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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Hybrid Analog-Digital Precoding Design forSecrecy mmWave MISO-OFDM Systems

Yahia R. Ramadan, Student Member, IEEE, Hlaing Minn, Fellow, IEEE, Ahmed S. Ibrahim, Member, IEEE

Abstract—Millimeter-wave large-scale antenna systems typi-cally apply hybrid analog-digital precoders to reduce hardwarecomplexity and power consumption. In this paper, we designhybrid precoders for physical-layer security under two types ofchannel knowledge. With full channel knowledge at transmitter,we provide sufficient conditions on the minimum number of RFchains needed to realize the performance of the fully digitalprecoding. Then, we design the hybrid precoder to maximizethe secrecy rate. By maximizing the average projection betweenthe fully digital precoder and the hybrid precoder, we proposea low-complexity closed-form hybrid precoder. We extend theconventional projected maximum ratio transmission scheme torealize the hybrid precoder. Moreover, we propose an iterativehybrid precoder design to maximize the secrecy rate. With partialchannel knowledge at transmitter, we derive a secrecy outageprobability upper-bound. The secrecy throughput maximizationis converted into a sequence of secrecy outage probabilityminimization problems. Then, the hybrid precoder is designed tominimize the secrecy outage probability by an iterative hybridprecoder design. Performance results show the proposed hybridprecoders achieve performance close to that of the fully digitalprecoding at low and moderate signal-to-noise ratios (SNRs), andsometimes at high SNRs depending on the system parameters.

Index Terms—Millimeter-wave, hybrid precoding, physicallayer security, partial channel knowledge, outage.

I. INTRODUCTION

NEXT generation wireless communication systems de-mand an exponential increase in data rate. The spectrum

available in the microwave band is too scarce to answersuch data rate need. This leads to a potential use of theunderutilized millimeter-wave (mmWave) band. Millimeter-wave communications can support multiple Gbps data rates,but since the carrier frequencies are so high, mmWave linkssuffer higher propagation path loss. Antenna arrays can beused to compensate such losses [1]. Tens of antennas canbe packed into a small area in mmWave transceivers due tothe tiny wavelength. However, implementing a separate radio-frequency (RF) chain for each antenna is impractical due to

Manuscript received March 6, 2016; revised June 4, 2017 and July 20,2017; accepted July 21, 2017. The work of Y. R. Ramadan and H. Minnis supported in part by National Science Foundation under Award No. AST-1547048. The work of A. S. Ibrahim is supported in part by National ScienceFoundation under Award No. CNS-1618692. The associate editor coordinatingthe review of this paper and approving it for publication was Zhiguo Ding.(Corresponding author: Hlaing Minn.)

Y. R. Ramadan and H. Minn are with the Department of Electrical andComputer Engineering, University of Texas at Dallas, Richardson, TX 75080,USA, e-mails: {yahia.ramadan, hlaing.minn}@utdallas.edu. A. S. Ibrahimis with the Department of Electrical and Computer Engineering, FloridaInternational University, Miami, FL 33174, USA, e-mail: [email protected].

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2017.0000000

the high cost and power of mixed-signal devices. An efficientsolution to reduce the hardware complexity and the powerconsumption is the hybrid analog-digital precoding, where theantenna array with NT elements is connected via an analogRF precoder to NRF RF chains (NRF < NT) which processthe digitally-precoded transmitted stream [2].

Due to the broadcast nature of wireless links, wirelesscommunication is susceptible to eavesdropping. As a result,physical layer security has recently gained a lot of interest inthe literature especially for multiple-antenna systems and 5Gwireless communication networks [3]. The spatial degrees offreedom can be exploited to enhance the main channel anddegrade the channels to eavesdroppers (Eves). This enhancesthe secrecy rate which is defined as the minimum differencebetween the achievable rate of the main channel and eachachievable rate of the channels to Eves [4]. This paper de-velops hybrid precoding designs for physical layer security inlarge-scale mmWave systems.

A. Related Works

With full channel knowledge at the transmitter (Alice),the beamforming strategy was proven to achieve the secrecycapacity when the intended receiver (Bob) has a single antennain presence of a single Eve [5]. A semidefinite programming(SDP) framework was developed in [6] to maximize thesecrecy rate with perfect or imperfect channel knowledge andmultiple eavesdroppers with multiple antennas. The generationof artificial noise (AN) on the null space of the main channelwas also introduced to degrade only the channels to Eves. IfAlice does not have any knowledge of the channels to Eves,AN is uniformly spread on the null space of the main channel(isotropic AN) [7]. With partial channel knowledge at Alice,spatially-selective AN on the null space of the main channelis generated to effectively degrade the channels to Eves [8],[9]. We note that the works in [5]–[9] were restricted to securebaseband precoding with full RF chains.

As for secure RF precoding, previous works have focusedon directional modulation (DM). In [10], [11], the RF precoderis designed such that the transmitted symbol is correctlymodulated along the desired direction while the signal con-stellation is distorted along the other directions. The antennasubset modulation (ASM) proposed in [12] adopts the sameidea besides choosing a different subset of antennas at eachsymbol. Using this approach, an additional randomness in theconstellation along the other directions is introduced. In [13],two different transmission modes were used to analyze thesecrecy throughput, and the impact of large antenna arrays onthe secrecy throughput was also examined. In [14], the impact

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2 IEEE TRANSACTIONS ON COMMUNICATIONS

of random blockage, mmWave frequencies, and number oftransmit antennas on the secrecy rate was analyzed. However,previous works in [10]–[14] consider only the case that Alicehas a single RF chain and the channels are line-of-sight (LOS)or dominated by a single propagation path while neglectingthe small-scale fading. The first work to consider secure RFprecoding for frequency-selective mmWave channels was in[15], where the RF precoder is designed to maximize thesecrecy rate with full or partial channel knowledge at Alice.However, the work in [15] considers only the case that Alicehas a single RF chain, and lacks designing the hybrid precoderif Alice has multiple RF chains.

With partial channel knowledge at Alice, the sparse struc-ture of mmWave channels was exploited in [16] to generatespatially-selective AN to minimize the secrecy outage proba-bility. However, the work in [16] was also restricted to securebaseband precoding with full RF chains. The only works thatconsider secure hybrid precoding are in [3], [17], where thebaseband preocder is designed to generate isotropic AN, andthe RF precoder is designed to enhance the main channel.However, the works in [3], [17] consider only the case thatAlice does not have any knowledge of the channels to Evesand the channels are flat fading. The security aspect for two-way relaying was considered in [18], where the source nodeshave multiple antennas connected to a single RF chain whilethe relay has multiple antennas connected to two RF chains,and the secure RF precoders are designed to maximize thesecrecy rate. The works in [3], [16]–[18] were restricted toflat fading channels, but mmWave channels are likely to befrequency-selective due to the large transmission bandwidth[19]. To the best of our knowledge, secure hybrid precodingfor frequency-selective mmWave channels with full or partialchannel knowledge has not been developed in the literature.For flat fading channels, the hybrid precoder can be designedper symbol. However, for frequency-selective channels, the RFprecoder has to be fixed across the subcarriers of orthogonalfrequency division multiplexing (OFDM) symbol as it isapplied in the time domain, while the baseband precoder isdesigned per subcarrier [20], thus creating a different designproblem. Since we assume that Alice has some knowledge ofthe channels to Eves, we consider secure hybrid precodingusing the beamforming strategy. Incorporation of AN in ourframework requires a different development than the existingAN works, and hence we leave it as our future work.

B. ContributionsIn this paper, we investigate the hybrid precoder design

in mmWave multiple-input single-output (MISO) systems forphysical layer security. We consider two types of channelknowledge at the transmitter. With full channel knowledge atAlice, we design the hybrid precoder to maximize the secrecyrate. With partial channel knowledge at Alice, we design thehybrid precoder to maximize the secrecy throughput. Our maincontributions are summarized as follows:• With full channel knowledge at Alice, we provide suffi-

cient conditions on the minimum number of RF chainsneeded to realize the performance of the fully digitalprecoding.

• By maximizing the average projection between the fullydigital precoder and the hybrid precoder, we proposea low-complexity closed-form hybrid precoder designwhich achieves exactly the same performance of thealternating minimization algorithms in [21].

• We extend the conventional projected maximum ratiotransmissions (P-MRT) scheme to realize the hybridprecoder. We propose two P-MRT schemes. The first P-MRT scheme nulls the channels to Eves at time domain(TD-PMRT), while the second P-MRT scheme nulls thechannels to Eves at frequency domain (FD-P-MRT). Thetwo schemes TD-P-MRT and FD-P-MRT have differentregions of feasibility. We define our P-MRT as an adap-tive scheme that applies the one with higher secrecy ratebetween TD-P-MRT and FD-P-MRT for each channelrealization.

• We propose an iterative hybrid precoder design to max-imize the secrecy rate. The optimal baseband preocderis obtained as a function of the RF precoder. As aresult, we write the secrecy rate as a function of the RFprecoder only. Then, we propose a simple gradient ascentalgorithm to design the RF precoder.

• With partial channel knowledge, where Alice has fullknowledge of the channel to Bob but has knowledge onlyof the angles of departure (AoDs) of the propagationpaths to Eves, we derive a secrecy outage probabilityupper bound. We convert the secrecy throughput max-imization problem into a sequence of secrecy outageprobability minimization problems, each is solved for afixed target secrecy rate. Then, we propose an alternatingminimization algorithm, based on gradient descent, tominimize the secrecy outage probability.

• We present extensive simulation results to show that theproposed hybrid precoding designs achieve performanceclose to that of the fully digital precoding at low andmoderate signal-to-noise ratios (SNRs), and sometimesat high SNRs depending on the system parameters.

C. Notations and Organization

We use the following notation throughout this paper: Ais a matrix, a is a vector, ‖a‖ is its l2-norm, and a is ascalar, whereas (·)T and (·)H are the transpose and conjugatetranspose operators respectively. IN is the identity matrixof order N . Tr [A] denotes the trace of A, Emax [A] isthe principal eigenvector of A, and λmax [A] is the corre-sponding maximum eigenvalue, while E1:N [A] is the first Nprincipal eigenvectors of A. N [A] returns the orthonormalbasis of the null space of A. diag (a1, a2, . . . , aN ) returnsthe diagonal concatenation of elements a1, a2, . . . , aN , whileblkdiag (A1,A2, . . . ,AN ) returns the block diagonal con-catenation of matrices A1,A2, . . . ,AN . gcd (a1, a2) is thegreatest common divisor of a1 and a2. P (x) and E (x)denote the probability and expectation of x. We use MATLABnotations, where a (i : j) consists of the ith to the jth elementsof a, A (i, j) denotes the (i, j)

th element of A, A (i : j,m)consists of the ith to the jth elements of the mth column ofA, A (i, :) consists of the ith row of A, and A (i : j, :) and

RAMADAN et al.: HYBRID PRECODING DESIGN FOR SECRECY MMWAVE MISO-OFDM SYSTEMS 3

Cyclic

Prefix

RF

ChainFFT

Cyclic

Prefix

RF

ChainFFT

Cyclic

Prefix

RF

ChainFFT

K-1 Eavesdroppers

(Eves)

Cyclic

Prefix

RF

ChainIFFT

Cyclic

Prefix

RF

ChainIFFT

NTNRF

Digital

Baseband

Precoder

fBB,n

Analog

RF

Precoder

Legitimate Receiver

(Bob)

FRF

Transmitter

(Alice)

hn,1

hn,2

hn,K

Fig. 1. Secrecy mmWave massive MISO-OFDM system with K − 1 Eves.

