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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 11, NO. 2, MARCH 2017 295

The Random Frequency Diverse Array: A NewAntenna Structure for Uncoupled Direction-Range

Indication in Active SensingYimin Liu, Member, IEEE, Hang Ruan, Lei Wang, and Arye Nehorai, Fellow, IEEE

Abstract—In this paper, we propose a new type of array an-tenna, termed the random frequency diverse array (RFDA), foran uncoupled indication of target direction and range with lowsystem complexity. In RFDA, each array element has a narrowbandwidth and a randomly assigned carrier frequency. The beam-pattern of the array is shown to be stochastic but thumbtack-like,and its stochastic characteristics, such as the mean, variance, andasymptotic distribution are derived analytically. Based on thesetwo features, we propose two kinds of algorithms for signal pro-cessing. One is matched filtering, due to the beampattern’s goodcharacteristics. The other is compressive sensing, because the newapproach can be regarded as a sparse and random sampling oftarget information in the spatial-frequency domain. Fundamentallimits, such as the Cramer–Rao bound and the observing matrix’smutual coherence, are provided as performance guarantees of thenew array structure. The features and performances of RFDA areverified with numerical results.

Index Terms—Array antenna, compressive sensing, Cramer–Rao bound, frequency diverse.

I. INTRODUCTION

THE array antenna is very popular in active sensing tech-niques, such as radar, sonar, and ultrasonic detection[2],

for reasons such as flexibility in beam steering over a range ofdirections, and its efficient system resource management [3].In most conventional array structures, only the directions of thebeampattern can be formed. But the range (or delay) features arealso usually important, especially in target indication or activeimaging applications [3].

A promising new array structure, named the Frequency Di-verse Array (FDA)[4] was proposed in 2006, and has attracted

Manuscript received March 27, 2016; revised August 15, 2016; acceptedOctober 6, 2016. Date of publication November 9, 2016; date of current versionFebruary 13, 2017. The work of Y. Liu was supported by the National NaturalScience Foundation of China under Grant 61571260. The work of A. Nehoraiwas supported by the AFSOR under Grant FA9550-11-1-0210. The results ofthis paper were presented in part at the International Conference on Acoustics,Speech, and Signal Processing, Shanghai, China, March 2016. The guest editorcoordinating the review of this paper and approving it for publication was Prof.Shannon David Blunt.

Y. Liu, H. Ruan, and L. Wang are with the Department of Electronic Engineer-ing, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]; [email protected]).

A. Nehorai is with the Department of Electrical and Systems Engineer-ing, Washington University, St. Louis, MO 63130 USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/JSTSP.2016.2627183

more and more attention [5] in recent years. The FDA can pro-duce a beampattern with controllable direction and range bylinearly shifting the carrier frequencies across the array1. Thisnew feature provides many advantages in active sensing, such aslow probability of interception and jamming resistance [6], res-olution enhancement in Synthetic Aperture Radar (SAR) imag-ing [7], and ambiguous range clutter suppression in space-timeadaptive processing [8].

However, as discussed in [9] and [10], the direction and rangeof a FDA’s beampattern are coupled, which means there mightbe a group of direction-range pairs that match well with theecho from a point target, hence introducing aliases in targetindication. To overcome this problem, several methods havebeen introduced recently. In [11], one straightforward method,named “double-pulse”, can effectively remove the coupling bysuccessively transmitting two pulses. The direction and rangeindication are accomplished separately in each pulse, but thedata rate drops to half.

The subarray, introduced by [12] and [13], is an alterna-tive. In this approach, the FDA is divided into serval subarrays,each with a different carrier frequency increment. Uncoupleddirection and range can be estimated, and the correspondingaccuracy can be improved by optimizing the coordinates of thearray elements and the frequency increments. However, only asingle-target scenario is considered in these works.

Multi-Input-Multi-Output (MIMO) radar [14] is a powerfultechnique with notable improvements, including the antennaaperture, spatial diversity, degrees of freedom. In [15], a newstructure named Frequency Diverse MIMO (FD-MIMO) wasproposed by combining MIMO and FDA. In FD-MIMO,each receiving element can demodulate all the signals fromevery transmitting element, each of which has a differentcarrier frequency. This design enables forming uncoupleddirection-range beampattern by combining all the basebandsamples, and dramatically increases the system’s degrees offreedom. The FD-MIMO approach locates both single andmultiple targets very well. However, the system complexity isalso seriously increased due to the need for multiple receiverchannels in each receiving element.

