

for RIS-aided D2D communication system with hardware

impairments is depicted in Section II. We derive the sum

achievable rate’s expression and maximize it in Section III.

Numerical results are presented in Section IV. And conclusions

are drawn in Section V.

γi=pigT

biΘΦgai

2

K

P

j=1,j6=ipjgT

biΘΦgaj

2

+|gT

biΘΦGa√Ληt|2+Υri +σ2

i

.(10)

Ri≈log2

1+ piαbi αai

εiβi˜

Γi,i+L(εi+βi)+L

(εi+1)(βi+1)

(1 + κr) (1 + κt)

K

P

j=1 pjαbiαaj

εiβj˜

Γi,j +L(εi+βj)+L

(εi+1)(βj+1) −piαbiαai

εiβi˜

Γi,i+L(εi+βi)+L

(εi+1)(βi+1) +σ2

i

(13)

2

RN-RIS-HWIs

i≈log2

1+ piαbi αai

εiβiΓi,i+L(εi+βi)+L

(εi+1)(βi+1)

(1 + κr) (1 + κt)

K

P

j=1 pjαbiαaj

εiβjΓi,j+L(εi+βj)+L

(εi+1)(βj+1) −piαbiαai

εiβiΓi,i+L(εi+βi)+L

(εi+1)(βi+1) +σ2

i

(15)

RIS

1a

g

ai

g

aK

g

bi

g

bK

g

DT

i

DT

K

1

DR

DR i

DRK

1

DT

1b

g

Fig. 1. System model for RIS-aided D2D communications.

II. SY S TE M MO DE L

We consider an RIS-aided D2D communication system,

where an RIS assists Kpairs of devices to exchange in-

formation, as shown in Fig. 1. It is assumed that there

are a total of KD2D links in the network, where the

ith single-antenna transmitter (receiver) is denoted by DTi

(DRi) for i= 1,···, K. In the overlay mode, the radio

resources occupied by D2D links are orthogonal to that of

the cellular links, which ensures no interference between

D2D links and cellular links. The RIS includes Lscattered-

reﬂection elements. And the phase shift matrix is denoted by

Θ=diag(ejθ1,···, ejθℓ,···, ejθL), where θℓis the phase

shift of the ℓth scattered-reﬂection element.

The channel between DTiand the RIS (the RIS and DRi)

can be written as gai =√αaihai ,(1)

gbi =√αbihbi ,(2)

where αai and αbi denote the large-scale fading coefﬁcients,

and hai ∈CL×1and hbi ∈CL×1denote Rician fading

channels, which can be expressed as

hai =rεi

εi+ 1hai +r1

εi+ 1 ˜

hai,(3)

hbi =sβi

βi+ 1 hbi +r1

βi+ 1 ˜

hbi,(4)

where εiand βidenote the Rician factors, ˜

hai ∈CL×1and

˜

hbi ∈CL×1are non-line-of-sight components, whose entries

are standard Gaussian random variables with independent and

identical distribution, i.e., CN(0,1), and hai ∈CL×1and

hbi ∈CL×1are line-of-sight (LoS) components. Particularly,

hai and hbi can be written as

hai (ϕa

i, ϕe

i)="1,···, ej2πd

λ(xsin ϕa

isin ϕe

i+ycos ϕe

i),···,

ej2πd

λ((√L−1)sin ϕa

isin ϕe

i+(√L−1)cos ϕe

i)

#T

,(5)

hbi (ςa

i, ςe

i)="1,··· , ej2πd

λ(xsin ςa

isin ςe

i+ycos ςe

i),···,

ej2πd

λ((√L−1)sin ςa

isin ςe

i+(√L−1)cos ςe

i)

#T

,(6)

where 06x, y 6√L−1,ϕa

i, ϕe

i(ςa

i, ςe

i)respectively denote

the ith pair of devices’ azimuth and elevation angles of arrival

(angle of departure). We assume d=λ

2in our letter [1].

The signal transmitted from DTiis given by [11]

xi= ˆxi+ηti ,(7)

where ˆxi∼ CN(0,1) represents the data symbol, ηti ∼

CN 0, κtE|ˆxi|2denotes the impact of impairments in the

transmitter hardware, and κt∈(0, 1) is the coefﬁcient which

characterizes the impairments at the transmitter.

