A Joint Beamforming Design and Integrated CPM-LFM Signal ...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. X, AUG. 2021 1

A Joint Beamforming Design and IntegratedCPM-LFM Signal for Dual-functional

Radar-communication SystemsYu Cao, Student Member, IEEE, Qi-Yue Yu, Senior Member, IEEE

Abstract—The dual-functional radar-communication (DFRC)system is an attractive technique, since it can support bothwireless communications and radar by a unified hardwareplatform with real-time cooperation. Considering the appealingfeature of multiple beams, this paper proposes a precodingscheme that simultaneously support multiuser transmission andtarget detection, with an integrated continuous phase modulation(CPM) and linear frequency modulation (LFM) signal, based onthe designed dual mode framework. Similarly to the conceptionof communication rate, this paper defines radar rate to unify theDFRC system. Then, the maximum sum-rate that includes boththe communication and radar rates is set to be the objectivefunction. Regarding as the optimal issue is non-convex, theoptimal problem is divided into two sub-issues, one is the userselection issue, and the other is the joint beamforming design andpower allocation issue. A successive maximum iteration (SMI)algorithm is presented for the former issue, which can balancethe performances between the sum-rate and complexity; andmaximum minimization Lagrange multiplier (MMLM) iterationalgorithm is utilized to solve the latter optimal issue. Moreover,we deduce the spectrum characteristic, bit error rate (BER) andambiguity function (AF) for the proposed system. Simulationresults show that our proposed system can provide appreciatedsum-rate than the classical schemes, validating the efficiency ofthe proposed system.

Index Terms—Dual-functional radar-communication (DFRC),precoding, continuous phase modulation (CPM), linear frequencymodulation (LFM), maximum minimization Lagrange multiplier(MMLM), user selection.

I. INTRODUCTION

CURRENTLY, the joint communication and radar (JCR)system has been widely researched in both academic and

industry [1], since it can simultaneously satisfy the data trans-mission and target detection, providing real-time informationsharing. Therefore, the JCR system has been applied into in-door positioning [2], UAV networks [3], vehicle networks [4],and etc. Especially, the JCR system is an appealing choice forseamless connectivity to serve multiple communication usersand achieve the target information. Compared to the classicalsingle radar or communication equipment, it has advantages interms of cost, size, power consumption, spectrum usage, andetc [5] [6].

The work presented in this paper was supported by the National NaturalScience Foundation of China under Grand No. 62071148, and partly supportedby the Natural Science Foundation of Heilongjiang Province of China underGrand No. YQ2019F009.(Corresponding author: Qiyue Yu).

Y. Cao and Q.-Y. Yu are with the Communication Research Center, HarbinInstitute of Technology, China. (email: {yucao, yuqiyue}@hit.edu.cn).

The concept of JCR was first proposed by Space Shuttleprogram of National Aeronautics and Space Administration(NASA) in the late 1970’s [7]. The initial goal was to integratethe electronic warfare (EW) of communication and radarfunctions into the antenna array (AA) for resource sharing,mainly used in military scenarios. Slowly, JCR is applied intocivil area, e.g., advanced driver assistance systems (ADAS)[8]. In airborne areas, [9] merges the communication systemand the synthetic aperture radar (SAR) by the space timecoding (STC) scheme.

The prospective research directions of JCR mainly includeradar-communication coexistence system (RCC) and dual-functional radar-communication (DFRC) [10]. The RCC sys-tem emphasizes on the coexistence of communication andradar system that mainly develops efficient interference man-agement techniques, including null space signal projection,transmit waveform design, power-efficient pattern design anddynamic spectrum allocation [11]-[14]. [11] proposes a nullspace projection method to eliminate the interferences ofthe RCC system, with the knowledge of the channel stateinformation (CSI). [12] designs transmit waveforms by space-time code based on the controllable degrees of freedom toreduce the self-interferences of the RCC receiver. For thesignal side-lobe interference control aspect, [13] presents aRiemann gradient conjugate power budget waveform designmethod to reduce the influence of side-lobe interference ofa multiple input multiple output (MIMO) RCC system. In[14], the authors aim to minimize the downlink multiuserinterference by using branch-and-bound dynamic spectrumallocation method.

On the other hand, the DFRC focuses on the cooper-ation of communication and radar systems, and generallyadopts integrated signal to improve the performances of bothcommunication and radar [15]. There are many prospectiveresearch directions of DFRC, especially, beamforming designand integrated signal construction are two of the appealingones [16].

There are many works on beamforming design of the DFRCsystem [17]- [21]. In [17], the authors present an optimalbeamforming method to minimize the downlink multiuserinterference, and further propose a weighted beam-pattern fora flexible trade-off between the performance of radar andcommunication. In [18], the authors design a beamformer thatmatches the radar’s beam-pattern; moreover, it satisfies thecommunication performance relied on the null-space criterionwith the constraint of signal-to-interference-plus-noise ratio

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(SINR). [19] presents a predictive beamforming scheme toenhance angle estimation accuracy by the maximum pos-teriori probability criterion, which can acquire the locationand track the information of a vehicular system. In [20], theauthors present a hybrid beamformer based on the minimummean square error (MMSE) criterion, which can enhance theachievable sum-rate significantly. [21] presents the transmitbeamforming scheme based on the maximum SINR criterion,which optimizes both the radar transmit beam pattern andcommunication rate.

Besides the beamforming topic, signal waveforms designingis another hot topic. To combine the communication andradar waveforms, there are generally three major categories,they are orthogonal frequency division multiplexing (OFDM),frequency modulation (FM) and spread spectrum (SS) [22].In [23], the author proposes an OFDM queue schedulingmodel, taking both the network stability and radar detectionperformance into consideration. [24] proposes a novel code-division OFDM system to provide high spectrum efficiencyand robust radar sensing performances. The linear frequencymodulation (LFM) is widely considered in DFRC systems,since it provides a larger detection range and a higher rangeresolution for radar detection [25]. Phase-coded frequencymodulation can mitigate the impact of Doppler shift, thusit is suitable for high speed target detection [26]. In [27],the authors combine the low density parity check (LDPC)codes with LFM signals for a DFRC system, which canprovide a better BER than the LFM signal. The combinationof continue phase modulation (CPM) and LFM can reducethe influence of side lobes [28], thus improve the power andspectrum efficiency. As for spread spectrum signal, the directsequence (DS) can provide well-behaved auto- and cross-correlation properties to separate the radar and communicationinformation perfectly [29].

Obviously, a joint design of beamforming and signal-formfor a DFRC system is a prospective research topic, since itcan take signal processing gains to avoid interferences andimprove the spectrum efficiency. [30] considers the Hadamard-Walsh orthogonal codes with minimum variance distortion-less response (MVDR) beamforming algorithm to achieve abetter jamming and interference mitigation capability. [31]presents a zero-forcing beamforming algorithm to transmitthe independent radar waveforms and communication symbolssimultaneously.