In this work, we regard the acquiring of a target’s direction-range information by FDA as a sampling procedure in both the

1For abbreviation, we named it as Linear Frequency Diverse Array (LFDA)in the following discussion.

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296 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 11, NO. 2, MARCH 2017

spatial and frequency domains. Then we propose a new systemstructure for a FDA, named Random Frequency Diverse Array(RFDA), which randomly assigns the carrier frequency of eacharray element. Inspired by the basic ideas of compressive sens-ing, RFDA can get the target information from both the spatialand frequency domains in a single glance. Through two-dimensional (2D) random sparse sampling, the new structureprovides a thumbtack-like 2D beampattern, and can simulta-neously achieve good resolution and estimation accuracy forboth direction and range. The mutual coherence [16] of RFDA’sobserving matrix is also quite small, which implies the abilityof uncoupled target location in both single and multiple targetscenarios. Moreover, each element in RFDA needs only onereceiver channel, hence the above advantages can be attainedwithout increasing the system complexity. The main contribu-tions of this paper are as follows.

1) We propose a new array structure, named RFDA, whichrandomly assigns the carrier frequencies of elements in aUniform Linear Array (ULA). The new structure is shownto have the ability to simultaneously indicate target’s di-rection and range without coupling.

2) We derive the analytic form of RFDA’s beampattern, andprovide its stochastic characteristics, such as the mean,variance, and asymptotic distribution, to reveal the reso-lution and sidelobe level of this new array structure.

3) We propose two signal processing methods for RFDA,matched filtering and compressive sensing, to detect thetargets and indicate their locations. Matched filtering hasthe highest output signal-to-noise-ratio (SNR), while com-pressive sensing is more suitable for multi-target scenar-ios. Moreover, we give an approximately equivalent ap-proach for matched filtering, to reduce the computationload.

4) We derive the performance limits for RFDA, includingthe Cramer-Rao Bound (CRB) and mutual coherence, toquantify its direction/range estimation and target indica-tion abilities.

The rest of this paper is organized as follows. Section IIpresents a system sketch and constructs the signal model. InSection III, the beampattern and stochastic characteristics ofRFDA are derived. In Section IV, we introduce two signal pro-cessing methods, matched filtering, and compressive sensing.Section V investigates the performance limits of RFDA, such asthe CRB and mutual coherence. Finally, numerical illustrationsare given in Section VI, and conclusions are drawn in the lastsection.

Notations: ‖ · ‖0 is the non-zero entry number of an argu-ment; ‖ · ‖2,0 is the number of rows with non-zero l2 normsin an argument; E†{·} and V†{·} are the expectation and vari-ance of arguments with respect to (w.r.t.) the random vector (orvariable) †, respectively; [·]i,n is the ith row, jth column of anargument; [·]n is the nth element (or column) of an argument,while the argument is a vector (or matrix); [·]∗, [·]T , and [·]Hare the conjugation, transpose, and conjugate transpose of ar-guments, respectively; A � B implies that A − B is positivesemi-definite; �{·} and �{·} are the real and imaginary partof an argument; The operator ⊗ means the Kronecker product,while � means the Hadamard product.

Fig. 1. System sketch of RFDA.

II. SYSTEM SKETCH AND SIGNAL MODEL

In this section, we will introduce a system sketch of the ran-dom frequency diverse array radar, and then propose the signalmodel of this new kind of array.

A brief system sketch of the RFDA is shown in Fig. 1. Thereare N elements in a linear array, with a constant inter-elementdistance d. These array elements are located symmetricallyalong the x-axis, and the coordinate of the nth one is

xn =(−N − 1

2+ n

)d, n = 0, 1, . . . , N − 1. (1)

Each element is connected to a narrow band transceiver. Thetransmitted waveform of each element is monotone sinusoid, butthe carrier frequencies of the different elements are randomlyassigned. The carrier frequency of the nth element is

fn = fc + mnΔf, (2)

where fc is the center frequency, Δf is the frequency increment,and mn is a random variable. In this work, all the mn s are chosenas i.i.d. random variables, and can be arranged into a randomvector, m = [m0 ,m1 , . . . ,mN −1 ]T .