The RIS’s hardware impairments can take the form of phase

noise because the RIS is a passive device and highly accurate

reﬂection adjustment is impractical. The phase noise matrix

is denoted by Φ=diag(ej∆θ1,··· , ej∆θℓ,··· , ej∆θL), for

ℓ= 1,2,···, L.∆θℓis random phase error that follows

the Von Mises distribution with zero mean and concentration

parameter κ∆θ . Additionally, ∆θℓhas a characteristic function

χ=I1(κ∆θ)

I0(κ∆θ)with Ip(κ∆θ )being the modiﬁed Bessel function

of the ﬁrst kind and order p[12].

We exploit statistical channel state information (CSI), which

varies much slowly than the instantaneous CSI and can be

readily obtained. The signal received at DRiis given by

yi=gT

biΘΦ

K

X

j=1

√pjgajxj+ηri +ni

=gT

biΘΦGa√Λx +ηri +ni,(8)

where Ga,[ga1,···,gaK ],x,[x1,···, xK]T,pjdenotes

the transmit power of UAj with Λ,diag (p1,···, pK),ηri ∼

CN (0, Υr i)denotes the impact of impairments in the receiver

hardware with Υri =κrEn|gT

biΘΦGa√Λx|2o,κr∈(0, 1)

is the coefﬁcient which characterizes the impairments at the

receiver, and ni∼ CN(0, σ2

i)is the additive white Gaussian

noise at DRi, for i= 1,···, K.

To analyze the achievable rates, we consider the expansion

of (8) as follows

yi=√pigT

biΘΦgai ˆxi+

K

X

j=1,j6=i

√pjgT

biΘΦgaj ˆxj

+gT

biΘΦGa√Ληt+ηri +ni,(9)

where ηt,[ηt1,···, ηtK ]T.

From (9), we obtain the signal-to-interference plus noise

ratio at DRiin (10) shown at the bottom of this page.

Therefore, the achievable rate for DRican be expressed as

Ri=E{log2(1 + γi)},(11)

and the sum achievable rate can be written as

C=

K

X

i=1

Ri.(12)

III. ACHI EVABLE RATE ANA LYS IS

In this section, we ﬁrst derive an approximate expression for

the sum achievable rate in the following theorem. Then, we

consider two special cases of N-RIS-HWIs and N-T-HWIs.

Finally, we maximize the achievable rate with GA (genetic

algorithm) method.

3

Theorem 1. In a D2D communication system aided by RIS,

the ergodic sum achievable rate of DRican be approximated

as (13), where ˜

Γi,i and ˜

Γi,j can be given by

˜

Γi,h =E

L

X

ℓ=1

ej[(θℓ+∆θℓ)+πT n,m

i,h ]

2

=L+ 2χ2X

1≤m<n≤L

cos θn−θm+πT n,m

i,h , h ∈ {i, j}(14)

where Tn,m

i,h = (xn−xm)pi,h + (yn−ym)qi,h, with xz=

⌊(z−1)

√L⌋,yz= (z−1) mod√L,z∈ {m, n},pi,h =

sin ϕa

isin ϕe

i+ sin ςa

hsin ςe

h, and qi,h = cos ϕe

i+ cos ςe

h.

Proof: See Appendix A. ✷

Corollary 1. Assuming that the RIS hardware is ideal and thus

there is no phase noise, i.e., ∆θℓ= 0, for i= 1,2,···, N .

It follows Φ=IL. The rate of DRion N-RIS-HWIs can be

approximated as (15), where Γi,i and Γi,j can be deﬁned as

Γi,h =L+ 2 X

1≤m<n≤L

cos θn−θm+πT n,m

i,h , h ∈ {i, j}.(16)

Proof: When Φ=IL, we can use [9, Eq. (20) Eq. (21)],

and write EnhT

biΘhaj

2oand EnhT

biΘhai

2oas

EnhT

biΘhah

2o=εiβhΓi,h +L(εi+βh) + L

(εi+ 1)(βh+ 1) , h ∈ {i, j },

(17)

where Γi,h is deﬁned in (16).

By substituting (17) into (30), we are able to obtain the ﬁnal

result in (15). This completes the proof. ✷

We aim to solve the achievable rate maximization problem

by optimizing the phase shifts matrix Θin the special case

with only one pair of users, i.e., K= 1. Without loss of

generality, the user pair is referred to as user pair i. We can

rewrite the achievable rate as

RN-RIS-HWIs

i≈

log2

1+

piαbiαai

(εi+1)(βi+1) (εiβiΓi,i +L(εi+βi)+L)

(κtκr+κt+κr

)piαbiαai

(εi+1)(βi+1)(εiβiΓi,i+L(εi+βi)+L)+σ2

i!.