Considering the attractive feature of the joint design, thispaper presents a joint beamforming and integrated CPM-LFMsignal for a DFRC system. The major advantages of CPM aretwo aspects: one is the continuity of signal phase that improvesthe spectrum; and the other is that the Viterbi decoding (ordemodulation) provides better BER performance. Furthermore,LFM signal is widely used in radar detection, since it haslarge time-bandwidth product, which can effectively solvethe contradiction between resolution and measuring accuracy.Thus, the integrated CPM-LFM signal is an appealing attemptfor a JCR system. To further improve the performance ofCPM-LFM, antenna array is jointly considered in this paper,and the precoder is designed based on the maximum sum-rate that includes both communication and radar rates as our

objective function. The contributions of this paper are mainlysummarized as three aspects:

1) We present a joint beamforming design and integratedCPM-LFM signal for JCR systems. The designed dualmode framework that is consisted of static and dynamicbeams can simultaneously support multiuser transmis-sion and target detection;

2) Based on the definition of radar rate, we set maximumsum-rate as our objective function that can be provedto be a non-convex optimal issue. To find the opti-mal solution, the entire problem is divided into twosub-issues, which are user selection and beamformingweights design with power allocation. A successivemaximum iteration algorithm is proposed for the userselection, and the maximum minimization Lagrangemultiplier (MMLM) is proposed to solve the latter one;and

3) Theoretical analysis are presented for the proposedsystem, including spectrum, bit error rate (BER) andambiguity function (AF) analysis. Numerical resultsverify the validity of the proposed system.

The remainder of this paper is organized as follows. SectionII presents the system model and the dual mode frameworkof the proposed system. In Section III, the transmitter andreceiver of the proposed system is described in details. Anobjective function and its solutions are deduced in SectionIV. Section V discusses the spectrum characteristic, BER andAF. Simulation results are given in Section VI, following withsome conclusion remarks in Section VII.

II. PRELIMINARY

In this paper, denote B and C by binary field and complexfield. Let x, x and X be a scalar, a vector and a matrix, respec-tively. The transposition, conjugate and conjugate transpose ofx are donated by xT, x∗ and xH respectively. Define ∗ asconvolution operation, and ‖x‖2 represents vector Euclideannorm. CN (µ, σ2) indicates the complex Gaussian distributionwith mean µ and covariance σ2. Re(x) means the real partof the complex number x. E(x) means the expectation ofa random sequence x. In is n-dimensional identity matrix,Det(X) represents the determinant of X, tr(X) means the atrace of X, Diag(X) means a diagonal matrix that is composedof the diagonal element of X, and vec(X) indicates convertinga matrix X to a vector by vectorizing columns from the leftside to the right side of X.

A. System model

Assume the base station (BS) is equipped with Nt transmitantennas and Nr received antennas, which form transmit andreceived antenna arrays for both communication and radarfunctions. There are totally U user equipments (UEs), whereU � Nt, and each UE holds a single antenna. Since the BScannot serve all the UEs simultaneously, we assume that KUEs are selected from U , where K ≤ Nt. The locations ofall the U UEs are assumed to be available at the BS. Let therange, speed, elevation and azimuth angle of the target (TA)be dT , vT and (θT , ϕT ) respectively, which will be estimated


Subframe 2

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1Tc 2Tc TLc 1Tc 2Tc TLc 1Tc 2Tc TLc

Subframe 1 Subframe M Subframe 2

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Subframe 1 Subframe M

Time domain Time domain

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Echo i

nte

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1 1( , ) 2 2( , ) ( , )M M

1 1( , )

2 2( , )

( , )M M

UE 1 UE 2 UE KTA

... ...

...

...

...... ......

...

...

.........TNc

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Fig. 1: A dual mode framework of a DFRC system, including time-sharing scanning mode given in (a) and fixed-directionmode given in (b).

by the radar at the BS. The ranges of elevation θT and azimuthϕT are respectively (0, π/2) and (0, 2π).

The transmit antennas of the BS is set to be a uniformrectangular array (URA) [32], whose element owns a separatephase shifter, providing a three-dimensional (3D) beamform-ing from elevation and azimuth dimensions. Suppose thenumbers of the elements of the transmit URA placed alongthe X-axis and Y-axis are respectively Ntx and Nty , indicatingNt = Ntx ×Nty . Let the distances between adjacent antennaelements of X-axis and Y-axis be dx and dy respectively. De-note the label of each transmit antenna element by (ntx, nty),where 1 ≤ ntx ≤ Ntx and 1 ≤ nty ≤ Nty . The label of thentth element can be expressed as nt = (nty − 1)Ntx + ntx,where 1 ≤ nt ≤ Nt.

Similarly, the received antenna array of the BS is also setto be a URA, and the elements of the received URA arerespectively Nrx and Nry, i.e., Nr = Nrx×Nry . The label ofthe nrth received antenna is nr = (nry − 1)Nrx+nrx, where1 ≤ nrx ≤ Nrx, 1 ≤ nry ≤ Nry and 1 ≤ nr ≤ Nr.

B. Framework

The transmit URA of the BS should simultaneously pro-vide different beams for both UEs’ communication and TA’sdetection. The UEs’ angle information are available at theBS, and the beamforming algorithms are determined by theseangle information. On contrast, the TA’s angle information isunknown, which needs to be estimated. Thus, the transmitURA of the BS is consist of two types of beams, one isdynamic beams for TA’s searching, and the other is staticbeams for UEs’ communications, as shown in Fig. 1.

Through dynamic beam scanning, both the position andspeed information of the TA are expected to be estimated.Meanwhile, UEs are served by the static beams. Because ofthe different roles of dynamic and static beams, this paperproposes a dual mode framework that is consisted of dynamictime-sharing scanning mode and static fixed-direction mode.

Assume one frame is consisted of M blocks and each blockincludes N symbols. It is noted that M is equal to the productof Me and Ma, i.e., Me = π

2δθand Ma = 2π

δϕ, where δθ

and δϕ are beamwidth in elevation and azimuth dimensions.Therefore, the TA detection precision relays on the value of M .A large M indicates a higher precision, at the cost of longerscanning time. The dynamic beam alters its direction everyblock time, thus named as “dynamic”. During one frame time,there are totally M dynamic beams. Conversely, the staticbeams keep the directions during the entire frame time, as if“static”. Dynamic beam utilize time-sharing scanning mode todetect the position and speed of a moving TA. The scanningdirection (θm, ϕm) of the mth block, where 1 ≤ m ≤ M ,keeps as a constant during the mth block. If the reflectedecho signal of the mth dynamic beam that is larger than thethreshold, the TA is estimated to be at the direction (θm, ϕm).If the reflected echo signal of the mth dynamic beam is smallerthan the threshold, there is no TA exists.

III. DESIGN OF JOINT BEAMFORMING AND CPM-LFMINTEGRATED WAVEFORM

This section presents a joint beamforming and CPM-LFMintegrated system that is designed based on the dual modeframework, as shown in Fig. 2.

A. Transmit signals of the BS

To detect whether there is a TA or not, a pilot sequenceis transmitted by the mth dynamic beam, and defined byaT = [aT,1, aT,2, ..., aT,N logM] ∈ B1×N logM, where M ismodulation order of a M -ary amplitude shift keying (MASK),and the subscript “T” indicates the sequence (or signal) forthe TA detection. Suppose K selected UEs are simultane-ously served by the static beams, where 1 ≤ K ≤ U ,define the transmit binary data sequence of the kth UE byak = [ak,1, ak,2, ..., ak,N logM] ∈ B1×N logM, where 1 ≤


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Fig. 2: A diagram of the transmitter and receiver of our proposed DFRC system, where “DeMod” indicates demodulation.There are Nt transmit antennas, and Nr received antennas at the BS. Each UE is equipped with one antenna.

k ≤ K. The binary data sequences are firstly passed tothe MASK mapper, and achieved complex signals bT =[bT,1, bT,2, ..., bT,N ] ∈ C1×N and bk = [bk,1, bk,2, ..., bk,N ] ∈C1×N .