In this paper, g(mn ), which is assumed to be an even func-tion, is defined as the frequency diverse distribution (the prob-ability density function (PDF) of mn ). And in the followinginvestigation, we will take three common PDFs as examples:

1) Gaussian: g(mn ) = 1/√

2πσ2 · e−m 2

n2 σ 2 , where σ2 is the

variance. The larger the σ2 is, the wider the bandwidth ofRFDA becomes.

2) Continuous uniform: g(mn ) = 1/M , where M is a pos-itive real number, and mn ∈ [−M/2,M/2]. The totalbandwidth of RFDA is MΔf .

3) Discrete uniform: g(mn ) = 1/M , where M is a pos-itive integer, and mn ∈ {−(M − 1)/2,−(M − 1)/2 +1, . . . , (M − 1)/2}. The total bandwidth of RFDA is alsoMΔf .

Besides the above three, there can be other kinds of distri-butions for the frequency assignments of RFDA. With these

LIU et al.: RFDA: A NEW ANTENNA STRUCTURE FOR UNCOUPLED DIRECTION-RANGE INDICATION IN ACTIVE SENSING 297

definitions, the transmitted waveform of the nth element is

sn (t) = ej2π (fc +mn Δf )t . (3)

A. Definition and Basic Results

A simple diagram of RFDA’s waveform is illustrated in thelower part of Fig. 1. The blue blocks illustrate the waveformdistribution in the spatial-frequency domain. In this antenna,the original point of the x-axis, O, is selected as the phasecenter of the array. If there exists an ideal unit point target withdirection θ and range r, then with the far-field assumption, thereceived radio-frequency (RF) echo of the nth element is

rn (t; θ, r) = sn

(t − 2

r + xn sin θ

c

)

= ej2π (fc +mn Δf )(t−2 r + x n s in θc ), (4)

where c is the wave propagation speed. In each receiver, the RFecho is demodulated with its transmitting carrier frequency, sothe baseband signal of the nth element is

bn (θ, r) = e−j 4 πc (fc +mn Δf )[r+(−N −1

2 +n)d sin θ]. (5)

The baseband samples bn (θ, r) of all elements (n =0, 1, . . . , N − 1) can be arranged in order to formulate adirection-range dependent steering vector:

b(θ, r) = [b0(θ, r), b1(θ, r), . . . , bN −1(θ, r)]T

= bD (θ) � bR (r), (6)

where

bD (θ) = [e−j4 π f c ( 1 + δ m 0 )d s in θ

c (−N −12 ) ,

e−j4 π f c ( 1 + δ m 1 )d s in θ

c (1−N −12 ) ,

. . . , e−j4 π f c ( 1 + δ m N −1 )d s in θ

c ( N −12 ) ]T , (7)

δ = Δf/fc , and

bR (r) = e−j 4 πc fc r [e−j 4 π Δ f

c m 0 r , e−j 4 π Δ fc m 1 r ,

. . . , e−j 4 π Δ fc mN −1 r ]T . (8)

For the multi-target, multi-snapshot, and noisy scenario, sup-pose the direction and range of the ith target is {θi, ri}(i = 1, 2, . . . , P , P is the target number), and the complex re-flection amplitude of the ith targets at the lth snapshot is a(l)(l = 0, 1 . . . , L − 1, L is the snapshot number.). Then the base-band echo vector of the lth snapshot is the superimposition ofechoes from all targets:

r(l) =P∑

i=1

ai(l) · bD (θi) � bR (ri) + v(l), (9)

where v(l) is the N × 1 additive receiver noise vector. In (9),ai(l) can vary from snapshot to snapshot, because of the target’sfluctuation [3].

For moving target scenarios, the relative radial velocities be-tween the array antenna and the targets will introduce Dopplershifts to the echo [3]. In this case, a Doppler filter bank should befirst applied on the receiver of each array element. Then for all

the filter banks, the outputs which correspond to the same targetvelocity, can be gathered together to achieve steering vectorswith identical expressions in (6)-(8).

III. THE BEAMPATTERN OF RFDA

The system and signal model of RFDA were formulated inthe last section. In this section, we will discuss its beampattern,which is an important characteristic of an array antenna [3]. Inarray processing [2], the beampattern is the system response ofan array beamformed in one direction to a unit amplitude targetlocated in another direction. According to (5), the received echodepends on both the direction and the range simultaneously.So the corresponding beampatterns of RFDAs are functions ofboth direction and range. The following discussion will showthat, with signal processing, RFDA can indicate both the target’sdirection and range without coupling.