A (:, i : j) consist of the ith to the jth rows and columns of Arespectively. Γ (m) =

∫∞0tm−1e−tdt is the Gamma function,

Γ (m,x) =∫∞xtm−1 exp (−t) dt is the upper incomplete

Gamma function, and Υ (m,x) =∫ x0tm−1 exp (−t) dt is the

lower incomplete Gamma function.The rest of this paper is organized as follows. In section II,

we describe the system and the channel models. In section III,with full channel knowledge at Alice, the hybrid precoder isdesigned to maximize the secrecy rate. In section IV, withpartial channel knowledge at Alice, the hybrid precoder isdesigned for secrecy throughput maximization. In section V,we present the numerical results. Finally, section VI concludesthe paper.

II. SYSTEM AND CHANNEL MODELS

A. System Model

We consider a secrecy mmWave MISO-OFDM system withK single-antenna receivers as shown in Fig. 1. The transmitter(Alice) sends a confidential message to the first receiver (Bob),while the rest K − 1 receivers are eavesdroppers (Eves). Weassume that the transmitter is equipped with a uniform lineararray (ULA) with NT antennas (NT � K). The spacing be-tween antennas is half the wavelength. To reduce the hardwarecomplexity and the power consumption, the uniform linearantenna array is connected via an analog RF precoder to NRF

RF chains (NRF < NT) which process the digitally-precodedtransmitted stream.

Due to the large transmission bandwidth of mmWave com-munications, mmWave channels are likely to be frequency-selective. Hence, OFDM is one of the most appropriatemodulation techniques as it can convert the frequency-selectivefading channel into a number of parallel flat fading subchan-nels. The received signal yn,k at the nth subcarrier and kth

receiver is given byyn,k = hn,kFRFfBB,nsn + zn,k, (1)

where hn,k ∈ C1×NT is the mmWave frequency domainchannel at the nth subcarrier to the kth receiver, FRF ∈CNT×NRF is the analog RF precoder, fBB,n ∈ CNRF×1 andsn are the digital baseband precoder and the transmitted codedconfidential symbol with E

[|sn|2

]= PT at the nth subcarrier

respectively, and zn,k is the zero-mean additive white com-plex Gaussian noise with variance σ2 at the nth subcarrierand the kth receiver. Since the analog RF precoder FRF

is applied in time-domain, it is fixed across the subcarriersof OFDM symbol. On the other hand, the digital baseband

precoder fBB,n is designed per subcarrier since it is appliedin frequency-domain. The RF precoder FRF and basebandprecoder fBB,n have to be designed jointly due to the coupledpower constraint,

∑NC

n=1 ‖FRFfBB,n‖2 = NC, where NC isthe number of subcarriers.

B. Channel Model

Millimeter-wave channels are expected to have limited scat-tering [22], [23]. We adopt a sparse geometric multipath chan-nel model. The discrete-time channel vector ht,k ∈ C1×NT attime instant t to the kth receiver is given by

ht,k =

L∑l=1

αl,kaHl,kδ (t− τl,k) , (2)

where L is the number of paths, αl,k and τl,k are the complexchannel gain and delay (in samples) of the the lth path tothe kth receiver, similar to [24]–[28] {αl,k} are independentcomplex random variables representing a multipath Nakagami-m fading channel [29] with shape parameter of m and scaleparameters of

{ρlm

}where {ρ1, ρ2, . . . , ρL} is the power delay

profile, al,k is transmit steering vectors of the lth path to thekth receiver with angle of departure (AoD) of ϕl,k,

al,k =[1, e−j

2πdλc

cos(ϕl,k), . . . , e−j2πdλc

(NT−1) cos(ϕl,k)]T,

(3)where d = λc

2 is the spacing between antennas and λc is thewavelength, and δ (t) is the Dirac delta function. Assumingperfect synchronization, the frequency domain channel vectorhn,k at the nth subcarrier to the kth receiver is given by

hn,k =

L∑l=1

αl,kaHl,kωn,l,k, (4)

where ωn,l,k is defined as ωn,l,k = exp(−j2π(n−1)τl,k

Nc

). We

can write hn,k in a compact form ashn,k = wn,kDkA

HT,k, (5)

where wn,k = [ωn,1,k, ωn,2,k, . . . , ωn,L,k] ∈ C1×L, Dk =diag (α1,k, α2,k, . . . , αL,k) ∈ CL×L, and AT,k ∈ CNT×L istransmit array response matrix to the kth receiver given by

AT,k = [a1,k,a2,k, . . . ,aL,k] . (6)

C. Analog RF Precoder Structures

The analog RF preocder FRF is usually implemented usinganalog phase shifters and analog combiners. Four structuresfor the analog RF precoder are shown in Fig. 2. The fully-connected structure F1 requires 2NRF(NT−NRF +1) analogphase shifters and NRF(NT −NRF) +NT analog combiners[30], while the fully-connected structure F2 requires NTNRF

analog phase shifters and NT analog combiners. The subarraystructure S1 requires 2NT analog phase shifters and NT analogcombiners, while the subarray structure S2 requires only NT

analog phase shifters.In [30], it was shown that the fully-connected structure

F1 has no constraints on the entries of FRF. Each non-zeroentry of FRF can be expressed as a sum of two analogphase shifters. The other three structures have constraintson the entries of FRF. For the fully-connected structureF2, we have |FRF (l,m)| = 1/

√NT ∀l,m. For the sub-

array structure S1, FRF has to be expressed as FRF =


RFF

RFN

Analog RF Precoder

RFN

1

1RF�N

TN

(a) Fully-connected structure F1

TN

TN

RFN

Analog RF Precoder

RFF

TN

(b) Fully-connected structure F2

RFT/NN

RFN

Analog RF Precoder

RFF

RFT/NN

(c) Subarray structure S1

RFT/NN

RFT/NN

RFN

Analog RF Precoder

RFF

(d) Subarray structure S2

Fig. 2. Analog RF precoding structures.

blkdiag (fRF,1, fRF,2, . . . , fRF,NRF) where fRF,r ∈ CNTNRF

×1

∀r ∈ {1, 2, . . . , NRF}. Similarly, each entry of fRF,r canbe expressed as a sum of two analog phase shifters. Thesubarray structure S2 has the same constraint of subarraystructure S1 and an additional constraint that |fRF,r (l)| =1/√NT/NRF ∀l. Let us denote by FF1

RF the set of allNT × NRF complex matrices and by FF2

RF the set of ana-log RF precoders satisfying the constraint of fully-connectedstructure F2, while by FS1

RF and FS2RF the sets of analog RF

precoders satisfying the constraints of subarray structure S1and S2 respectively. Note that FS2

RF ⊂ FS1RF ⊂ FF1

RF andFS2

RF ⊂ FF2RF ⊂ FF1

RF.

D. Hybrid Precoding Design ProblemsThe achievable rate Rk of the kth receiver is given by

Rk =1

NC

NC∑n=1

log2

(1 + γ |hn,kFRFfBB,n|2

), (7)

where γ = PT/σ2 is the transmit SNR per subcarrier.

Throughout the paper, we assume that the secure coding isapplied jointly across all subchannels (coding across sub-messages [31]). With full knowledge of all channels at thetransmitter, maximizing the secrecy rate Rsec given by [4]

Rsec = mink{R1 −Rk}Kk=2 (8)

is preferred. With full knowledge of the channel to Bob andpartial knowledge of the channels to Eves at the transmitter,maximizing the secrecy throughput ηsec given by [32], [33]

ηsec = Rsec (1− εsec) (9)is preferred, where εsec is the secrecy outage probability.

The subsequent parts of the paper focus on designing thesecure hybrid precoder for the the aforementioned two typesof channel knowledge at the transmitter:

1) With full knowledge of all channels at the transmitter,section III focuses on designing the hybrid precoder to maxi-mize the secrecy rate Rsec,

arg maxFRF,{fBB,n}

Rsec,

s.t. FRF ∈ FRF,

NC∑n=1

‖FRFfBB,n‖2 = NC. (10)

2) With full knowledge of the channel to Bob and partialknowledge of the channels to Eves at the transmitter, section

IV focuses on designing the hybrid precoder to maximize thesecrecy throughput ηsec,

arg maxFRF,{fBB,n},Rsec

ηsec,

s.t. FRF ∈ FRF,

NC∑n=1


Note that for the two hybrid precoder design problems in(10) and (11), FRF can be FF1

RF, FF2RF, FS1

RF, or FS2RF according

to the used structure. We consider all the four structures.

III. HYBRID PRECODER DESIGN FOR SECRECY RATEMAXIMIZATION WITH FULL CHANNEL KNOWLEDGE

We design the hybrid precoder to maximize the secrecyrate Rsec for a given transmit SNR per subcarrier γ. Weassume that Alice has perfect full knowledge of the channelsto Bob and to Eves. Bob and Eves have full knowledge oftheir channels to Alice. Furthermore, we assume that Eves donot cooperate. These assumptions become realistic if Eves areactive nodes which have communicated with Alice [34].

To decouple the hybrid precoder design and the powerallocation, we sub-optimally divide the problem into twosub-problems. In the first sub-problem, we relax the powerconstraint to ‖FRFfBB,n‖2 = 1 ∀n (which satisfies∑NC

n=1 ‖FRFfBB,n‖2 = NC) and then design the hybridprecoder (as will be presented in this section). In the sec-ond sub-problem, based on the equivalent single-input-single-output (SISO) channels {hn,kFRFfBB,n} and the constraint∑NC

n=1 ‖FRFfBB,n‖2 = NC, the power allocation across sub-carriers is obtained in a closed-form solution as in [35].

A. Fully Digital Precoding Design and The Minimum Numberof RF Chains to Realize It

When NRF = NT, Alice applies fully digital precodingto maximize the secrecy rate Rsec. The optimal basebandprecoder can be obtained by solving the optimization problemin (10) using semidefinite programming (SDP) as in [6]. Toobtain a closed-form solution, we apply an approximation bytreating the K−1 Eves as one Eve with K−1 antennas. Thisapproximation gives a secrecy rate lower bound Rsec for the


system with K − 1 Eves, which can be written as

Rsec =1

NC

NC∑n=1

log2

(1 + γ |hn,1fopt,n|2

1 + γ ‖Hnfopt,n‖2

), (12)

where Hn =[hTn,2,h

Tn,3, . . . ,h

Tn,K

]T ∈ C(K−1)×NT andfopt,n ∈ CNT×1 is the fully digital precoder. Satisfying thepower constraint ‖fopt,n‖2 = 1, fopt,n maximizing (12) isobtained using the generalized eigenvector decomposition asfopt,n = Emax

[(INT + γHH

n Hn

)−1 (INT + γhHn,1hn,1

)],

(13)and the corresponding Rsec is given by

Rsec =1

Nc

Nc∑n=1

log2

(λmax

[(INT

+ γHHn Hn

)−1 (INT

+ γhHn,1hn,1)])

.