Then bT and bk are passed to the CPM modulator. In orderto ensure the phase continuity, the phase of the baseband signalof TA and the kth UE at the nth symbol duration Ts arecalculated as

βT,n =

n∑i=1

bT,ihπ,

βk,n =

n∑i=1

bk,ihπ,

(1)

where 1 ≤ n ≤ N and the parameter h is modula-tion index [33]. The corresponding phase vectors of theTA and the kth UE are βT = [βT,1, βT,2, ..., βT,N ] andβk = [βk,1, βk,2, ..., βk,N ], respectively. Thus, the basebandsignals of CPM are obtained as cT = ejβT ∈ C1×N andck = ejβk ∈ C1×N , forming the transmit data matrixC =

[cTT , c

T1 , ...c

TK

]T ∈ C(K+1)×N .In this paper, the transmit URA adopts fully con-

nection structure. Suppose that the beamforming weightvectors of the TA and the kth UE are defined bywT = [w1,T , ..., wnt,T , ..., wNt,T ] ∈ CNt×1 and wk =[w1,k, ..., wnt,k, ..., wNt,k] ∈ CNt×1, thus the precoding matrixcan be represented as W = [wT ,w1, ...,wK ] ∈ CNt×(K+1).Assume the transmit power of the TA and kth UE are PTand Pk respectively, and the corresponding power allocationmatrix is P = Diag (PT , P1, ..., PK) ∈ C(K+1)×(K+1). Afterprecoding and power allocation, the transmit data on the ntthantenna is represented as

xnt =√PT wnt,T cT +

K∑k=1

√Pk wnt,k ck, (2)

where xnt = [xnt,1, ..., xnt,n, ..., xnt,N ] ∈ C1×N and 1 ≤nt ≤ Nt. The baseband signal xnt is passed to the filter,and modulated to the carrier frequency fc to form LFMwaveforms, then the corresponding integrated bandpass sig-

...

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(a) The relationship between signal frequency and symbol duration.

(b) The relationship between beat frequency and symbol duration.

Fig. 3: The frequency variations of the carrier, echo signal anddifference signal during M blocks, where τ is the latency, fcis the center frequency and Bw is the sweep bandwidth.

nal is s(t) = [s1(t), ..., snt(t), ..., sNt(t)]T, where snt(t) is

transmitted on the ntth antenna, which is given by

snt(t) =

N∑n=1

xnt,n g (t− (n− 1)Ts) ej(2πfct+πµt2), (3)

where (n− 1)Ts ≤ t ≤ nTs. g(t) is the transmit pulse, whichcan be rectangular pulse, raised cosine pulse, Gaussian pulse,and etc [34]. µ is the chirp rate of the bandpass integratedsignal [35]. If µ > 0, it is a up-chirp signal; otherwise it is adown-chirp signal.

Due to the feature of the triangular wave, the chirp rate isa constant value during one block time TB , i.e., TB = NTs,and the chirp rate changes to its opposite value during the nextblock time. As shown in Fig. 3 (a), there are M blocks, and thechirp rates of the odd and even blocks are respectively µ and−µ, which are also called up-chirp and down-chirp blocks.

According to the chirp rate µ and block time TB , the sweepbandwidth Bw is given by

Bw = µTB = µNTs, (4)


atx(θk, ϕk) =1√Ntx

[1, ej2π

dxλ cos θk cosϕk , ..., ej2π(Ntx−1) dxλ cos θk cosϕk

],

aty(θk, ϕk) =1√Nty

[1, ej2π

dyλ cos θk sinϕk , ..., ej2π(Nty−1)

dyλ cos θk sinϕk

].

(6)

which indicates the highest frequency and lowest frequency ofthe transmit signals are respectively fc+Bw/2 and fc−Bw/2.

B. Received signals at each UE

The integrated signal is then transmitted to the fadingchannel. Assume that the elevation and azimuth of the kthUE θk and ϕk are perfectly known at the BS, and the rangesof elevation θk and azimuth ϕk are respectively (π/2, π) and(0, 2π). Thus, the CSI hk ∈ C1×Nt between the kth UE andBS is modeled as

hk =

√(dk/d0

)−αδk at(θk, ϕk), (5)

where dk is the distance from the BS to the kth UE, d0

is the reference distance, dk ≥ d0, α is the path lossexponent, δk indicates shadowing effect that is a zero meanand σδ variance log-normal random variable. at(θk, ϕk) =

vec(atx(θk, ϕk)

Taty(θk, ϕk)

)∈ C1×Nt is the transmit steer-

ing vector from the BS to the kth UE [36], which are givenby (6), where λ = c

fcstands for the wavelength.

At the receiver of the kth UE, the received bandpass signalrk(t) can be given as

rk(t) = hk s(t) + nk(t). (7)

The received bandpass signal rk(t) is coherent demodulatedand passed through the lowpass filter, the equivalent basebandsignal yk ∈ C1×N is

yk =√Pk hkwkck +

√PT hkwT cT

+

K∑i=1,i6=k

√Pi hkwici + nk, (8)

where nk is the complex additive white Gaussian noise(AWGN) obeying CN (0, σ2

k) with variance σ2k. The first item

is our expected signal, the second and the third items are bothinterferences, which can be eliminated by designing precodingmatrix X, and the last one is noise.

C. Received signals at the BS

If there is a TA at the location of (θm, ϕm) dur-ing the mth block time, an echo signal is feedback tothe BS. The received URA of the BS is used to ac-quire the echo signals of the TA. The steering vectorsfrom the BS to the TA is denoted by at(θm, ϕm) =vec(aTtx(θm, ϕm)aty(θm, ϕm)

)∈ C1×Nt , and from the TA

to BS is ar(θm, ϕm) = vec(aTrx(θm, ϕm)ary(θm, ϕm)

)∈

C1×Nr . Let A = aTr (θm, ϕm)at(θm, ϕm) ∈ CNr×Nt , and

AK =[aTr (θ1, ϕ1),aT

r (θ2, ϕ2), ...,aTr (θK , ϕK)

]∈ CNr×K ,

where aTr (θk, ϕk) is the steering vector from the kth UE to

the BS.

Thus the received bandpass signal at the BS is

r(t) = LA s(t− τ)ej2πfdt + AKIK(t) + nT (t), (9)

where r(t) = [r1(t), ..., rnr (t), ..., rNr (t)]T, L =

√LT ηT is

the gain of echo signal, LT is the propagation attenuationand ηT represents radar cross section (RCS). τ and fd arerespectively latency and Doppler frequency of the TA, whichare need to be estimated. Let IK = [I1(t), I2(t), ..., IK(t)],where Ik(t) is the signal of the kth UE, which is viewedas interferences for the TA detection, the power of whichsatisfy E

(|Ik(t)|2

)= 1. nT (t) is the complex AWGN obey

CN (0, σ2) with variance σ2.Define the processing vector of the BS by V ∈ C1×Nr ,

thus, we have

y(t) = V r(t),

= LVAWC · ej[2πfc(t−τ)+πµ(t−τ)2+2πfdt] (10)+ VAKIK(t) + n(t),

where n(t) = V nT (t) is the equivalent noise after signalprocessing.