(18)

The rate depends on the phase shifts matrix Θonly

through the intermediate variable Γi,i. Since Γi,i =

PL

ℓ=1 ej[θℓ+πT n,m

i,h ]

26PL

ℓ=1 |ej[θℓ+πT n,m

i,i ]|2=L2and

Γi,i ≫0, the optimization problem can be formulated as

follows max

Γi,i

RN-RIS-HWIs

i(19a)

s.t. 06Γi,i 6L2.(19b)

The expression for the ﬁrst-order derivative of SINRi(Γi,i )

with respect to Γi,i is

∂SINRi(Γi,i)

∂Γi,i

=piαbiαai

(εi+1)(βi+1) εiβiσ2

i

h(κtκr+κt+κr

)piαbiαai

(εi+1)(βi+1) (εiβiΓi,i +L(εi+βi)+L)+ σ2

ii2

>0.(20)

Thus, the optimal design for Θcorresponds to setting Γi,i =

L2, where the optimal phase shifts of all the RIS elements

are given by θℓ=−πT n,m

i,h +C0,∀ℓ, and C0is an arbitrary

constant.

Considering this single-user system with optimal phase

shift, we assume the transmit power is scaled as pi=Eu

L2

and pi=Eu

Lwith L→ ∞. If the transmit power is scaled as

pi=Eu

L2, the rate becomes

RN−RIS−HWIs

i→(21)

log2

1+

Euαbiαai εiβi

(εi+1)(βi+1)

(κtκr+κt+κr

)Euαbiαai εiβi

(εi+1)(βi+1) +σ2

i!,as L→ ∞.

If the transmit power is scaled as pi=Eu

L, the rate becomes

RN−RIS−HWIs

i→log2

1+ 1

κtκr+κt+κr,as L→ ∞.(22)

In this case, the achievable rate only depends on the

impairment coefﬁcients at the transceiver when Lgrows into

inﬁnity.

Corollary 2. Assuming that the transceiver hardware is ideal,

i.e., κt=κr= 0. The rate of DRiwith N-T-HWIs can be

approximated as

RN-T-HWIs

i≈

log2

1+ piαbi αai

εiβi˜

Γi,i+L(εi+βi)+L

(εi+1)(βi+1)

K

P

j=1,j6=ipjαbiαaj

εiβj˜

Γi,j +L(εi+βj)+L

(εi+1)(βj+1) +σ2

i

.(23)

We design the phase shift with one pair of users, the

achievable rate is

RN-T-HWIs

i≈log2

1+ piαbiαai

εiβi˜

Γi,i+L(εi+βi)+L

(εi+1)(βi+1)

σ2

i

,(24)

which is a monotonically increasing function of ˜

Γi,i. The

optimal design for Θcorresponds to setting ˜

Γi,i =L2, where

the optimal phase shifts of all the RIS elements are given by

θℓ=−πT n,m

i,h +C1,∀ℓ, and C1is an arbitrary constant.

Considering this single-user system with optimal phase

shift, we assume the transmit power is scaled as pi=Eu

L2

with L→ ∞. If the transmit power is scaled as pi=Eu

L2, the

rate becomes

RN−T−HWIs

i→log2

1+ Euαbi αai εiβi

σ2

i(εi+ 1) ( βi+ 1),as L→ ∞.(25)

Due to the ﬁrst-order derivative of the sum rate with respect

to the phase shift Θis quite hard to obtain, we maximize the

rate with the GA method adopted in [9] by optimizing the

phase shifts. The complexity of the proposed GA algorithm is

Nt∗n, where Ntis the population size, and nis the number of

generations evaluated. We take into account both continuous

phase shifts (CPSs) and discrete phase shifts (DPSs). In

practice, only a limited number of phase shifts can be used.

We assume that the phase shift of the reﬂecting elements is

quantized with Bbits when considering DPS, and thus 2B

phase shifts can be chosen for each reﬂecting element.

IV. NUM ERI CA L RE SU LTS

We evaluate the impact of various parameters on the data

rate performance. We set SNR = pi,εi= 10 dB, κ=κt=κr,

κ∆θ = 4, σ2

i= 1, and ϕa

i=ϕe

iand ςa

i=ςe

iare respectively

set as ϕiand ςiin [9], for i= 1,···, K . Other parameters

are set the same as [9].