If the target is located at {θ1 , r1}, and the RFDA isbeamformed to another location, named {θ2 , r2}, the arrayresponse is

β({θ1 , r1}, {θ2 , r2}) =< b(θ1 , r1),b(θ2 , r2) >

‖b(θ1 , r1)‖2‖b(θ2 , r2)‖2. (10)

Substituting (5) and (6) into (10), the beampattern of a RFDAis

β({θ1 , r1}, {θ2 , r2})

=1N

[bD (θ1) � bR (r1)]H [bD (θ2) � bR (r2)]

=1N

[b∗D (θ1) � bD (θ2)]T [b∗

R (r1) � bR (r2)]

=1N

ej2π pδ

N −1∑n=0

ej2π (n−N −12 )q · ej2πmn [p+δq(n−N −1

2 )]

= β(q, p), (11)

where [·]∗ indicates element-wise conjugation. The variablesq and p are defined as q = 2(sin θ1 − sin θ2)fcd/c, and p =2(r1 − r2)Δf/c. According to (11), the RFDA beampatterndepends only on the difference of direction sine, sin θ1 − sin θ2 ,and the difference of range , r1 − r2 . But it is independent ofthe absolute value of the target’s direction or range.

Meanwhile, for traditional LFDAs [4][15], the transmittedfrequencies are linearly shifted along the element. Hence thesignal models of LFDA can be achieved by alternating mn withn − N −1

2 in (11),

β(q, p) =1N

ej2π pδ

N −1∑n=0

ej2π (p+q)(n−N −12 ) · ej2πδq(n−N −1

2 ) .

(12)Fig. 2 shows the beampatterns of different kinds of FDA an-

tennas. Subfigure (a) is the beampattern of an LFDA. As shown,this beampattern has high sidelobe ridges, which implies thatthe direction and range are coupled, and the indication of tar-get location is ambiguous. The other three subfigures are thebeampatterns of RFDA antennas. Subfigure (b) is the beam-pattern when the carrier frequency of each element is discreteuniform distributed, while the beampatterns of (c) and (d) are


Fig. 2. Examples of beampatterns of different frequency diverse arrays. (a) LFDA; (b) RFDA, discrete uniform distributed; (c) RFDA, continuous uniformdistributed; (d) RFDA, Gaussian distributed.

continuous uniform and Gaussian, respectively. However, re-gardless of the frequency distribution, the RFDAs’ beampat-terns are thumbtack-like, and all the peaks are located wherep = 0 and q = 0, which implies that the direction and rangehave successfully been decoupled and the target location can beuniquely and correctly indicated.

In (11), it is shown that the beampattern depends on the dis-tribution of mn and can be regarded as a stochastic process w.r.t.q and p. Hence an analysis about its stochastic features will bevery helpful for a more comprehensive study of RFDA’s charac-teristics. In the following subsections, we will explicitly deriveexpressions of the mean, variance, and asymptotic distributionof the beampattern.

A. Mean

The mean of the beampattern is derived in this subsection.Because the transmitted frequencies of each element are i.i.d.,the mean of the beampattern can be calculated as

β(q, p) � Em {β(q, p)}

=∫

mn ∈Mβ(q, p)g(mn )dmn

=1N

ej2π pδ

N −1∑n=0

ej2π (n−N −12 )q

·∫

g(mn )ej2πmn [p+δq(n−N −12 )]dmn

=1N

ej2π pδ

N −1∑n=0

ej2π (n−N −12 )q

·Φ(

p + δq

(n − N − 1

2

))

=1N

ej2π pδ

N −1∑n=0

ej2π (n−N −12 )q

·[Φ(p) +

∞∑k=1

1k!

Φ(k)(p)δk qk

(n − N − 1

2

)k]

= β(q, p) + Δβ (q, p). (13)

In (13), M is the sample space of mn , Φ(x) is the moment-generating function of mn ,

Φ(x) =∫

mn ∈Mg(mn )ej2πmn xdmn , (14)

and Φ(k)(x) denotes the kth derivative of Φ(·) evaluated at thepoint x. Moreover, the definitions of β(q, p) and Δβ (q, p) are

β(q, p) =1N

ejαSNa (q)Φ(p), (15)

and

Δβ (q, p) =1N

ejα∞∑

k=1

δk

k!Φ(k)(p)ΨN

k (q), (16)

where α = 2πp/δ, SNa (x) = (sin Nπx)/(N sin πx), and

ΨNk (x) =

N −1∑n=0

ej2π (n−N −12 )x

[(n − N − 1

2

)x

]k

. (17)

For the three carrier frequency distribution instances, whenmn is Gaussian distributed, Φ(x) = e−2π 2 σ 2 x2

(σ2 is the


variance), and the 1st-order Taylor approximation ofβ(q, p) is

βG (q, p) ≈ 1N

ejα · e−2π 2 σ 2 p2 [SN

a (q) − 4δπ2σ2pΨN1 (q)

].