(14)Let us define Fopt =[fopt,1, fopt,2, . . . , fopt,NC

]∈ CNT×NC .Next, we provide sufficient conditions on the number ofRF chains needed for the hybrid precoder to realize theperformance of fully digital precoding (i.e., expressing Fopt

as Fopt = FRF [fBB,1, fBB,2, . . . , fBB,NC ]).

Proposition 1. To realize the performance of fully digitalprecoding, it is sufficient for the hybrid precoding utilizingthe fully-connected structure F1 that NRF ≥ KL. For thehybrid precoding utilizing the fully-connected structure F2,the sufficient condition becomes NRF ≥ 2KL, and it reducesto NRF ≥ KL only if all the channels follow the mmWavechannel model in (2).

Proof: See Appendix A.

Proposition 2. For the subarray structures, there is no suffi-cient condition depending only on the number of RF chainsto realize the performance of fully digital precoding.

Proof: See Appendix B.For practical system parameters, the above sufficient condi-tions are not likely to be satisfied. Next, we provide differenthybrid precoder designs to maximize the secrecy rate.

B. Low-Complexity Secrecy Hybrid Precoding Desgins

1) Approximating The Fully Digital Precoding (App-FD):Traditionally, the hybrid precoder is designed to approximatethe fully digital precoder by minimizing the average Euclideandistance between the fully digital precoder and the hybridprecoder [21], [36], [37]. Different from the average Euclideandistance criterion, we design the hybrid precoder to approx-imate the fully digital precoder by maximizing the averageprojection between the fully digital precoder and the hybridprecoder. Interestingly, the two criteria are related to eachother, and they have similarity in the design of basebandprecoder (see Appendix C).

In the following, we obtain closed-form solutions for thehybrid precoder maximizing the average projection betweenthe fully digital precoder and the hybrid precoder. In otherwords, the hybrid precoder is designed as

arg maxFRF,{fBB,n}

NC∑n=1

∥∥fHopt,nFRFfBB,n

∥∥2 ,s.t. FRF ∈ FRF, ‖FRFfBB,n‖2 = 1 ∀n. (15)

After applying the power constraint ‖FRFfBB,n‖2 = 1 into theobjective function and dropping the constraint on the entriesof the RF precoder, we have

arg maxFRF,{fBB,n}

NC∑n=1

fHBB,n

(FHRFfopt,nfHopt,nFRF

)fBB,n

fHBB,n

(FHRFFRF

)fBB,n

. (16)

Using the generalized eigenvector decomposition, fBB,n max-imizing (16) is given by

fBB,n = κnEmax

[(FHRFFRF

)−1FHRFfopt,nfHopt,nFRF

]=

(FHRFFRF

)−1FHRFfopt,n∥∥∥(FHRFFRF

)− 12 FHRFfopt,n

∥∥∥ , (17)

where κn is a scaling factor to have ‖FRFfBB,n‖2 = 1,and the second equality holds since

(FHRFFRF

)−1FHRFfopt,n

fHopt,nFRF is a rank-one matrix. Substituting (17) into (16),we get

arg maxFRF

NC∑n=1

λmax

[(FHRFFRF


]= arg max

FRF

Tr

[(FHRFFRF

)− 12 FHRF

(NC∑n=1

fopt,nfHopt,n

)

× FRF

(FHRFFRF

)− 12

]

= E1:NRF

[ NC∑n=1

fopt,nfHopt,n

]. (18)

For the fully-connected structure F1, as there are no con-straints on the entries of FRF, the RF precoder is simply givenby (18). Since FHRFFRF = INRF

due to (18), fBB,n in (17) issimplified to

fBB,n =FHRFfopt,n∥∥FHRFfopt,n

∥∥ . (19)

For the fully-connected structure F2, we obtain FRF as

FRF =1√NT

exp(j∠E1:NRF

[ NC∑n=1

fopt,nfHopt,n

]), (20)

which satisfies the modulus constraint and is a good ap-proximation to (18). Then, we obtain fBB,n as in (17) sinceFHRFFRF 6= INRF

.

For the subarray structures, FHRFFRF = diag(‖fRF,1‖2 ,

. . . , ‖fRF,NRF‖2)

, and hence, (18) can be solved for eachfRF,r as

fRF,r = arg maxfRF,r

fHRF,r

(∑NC

n=1 fopt,n,rfHopt,n,r

)fRF,r

fHRF,rfRF,r, (21)

where fopt,n,r = fopt,n

((r − 1) NT

NRF+ 1 : r NT

NRF

)∈ C

NTNRF

×1

∀r ∈ {1, 2, . . . , NRF}. Therefore, fRF,r is obtained in aclosed-form for the subarray structure S1 as

fRF,r = Emax

[ NC∑n=1

fopt,n,rfHopt,n,r

], (22)

while fRF,r for the subarray structure S2 is obtained as

fRF,r =1√

NT/NRF

exp(j∠Emax

[ NC∑n=1

fopt,n,rfHopt,n,r

]).

(23)


Since FHRFFRF = INRFfor the subarray structures due to (22)

and (23), fBB,n in (17) is simplified to

fBB,n =

[fHRF,1fopt,n,1, . . . , f

HRF,NRF

fopt,n,NRF

]T√∑NRF

r=1

∣∣∣fHRF,rfopt,n,r

∣∣∣2 . (24)

Although the average projection criterion and the averageEuclidean distance criterion are slightly different (as shownin Appendix C), we observe in our numerical results thatour closed-form hybrid precoder in this subsection achievesexactly the same performance of the hybrid precoder obtainedin [21] which applies two nested iterative algorithms to designthe hybrid precoder. The computational complexity of App-FDis O

(N3

TNC +N2TNCK

).

2) Projected Maximum Ratio Transmission (P-MRT): Themain idea of P-MRT is to maximize the average SNR of Bob(SNRB

)in the null space of channels to Eves [6]. Generally,

P-MRT is suboptimal at low and moderate SNRs but optimalat high SNRs. We will show how to design P-MRT using thehybrid precoder. Nulling the channels to Eves can be doneusing the analog RF precoder (at time domain TD-P-MRT) orusing the digital baseband precoder (at frequency domain FD-P-MRT). The two schemes TD-P-MRT and FD-P-MRT havedifferent regions of feasibility (as will be shown). As a result,our P-MRT adaptively selects the better scheme from TD-P-MRT and FD-P-MRT depending on the system parameters andchannel realizations. This will yield higher secrecy rate.

a) TD-P-MRT: First, we consider the fully-connectedstructures F1 and F2. For both structures, it is necessary thatNT > (K − 1)L to apply TD-P-MRT. We express FRF as

FRF = URFFRF, (25)where URF = N [AT,Eves] ∈ CNT×(NT−(K−1)L) is a semi-unitary matrix in the null space of the channels to Eves, whereAT,Eves ∈ CNT×(K−1)L is given by

AT,Eves = [AT,2,AT,3, . . . ,AT,K ] . (26)We design FRF and fBB,n to maximize SNRB given by

SNRB =γ

NC

NC∑n=1

∣∣∣hn,1URFFRFfBB,n

∣∣∣2∥∥∥URFFRFfBB,n

∥∥∥2=

γ

NC

NC∑n=1

fHBB,n(FHRFUHRFhHn,1hn,1URFFRF)fBB,n

fHBB,n(FHRFFRF)fBB,n

,

(27)where UH

RFURF = INT−(K−1)L. Using the generalized eigen-vector decomposition, fBB,n maximizing (27) is given by

fBB,n= κnEmax

[(FHRFFRF

)−1FHRFUH

RFhHn,1hn,1URFFRF

]

=

(FHRFFRF

)−1FHRFUH

RFhHn,1∥∥∥∥∥(FHRFFRF

)− 12

FHRFUHRFhHn,1

∥∥∥∥∥, (28)

where κn is a scaling factor to have ‖FRFfBB,n‖2 = 1,

the second equality holds since(FHRFFRF

)−1FHRFUH

RFhHn,1

hn,1URFFRF is a rank-one matrix, and the corresponding

SNRB is expressed as

SNRB =γ

NC

NC∑n=1

λmax

[(FHRFFRF

)−1FHRFUH

RFhHn,1

× hn,1URFFRF

]

=γ

NCTr

[(FHRFFRF

)− 12

FHRF

(NC∑n=1

UHRFhHn,1hn,1URF

)× FRF

(FHRFFRF

)− 12

]. (29)

For the fully-connected structure F1, FRF maximizing (29)is obtained as

FRF = E1:NRF

[ NC∑n=1

UHRFhHn,1hn,1URF

]. (30)

Since FHRFFRF = INRFdue to (30), fBB,n in (28) is simplified

to

fBB,n =FHRFUH

RFhHn,1∥∥∥hn,1URFFRF

∥∥∥ , (31)

which is the well-known MRT precoder for the equivalentchannel hn,1URFFRF. For the fully-connected structure F2,we need URFFRF to satisfy the modulus constraint. Conse-quently, we obtain FRF using only half of the number of RFchains as

FRF = E1:

NRF2

[ NC∑n=1

UHRFhHn,1hn,1URF

], (32)

and fBB,n =FHRFUH

RFhHn,1

‖hn,1URFFRF‖ ∈ CNRF

2 ×1. Then, URFFRF

is decomposed (as described in proof of Proposition 1) asURFFRF = QRFRBB where QRF ∈ CNT×NRF is withunit modulus entries and RBB ∈ RNRF×

NRF2 . Finally, we set

FRF = 1√NT

QRF and fBB,n =√NTRBBfBB,n ∈ CNRF×1.

For the subarray structure S1, each fRF,r has to null thechannels to Eves. As a result, we need NT > NRF (K − 1)L.However, we may have NT > (K − 1)L but NT ≤NRF (K − 1)L. Therefore, we divide the RF chains intoNRF distinct groups, each group has NRF

NRFRF chains which

process the same digitally-preocded symbols, where NRF =

gcd(

min(⌊

NT−1(K−1)L

⌋, NRF

),NRF

). Therefore, we should

have NT > NRF (K − 1)L provided that NT > (K − 1)L.Equivalently, we proceed assuming that we have NRF chains,each is connected to NT

NRFantennas such that NT >

NRF (K − 1)L. We express fRF,r asfRF,r = URF,r fRF,r, (33)

where URF,r = N [AT,Eves,r] ∈ CNTNRF

×(NTNRF

−(K−1)L)

is a semi-unitary matrix in the null space of the channelsto Eves seen by the rth RF chain group, AT,Eves,r =AT,Eves

((r − 1) NT

NRF+ 1 : r NT

NRF, :

), and fRF,r ∈

C(NTNRF

−(K−1)L)×1

. Let URF = blkdiag(URF,1, . . . ,

URF,NRF

)∈ CNT×(NT−NRF(K−1)L) and FRF = blkdiag(

fRF,1, . . . , fRF,NRF

)∈ C(NT−NRF(K−1)L)×NRF , then we

have FRF = URFFRF, and UHRFURF = INT−NRF(K−1)L.