4

-10 -5 0 5 10 15 20 25

SNR (dB)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sum Rate (bits/s/Hz)

N-RIS-HWIs

= 4

= 2

= 1

= 0

simulation

rand scheme ( = 0)

Fig. 2. Sum achievable rate versus SNR with L= 16, K= 6, κ= 0.05,B

= 2.

-10 -5 0 5 10 15 20 25

SNR (dB)

0

0.5

1

1.5

2

2.5

3

3.5

4

Sum Rate (bits/s/Hz)

GA(N-T-HWIs, = 0)

exhaustion (N-T-HWIs, = 0)

GA( = 0.01)

exhaustion( = 0.01)

GA( = 0.05)

exhaustion( = 0.05)

rand scheme

Fig. 3. Sum achievable rate versus SNR with L= 9, K= 2 and B= 2.

In Fig. 2, we depict the rate in (12) versus the SNR obtained

from the approximate expression in (30) and Monte-Carlo

simulations from (11), when L= 16, K= 6, κ= 0.05, B

= 2. The Monte-Carlo simulation results match the analytical

expressions well, which veriﬁes the data rate performance.

Furthermore, the rate with N-RIS-HWIs performs better than

the case with RIS-HWIs. As κ∆θ decreases, the rate decreases.

Additionally, when κ∆θ = 0, the Von Mises distribution de-

generates into the uniform distribution, i.e., ∆θℓ∼ U[−π, π),

for ℓ= 1,2,···, L. In this case, the random scheme curve

has the same performance as that of our derived results, which

demonstrate that there is no need to optimize the phase shift.

The phenomenon is consistent with the result in [12].

Fig. 3 shows the sum rate versus the SNR when L= 8, K

= 2, and B= 2. We can reduce the harmful impact of the

T-HWIs by tuning the phase shifts of the RIS. Compared with

the random scheme, the optimal phase shift can achieve higher

rate. It is worth noting that the proposed GA method achieves

similar performance to the exhaustive search, which implies

that our proposed algorithm can achieve almost the globally

optimal solution. Moveover, we observe that different levels of

hardware impairment obtain different rate: the higher the level

of the hardware impairment, the worse the performance of the

rate. The rate with the ideal transceiver hardware (N-T-HWIs,

κ= 0) performs the best, as expected.

Fig. 4 illustrates the rate versus the Rician factor when SNR

= 20 dB and B= 2. In all cases, with different values of L

and K, the rates approach the ﬁxed value as Rician factor

εi→ ∞. This is because the channels are mainly inﬂuenced

by LoS component when Rician factor is large.

Fig. 5 shows the sum rate versus B. The ﬁgure shows

that, under the condition of DPS, three quantization bits are

-10 0 10 20 30 40 50 60

Rician factor (dB)

0.5

1

1.5

2

2.5

3

3.5

Sum Rate (bits/s/Hz)

= 0.01 (L = 16,K = 4)

= 0.05 (L = 16,K = 4)

= 0.01 (L = 9,K = 2)

= 0.05 (L = 9,K = 2)

simulation

Fig. 4. Sum achievable rate versus Rician factor with B= 2 and SNR = 20

dB.

123456

Number of coding bits B

1

1.5

2

2.5

3

3.5

Sum Rate (bits/s/Hz)

DPSs ( = 0.01,L = 16,K = 4)

DPSs ( = 0.05,L = 16,K = 4)

DPSs ( = 0.01,L = 9,K = 4)

DPSs ( = 0.05,L = 9,K = 4)

CPSs

Fig. 5. Sum achievable rate versus Bwith εi= 10 dB and SNR = 20 dB.

sufﬁcient, which provides useful insight into the engineering

design of systems aided by RISs. In addition, to resolve

the issue caused by imperfect hardware, we can increase the

number of low-cost reﬂection elements.

V. CO NCL US I ON

In this letter, we investigated an RIS-aided D2D com-

munication system over Rician fading channels, considering

hardware impairments both at the terminals and the RISs. We

considered two special cases of N-RIS-HWIs and N-T-HWIs.

We derived closed-form expressions for the achievable rate of

different cases. The impact of the hardware impairment on the

system performance have been observed. To resolve the issue

caused by imperfect hardware, we can increase the number of

low-cost reﬂection elements. Additionally, three quantization

bits are sufﬁcient for the DPSs setup, which provides useful

insight into the engineering design of systems aided by RISs.