(18)Besides, for continuous and discrete uniform distributions, the1st-order Taylor approximations of β(q, p) can be achievedthrough substituting g(mn ) = 1/M , mn ∈ [−M/2,M/2],and g(mn ) = 1/M , mn ∈ {−(M − 1)/2,−(M − 1)/2 +1, . . . , (M − 1)/2] into (13-16), respectively as

βC (q, p) ≈ 1N

ejα

[SN

a (q) · sin(Mπp)Mπp

+

Mπp cos(Mπp) − sin(Mπp)Mπp2 δΨN

1 (q)]

(19)

and

βD (q, p) ≈ 1NM

ejα

[SN

a (q) · 1M

SMa (p)+

Mπ cos(Mπp) sin(πp) − π sin(Mπp) cos(πp)M sin2(πp)

· δΨN1 (q)

]. (20)

Definition 1: The Kth-order-approximation condition (AC)is

∀k ≥ K, (Nδ)kΦ(k)(x) ≈ 0 and[NδΦ(1)(x)

]k ≈ 0. (21)

As usually supposed in previous studies, such as [15], thefrequency increment Δf is far less than fc , which means Nδ isusually less than 1. Besides, the moment generation functionsof common frequency diverse distributions changes moderately,which means Φ(k)(p) is not very large. For the mean of thebeampattern, if the 1st-order-AC is satisfied, |Δβ (q, p)| ≈ 0.Then according to (13), the direction variable q and range vari-able p can regarded as decoupled. Moreover, it is known that thedirection beampattern of a traditional N -element linear array is

β(q) =sin Nπq

N sin πq= SN

a (q),

and the range response (the “range beampattern”) of a randomfrequency radar is

β(p) =∫

mn ∈Mg(mn )ej2πmn pdmn = Φ(p).

Hence the mean of an RFDA’s beampattern can be approxi-mately regarded as the Cartesian product of the direction beam-pattern and range beampattern, which also implies that theRFDA has the ability to resolve the target direction and rangesimultaneously. Moreover, the direction and range resolution ofRFDA can be respectively evaluated by SN

a (q) and Φ(p).Fig. 3 gives an example of the beampattern mean of RFDAs

with different kinds of distributions. The averaged beampatternof 10,000 Monte Carlo trials (in each trial, the carrier frequen-cies of all the array elements are independent samples from thecorresponding distribution.) are listed in the left column, andthe theoretical results are shown in the right column. The com-parison shows that, for all the Gaussian (upper row and (18)),

Fig. 3. The mean of beampatterns (in decibel). The figures in the left columnare the averaged results of 10,000 Monte Carlo trials. The right column showsthe theoretical values which are plotted following the expressions in (18-20)(by the 2nd-order Taylor approximation). The upper, middle, and lower rowsare for the Gaussian, discrete, and continuous uniformly distributed carrierfrequencies, respectively

discrete uniform (lower row and (20)), and continuous uniform(middle row and (19)) distributions, the simulation results matchwell with the theoretical expressions.

B. Variance

Fig. 2 shows that the beampatterns of RFDA antennas havea noise-like sidelobe base. In single target scenarios, the side-lobe base is immaterial for target detection and direction-rangeindication. But in multi-target scenarios, the sidelobe base ofdominant targets may conceal weak targets. Hence it is neces-sary to analysis the peak-sidelobe-base-ratio (PSBR).

As introduced in Subsection II-A, the beampattern can beregarded as a random process. Hence the PSBR correspondingto each {q, p} pair can be measured by the inverse proportionof the beampattern’s variance, whose explicit expression can beprovided by the following derivation.

σ2β (q, p) � Vm {β(q, p)}

= Em {β∗(q, p)β(q, p)} − |β(q, p)|2

=1N

− 1N 2

N −1∑n=0

∣∣∣∣Φ(

p + δq

(n − N − 1

2

))∣∣∣∣2

= σ2β (q, p) + Δσ 2

β(q, p), (22)

where

σ2β (q, p) =

1N

− 1N

|Φ(p)|2 , (23)

and

Δσ 2β(q, p) = − 1

N 2

∞∑k=1

1(2k)!