Define the received difference signal dr(t) by the low-passsignal of the product of the echo signal y(t) and the coherencefrequency carrier, i.e., dr(t) = LPF

[y(t) · ej(2πfct+πµt

2)],where LPF represents low-pass filter. The frequencies of dr(t)are defined by beat frequencies, including fup and fdown,which are respectively calculated in the up-chirp and down-chip stages, as shown in Fig. 3 (b). In each up-chirp block, thefrequency of the echo signal is lower than the carrier owing tothe influence of latency and Doppler frequency, while in thedown-chirp block, the frequency of the echo signal is higherthan the carrier.

Thereby, during the up-chirp stage, the received differencesignal dr,up(t) can be expressed as

dr,up(t) = LPF[Re (y(t)) · Re(ej(2πfct+πµt

2))],

= LVAWC · ej[2π(µτ−fd)t+2π(fcτ+ 12µτ)τ)]

+ VAKIK(t) + n(t), (11)

where the beat frequency of the up-chirp stage is fup = µτ −fd. Similarly, it is able to derive the received difference signalof the down-chirp stage as

dr,down(t) = LPF[Re (y(t)) · Re(ej(2πfct−πµt2))

],

= LVAWC · ej[2π(µτ+fd)t−2π(fcτ+ 12µτ)τ)]

+ VAKIK(t) + n(t), (12)

with the beat frequency of the down-chirp stage as fdown =µτ + fd.


The latency and distance between the BS and TA can becalculated as τ =

fup+fdown

µ and dT = cτ2 , respectively.

The Doppler frequency is equal to fd =fdown−fup

2 , and thecorresponding radial velocity of the TA is vT = λfd

2 .Obviously, the latency and Doppler frequency can be exactly

estimated without estimation error, if the sweep bandwidthand transmit pulse bandwidth satisfy the resolutions of therequirements. However, the matrix W and vector V affect thereceived power of TA, how to design W and V is presentedin the following section.

IV. A JOINT BEAMFORMING AND RESOURCE ALLOCATIONALGORITHM

To design the precoding matrix W and processing vectorV, this section sets the maximum sum-rate as our objectivefunction. Actually, the sum-rate consists of both communica-tion rate and radar rate. It is important to denote a unifiedsum-rate of a DFRC system. As known, communication sum-rate (or channel capacity) is widely used in communicationsystems, which indicates the quality of a communicationsystem. To unify the communication and radar systems, wepresent a concept of radar rate, similarly to the conception ofcommunication rate. Thus, this paper exploits the maximumsum-rate that includes both communication and radar rates asour objective function.

A. Communication rate and radar rate

This paper only considers the downlink multiuser channelcapacities between the BS and the UEs as the communicationrate. Recalling (8), the SINR of the kth UE can be expressedas

γk =Pk |hkwk|2∣∣∣√PT hkwT +∑Ki=1,i6=k

√Pi hkwi

∣∣∣2 + σ2k

, (13)

where Pk |hkwk|2 and∣∣∣√PT hkwT +

∑Ki=1,i6=k

√Pi hkwi

∣∣∣2are respectively the power of received signal and interferenceof the kth UE. Thus, the communication rate of all the UEsis equal to

Rcom =

K∑k=1

log2 (1 + γk) , (14)

where the subscript “com” of Rcom stands for communicationsystem.

Whether a TA exists or not, it is determined by the receivedpower of TA at the BS. According to the definition of radarequation [37], if the received power of TA is greater than theminimum detection power, i.e. Pth, the TA can be detected.According to (11), the SINR of the TA is calculated as

γT =EradVZWWHZHVH

VAKAHKVH + VVH σ2

, (15)

where Z = LA ej(2πfcτ+πµτ2), Erad = E[CCH] is the powerof radar transmit pilot sequence, and the term VAKAH

KVH

indicates the interferences of other UEs.

Similarly to the concept of communication rate, we definethe radar rate by

Rrad = log2 (1 + γT ) , (16)

where the subscript “rad” stands for radar, and the radar rateis affected by V and W. Obviously, the defined radar rate canreflect the target detection performance, a large Rrad standsfor a better detection performance.

B. Objective function

In order to ensure both the communication and radar per-formances, the optimal function is to maximize the sum-rateof the system, with the constraints given by

max Rsum =

K∑k=1

log2 (1 + γk) + log2 (1 + γT )

= log2 Det (IK+1 + ΓK+1) . (17)s.t. C1 : γk ≥ 2ρk − 1,∀k,

C2 : γT ≥ 2ρT − 1,

C3 : PT +

K∑k=1

Pk ≤ Ptot,

where Ptot is the total transmit power of the BS, ρk and ρTare respectively the thresholds of the required SINR of UE andTA. ΓK+1 = Diag[γ1, γ2, ..., γK , γT ] is a diagonal matrix.

There are three constraints. C1 is to certify the communi-cation the quality of service (QoS) requirement of each UE.C2 is used to ensure the TA detection, and the radar rate Rrad

should be larger than the threshold ρT . C3 is the total powerconstraint.

Obviously, the optimization issue is non-convex. Therefore,we propose a sub-optimal joint resource allocation algorithm,including two parts: user selection, and precoding & process-ing design with power allocation. It is noted that the UE’sweight wk is constant, while the TA’s weight wT is variousduring different M blocks. The user selection part is to selectK UEs from the user set, so that to maximize Rcom. Based onthe selected UEs, the power allocation algorithm is designedto acquire W and V to obtain the sub-optimal solution of(17).

C. Sub-issue: user selection

This part is discussed based on a given W. We adopt MRTprecoding as the initial iteration value, where WMRT = HH

and the subscript “MRT” indicates maximum ratio transmis-sion (MRT) [38]. Note that the user selection result will notaffect the design of W. Actually, γk is mainly determined byeach UE’s θk and ϕk. Define Ψ = {q1, q2, ..., qu, ...qU} andψ = {qi1 , qi2 , ..., qik , ..., qiK} by the sets of all the UEs andthe selected served UEs respectively, where 1 ≤ i1 < i2 <... < iK ≤ U . Obviously, ψ is a subset of Ψ, i.e., ψ ⊆ Ψ. Theoptimal user selection algorithm is traversal algorithm, and itscomplexity is defined by the size of user set, which is O(UK)complex multiplications. If U and K are small, the traversalalgorithm is appreciated. However, when U and K are large,


Algorithm 1 Successive maximum iteration user selectionalgorithm

Input: Ψ, Gu, du, α, δu, at(θu, ϕu)Output: ψ

1: Initialization K and M .2: for m = 1 : M3: wT = vec

(aTtx(θm, ϕm)aty(θm, ϕm)

)∈ C1×Nt

4: when K < U

5: Calculate hu =√(

du/d0

)−αδu at(θu, ϕu).

6: Let wu = hHu .

7: for k = 1 : K8: for qu ∈ Ψ9: Let ψ = ψ + {qu}.

10: Calculate ck = max∑ψ

log2 (1 + γu).

11: end12: if ck > c∆13: Select the UE qik = arg ck.14: Update c∆ = ck, Ψ = Ψ − {qik} and ψ =

ψ + {qik}.15: else16: Output the selected UE set ψ.17: end18: end19: end

the complexity will increase dramatically. Thus, it is importantto present an algorithm to reduce the algorithm complexity.