Therefore, SNRB and fBB,n can also be given by (27) and(28) respectively. Furthermore, SNRB can be simplified to

SNRB =γ

NC

NRF∑r=1

fHRF,r

(∑NC

n=1 UHRF,rh

Hn,1,rhn,1,rURF,r

)fRF,r

fHRF,r fRF,r

,

(34)

where hn,1,r = hn,1

((r − 1) NT

NRF+ 1 : r NT

NRF

)∀r. There-

fore, we obtain fRF,r which maximizes (34) as

fRF,r = Emax

[ NC∑n=1

UHRF,rh

Hn,1,rhn,1,rURF,r

]. (35)

Since FHRFFRF = INRFdue to (35), fBB,n is simplified to

fBB,n =

[fHRF,1U

HRF,1h

Hn,1,1, . . . , f

HRF,NRF

UHRF,NRF

hHn,1,NRF

]T√∑NRF

r=1

∣∣∣hn,1,rURF,r fRF,r

∣∣∣2 ,

(36)which is the well-known MRT precoder for the equivalentchannel hn,1URFFRF. Note that entries of URF,r fRF,r arenot likely to satisfy the modulus constraint. Moreover, approxi-mating URF,r fRF,r by exp

(j∠(URF,r fRF,r

))/√NT/NRF

results in losing the null space property. Therefore, TD-P-MRTis not applicable for the subarray structure S2. The computa-tional complexity of TD-P-MRT is O

(N3

T +NTNRFNC

).

b) FD-P-MRT: For the four structures, it is necessarythat NRF ≥ K to apply FD-P-MRT. We express the basebandprecoder fBB,n as

fBB,n = UBB,nfBB,n, (37)where UBB,n = N [HnFRF] ∈ CNRF×(NRF−(K−1)) is asemi-unitary matrix in the null space of the equivalent fre-quency domain channels to Eves HnFRF. We design FRF

and fBB,n to maximize SNRB given by

SNRB =γ

NC

NC∑n=1

∣∣∣hn,1FRFUBB,nfBB,n

∣∣∣2∥∥∥FRFUBB,nfBB,n

∥∥∥2=

γ

NC

NC∑n=1

fHBB,n(UHBB,nFHRFhHn,1hn,1FRFUBB,n)fBB,n

fHBB,n(UHBB,nFHRFFRFUBB,n)fBB,n

.

(38)Using the generalized eigenvector decomposition, fBB,n max-imizing (38) is given by

fBB,n =κnEmax

[ (UH

BB,nFHRFFRFUBB,n

)−1×UH

BB,nFHRFhHn,1hn,1FRFUBB,n

]

=

(UH

BB,nFHRFFRFUBB,n

)−1UH

BB,nFHRFhHn,1∥∥∥∥(UHBB,nFHRFFRFUBB,n

)−− 12

UHBB,nFHRFhHn,1

∥∥∥∥ ,(39)

where κn is a scaling factor to have ‖FRFfBB,n‖2 = 1,the second equality holds since (UH

BB,nFHRFFRFUBB,n)−1

UHBB,nFHRFhHn,1hn,1FRFUBB,n is a rank-one matrix, and the

corresponding SNRB is expressed as

SNRB =γ

NC

NC∑n=1

hn,1FRFUBB,n

(UH

BB,nFHRFFRFUBB,n

)−1×UH

BB,nFHRFhHn,1. (40)Note that UBB,n is a function of FRF, and FRF is fixed acrosssubcarriers while UBB,n is not. Even if we fix UBB,n, wecannot get FRF in a closed-form. Consequently, we cannotget FRF in a closed-form or by an alternately optimizingalgorithm. As suboptimal solutions, we obtain FRF for thefully-connected structures F1 and F2 as in (18) and (20)respectively, and for the subarray structures S1 and S2 asin (22) and (23) respectively. Then, fBB,n is obtained asin (37). The computational complexity of FD-P-MRT isO(N3

TNC +N3RFNC

).

C. Iterative Secrecy Hybrid Precoding DesignThe drawback of the aforementioned solutions is that they

do not directly consider the original problem in (10). In thissubsection, we propose an iterative hybrid precoding designto maximize the secrecy rate lower bound Rsec, which gives agood solution to the problem in (10) (as will be shown). Thereason why we choose Rsec to maximize is that we can writeRsec as a function of FRF only.

Similar to (12), Rsec can be written as a function of FRF

and fBB,n as

Rsec =1

NC

NC∑n=1

log2

(1 + γ |hn,1FRFfBB,n|2

1 + γ ‖HnFRFfBB,n‖2

). (41)

Applying the power constraint ‖FRFfBB,n‖2 = 1 into (41),we get

Rsec =1

NC

NC∑n=1

log2

(fHBB,n(FHRFFRF + γFHRFhHn,1hn,1FRF)fBB,n

fHBB,n(FHRFFRF + γFHRFHHn HnFRF)fBB,n

).

(42)Using the generalized eigenvector decomposition, fBB,n max-imizing (42) is given by

fBB,n =κnEmax

[(FHRFFRF + γFHRFHH

n HnFRF)−1

× (FHRFFRF + γFHRFhHn,1hn,1FRF)

](43)

where κn is a scaling factor to have ‖FRFfBB,n‖2 = 1, andthe corresponding Rsec is expressed as

Rsec =1

NC

NC∑n=1

log2

(λmax


n HnFRF)−1


]). (44)

Now, we can write the hybrid precoding design problem as afunction of FRF only as

arg maxFRF

1

NC

NC∑n=1

log2

(λmax


n HnFRF)−1


]),

s.t. FRF ∈ FRF. (45)We know that Rsec is a non-convex function of FRF. More-over, the constraint is non-convex (except for FRF = FF1

RF).


Algorithm 1 Hybrid precoder design for secrecy rate maximization

Initialization: Obtain F(0)RF as in App-FD or P-MRT (choose the scheme that

achieves higher secrecy rate), p = 0.while p ≤ P (or any other appropriate stopping criterion)

Calculate the gradient ∇(p)FRF

using (46).Updating rule:

For fully-connected structure F1,[F

(p+1)RF , α

]= argmax

FRF,αRsec (FRF),

s.t. FRF = F(p)RF + α∇(p)

FRF.

For fully-connected structure F2,[F

(p+1)RF , α

]= argmax

FRF,αRsec (FRF),

s.t. FRF = exp(j∠(F

(p)RF + α∇(p)

FRF

))/√NT.

For subarray structure S1,[F

(p+1)RF , α

]= argmax

FRF,αRsec (FRF),

s.t. FRF = M(F

(p)RF + α∇(p)

FRF

).

For subarray structure S2,[F

(p+1)RF , α

]= argmax

FRF,αRsec (FRF),

s.t. FRF = M(exp

(j∠(F

(p)RF + α∇(p)

FRF

)))/√

NTNRF

.p = p+ 1.

end whileObtain the baseband precoders

{fBB,n

}using (43).

We propose a suboptimal gradient ascent algorithm to designFRF. However, the maximum eigenvalue function λmax [X]is non-differentiable. Applying the inequality λmax [X] ≥

1rank[X]Tr [X] into (44), we obtain a differentiable lower boundwith gradient ∇FRF

given by [38]

∇FRF =

NC∑n=1

(INT−CnFRF(FHRFCnFRF)−1FHRF

)BnFRF(FHRFCnFRF)−1

NRFNC ln 2 log2

(1

NRFTr[(FHRFCnFRF)−1FHRFBnFRF

]) ,

(46)where Cn = INT + γHH

n Hn and Bn = INT + γhHn,1hn,1.Using the gradient ∇FRF

in (46), we obtain the RF pre-coder FRF by Algorithm 1, where P is the number ofiterations, M (X) = blkdiag (x1,x2, . . . ,xNRF

) ∈ CNT×NRF

and xr = X(

(r − 1) NT

NRF+ 1 : r NT

NRF, r)∈ C

NTNRF

×1

∀r ∈ {1, 2, . . . , NRF}. The step size α is obtained by abacktracking line search [39].

It is well-known that gradient ascent algorithm is highly af-fected by the initial solution since it is a local solver. As an ini-tialization, we propose to use the hybrid precoder of App-FDor P-MRT, where we choose the scheme that achieves highersecrecy rate. The computational complexity of Algorithm 1 isO(N3

TNC +NTNRFNC +(N3

RF +NTN2RF

)NCP

).

IV. HYBRID PRECODER DESIGN FOR SECRECYTHROUGHPUT MAXIMIZATION WITH PARTIAL CHANNEL

KNOWLEDGE

We design the hybrid precoder to maximize the secrecythroughput defined as ηsec = Rsec (1− εsec) [32], [33], whereεsec is the secrecy outage probability. We assume that Alice hasperfect full knowledge of the channel to Bob but has partialknowledge of the channels to Eves. Similar to [15] and [16],Alice has knowledge only of the AoDs of the paths to Eves.Bob and Eves have full knowledge of their channels to Alice.Furthermore, we assume that Eves do not cooperate.

Due to the limited scattering nature of mmWave channelsand the large number of antennas at the transmitter, the

mmWave channel is typically estimated by estimating thechannel gains and AoDs of the propagation paths basedon compressed sensing (e.g., [40]–[42]). The assumption ofpartial knowledge of the channels to Eves can be viewed intwo different scenarios as follows:

Scenario-1: The transmission is performed based onfrequency-division-duplexing (FDD). The uplink and down-link channels are different. However, the AoDs are invariantwith frequency [43]–[46], and hence without any feedbackAlice has partial knowledge (knowledge of the AoDs only) ofchannels to Bob and Eves. Alice receives a channel feedbackfrom Bob to complete the channel knowledge, but Alicedoes not receive any channel feedback from Eves. Therefore,Alice uses the AoDs of the paths to Eves as partial channelknowledge.

Scenario-2: The transmission is performed based on time-division-duplexing (TDD). The uplink and downlink channelsare reciprocal. Since we assume that Eves are active nodesin the system (Bob and Eves play interchangeable roles), thechannel knowledge of Bob and Eves may not be up-to-date.Alice re-estimates the channel to Bob. Since the coherencetime of the AoDs is much longer than that of the channelgains [15], [16], [43]–[45], [47], Alice uses the estimates ofthe AoDs of the paths to Eves as partial channel knowledge(assuming that the AoDs remain almost unchanged), and doesnot re-estimate the channel gains to Eves.1

In the following, we derive a secrecy outage probabilityupper bound εsec, which results in a secrecy throughput lowerbound ηsec = Rsec(1− εsec). Then, we design the hybrid pre-coder to maximize the secrecy throughput lower bound ηsec.