Moreover, the extension to jointly optimizing power allocation

and the phase shifts of the RIS will be left for our future work.

APP EN D IX A

PROO F OF THE ORE M 1

Using Lemma 1 in [13], Riin (11) is approximated as (23).

Next, we derive EngT

biΘΦgai

2o,EngT

biΘΦgaj

2o,

En|gT

biΘΦGa√Ληt|2o, and Υr j . To begin with, we have

EngT

biΘΦgah

2o=αbiαah EnhT

biΘΦhah

2o, h ∈ {i, j}.

(24)

Both En|gT

biΘΦGa√Ληt|2oand Υrj contain the terms of

EnhT

biΘΦhai

2oand EnhT

biΘΦhaj

2o. We derive these

items later.

5

Ri≈log2

1 +

piEngT

biΘΦgai

2o

K

P

j=1,j6=ipjEngT

biΘΦgaj

2o+En|gT

biΘΦGa√Ληt|2o+Υri +σ2

i

(23)

Υrj =κrEn|gT

biΘΦGa√Λx|2o=κrEngT

biΘΦGa√Λ(ˆx +ηt)ˆxH+ηH

t√ΛHGH

aΦHΘHg∗

bio

(a)

=κrEngT

biΘΦGa√ΛEnˆxˆxH+ˆxηH

t+ηtˆxH+ηtηH

to√ΛHGH

aΦHΘHg∗

bi

o(b)

=κr(1+ κt)EngT

biΘΦGaΛGH

aΦHΘHg∗

bi

o(26)

Ri≈log2

1+

piαbiαai EnhT

biΘΦhai

2o

K

P

j=1,j6=ipjαbi αaj EnhT

biΘΦhaj

2o+ (κtκr+κt+κr)αbi

K

P

j=1 αaj pjE|hT

biΘΦhaj |2+σ2

i

= log2

1+

piαbiαai EnhT

biΘΦhai

2o

(1 + κr) (1 + κt)

K

P

j=1 pjαbiαaj E|hT

biΘΦhaj |2−piαbiαaiEnhT

biΘΦhai

2o+σ2

i

(30)

Then, En|gT

biΘΦGa√Ληt|2ocan be written as follows

En|gT

biΘΦGa√Ληt|2o=κtEgT

biΘΦGaΛGH

aΦHΘHg∗

bi.

(25)

Υrj also contains the terms of

EgT

biΘΦGaΛGH

aΦHΘHg∗

bi, which we will derive

later.

Moreover, Υrj is calculated in (26) at the top of this page.

Equality in (a) is obtained because ˆx is independent of both

gbi and Ga;ηtis also independent of both gbi and Ga.

Additionally, equality in (b) uses the following results

EnˆxˆxHo=IK,EηtηH

t=κtIK,(27)

EˆxηH

t=0,EηtˆxH=0.(28)

We can ﬁnd En|gT

biΘΦGa√Ληt|2oand Υrj contain the

item EngT

biΘΦGaΛGH

aΦHΘHg∗

bio, given by

EngT

biΘΦGaΛGH

aΦHΘHg∗

bio

(c)

=αbiEnhT

biΘΦHapDaΛpDa

HHH

aΘΦHh∗

bio

=αbiE

K

X

j=1

αaj pj|hT

biΘΦhaj |2

=

K

X

j=1

αbiαaj pjE|hT

biΘΦhaj |2.(29)

We obtain equality in (c) by deﬁning Da,

diag (αa1,···, αaK ).

Substituting (24), (25), (26) and (29) into (23), we obtain

(30). Employing the results of [9, Eq. (20) Eq. (21)] one

obtains

EnhT

biΘΦhah

2o=εiβh˜

Γi,h +L(εi+βh) + L

(εi+ 1)(βh+ 1) , h ∈ {i, j },

(31)

where ˜

Γi,h is deﬁned by

˜

Γi,h=L+2 X

1≤m<n≤L

Encos h(θn+∆θn)−(θm+∆θm) + πT n,m

i,h io.

(32)

With the help of

Encos h(θn+∆θn)−(θm+∆θm) + πT n,m

i,h io

=χ2cos hθn−θm+πT n,m

i,h i,(33)

˜

Γi,h can be further given in (14).

By substituting (31) into (30), we are able to obtain the ﬁnal

result in (13). The proof of theorem 1 is completed.

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