This paper proposes a low complexity sub-optimal userselection algorithm, named as successive maximum iteration(SMI) user selection algorithm. The main idea behind SMI isto select the UE who can provide the maximum Rcom duringeach iteration. Then, the selected UE is removed from theuser set, and goes to the next selection step. The iterationis repeated, until all the K UEs are selected. Thus, thecomplexity of SMI is O(UK), which is much lower than thatof the traversal method. The SMI is shown in Algorithm 1.

D. Sub-issue: precoder and processor design with powerallocation

In the following discussion, assume the UEs have beenselected, and the CSI of the selected K UEs are known.Moreover, suppose there exists a TA at the location of themth block time. It is well known that the Lagrange multipliermethod (LM) [39] is one of the most popular algorithm tosolve the optimization issue. However, the proposed issue isnon-convex, thus, we explore the majorization minimizationalgorithm (MM) [40] with the Lagrange multiplier method,i.e. maximum minimization Lagrange multiplier algorithm(MMLM), to solve our problem.

The MM procedure consists of two stages. The first stageis to find a surrogate function, which is simple and solvable.Then, we minimize the gap between the original function andsurrogate function at a specific point. Besides, the surrogatefunction is an upper bound of the objective function. Thesecond stage is to minimize the surrogate function, and find

the optimal value, bringing the derived solutions for the nextMM iteration. The procedure is repeated until reaching themaximum iteration number.

The key issue of applying MM is to find a surrogatefunction. In [39], it shows that the function log [Det(Σ)] canbe given as

log [Det(Σ)] ≤ log [Det(Σ0)] + tr(Σ−1

0 (Σ−Σ0)), (18)

where Σ0 is a constant matrix, and the inequality strictly holdswith Σ = Σ0. The problem of (17) can reformulate as

min −Rsum = log2

[Det (IK+1 + ΓK+1)

−1]. (19)

s.t. C1, C2, C3.

Thus, (19) becomes

log2

[Det (IK+1 + ΓK+1)

−1]≤ log2

[Det (IK+1 + ΓMRT )

−1]

+ tr[(IK+1 + ΓMRT )(ΓK+1 − ΓMRT )−1

], (20)

where ΓMRT is the diagonal matrix as aforementioned.Our goal is to minimize the right side of (20). Since ΓMRT

is a constant matrix, the problem is equivalent to maximizethe trace of ΓK+1, as

max tr(ΓK+1) =

K∑k=1

γk + γT . (21)

s.t. C1, C2, C3.

Obviously, (21) is a convex function, which can be solved byLagrange multiplier method. Define Lagrange function as

L (W,V,η) =

K∑k=1

[(1 + ηk) γk + ηk(1− 2ρk)

]+ (1 + ηT )γT + ηT (1− 2ρT ) + ηP

( K∑k=1

Pk + PT − Ptot),

(22)

where η = [η1, ..., ηK , ηT , ηP ] ∈ C1×(K+2) is a vector ofnon-positive Lagrange multipliers. The Karush-Kuhn-Tucker(KKT) conditions [41] of (22) are

∂L (W,V,η)

∂W= 0, (22a)

∂L (W,V,η)

∂V= 0, (22b)

ηυ ≤ 0, υ ∈ {1, 2, ...,K, T, P}, (22c)γυ + 1− 2ρυ ≥ 0, (22d)ηυ(γυ + 1− 2ρυ ) = 0, (22e)ηP ≤ 0, (22f)K∑k=1

Pk + PT − Ptot ≤ 0, (22g)

ηP

( K∑k=1

Pk + PT − Ptot)

= 0, (22h)


where (22a) and (22b) can be extended as

∂L (W,V,η)

∂wk=

2(1 + ηk)Pk |hkwk|hk∣∣∣√PT hkwT +∑Ki=1,i 6=k

√Pihkwi

∣∣∣2 + σ2k

−K∑i=1i6=k

2(1 + ηi)Pi |hiwi|2|hiwk|hi(∣∣∣√PT hiwT +∑Kj=1,j 6=i

√Pj hiwj

∣∣∣2+σ2i

)2

+ (1 + ηT ) ·EradVZ ∂(WWH)

∂wkZHVH

VAKAHKVH + VVHσ2

, (23)

∂L (W,V,η)

∂V=

[(2ZWWHZHVH)(VAKAH

KVH + VVH σ2)

(VAKAHKVH + VVH σ2)2

− (VZWWHZHVH)(2AKAHKVH + 2VH σ2)

(VAKAHKVH + VVH σ2)2

]· (1 + ηT ) Erad. (24)

To obtain the solutions of W and V in (22), we adoptthe alternating direction method of multipliers (ADMM). Theinitialized W by MRT is denoted by W(0), where “0”

indicates the initial step. Let∂L(W(0),V,η)

∂V = 0, then it isderived that

ZW(0)(W(0))HZHVH(VAKAH

KVH + VVH σ2)

=

VZW(0)(W(0))HZHVH(AKAH

KVH + VH σ2). (25)

By solving (25), we can obtain V(1) and then bring it to (22a).

Set∂L(W,V(1),η)

∂wk= 0, and achieve W(1). According to

the derived V(1) and W(1), do water filling power allocationalgorithm to maximum the trace

tr(Γ

(1)K+1

)=

K∑k=1

Pk

∣∣∣hkw(1)k

∣∣∣2∣∣∣√PT hkw(1)T +

∑Ki=1,i 6=k

√Pi hkw

(1)i

∣∣∣2+σ2k

+EradV(1)ZW(1)W(1)HZHV(1)H

V(1)AKAHKV(1)H + V(1)V(1)Hσ2

, (26)

and the optimal power allocation results P (1)k and P (1)

T can beobtained.

Following, take V(1) and W(1) into (20), and obtain thenext surrogate function. By optimizing the new surrogatefunction, we can solve V(2) and W(2) by KKT conditions,and then do water filling power allocation to maximizetr(Γ

(2)K+1

). Repeat this process, until the iteration number i

reaches the maximum iteration number νmax, or satisfying‖W(i)−W(i−1)‖2 < ε and ‖V(i)−V(i−1)‖2 < ε, where ε isa small position number, i.e., 10−3. Finally, we can obtain thesolutions W = W(νmax) and V = V(νmax), and the powerallocation results P (νmax)

k and P(νmax)T , where νmax is the

final iteration times. The MMLM is presented in Algorithm 2.As a summary, the joint resource allocation algorithm

includes user selection, and precoder and processor designwith power allocation. During the initialization, we firstly doSMI to find the UE set ψ, then utilize MMLM to calculateW, V, PT and Pk. Overall, the whole algorithm needs at least(UKνmax) complex multiplications.

Algorithm 2 Maximum minimization Lagrange multiplieralgorithm for designing W and VInput: ψ, γk, γT , ρk, ρT , ε, νmaxOutput: W and V

1: Initialization W(0) by MRT precoding.2: Set i = 1, ε and νmax.3: while i < Nite & ‖W(i) −W(i−1)‖2 ≥ ε & ‖V(i) −

V(i−1)‖2 ≥ ε4: Find the surrogate function according to (20) and opti-

mize the surrogate function.5: Construct Lagrange function L

(W(i),V(i),η

).

6: Calculate KKT conditions of (22).7: Calculate V(i) by

∂L(W(i−1),V(i),η)∂V(i) = 0 and bring it

back to (23).8: Calculate w

(i)k by

∂L(W(i),V(i),η)∂w

(i)k

= 0 and update KKTconditions.