A. Secrecy Outage Probability

The secrecy outage probability εsec is defined as [31], [51]

εsec = P(

mink{R1 −Rk}Kk=2 < Rsec

). (47)

It can be rewritten as

εsec = 1− P(

maxk{Rk}Kk=2 < Ro

)= 1−

K∏k=2

P (Rk < Ro) ,

(48)where Ro = R1−Rsec. We derive a secrecy outage probabilityupper bound εsec. Applying Jensen’s inequality into (7), we get

Rk ≤ log2

(1 + γ

1

NC

NC∑n=1

|hn,kFRFfBB,n|2)

= log2

(1 + γ

1

NC

NC∑n=1

∣∣∣∣∣L∑l=1

αl,kaHl,kωn,l,kFRFfBB,n

∣∣∣∣∣2)

.

(49)

1For link adaptation or channel knowledge exploitation, the key parameter isthe coherence time of the relevant channel parameters divided by transmissiontime interval (TTI). For mmWave systems, although the channel coherencetime can become much shorter than that of current systems (e.g., 10 timesshorter if 30 GHz versus 3 GHz with the same mobile speed), TTI formmWave systems are also much shorter than that of current systems due tomuch smaller latency requirement (e.g., 125 µs TTI for mmWave system [48]versus 1 ms TTI for long-term evolution (LTE) [49]). The relevant channelparameter for our case is AoD, which has coherence time much longer (tensor hundreds times as reported in [50]) than that of the channel gains [15],[16], [43]–[45], [47]. Thus, this justifies our assumptions.


Using the triangle inequality and the arithmetic-quadraticmean inequality, we have∣∣∣∣∣ 1L

L∑l=1

αl,kaHl,kωn,l,kFRFfBB,n

∣∣∣∣∣2

≤ 1

L

L∑l=1

|αl,k|2∣∣aHl,kFRFfBB,n

∣∣2 .(50)

Applying (49) and (50) into (48), we get

εsec ≤ 1−K∏k=2

P

(L∑l=1

|αl,k|2NC∑n=1

∣∣aHl,kFRFfBB,n

∣∣2 < δ

), εsec, (51)

where δ =(2Ro−1)NC

γ L . The term∑Ll=1 |αl,k|

2∑NC

n=1∣∣∣aHl,kFRFfBB,n

∣∣∣2 represents a sum of independent Gamma-distributed random variables with shape parameter of mand scale parameters of

{ rl,km

}, where rl,k = ρl

∑NC

n=1∣∣∣aHl,kFRFfBB,n

∣∣∣2. Using [52, Eq. 15], we can write the secrecyoutage probability upper bound εsec in (51) as

εsec = 1−K∏k=2

[L∏l=1

(rmin,k

rl,k

)m] ∞∑t=0

vtΥ(mL+ t, mδ

rmin,k

)Γ (mL+ t)

,

(52)where rmin,k = min

l{rl,k}, and the coefficients {vt} can be

obtained recursively as

vt =m

t

t∑i=1

L∑j=1

(1− rmin,k

rj,k

)i vt−i, ∀t ≥ 1, (53)

where v0 = 1. For integer values of m, εsec in (52) issimplified (with the help of [53, Sec. II]) to

εsec = 1−K∏k=2

[ L∏l=1

(−rl,km

)m ] L∑l=1

m∑t=1

(−1)tvl,t,k

Υ(t, mδrl,k

)Γ (t)

,

(54)where the coefficients {vl,t,k} can be obtained as

vl,t,k = lims→ m

rl,k

1

(m− t)!∂m−t

∂m−ts

[ L∏i=1, i 6=l

(s− m

ri,k

)−m].

(55)For the special case m = 1, εsec in (54) can be simplified to

εsec = 1−K∏k=2

(1−

L∑l=1

e− δrl,k

∏n 6=l

rl,krl,k − rn,k

). (56)

Now, we obtain the secrecy throughput lower bound ηsec asηsec = Rsec (1− εsec).

B. Low-Complexity Secrecy Hybrid Precoding Designs

In this subsection, we investigate if the low-complexitysecrecy hybrid precoding strategies in subsection III-B areapplicable in case of partial channel knowledge. Examiningthe secrecy outage probability upper bound εsec in (59) withNRF = NT (FRF = INT ), we know that the fully digitalprecoders {fBB,n} have to be designed jointly and do not haveclosed-form expressions, in contrast to maximizing the secrecyrate in subsection III-A where the fully digital precodersare designed separately and have closed-form expressions.As a result, we cannot have sufficient conditions on thenumber of RF chains needed to realize the performance of thefully digital precoding in case of partial channel knowledge.Furthermore, it is not tractable to design the hybrid precoder

by approximating the fully digital precoders in case of partialchannel knowledge.

Since TD-P-MRT described in subsection III-B2 requiresonly the knowledge of the AoDs to Eves which is availableat Alice in case of partial channel knowledge, TD-P-MRT canbe applied also in case of partial channel knowledge with thesame constraints on NRF. On the contrary, FD-P-MRT cannotbe applied since it requires the knowledge of frequency domainchannels to Eves which is not available at Alice in case ofpartial channel knowledge.

C. Iterative Secrecy Hybrid Precoding Design

Following the derived secrecy outage probability upperbound εsec in (59), the hybrid precoder is designed to maxi-mize the secrecy throughput lower bound ηsec as

arg maxFRF,{fBB,n},Rsec

Rsec (1− εsec (FRF, {fBB,n} , Rsec)) ,

s.t. 0 ≤ Rsec ≤ RMRT, FRF ∈ FRF,NC∑n=1

‖FRFfBB,n‖2 = NC, (57)

where RMRT is the maximum achievable rate of Bob by themaximum ratio transmission (MRT) scheme in [54] wherethe hybrid precoder is designed to maximize the rate of Bobwhile ignoring Eves. The optimization problem in (57) is non-convex, and cannot be solved directly. In the following, we willfocus on the secrecy outage probability minimization problemwritten as

arg minFRF,{fBB,n}

εsec (FRF, {fBB,n} , Rsec) ,

s.t. FRF ∈ FRF,

NC∑n=1


Then, we will show how to use the secrecy outage probabilityminimization problem to maximize the secrecy throughput.

We know that εsec in (52) is a non-convex and non-differentiable function of the hybrid precoder. Applying theaverage-max inequality onto (51), we have

εsec ≤ 1−K∏k=2

P(

maxl

{|αl,k|2

NC∑n=1

∣∣aHl,kFRFfBB,n

∣∣2 } < δ

L

),

= 1−K∏k=2

L∏l=1

P(|αl,k|2

NC∑n=1

∣∣aHl,kFRFfBB,n

∣∣2 < δ

L

)= 1−

K∏k=2

L∏l=1

1

Γ (m)Υ(m,

mδ

Lrl,k), (59)

where the upper bound in (59) is a non-convex but differen-tiable function of the hybrid precoder. Next, we propose analternating algorithm which designs the RF precoder and thebaseband precoders alternately using the gradient of the upperbound in (59).

1) RF Precoding Design: First, we fix the baseband pre-coders {fBB,n} and optimize over the RF precoder FRF tominimize the secrecy outage probability upper bound εsec. Wepropose a suboptimal gradient descent algorithm to design theRF precoder. The gradient ∇FRF

of the the upper bound in(59) with respect to the RF precoder is obtained (with the help


Algorithm 2 RF precoding design for secrecy outage probability minimization

Input: F(0)RF,

{f(0)BB,n

}, Rsec.

p = 0.while p ≤ PRF (or any other appropriate stopping criterion)

Calculate the gradient ∇(p)FRF

using (60).Updating rule:For fully-connected structure F1,[F

(p+1)RF ,

{f(p+1)BB,n

}, α]= argmin

FRF,{fBB,n},αεsec

(FRF,

{fBB,n

}, Rsec

),

s.t. FRF = F(p)RF + α∇(p)

FRF,

fBB,n = f(p)BB,n/

√∑NCn=1

∥∥∥FRFf(p)BB,n

∥∥∥2 ∀n.For fully-connected structure F2,[F

(p+1)RF ,

{f(p+1)BB,n

}, α]= argmin

FRF,{fBB,n},αεsec

(FRF,

{fBB,n

}, Rsec

),

s.t. FRF = exp(j∠(F

(p)RF + α∇(p)

FRF

))/√NT,

fBB,n = f(p)BB,n/

√∑NCn=1


∥∥∥2 ∀n.For subarray structure S1,[F

(p+1)RF ,

{f(p+1)BB,n

}, α]= argmin

FRF,{fBB,n},αεsec

(FRF,

{fBB,n

}, Rsec

),

s.t. FRF = M(F

(p)RF + α∇(p)

FRF

),

fBB,n = f(p)BB,n/

√∑NCn=1


∥∥∥2 ∀n.For subarray structure S2,[F

(p+1)RF ,

{f(p+1)BB,n

}, α]= argmin

FRF,{fBB,n},αεsec

(FRF,

{fBB,n

}, Rsec

),

s.t. FRF = M(exp

(j∠(F

(p)RF + α∇(p)

FRF

)))/√

NTNRF

,

fBB,n = f(p)BB,n/

√∑NCn=1


∥∥∥2 ∀n.p = p+ 1.

end whileOutput: F(PRF+1)

RF ,{f(PRF+1)BB,n

}.

of Leibniz integral rule [55, Eq. A.2-1]) as

∇FRF =

K∑k=2

L∑l=1

m(

mδLrl,k

)m−1exp

(− mδLrl,k

)LΥ

(m, mδ

Lrl,k

)r2l,k

× (δ∇FRF (rl,k)− rl,k∇FRF (δ)) , (60)where ∇FRF (rl,k) = ρlal,ka

Hl,kFRF

∑NC

n=1 fBB,nfHBB,n and

∇FRF(δ) =

∑NC

n=1

2RohHn,1hn,1FRFfBB,nfHBB,n

L(1+γ|hn,1FRFfBB,n|2). Using the gra-

dient ∇FRFin (60), we obtain the RF precoder FRF by

Algorithm 2, where PRF is the number of iterations, and thestep size α is obtained by a backtracking line search.

2) Baseband Precoding Design: Now, we fix the RF pre-coder FRF and optimize over the baseband precoders {fBB,n}to minimize the secrecy outage probability upper bound εsec.We propose a suboptimal gradient descent algorithm to designthe baseband precoders. The gradient ∇fBB,n

of the the upperbound in (59) with respect to the baseband precoder is obtained(with the help of Leibniz integral rule [55, Eq. A.2-1]) as

∇fBB,n =

K∑k=2

L∑l=1

m(

mδLrl,k

)m−1exp

(− mδLrl,k

)LΥ

(m, mδ

Lrl,k

)r2l,k

×(δ∇fBB,n

(rl,k)− rl,k∇fBB,n(δ)), (61)

where ∇fBB,n (rl,k) = ρlFHRFal,ka

Hl,kFRFfBB,n and

∇fBB,n(δ) =

2RoFHRFhHn,1hn,1FRFfBB,n

L(1+γ|hn,1FRFfBB,n|2). Using the gradient

∇fBB,n in (61), we obtain the baseband precoders {fBB,n} byAlgorithm 3, where PBB is the number of iterations, and thestep size α is obtained by a backtracking line search.