Φ2(2k)(p)δ2k q2k

N −1∑n=0

(n − N − 12

)2k , (24)


Fig. 4. The variances of beampatterns (in decibel). All the subfigures in the leftcolumn are counted from 10,000 Monte Carlo trials. And the right column showsthe theoretical results provided in (22) (by the 2nd order Taylor approximation).The upper, middle and lower rows are the results for Gaussian, discrete, andcontinuous uniformly distributed carrier frequencies, respectively.

where Φ2(k)(x) denotes the kth derivative of Φ2(·) evaluated atthe point x.

For (24), it is easy to verify that if the 2nd-order-AC is satis-fied, |Δσ 2

β(q, p)| approaches zero. Thus

σ2β (q, p) ≈ σ2

β (q, p) =1N

− 1N

|Φ(p)|2 . (25)

Equation (25) shows that the PSBR of an RFDA dependsmainly on the element number, N , and the moment-generatingfunction, Φ(p). Furthermore, this equation also implies that,under the 2nd-order-AC,

1) The variance σ2β (q, p) is a function of p, which means

that the PSBR is determined only by the range. Differentdirections with the same range have the same PSBR;

2) When |Φ(p)| is sufficiently small, the PSBR approachesN , which means the larger the element number, the largerthe PSBR. It will also be easier to unmask weak targetsfrom sidelobes of dominant targets.

3) The variance σ2β (q, p) is equal to zero when p = 0, which

means the beampattern is deterministic when the rangedifference is zero, and

β(q, 0) ≈ β(q, 0) ≈ 1N

SNa (q). (26)

The variances of beampatterns are illustrated in Fig. 4. Allthe simulated variances, which are counted from the results of10,000 Monte Carlo trials, are displayed in the left column,and the theoretical values given by (22) are on the right. Thisillustration shows that, for the variances of beampatterns, thesimulation results also match the theoretical expressions well.

C. Asymptotic Distribution

In subsection III-A and III-B, the mean and variance of thebeampattern are given to evaluate the resolution and PSBR of

RFDA. An exact distribution of the beampattern is preferred ifmore properties are required. However, it is quite difficult togive an explicit form of the beampattern’s distribution. Alterna-tively, we will give an asymptotic distribution of the beampat-tern, which is an acceptable approximation when the number ofarray elements is sufficiently large. Some of the derivations areinspired by [17], and we expand the approach to a 2D case forRFDA.

According to (11), β(q, p) can be interpreted as a sum of Nindependent random variables yn (q, p), where

yn (q, p) =1N

ejαδ ej2π (n−N −12 )q ej2πmn [p+δq(n−N −1

2 )], (27)

and

β(q, p) =N −1∑n=0

yn (q, p). (28)

Then the mean and variance of yn (q, p) are

Emn{yn (q, p)} =

∫mn ∈M

yn (q, p)g(mn )dmn

=1N

ejαej2π (n−N −12 )q

·Φ(

p + δq

(n − N − 1

2

)), (29)

Vmn{yn (q, p)} = Emn

{|yn (q, p)|2} − |Emn{yn (q, p)}|2

=1

N 2 − 1N 2 |Φ(p)|2 −

1N 2

∞∑k=1

(δq)2k

(2k)!Φ2(2k)(p)

(n − N − 1

2

)2k

.

(30)

It can be validated that yn (q, p) can satisfy the sufficientcondition of the Lyapunov central limit theorem [18]. Hence thenormalized sum of yn (q, p), given by

N −1∑n=0

yn (q, p) − Emn{yn (q, p)}√

Vmn{yn (q, p)} ,

approaches the standard normal distribution when the elementnumber N is sufficiently large.

So according to the Lyapunov central limit theorem, theasymptotic distribution of β(q, p) can be regarded as complexGaussian. Denote β1(q, p), β2(q, p) as the real and imaginaryparts of the β(q, p), and we have the following theorem.