9: Do water filling algorithm to maximize the tracetr(Γ

(i)K+1

), acquire the optimal power allocation results

P(νmax)k and P (νmax)

T .10: end11: W = W(νmax) and V = V(νmax)

V. THEORETICAL ANALYSIS

In this section, we derive the spectrum, BER and AF of theproposed system.

A. Spectrum

Actually, the spectrum of the integrated signal is determinedby the transmit signal snt(t). Since the spectrum of the band-pass and baseband signals are equivalent, we only consider theequivalent baseband signal snt(t) of (3), which is given by

snt(t) =

N∑n=1

xnt,n g (t− (n− 1)Ts) ejπµt2

,

=

N∑n=1

(wnt,T ej

∑ni=1 bT,ihπ +

K∑k=1

wnt,k ej∑ni=1 bk,ihπ

)· g (t− (n− 1)Ts) ejπµt

2

. (27)

Since the spectrum of different transmit antennas are thesame, we take the ntth antenna as an example for analyzing.Define the Fourier transform of snt(t) by S(f), which can beexpressed as

S(f) =∫

+∞

−∞

N∑n=1

(wnt,T ej

∑ni=1

bT,ihπ+

K∑k=1

wnt,k ej∑ni=1

bk,ihπ)

· g (t− (n− 1)Ts) ejπ(µt2−2ft) dt,

=

N∑n=1

(wnt,T ·

n∏i=1

ejbT,ihπ +

K∑k=1

wnt,k ·n∏i=1

ejbk,ihπ)

·∫ TB

0

g(t) ejπ(µt2−2ft) dt. (28)


∣∣∣GLFM (f)∣∣∣ =

√[C

(√µTB −

f√µ

)+ C

(f√µ

)]2

+

[S

(√µTB −

f√µ

)+ S

(f√µ

)]2

. (33)

As mentioned in Section III, bk,i and bT,i are complexsignals. Actually, the nth symbol of the ntth transmit antennaxnt,n is a random variable, thus the mean value of xnt,n is

E [|xnt,n|] = E

[|wnt,T

n∏i=1

ejbT,ihπ +

K∑k=1

wnt,k

n∏i=1

ejbk,ihπ|

],

=

∣∣∣∣∣wnt,T +

K∑k=1

wnt,k

∣∣∣∣∣ n

M

(M−1)∑ξ=−(M−1)ξ is odd

ejξMhπ, (29)

where ξ represents one of the M values of the modulatedindependent symbols bk,i and bT,i, and the prior probabilityof which is 1

M . From (29), it is found that different precodingschemes W and modulation orders M can affect the ampli-tude of the spectrum. Compared with MRT precoding, thespectrum of our scheme has a larger amplitude, and a smallerM derives a lower amplitude. Assume

G(f) =

∫ TB

0

g(t) ejπ(µt2−2ft) dt,

= Gu(f) ∗ GLFM (f), (30)

where Gu(f) is the Fourier transform of transmit pulse g(t),and Bg is the bandwidth of g(t). If we adopt rectangular pulse,i.e., Bg = 1/Ts or raised-cosine pulse i.e., Bg = (1+αg)/2Ts,where αg is the roll-off factor.

Define GLFM (f) by the Fourier transform of LFM signal,given by

GLFM (f) =

∫ TB

0

ejπ(µt2−2ft) dt,

= e−jπf2

µ

∫ √µTB− f√µ

− f√µ

[cos(πt

′2) + j sin(πt

′2)]

dt′,

(31)

where t′

=√µt − f√

µ , C(x) =∫ x

0cos(πα2) dα and

S(x) =∫ x

0sin(πα2) dα are cosine and sine Fresnel integral

respectively. Since both C(x) and S(x) are odd function, theamplitude of GLFM (f) can be expressed in (33).

According to (30), it is found that the integrated signalbandwidth is equal to B = Bw + Bg , i.e. B = µTB + 1/Tsfor rectangular pulse or B = µTB + (1 +αg)/2Ts for raised-cosine pulse. To improve the spectrum efficiency, we can adoptbandwidth-saving pulse.

B. BERActually, the BER of the proposed system is independent

of LFM. The optimum detector of a CPM signal is realizedby Viterbi algorithm [42]. Considering the proposed system,the BER of the kth UE becomes

Pe,k = 2

∫ +∞

0

Q

(√log2M

(1− sin 2πh

2πh

)γk

)p(γk) dγk,

(34)

where Q(x) =∫ +∞x

1√2π

e−t2

2 dt, p(γk) is the probabilitydensity function (PDF) of γk. For mathematic convenience, weutilize W(0) that is obtained by MRT instead of the iterationoptimal W(νmax) to calculate p(γk). The expression of γk in(13) is

γk ≤Pk(dk/d0

)−αδkNt∣∣∣√PT hkwT +

∑Ki=1,i6=k

√Pi hkwi

∣∣∣2 + σ2k

, (35)

where the path loss d−αk and shadowing effect δk obey powerlaw and log-normal distributions, the PDF of δk is p(δk) =

1δk√

2πσδke− ln(δk)2

2σδ , where σδ is the variance of δk.

Let Ik =√PT hkwT +

∑Ki=1,i6=k

√Pi hkwi be the in-

terferences, and it is the sum of several independent andidentically (i.i.d) distributed random variables. According tothe central limit theorem, Ik can be approximately viewedas complex Gaussian distribution, i.e., N (µI , σI), which issimulated by Monte Carlo. The mean value of Ik can bederived by µI =

√PT µT +

∑Kk=1

√Pi µk, where µT and µk

are respectively the mean value of hkwT and hkwi. It can beseen that a larger K leads to a larger mean value µI when thepower allocation factor is fixed. Moreover, for a given K, e.g.,K = 4, PT shows a significant influence on the µI , indicatingthat the interference increases with the increased of PT . Thevariance of Ik can be expressed as σI = PT σT +

∑Kk=1 Pi σk,

where σT and σk are respectively the variance of interferencehkwT and hkwi. Thus, the total interference plus noise obeyschi-square distribution with degree of freedom 2, given by

p(Ik) =1

2σ2I

e− Ik

2σ2I . (36)

Assume zk =d−αk δkIk

, according to the relationship ofproduct and quotient of random variables, the PDF of γk is

p(γk) =(2(σI + σk)2)−

1α e

σ4δα2

α(dk,2 − dk,1)Q(−σδ

2

)Γ

(1− 1

α

)γ− 1α−1

k ,

(37)where Γ(x) =

∫ +∞0

tx−1e−t dt, dk,1 and dk,2 are respectivelythe minimum and maximum range from the BS to the kth UE,i.e. dk,1 ≤ dk ≤ dk,2.

Taking (37) into (34), we can derive the upper bound of theBER expression. Unfortunately, the closed-form solution cannot be deducted since the integral of the Q function can beexpressed by the analytical form.

C. AF

The ambiguity function is defined as the time-frequencyresponse observed at the matched filter of the BS receiver,which is given by [43]

|χ(τ, fd)|2 =

∣∣∣∣∫ +∞

−∞snt(t) s

∗nt(t− τ) ej2πfdt dt

∣∣∣∣2 , (38)


|χ(τ, fd)|2 =

∣∣∣∣∣e−jπµτ2N∑n=1

N∑i=1

∫ +∞

−∞xnt,n x

∗nt,i g (t− (n− 1)Ts) g

∗ (t− (i− 1)Ts − τ) ej2π(µτ+fd)t dt

∣∣∣∣∣2

. (41)

where snt(t) is the complex envelope of baseband signalsnt(t), substituting (27) into (38) yields (41).