Algorithm 3 Baseband precoding design for secrecy outage probabilityminimization

Input: FRF,{f(0)BB,n

}, Rsec.

p = 0.while p ≤ PBB (or any other appropriate stopping criterion)Calculate the gradient

{∇(p)

fBB,n

}using (61).

Updating rule:[{f(p+1)BB,n

}, α]= argmin{fBB,n},α

εsec(FRF,

{fBB,n

}, Rsec

), s.t. fBB,n =

(f(p)BB,n+α∇

(p)fBB,n

)/

√∑NCn=1

∥∥∥FRF

(f(p)BB,n+α∇

(p)fBB,n

)∥∥∥2 ∀n.p = p+ 1.

end whileOutput:

{f(PBB+1)BB,n

}3) Initial Hybrid Precoder: Generally, the average SNR

of Bob SNRB is expressed as SNRB = γNC

∑NC

n=1|hn,1FRFfBB,n|2

‖FRFfBB,n‖2. For any RF precoder FRF, the baseband

precoder fBB,n which maximizes SNRB is obtained as

fBB,n =

(FHRFFRF

)−1FHRFhHn,1∥∥∥(FHRFFRF

)− 12 FHRFhHn,1

∥∥∥ , (62)

and the corresponding SNRB = γNC

∑NC

n=1 hn,1FRF

(FHRFFRF)−1FHRFhHn,1. As an initial solution, we design theRF precoder FRF to maximize the average received SNR ofBob in the null space of the K−1 expected strongest directionsto Eves.

For the fully-connected structure F1, we express FRF asFRF = URFFRF, (63)

where URF = N[Emax[

∑Ll=1 ρlal,2a

Hl,2], . . . ,Emax[

∑Ll=1 ρl

al,KaHl,K ]]∈ CNT×(NT−(K−1)) is a semi-unitary matrix in the

null space of the K− 1 expected strongest directions to Eves.Similar to (30), FRF which maximizes SNRB is obtained asFRF = E1:NRF

[∑NC

n=1 UHRFhHn,1 hn,1URF

]. For the fully-

connected structure F2, we obtain FRF as

FRF =1√NT

exp(j∠URFFRF

), (64)

which satisfies the modulus constraint and is a good approxi-mation to (63).

For the subarray structure S1, we express fRF,r asfRF,r = URF,r fRF,r, (65)

where URF,r = N[E[∑Ll=1 ρlal,2,ra

Hl,2,r], . . . ,Emax[

∑Ll=1 ρl

al,K,raHl,K,r]

]∈ C

NTNRF

×(NTNRF

−(K−1))

is a semi-unitary ma-trix in the null space of the K − 1 expected strongestdirections to Eves seen by the rth RF chain, andal,k,r = al,k

((r − 1) NT

NRF+ 1 : r NT

NRF

). Similar to

(35), fRF,r which maximizes SNRB is obtained as fRF,r

= Emax

[∑NC

n=1 UHRF,rh

Hn,1,rhn,1,rURF,r

]. For the subarray

structure S2, we obtain fRF,r as

fRF,r =1√

NT/NRF

exp(j∠URF,r fRF,r

), (66)

which satisfies the modulus constraint and is a good approx-imation to (65). The initial RF precoders in (63) and (64)require NT ≥ K, while the ones in (65) and (66) requireNT ≥ KNRF. These two conditions are likely to be satisfiedfor large-scale mmWave systems.

To solve the secrecy throughput maximization problem in(57), we convert it into a sequence of secrecy outage probabil-


TABLE I. Necessary conditions to apply P-MRT

Channel knwoledge Fully Digital Precoding Hybrid Analog-Digital Precoding(NRF = NT) (NRF < NT)

Structures F1, F2, and S1 Subarray structure S2

Full knowledge NT ≥ K (FD-P-MRT) NT > (K − 1)L (TD-P-MRT)NRF ≥ K (FD-P-MRT)or NRF ≥ K (FD-P-MRT)

Partial knowledge NT > (K − 1)L (TD-P-MRT) NT > (K − 1)L (TD-P-MRT) infeasible

Algorithm 4 Hybrid precoding design for secrecy throughput maximization

Initialization: Obtain F(0)RF as in (63), (64), (65), or (66) according to the

used structure, then{f(0)BB,n

}as in (62), p = 0.

R(0)sec = argmax

Rsec

Rsec

(1− εsec

(F

(0)RF,

{f(0)BB,n

}, Rsec

)),

s.t. 0 ≤ Rsec ≤ RMRT

while p ≤ P (or any other appropriate stopping criterion)[F

(p+1)RF ,

{fBB,n

}]= argmin

FRF,{fBB,n}εsec

(FRF,

{fBB,n

}, R

(p)sec

),

using Algorithm 2 with F(p)RF and

{f(p)BB,n

}as initial solutions.{

f(p+1)BB,n

}= argmin{fBB,n}

εsec(F

(p+1)RF ,

{fBB,n

}, R

(p)sec

),

using Algorithm 3 with{fBB,n

}as an initial solution.

R(p+1)sec = argmax

Rsec

Rsec

(1− εsec

(F

(p+1)RF ,

{f(p+1)BB,n

}, Rsec

)),

s.t. R(p)sec ≤ Rsec ≤ RMRT.

p = p+ 1.end while

ity minimization problems, each one is solved (as illustratedabove) for a fixed target secrecy rate Rsec. The secrecy rateRsec which maximizes the secrecy throughput is obtained byone dimensional search. The detailed algorithm is shown inAlgorithm 4, where P is the number of iterations. Note that forfully digital precoding, the throughput maximization problemcan also be solved using Algorithm 4 after excluding the RFprecoder design step. The computational complexity of Algo-rithm 4 is O

(N3

T+NTNRFNC+max(NC,KL

)(NRFNTPBB

+N2TPRF

)P)

.V. SIMULATION RESULTS

We evaluate the performances of the proposed hybrid pre-coders and compare them with the performances of the fullydigital precoder and MRT hybrid precoder by means of Monte-Carlo simulations. Regarding the simulation setup, we assumethat Alice has 64 antennas (NT = 64). The number ofRF chains NRF will be an adjustable parameter through thenumerical results. All channels follow the mmWave channelmodel described in subsection II-B with 12 propagation paths(L = 12), where the channel gains {αl,k} represents a multi-path Nakagami-m fading channel [29] with shape parameter ofm = 2 as used in [24], [26], [28] and exponential power delayprofile defined as

{ρl = ql−1(1−q)

(1−qL)

}where q = 0.36, and the

angles of departure (AoDs) {ϕl,k} are uniformly-distributedwithin [0 2π). The number of subcarriers NC is 256.

Note that if the system parameters NT , K, L, and NRF(do not) satisfy the sufficient conditions of Proposition 1,the proposed hybrid precoders (would not) achieve the sameperformance as the fully digital precoder with full channelknowledge. In the following numerical results, the systemparameters do not satisfy the the sufficient conditions ofProposition 1, thus the hybrid precoders can show some per-formance gap from the fully digital precoder. The performance

of the fully digital precoder of [6] and the performance of theMRT hybrid precoder of [54] will be presented as performanceupper and lower bounds respectively.

If P-MRT is feasible, the optimal precoding strategy athigh SNRs is to maximize the rate of Bob in the null spaceof channels to Eves, and hence we have limγ→∞

Rsec(γ)log2(γ)

=

limγ→∞R1(γ)log2(γ)

= 1 since Bob has a single antenna [56]. Thisresult means that the secrecy rate should have a unit slopeat high SNRs if P-MRT is feasible. Similarly, the secrecythroughput should have a unit slope if P-MRT is feasible.Table 1 summarizes the necessary conditions (mentioned insubsections III-B2 and IV-B) to apply P-MRT for fully digitalprecoding and hybid precoding with full or partial channelknowledge. If P-MRT is infeasible, the secrecy rate andsecrecy throughput will not have a unit slope at high SNRs.

A. Achievable Secrecy Rate with Full Channel Knowledge

Fig. 3 shows the achievable secrecy rate as a function ofthe transmit SNR with NRF = 4 and different numbers ofEves (K − 1). With two Eves (Fig. 3a), we observe thatAlgorithm 1 achieves a secrecy rate very close to that achievedby the fully digital precoding for the whole SNR range. Thefour RF precoder structures achieve slightly different secrecyrates due to the different hardware complexities. Algorithm1approaches P-MRT at high SNRs since P-MRT is optimal athigh SNRs. Note that FD-P-MRT is feasible for all structuressince NRF > K. With four Eves (Fig. 3b), the secrecy rate gapbetween the fully digital precoding and Algorithm 1 increasesas SNR increases. Algorithm 1 (F1, F2, and S1) approaches thecorresponding P-MRT with unit secrecy slope at high SNRssince P-MRT is optimal at high SNRs, while Algorithm 1 (S2)does not achieve the unit secrecy slope. The reason is that FD-P-MRT is infeasible for all structures since NRF < K, whileTD-P-MRT is feasible only for the structures F1, F2, and S1since NT > (K − 1)L.

For both Fig. 3a and Fig. 3b, we observe that App-FDachieve approximately the same secrecy rate as Algorithm 1 atlow SNRs. However, App-FD does not achieve the unit secrecyslope at moderate and high SNRs, and the secrecy rate gapbetween Algorithm 1 and App-FD increases as SNR increases.The proposed adaptive hybrid precoder max(App-FD, P-MRT)in Fig. 4b combines the low/moderate SNR advantage ofApp-FD and the high SNR advantage of P-MRT, and yieldsbetter secrecy rate performance than App-FD and P-MRT withlow computational complexity (both App-FD and P-MRT areobtained in closed forms as derived in Section III-B). Notethat max(App-FD, P-MRT) reduces to App-FD if P-MRT isinfeasible. On the contrary, MRT of [54] achieves the worstsecrecy rate at moderate and high SNRs due to ignoring Eves.


-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

20

Secrecy

Rate

Rsec(bits/s/Hz)

Fully Digital [6]Algorithm 1 Hybrid (F1)Algorithm 1 Hybrid (F2)Algorithm 1 Hybrid (S1)Algorithm 1 Hybrid (S2)P-MRT Hybrid (F1)P-MRT Hybrid (F2)P-MRT Hybrid (S1)P-MRT Hybrid (S2)App-FD Hybrid (F1)App-FD Hybrid (F2)App-FD Hybrid (S1)App-FD Hybrid (S2)MRT Hybrid (F1) [54]MRT Hybrid (F2) [54]MRT Hybrid (S1) [54]MRT Hybrid (S2) [54]

-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

20

Secrecy

Rate

Rsec(bits/s/Hz)

Fully Digital [6]Algorithm 1 Hybrid (F1)Algorithm 1 Hybrid (F2)Algorithm 1 Hybrid (S1)Algorithm 1 Hybrid (S2)P-MRT Hybrid (F1)P-MRT Hybrid (F2)P-MRT Hybrid (S1)P-MRT Hybrid (S2)App-FD Hybrid (F1)App-FD Hybrid (F2)App-FD Hybrid (S1)App-FD Hybrid (S2)max(App-FD, P-MRT) Hybrid (F1)max(App-FD, P-MRT) Hybrid (F2)max(App-FD, P-MRT) Hybrid (S1)max(App-FD, P-MRT) Hybrid (S2)MRT Hybrid (F1) [54]MRT Hybrid (F2) [54]MRT Hybrid (S1) [54]MRT Hybrid (S2) [54]

(a) Achievable secrecy rate with 2 Eves (K = 3) (b) Achievable secrecy rate with 4 Eves (K = 5)

Fig. 3. Achievable secrecy rate as a function of transmit SNR γ with NRF = 4 and different numbers of Eves (K − 1).