Theorem 1: Random vector β(q, p) � [β1(q, p), β2(q, p)]T

is asymptotically joint Gaussian distributed. The mean and co-variance matrix of β(q, p) are

Em {β(q, p)} =1N

[SN

a (q)Φ(p) +∞∑

k=1

δk

k!Φ(k)(p)ΨN

k (q)

]

·[

cos αsin α

], (31)


and

Mβ(q, p)

� Em

{[β(q, p) − E{β(q, p)}][β(q, p) − E{β(q, p)}]T

}

=[

A BB C

], (32)

where

A = σ2r cos2 α + σ2

i sin2 α − 12Δ� sin 2α, (33)

B = (σ2r − σ2

i ) sin α cos α +12Δ� cos 2α, (34)

C = σ2r sin2 α + σ2

i cos2 α +12Δ� sin 2α, (35)

and

σ2r =

12N

[1 − Φ2(p) − SN

a (2q)N

(Φ2(p) − Φ(2p)

)]

+12[Δσ 2

β(q, p) + Δ�(q, p)

], (36)

σ2i =

12N

[1 − Φ2(p) +

SNa (2q)N

(Φ2(p) + Φ(2p)

)]

+12[Δσ 2

β(q, p) − Δ�(q, p)

], (37)

Δ�(q, p) =1

N 2

∞∑k=1

δ2k

(2k)!ΨN

2k (2q)

·[Φ(2k)(2p) − 122k

Φ2(2k)(p)], (38)

Δ�(q, p) =1

N 2

∞∑k=1

δ2k−1

(2k − 1)!ΨN

2k−1(2q)

·[Φ(2k−1)(2p) − 122k−1 Φ2(2k−1)(p)

]. (39)

Proof: The proof can be found in Appendix. �With the asymptotic distribution of the beampattern, we can

derive Corollary 1 for the sidelobe level of the RFDA.Corollary 1: If an RFDA satisfies the 1st-order-AC, for a

direction difference q = 1/2N , the complementary cumulativedistribution function of the sidelobe magnitude at {q, p} satisfies

Pr{|β(q, p)| > r} ≈ Q1

(a

τ,r

τ

), (40)

where Q1(x, y) is the first-order Marcum Q-function,

Q1(x, y) =∫ ∞

b

te−t 2 + x 2

2 I0(xt)dt,

and I0(·) is the modified Bessel function of the first kind of zeroorder, and

a =1N

∣∣SNa (q)Φ(p)

∣∣ ,

τ =

√1 − Φ2(p)

2N.

Fig. 5. A brief illustration of the zero-padding-2DFFT method, where m0 =−(M − 3)/2, m1 = −(M − 1)/2, . . . , mN −1 = (M − 3)/2.

Proof: Under the 1st-order-AC, it can be verified that the mag-nitudes of Δ� and Δ� approximate to zero. Then the corollarycan be achieved directly by combining Theorem 1 and equation(2.18) of [19]. �

IV. SIGNAL PROCESSING METHODS

The beampattern provided in last section shows that, inRFDA, the target direction and range can be decoupled. In thissection, we will introduce two signal processing methods fortarget indication and direction/range estimation.

A. Matched Filtering

Matched filtering is an effective way to obtain the highestfiltered SNR and the intrinsic resolution of waveforms [20].According to the equivalence between matched filtering resultsand the beampattern [2], the uncoupled direction-range indica-tion and direction/range resolution can be achieved by matchingfiltering, especially in single target scenarios.

In this paper, we omit the original matching-filtering algo-rithm because it is straightforward. However, for the discreteuniform distribution of carrier frequencies, if the 1st-order-ACis satisfied, another computation effective alternative, namedzero-padding and 2D fast Fourier transformation (zero-padding-2DFFT), can be employed to implement of matched filtering.The main steps of this method are

1) Step 1: Formulate an all-zero M × N data matrix.2) Step 2: For all the n = 0, 1, . . . , N − 1, fill the

(mn +

(M − 1)/2)th row, nth column entry with the baseband

echo sample bn , and leave all the other entries zero.3) Step 3: Apply two 2DFFT on the filled data matrix, and

the output can approximate the matched filtering result. Afiner result will be achieved via FFTs with more points.

The zero-padding-2DFFT method and its results are illus-trated in Fig. 5. With the effectiveness of FFT, the new proposedalgorithm can reduce the computation load from O(MN 2) (dueto the inner products between the received signal vector r and thearray steering vectors of all the possible ranges and directions.)to O(NM · log M + MN · log N).

B. Compressive Sensing for Direction and Range Indication

As mentioned, in multi-target scenarios, the strong targets’sidelobe bases may mask weak targets. In this subsection, we


adopt the sparse recovery algorithms for compressive sensingto solve this problem.