Let t1 = t− (n− 1)Ts, q = n− i, (41) becomes

|χ(τ, fd)|2 =

∣∣∣∣e−jπµτ2N∑n=1

ej2π(µτ+fd)(n−1)Ts

·N∑i=1

χ1(τ − qTs, fd + µτ)

∣∣∣∣2, (42)

where

χ1(τ − qTs, fd + µτ) =

∫ +∞

−∞xnt,n x

∗nt,i

g (t1) g∗ (t1 − (τ − qTs)) ej2π(µτ+fd)t1 dt1, (43)

is the single pulse auto-correlation function.Let g(t) be a rectangular pulse, regarding as the symmetry

property, i.e., |χ(τ, fd)|2 = |χ(−τ,−fd)|2, (43) becomes

|χ(τ, fd)|2 =

∣∣∣∣∣(N−1)∑

q=−(N−1)

sin [π(fd + µτ)(N − |q|)Ts]sin [π(fd + µτ)Ts]

· χ1 (τ − qTs, fd + µτ)

∣∣∣∣∣2

, (44)

Obviously, the AF is a two-dimensional function of τ andfd, whose maximum value occurs at (τ, fd) = (0, 0). Thecut along in Doppler axis is by setting τ = 0, indicating theresolution in the Doppler dimension, which is

|χ(0, fd)|2 =

∣∣∣∣∣(N−1)∑

q=−(N−1)

sin [πfd(N − |q|)Ts]sin (πfdTs)

χ1 (−qTs, fd)

∣∣∣∣∣2

.

(45)

Set |χ(0, fd)|2 = 0, the first null in Doppler domain occursat fd,w = 1

NTs. Similarly, the cut along in latency axis is

obtained by setting fd = 0, indicating the resolution in thelatency dimension, which is

|χ(τ, 0)|2 =

∣∣∣∣∣∣(N−1)∑

q=−(N−1)

sin [πµτ(N − |q|)Ts]sin (πµτTs)

χ1 (τ − qTs, µτ)

∣∣∣∣∣∣2

.

(46)Let |χ(τ, 0)|2 = 0, the first null in latency domain occurs

at τw = 1µNTs

= 1Bw

. It can be seen that τw is completelydetermined by the LFM sweep bandwidth. Define κ by thecompression ratio of the transmit and received pulse duration,which is

κ =TBτw

= TB Bw. (47)

κ is the multiplication of block time and sweep bandwidth,which is also called time-bandwidth product [44]. The mini-mum detection distance is denoted by dmin = c

2Bw, indicating

the range resolution, and c is the speed of light. Similarly,

TABLE I: Simulation parameters

Significance Parameters ValuesNumber of UEs K 4

Number of transmit antennas Nt 16Number of received antennas Nr 4

Number of blocks M 324Number of symbols per block N 20

Symbol duration Ts 5 µsCarrier Central frequency fc 2.4 GHz

Chirp rate µ 1010

Pass loss factor α 3.0Transmit power Ptot 30 dBmReference range d0 100 m

the minimum detection velocity is defined by vmin = λ2TB

,reflecting the Doppler resolution.

According to dmin and vmin, the time-bandwidth productincreases with the increased sweep bandwidth, which is dif-ferent from the single carrier radar signal. Evidently, a largeκ can meet both the requirements of the range resolution andDoppler resolution.

VI. SIMULATION RESULTS

In this section, we evaluate the performance of our proposedsystem through numerical simulations. The equipped transmitand received antennas of the BS are respectively 16 and 4,which are respectively structured by a 4×4 and 2×2 URA. Allthe UEs are uniform distribution in the cell, and the simulationparameters are listed in Table I.

A. Spectrum

Fig. 4 shows the spectrum of the proposed integrated signal,with various chirp rates and pulses. It can be seen thatthe bandwidth of integrated waveforms are determined byboth sweep bandwidth Bw and transmit pulse bandwidth Bg ,independent of the center frequency fc. For a given µ, if Bw issmall, the ripples are very evident. On contrast, the spectrumtends to be a rectangular profile with the increasing of Bw.Obviously, the bandwidth of a raised cosine pulse is smallerthan that of a rectangular pulse, for a given Ts. Moreover, thechirp rate µ affects the bandwidth significantly, a larger µ maylead to a larger bandwidth, e.g., the bandwidth of the curvein µ = 5× 109 is half of the µ = 1× 1010. Furthermore, fora given g(t) and µ, the bandwidth of CPM is narrower thanthat of BPSK. From the spectrum, it is found that W has noeffect to the bandwidth of the proposed system, verifying theanalysis.

B. BER

The average BER performance is shown in Fig. 5. WhenK = 4, Nt = 16 and PT = 0.1Ptot, it can be seen that theproposed scheme achieves 10−3 when SNR is 20 dB. Thetheoretical curve is the upper bound, since γk can get the


(a) (b)

(c) (d) (e)

2399.0 2399.5 2400.0 2400.5 2401.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Rel

ativ

e am

pli

tud

e

Frequency (MHz)

CPM-LFM, rect, m=1010

CPM-LFM,rect, m=5×109

CPM-LFM, rc, m=1010

BPSK-LFM, rect, m=1010

2399.0 2399.5 2400.0 2400.5 2401.0

0.0

0.2

0.4

0.6

0.8

1.0R

elat

ive

amp

litu

de

Frequency (MHz)

BPSK-LFM, rect, m=1×1010

2399.0 2399.5 2400.0 2400.5 2401.0

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e am

pli

tud

e

Frequency (MHz)

CPM-LFM, rect, m=1×1010

2399.0 2399.5 2400.0 2400.5 2401.0

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e am

pli

tude

Frequency (MHz)

CPM-LFM, rc, m=1×1010

2399.0 2399.5 2400.0 2400.5 2401.0

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e am

pli

tud

e

Frequency (MHz)

CPM-LFM, rect, m=5×109

Fig. 4: Spectrum of the integrated signals with various chirp rates and pulses, where the subscript “rect” and “rc” representthe rectangular and raised-cosine pulse. (a) CPM-LFM signal with rectangular pulse, where µ = 1 × 1010; (b) CPM-LFMsignal with raised-cosine pulse, where µ = 1 × 1010; (c) CPM-LFM signal with rectangular pulse, where µ = 5 × 109; (d)BPSK-LFM signal with rectangular pulse, where µ = 1× 1010; (e) overall spectrum comparison.

0 5 1 0 1 5 2 0

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

BER

( d B )

P r o p o s e d s c h e m e , K = 4 , N t = 1 6 , P T = 0 . 1 P t o t P r o p o s e d s c h e m e , K = 4 , N t = 2 5 , P T = 0 . 1 P t o t P r o p o s e d s c h e m e , K = 8 , N t = 1 6 , P T = 0 . 1 P t o t P r o p o s e d s c h e m e , K = 4 , N t = 1 6 , P T = 0 . 5 P t o t B P S K , K = 4 , N t = 1 6 , P T = 0 . 1 P t o t B o u n d , C P M , K = 4 , N t = 1 6 , P T = 0 . 1 P t o t

2t o tP �

Fig. 5: The average BER versus Ptot/σ2 in different modu-lation types, precoding schemes, number of UEs and numberof transmit antennas.

maximum value. Moreover, when Nt is given, i.e., Nt = 16,a large K results in a worse BER performance because of theremaining interferences. In terms of the number of transmitantennas, a large Nt can eliminate the interferences effectively,providing a better BER performance. Furthermore, powerallocation result also has an impact on the BER performance,since a larger PT leads to the reduction of communicationpower Pk, indicating a deteriorated BER. As expected, whenK, Nt and PT are fixed, the proposed scheme provides betterBER performance than BPSK counterpart, since the Viterbialgorithm is exploited for the CPM signal.