-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

20

Secrecy

RateR

sec(bits/s/Hz)


-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

20

Secrecy

RateR

sec(bits/s/Hz)


(a) Achievable secrecy rate with NRF = 4 (b) Achievable secrecy rate with NRF = 8

Fig. 4. Achievable secrecy rate as a function of transmit SNR γ with 6 Eves (K=7) and different numbers of RF chains NRF.

With different number of RF chains NRF, Fig. 4 showsthe achievable secrecy rate as a function of the transmit SNRwith six Eves (K = 7). As expected, Algorithm 1 achievesthe highest secrecy rate among the hybrid precoding designs.With NRF = 4 (Fig. 4a), the secrecy rate gap between the fullydigital precoding and all hybrid precoding schemes increasesas SNR increases. Algorithm 1 does not achieve the unitsecrecy slope at high SNRs since P-MRT is infeasible for allstructures. On the other hand, with NRF = 8 (Fig. 4b), FD-P-MRT is feasible for all structures. As a result, Algorithm1 achieves a secrecy rate very close to that achieved by thefully digital precoding. From Fig. 4a and Fig. 4b, we canobserve the significant effect of the number of RF chains onthe achievable secrecy rate.B. Achievable Secrecy Throughput with Partial ChannelKnowledge

Fig. 5 shows the achievable secrecy throughput as a functionof the transmit SNR with NRF = 4 and different numbers ofEves (K − 1). With four Eves (Fig. 5a), we observe that thehybrid precoding with Algorithm 4 (F1, F2, and S1) achievesa secrecy throughput very close to that of the fully digitalprecoding with Algorithm 4, and they have a unit secrecy slopeat high SNRs since TD-PMRT is feasible for the fully digitalprecoding and the hybrid precoding structures F1, F2, and S1due to NT > (K − 1)L. On the contrary, Algorithm 4 (S2)

does not achieve the unit secrecy slope since TD-P-MRT is notfeasible for the subarray structure S2. With six Eves (Fig. 5b),the secrecy throughput gap between the fully digital precodingwith Algorithm 4 and the hybrid precoding with Algorithm 4increases as SNR increases. However, the hybrid precodingwith Algorithm 4 achieves good performance at low andmoderate SNRs. All precoding schemes (including the fullydigital precoding) do not achieve the unit secrecy slope sinceTD-P-MRT is infeasible. Similarly, MRT of [54] achievesthe worst secrecy throughput due to ignoring Eves. Note thatincreasing the number of RF chains (even if NRF = NT ) willnot achieve the unit secrecy slope at high SNRs since FD-P-MRT is infeasible in case of partial channel knowledge, andTD-P-MRT is infeasible due to NT < (K − 1)L.C. Tightness of Secrecy Rate and Throughput Lower Bounds

Fig. 6 shows an example for the tightness of the secrecy ratelower bound Rsec and the secrecy throughput lower bound ηsecwith NRF = 4 and 4 Eves (K = 5). The sold lines are forthe exact values while the dotted lines are for the lower boundvalues. We observe from Fig. 6a that the secrecy rate lowerbound is tight for the structures F1, F2, and S1, while thesecrecy rate lower bound predicts the performance behaviorof the structure S2 efficiently. From Fig. 6b, we observe thatthe proposed secrecy throughput lower bound is very tight forall structures. The difference between the exact values and the


-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

Secrecy

Throughputηsec(bits/s/Hz)

Algorithm 4 Fully DigitalAlgorithm 4 Hybrid (F1)Algorithm 4 Hybrid (F2)Algorithm 4 Hybrid (S1)Algorithm 4 Hybrid (S2)TD-P-MRT Hybrid (F1)TD-P-MRT Hybrid (F2)TD-P-MRT Hybrid (S1)TD-P-MRT Hybrid (S2)MRT Hybrid (F1) [54]MRT Hybrid (F2) [54]MRT Hybrid (S1) [54]MRT Hybrid (S2) [54]

-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

0

2

4

6

8

10

12

14

16

18

Secrecy

Throughputηsec(bits/s/Hz)

Algorithm 4 Fully DigitalAlgorithm 4 Hybrid (F1)Algorithm 4 Hybrid (F2)Algorithm 4 Hybrid (S1)Algorithm 4 Hybrid (S2)TD-P-MRT Hybrid (F1)TD-P-MRT Hybrid (F2)TD-P-MRT Hybrid (S1)TD-P-MRT Hybrid (S2)MRT Hybrid (F1) [54]MRT Hybrid (F2) [54]MRT Hybrid (S1) [54]MRT Hybrid (S2) [54]

(a) Achievable secrecy throughput with 4 Eves (K = 5) (b) Achievable secrecy throughput with 6 Eves (K = 7)

Fig. 5. Achievable secrecy throughput as a function of transmit SNR γ with NRF = 4 and different numbers of Eves (K − 1).

-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

2

4

6

8

10

12

14

16

18

20

Secrecy

Rate(bits/s/Hz)

Algorithm 1 Fully DigitalAlgorithm 1 Hybrid (F1)Algorithm 1 Hybrid (F2)Algorithm 1 Hybrid (S1)Algorithm 1 Hybrid (S2)

35 4014.6

15

15.4

15.8

16.2

16.616.8

Exact (Rsec)

Lower Bound (Rsec)

(a) Tightness of secrecy rate lower bound Rsec

-10 -5 0 5 10 15 20 25 30 35 40SNR γ (dB)

2

4

6

8

10

12

14

16

18

Secrecy

Through

put(bits/s/Hz)

Algorithm 4 Fully DigitalAlgorithm 4 Hybrid (F1)Algorithm 4 Hybrid (F2)Algorithm 4 Hybrid (S1)Algorithm 4 Hybrid (S2)

35 4012.712.812.9

1313.113.2

39.9 4016.6

16.8

17

17.2

17.4

17.6

Exact (ηsec)Lower Bound (ηsec)

(b) Tightness of secrecy throughput lower bound ηsec

Fig. 6. Tightness of secrecy rate and throughput lower bounds with NRF = 4, 4 Eves (K = 5), and transmit SNR γ = 15 dB.

lower bound values over the whole SNR range and all schemesis less than 0.0798 bits/s/Hz.

The reason of the tightness can be explained as follows.When the hybrid precoder is well designed (as the proposedschemes do) with NT � K (large-scale mmWave systems)and small L (limited scattering mmWave channels), the av-erage receive SNR of Eves will be very small compared tothe average receive SNR of Bob [3] (due to the capability ofgenerating very sharp beams avoiding, as much as possible, thedirections to Eves). Therefore, considering Eves as one Evewith K − 1 antennas (the approximation of Section III) willnot decrease the secrecy rate significantly. Similarly, applyingthe inequalities of Section IV will not increase the rates ofEves significantly.D. Convergence of Algorithms 1 and 4 and Effect of FiniteResolution Phase Shifters

Fig. 7 shows an example for the convergence of Algorithm 1and Algorithm 4 with NRF = 4, 4 Eves (K = 5), and transmitSNR γ = 15 dB. The secrecy rate and throughput achievedby Algorithms 1 and 4 after each iteration are plotted in Fig.7a and Fig. 7b respectively. We observe that both Algorithms1 and 4 converge in a small number of iterations, where 10iterations are sufficient for all RF precoder structures.

With finite resolution phase shifters, Fig. 8 shows the se-crecy rate and throughput achieved by Algorithms 1 and 4 ver-sus different numbers of quantization bits for the phase shifters

with NRF = 4, 4 Eves (K = 5), and transmit SNR γ = 15dB. After designing the hybrid precoder using Algorithm 1or Algorithm 4, the phases of the RF precoder are quantizedinto Q bits such that ∠FRF (i, j) ∈

{0, 2π

2Q, . . . , 2π2

Q−1

2Q

}∀FRF (i, j) 6= 0. We observe that 6 quantization bits aresufficient for the RF precoder structures F2, S1, and S2 withsecrecy rate/throughput loss less than 0.2 bits/s/Hz. On theother hand, the RF precoder structure F1 requires at least 10quantization bits to outperform the structure F2. The reason isthat the structure F1 uses approximately twice the number ofphase shifters of the structure F2. As a result, the quantizationerror of the structure F1 is larger than that of the remainingstructures.

VI. CONCLUSION

In this paper, we have designed the hybrid analog-digitalprecoder for physical layer security. With full channel knowl-edge at the transmitter, we provided sufficient conditionsfor the hybrid precoder to realize the performance of thefully digital precoding. If the sufficient conditions are notsatisfied, we design the hybrid precoder to maximize thesecrecy rate. By maximizing the average projection betweenthe fully digital precoder and the hybrid precoder, we proposeda low-complexity closed-form hybrid precoder design. Theconventional P-MRT scheme is extended to realize the hybrid


0 2 4 6 8 10 12 14 16 18 20Iteration Number

6.5

7

7.5

8

8.5

9

9.5

10

10.5

Secrecy

RateR

sec(bits/s/Hz)

Algorithm 1 Hybrid (F1)Algorithm 1 Hybrid (F2)Algorithm 1 Hybrid (S1)Algorithm 1 Hybrid (S2)

(a) Convergence of Algorithm 1

0 2 4 6 8 10 12 14 16 18 20Iteration Number

4

5

6

7

8

9

10

Secrecy

Through

putηsec(bits/s/Hz)


(b) Convergence of Algorithm 4

Fig. 7. Convergence of Algorithm 1 and Algorithm 4 with NRF = 4, 4 Eves (K = 5), and transmit SNR γ = 15 dB.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Number of Quantization Bits Q

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

Secrecy

RateR

sec(bits/s/Hz)


(a) Secrecy rate achieved by Algorithm 1 versus number ofquantization bits

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Number of Quantization Bits Q

0

1

2

3

4

5

6

7

8

9

10

Secrecy

Through

putηsec(bits/s/Hz)


(b) Secrecy throughput achieved by Algorithm 4 versus numberof quantization bits

Fig. 8. Effect of finite resolution phase shifters on the secrecy rate and throughput achieved by Algorithm 1 and Algorithm 4 with NRF = 4, 4 Eves (K = 5),and transmit SNR γ = 15 dB.

precoder. Two P-MRT schemes were presented. TD-P-MRTnulls the channels to Eves at time domain, and FD-P-MRTnulls the channels to Eves at frequency domain. Moreover, weproposed an iterative hybrid precoder design, based on gradi-ent ascent, which converges in a small number of iterationsand achieves secrecy rate close to that achieved by the fullydigital precoding.