C. AF

The AF of our proposed system is shown in Fig 6, thehorizontal and vertical axes stand for latency and Dopplerfrequency domains. When there is no TA exists, the peakof the AF is located at the (τ, fd) = (0, 0). The location ofthe peak value is shifted with a moving TA, and located inthe latency-Doppler domain. In (44), it can be seen that theAF is the combination of multiple spikes. The shape of eachspike is determined by the transmit pulse type and precodingalgorithm. Owing to the precoding methods, the peak of eachspike becomes relative sharper than classical LFM signal,which exhibits a better range resolution. The correspondingzero-latency and zero-Doppler cuts are presented in (45) and(46). When τ = 0 in (45) or fd = 0 in (46), a larger symbolnumber N leads to a narrower fd,w or τw, indicating a smallercircle outline of each pulse, i.e., the better range and Dopplerresolutions. Moreover, a larger µ leads to a larger sweepbandwidth Bw, which can increase the range resolution.

D. Sum-rate of the proposed DFRC system

The sum-rates of the proposed DFRC system are shownin Figs. 7-8. Fig. 7 shows the influence of W, with theassumption K = 4; and Fig. 8 focuses on the impact of userselection algorithm.

According to the simulation parameters, the bandwidth ofthe radar system is five times larger than communicationsystem, i.e. Bw = 5Bg . To make a fair comparison, we shouldeliminate the influence of signal bandwidth and consider thesum-rate per bandwidth. It can be seen from Fig. 7, theproposed scheme has the best sum-rate performance thanthe pure communication or radar system. This is reasonablesince there is a trade-off between the sum-rate per bandwidth


Doppl

er Freq

uency

(Hz)

L a t e n c y ( m s )0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

- 0 . 2 - 0 . 1 0 0 . 1 0 . 2- 1 0 0

- 5 0

0

5 0

1 0 0

Fig. 6: The AF contour of our proposed system in latency andDoppler frequency domains, where g(t) is rectangular pulse.

with the detection ability in radar system. Moreover, for theproposed JCR system, our designed W provides the largestsum-rate than other classical beamforming algorithms, i.e.,minimum mean square error (MMSE), followed by zero-forcing (ZF) and MRT. For example, when Ptot

σ2 = 10 dB,the sum-rate of our proposed scheme is 15.3 bps/Hz, which isapproximately 2.1 bps/Hz, 2.5 bps/Hz and 2.8 bps/Hz largerthan the MMSE, ZF and MRT algorithms.

From Fig. 8, it can be seen that the traversal user selec-tion algorithm always provides the larger sum-rate than theproposed SMI algorithm, since it can achieve the optimalsolution. Evidently, the complexity of SMI is much lowerthan the traversal algorithm, especially when U and K arelarge values. For example, when U = 30 and K = 4,the complexity of SMI is 120, while traversal algorithm is8.1×105. Moreover, M shows the impact on the TA detectionprecision. Let δϕ = 10◦, we compare the case δθ = 10◦ andδθ = 15◦, the corresponding M = 324 and 216. It can bederived that a larger M represents a higher sum-rate sinceit can enhance elevation or azimuth resolution, indicating theless interferences of each UE.

E. Trade-off between communication and radar performances

The relationship between the communication rate and de-tection range is shown in Fig. 9, where K = 4, 6 and RCSηT = 0.5, 0.8, 1.0 m2. When K and ηT are constant, thecommunication rate decreases with the increased maximumdetection range, since a larger detection range needs a largerPrad, leading to a smaller Pcom. It is remarkable that theinfluence of RCS can be ignored, when the detection distanceis less than a threshold, i.e., 0.3. In the case K = 4 andηT = 0.5, when the maximum detection range is 0.6 km, thecommunication rate is approximately 8.15 bps/Hz, while themaximum detection range is 0.8 km, the communication rateis 8.02 bps/Hz. The trade-off between the maximum detectionrange and the communication rate is mainly determined bythe power allocation, since the total transmit power Ptot is a

0 2 4 6 8 1 02468

1 01 21 41 61 8

2t o tP �

Sum-

rate (b

ps/Hz

)

( d B )

P r o p o s e d M M L M a l g o r i t h m , J C R M R T a l g o r i t h m , J C R Z F a l g o r i t h m , J C R M M S E a l g o r i t h m , J C R M M L M a l g o r i t h m , C o m m u n i c a t i o n M M L M a l g o r i t h m , R a d a r

Fig. 7: An illustration of the sum-rate versus Ptot/σ2 in

proposed MMLM, ZF, MMSE, MRT algorithms in DFRC,pure communication and radar systems.

2 0 4 0 6 0 8 0 1 0 0

4

6

8

1 0

1 2

1 4Su

m-rat

e (bps/

Hz)

U

T r a v e r s a l a l g o r i t h m , M = 3 2 4 T r a v e r s a l a l g o r i t h m , M = 2 1 6 P r o p o s e d S M I a l g o r i t h m , M = 3 2 4 P r o p o s e d S M I a l g o r i t h m , M = 2 1 6

Fig. 8: An illustration of the sum-rate versus U in traversaland proposed SMI algorithm, where M = 324 and 216.

constant value, a better radar performance leads to a smallercommunication rate.

VII. CONCLUSION

In this paper, we present a joint beamforming and CPM-LFM integrated signal, based on the proposed dual modeframework. The framework includes both dynamic mode andstatic mode, and the parameter M and N affect scanningprecision and detection resolutions respectively. Following, wepropose an objective function of sum-rate with constraints.To solve the non-convex issue, we give a sub-optimal jointbeamforming and resource allocation algorithm, i.e. SMI-MMLM, which includes successive maximum iteration foruser selection; and a maximum minimization Lagrange mul-tiplier algorithm for precoding & processing design, withwater filling algorithm to achieve the optimal power allocationresults. Then, we give a detail analysis of the proposed


0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 07 . 6

7 . 8

8 . 0

8 . 2

8 . 4

8 . 6Co

mmuni

cation

rate (b

ps/Hz

)

M a x i m u m d e t e c t i o n r a n g e ( k m )

K = 4 , � T = 0 . 5 m 2

K = 4 , � T = 0 . 8 m 2

K = 4 , � T = 1 . 0 m 2

K = 6 , � T = 0 . 5 m 2

K = 6 , � T = 0 . 8 m 2

K = 6 , � T = 1 . 0 m 2

Fig. 9: Comparison between the communication rate versusmaximum detection range in different user number K andRCS, where the chirp rate µ = 1010.

system, including the spectrum characteristic, BER and AF.Simulations results shows that our proposed CPM-LFM signalhas better spectrum efficiency and range resolution. Moreover,the proposed SMI algorithm shows lower complexity thanthe optimal traversal algorithm, and the proposed MMLMalgorithm performs larger sum-rate than the classical MRT,ZF and MMSE algorithms.

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