With partial channel knowledge, we derived a secrecyoutage probability upper bound. The secrecy throughput max-imization problem is converted into a sequence of secrecyoutage probability minimization problems. Then, the hybridprecoder is designed to minimize the secrecy outage probabil-ity by an iterative hybrid precoder design, based on gradientdescent, which converges in a small number of iterations andachieves secrecy throughput close to that achieved by the fullydigital precoding. With finite resolution phase shifters, weshowed that 6 quantization bits are sufficient for the structuresF2, S1, and S2. On the contrary, 10 quantization bits areneeded for the structure F1 to outperform the structure F2.

APPENDIX A (PROOF OF PROPOSITION 1)As shown in (13), the optimal fully digital pre-

coder fopt,n is the principal generalized eigenvector cor-responding to the maximum eigenvalue of the pencil(INT + γHH

n Hn, INT + γhHn,1hn,1). Among the NT gener-

alized eigenvalues, (NT −K) of them are equal to 1 andobtained using any vector that is orthogonal to the space

spanned by[hHn,1,H

Hn

]. The other K generalized eigenvec-

tors corresponding to the other K eigenvalues (including themaximum eigenvalue) lie completely in the space spanned by[hHn,1,H

Hn

]. Therefore, we can write fopt,n as

fopt,n = βnΠnhHn,1∥∥ΠnhHn,1

∥∥ +√

1− β2n

Π⊥nhHn,1∥∥Π⊥nhHn,1∥∥ , (67)

where βn =

∣∣∣hn,1ΠnEmax

[(INT

+γHHn Hn)

−1(INT

+γhHn,1hn,1)]∣∣∣

‖ΠnhHn,1‖,

Πn = HHn

(HnHH

n

)−1Hn ∈ CNT×NT denotes the or-

thogonal projection onto the space spanned by Hn, andΠ⊥n = INT −Πn denotes the projection onto its orthogonalcomplement. Equation (67) can be rewritten as

fopt,n = HTDWnpn, (68)where HTD ∈ CNT×KL is the time domain channel matrix tothe K receivers given by

HTD =[hHτ1,1,1, . . . , h

HτL,1,1, . . . , h

Hτ1,K ,K , . . . , h

HτL,K ,K

],

(69)Wn = blkdiag

(wHn,1,w

Hn,2, . . . ,w

Hn,K

)∈ CKL×K , and

pn =[µn, νnhn,1H

Hn

(HnHH

n

)−1]H CK×1, where µn =√

1−β2n

‖Π⊥n hHn,1‖and νn = βn

‖ΠnhHn,1‖−√

1−β2n

‖Π⊥n hHn,1‖. The perfor-

mance of fully digital precoding can be realized by settingFRF (:, 1 : KL) = HTD and fBB,n (1 : KL) = Wnpn ifNRF ≥ KL. Thus, to realize the performance of fully digitalprecoding, it is sufficient for the hybrid precoding utilizing the


fully-connected structure F1 that NRF ≥ KL. This completesthe proof of the first statement.

For the fully-connected structure F2, we have to satisfythe modulus constraint. In [57], it was shown that any vectorx ∈ CN×1 can be expressed as x = Xx, where X ∈ CN×2is with unit modulus entries and x ∈ R2×1. Following thisdecomposition, HTD can be expressed as HTD = QRFRBB,where QRF ∈ CNT×2KL has unit modulus entries obtained as

QRF (l, 2m− 1) = exp(j(∠HTD (l,m)− cos−1 (q+)

)),

(70)QRF (l, 2m) = exp

(j(∠HTD (l,m) + cos−1 (q−)

)), (71)

∀l ∈ {1, 2, . . . NT} and ∀m ∈ {1, 2, . . .KL}where q+ =

|HTD(l,m)|2+bmax,mbmin,m

|HTD(l,m)|2(bmax,m+bmin,m), q− =

|HTD(l,m)|2−bmax,mbmin,m

|HTD(l,m)|2(bmax,m−bmin,m), bmax,m = max

l|HTD (l,m)|,

bmin,m = minl|HTD (l,m)|, and RBB ∈ R2KL×KL is

obtained as RBB (2m− 1,m) = (bmax,m + bmin,m)/2,RBB (2m,m) = (bmax,m − bmin,m)/2 ∀m ∈ {1, 2, . . .KL}.Therefore, we can set FRF (:, 1 : 2KL) = 1√

NTQRF and

fBB,n (1 : 2KL) =√NTRBBWnpn if NRF ≥ 2KL. As

a result, it is sufficient that NRF ≥ 2KL for the hybridprecoding utilizing the fully-connected structure F2 to realizethe performance of fully digital precoding. By assuming thatall the channels follow the mmWave channel model in (2),equation (68) can be written as fopt,n = ATDTWnpn,where AT = [AT,1,AT,2, . . . ,AT,K ] ∈ CNT×KL andDT = blkdiag

(DH

1 ,DH2 , . . . ,D

HK

)∈ CKL×KL. Since AT

is with unit modulus entries, the sufficient condition for thehybrid precoding utilizing the fully-connected structure F2reduces to NRF ≥ KL by setting FRF (:, 1 : KL) = 1√

NTAT

and fBB,n (1 : KL) =√NTDTWnpn. This completes

the proof of the second statement. The obtained sufficientconditions vanish if NT ≤ KL since NRF < NT.

APPENDIX B (PROOF OF PROPOSITION 2)

In Proposition 1, it was shown that if NT ≤ KL, there isno sufficient condition depending only on the number of RFchains to realize the performance of fully digital precodingsince NRF < NT. Therefore, we consider the case that KL ≤NRF < NT. To realize the performance of the fully digitalprecoding, fBB,n has to be expressed as fBB,n = BBBWnpn,where BBB ∈ CNRF×KL is a mapping matrix. We have todesign FRF and BBB such that HTD = FRFBBB. Let NRF =NT − 1 ≥ KL which means that each RF chain is connectedto one antenna except only one RF chain that is connected totwo antennas. Without loss of generality, let the first RF chainbe the only RF chain that is connected to two antennas, weshould have

HTD (1, l) = fRF,1 (1) BBB (1, l) , (72)HTD (2, l) = fRF,1 (2) BBB (1, l) , (73)

where l ∈ {1, 2, . . . ,KL}. To satisfy (72) and (73), itis necessary that HTD (2, :) = cHTD (1, :) where c is aconstant, which is not guaranteed and depends on the channelrealizations. As a result, the hybrid precoding utilizing thesubarray structure S1 cannot generally realize the fully digitalprecoding for any NRF ≤ NT − 1. Since FS2

RF ⊂ FS1RF, we

arrive at the same conclusion for the subarray structure S2.This completes the proof of Proposition 2.

APPENDIX C (APPROXIMATING THE FULLY DIGITALPRECODING)

For the average Euclidean distance criterion, the hybridprecoder is designed to approximate the fully digital precodingas [21], [36], [37]

arg minFRF,{fBB,n}

NC∑n=1

∥∥∥∥fopt,n − FRFfBB,n

‖FRFfBB,n‖

∥∥∥∥2 . (74)

It is straightforward to show that for any FRF, fBB,n is theleast squares solution which can be expressed (after appropri-ate normalization) as

fBB,n =

(FHRFFRF

)−1FHRFfopt,n∥∥∥(FHRFFRF

)− 12 FHRFfopt,n

∥∥∥ , (75)

which is exactly the same as (17), and the corre-

sponding∑NC

n=1

∥∥∥fopt,n − FRFfBB,n

‖FRFfBB,n‖

∥∥∥2 = 2NC − 2∑NC

n=1√fHopt,nFRF

(FHRFFRF

)−1FHRFfopt,n. Thus, (74) can be writ-

ten as a function of FRF only as

arg maxFRF

NC∑n=1

√λmax

[(FHRFFRF


],

(76)which is similar to the first equality of (18) except for thesquare root. For (18), we have a closed- form solution ofFRF = E1:NRF

[∑NC

n=1 fHopt,nfopt,n

], while there is no a

closed-form solution for (76).

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Yahia R. Ramadan (S’14) received the B.S.(with highest honors) and M.S. degrees inElectronics and Communications Engineeringfrom Cairo University, Cairo, Egypt, in2011 and 2015, respectively. He is currentlypursuing his Ph.D. degree in ElectricalEngineering at the University of Texas atDallas, Richarson, TX, USA. His research

interests are in the areas of millimeter-wave communications,device-to-device communications, massive MIMO, physical layersecurity, cognitive radio networks, and spectrum access and sharing.

Hlaing Minn (S’99–M’01–SM’07–F’16) re-ceived B.E. degree in Electrical Engineer-ing (Electronics) from the Yangon Insti-tute of Technology, Yangon, Myanmar, in1995; M.Eng. degree in Telecommunica-tions from the Asian Institute of Technology,Pathumthani, Thailand, in 1997; and Ph.D. de-gree in Electrical Engineering from the Univer-sity of Victoria, Victoria, BC, Canada, in 2001.

During January–August 2002, he was a Postdoctoral Fellow with theUniversity of Victoria. He has been with the University of Texasat Dallas since 2002 and is currently a Full Professor. His researchinterests include wireless communications, signal processing, signaldesign, dynamic spectrum access and sharing, and next-generationwireless technologies.Dr. Minn serves as an Editor-at-Large since 2016 and served as anEditor from 2005 to 2016 for the IEEE Transactions on Communi-cations. He served as a Technical Program Co-Chair for the WirelessCommunications Symposium of the IEEE GLOBECOM 2014 andthe Wireless Access Track of the IEEE VTC, Fall 2009, as well as aTechnical Program Committee Member for several IEEE conferences.

Ahmed S. Ibrahim (M’09) is currently anAssistant Professor at the Electrical and Com-puter Engineering Department at Florida Inter-national University, Miami, FL, USA. He re-ceived the B.S. (with highest honors) and M.S.degrees in electronics and electrical commu-nications engineering from Cairo University,Cairo, Egypt, in 2002 and 2004, respectively.

He received the Ph.D. degree in electrical engineering from theUniversity of Maryland, College Park, MD, USA, in 2009. In thepast, Dr. Ibrahim was an Assistant Professor at Cairo University,Wireless Research Scientist at Intel Corporation, and Senior Engineerat Interdigital Communications Inc.Dr. Ibrahim’s research interests span various topics of next generationmobile networks and Internet of Things such as heterogeneousnetworks, vehicular networks, cooperative communications, Wi-Fioffloading, millimeter wave communications, and visible light com-munications. Dr. Ibrahim currently serves as an associate editor forthe Elsevier Digital Communications and Networks (DCN) Journal.He served as a Technical Program Committee Member for severalIEEE conferences.

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