First, the unambiguous extent of direction and unambiguousextent of range are uniformly divided into P and Q units, re-spectively (For the ambiguous direction and range problem, thereader can refer to [15] and [21] for more information.). Therewill be PQ direction-range pairs, {θi, ri}P Q

i=1 , in the unambigu-ous range-direction extent.

Define a PQ × 1 vector x(l), whose ith entry is the reflectionmagnitude of a target located at {θi, ri}, at the lth snapshot.The observing matrix Φ is N × PQ, and its ith column isdetermined by (6). Then in the single snapshot case (named thesingle measurement vector (SMV) in the compressive sensingrealm), the noiseless echo can be rewritten in a matrix form:

r(l) = Φx(l). (41)

And if the observations are noisy, an additive noise vector vneeds to be added on the right side of above equation, as

r(l) = Φx(l) + v. (42)

If there are L (L > 1) snapshots, the noisy echoes can beexpressed for the multiple measure vector (MMV) scenario as

R = ΦX + V, (43)

where R = [r(1), r(1), . . . , r(L)], X = [x(1),x(1), . . . ,x(L)],and V is the receiver noise matrix.

For the noiseless case, the compressive sensing can be ac-complished by basis pursuit (BP) [22] for sparse recovery of thetargets:

x(l) = argminx(l)

‖x(l)‖1 , s.t. y = Φx. (44)

And for the noisy case, the BP changes to quadratic constrainedbasis pursuit (QCBP) [23]:

x(l) = argminx(l)

‖x(l)‖1 , s.t. ‖y − Φx(l)‖2 < σl. (45)

Greedy methods, such as subspace pursuit (SP) [24], are at-tractive alternatives for reducing the computation load whilemaintaining the recovery quality. For the SMV scenario, thesparsest estimate of x(l) can be achieved by

min ‖x(l)‖0 , subject to ‖r(l) − Φx(l)‖2 ≤ σl. (46)

For the MMV scenario, we choose the ‖ · ‖2,0 norm to main-tain the consistency of the targets’ locations in all the snapshots.The estimate of X is

min ‖X‖2,0 , subject to ‖R − ΦX‖F ≤ σL , (47)

where ‖ · ‖F is the Frobenius norm. In (46) and (47), σl and σL

are the error tolerances determined by the noise power.In compressive sensing, correct recovery can be guaranteed

with high probability if the observing matrix Φ has a small mu-tual coherence [16]. Mutual coherence is defined as the maximalnormalized inner product between different columns of the ob-serving matrix

μ � maxi �=h

∣∣[Φ]Hi [Φ]h∣∣∥∥[Φ]i

∥∥2

∥∥[Φ]h∥∥

2

. (48)

As with (10), the mutual coherence is equal to the highest side-lobe in the beampattern. The result in Fig. 2 shows that, incomparison with the LFDA, the highest sidelobe level of theRFDA is fairly low. This property means that the RFDA, whichcan be regarded as a random sparse sampling in the spatial-frequency domain, is suitable for compressive sensing. An in-vestigation of the RFDA’s mutual coherence will be provided inSubsection V-B.

In this work, the SP, FOCUSS [25], and their MMV exten-sions, the Generalized Subspace Pursuit (GSP) [26] and the M-FOCUSS [27], are adopted to recover the targets in both SMVand MMV scenarios. The demonstration and performance com-parison of different sparse recovery algorithms will be given inSection VI.

V. PERFORMANCE ANALYSIS

In previous sections, it was shown that the direction and rangeare decoupled in the beampattern of an RFDA, and the direc-tions and ranges of targets can be estimated via matched fil-tering or compressive sensing. In this section, we will provideperformance bounds of the RFDA, by deriving the CRB of theestimation error and the mutual coherence of the observing ma-trix. The CRB can be used to evaluate the location accuracy, andthe mutual coherence is a helpful metric to determine the sparserecovery performance.

A. Cramer-Rao Bound

Because the mean square errors (MSEs) of the un-biasedestimation are lower bounded by the CRB [28], we will providethe CRB of the direction/range estimation, which can be adoptedas a performance guarantee of the RFDA.

Since the target number is P , the directions and ranges of allthe targets can be combined into a 2P × 1 parameter vector, ξ =[θ1 , r1 , θ2 , r2 . . . , θP , rP ]T . According to the signal model in(9), and assuming the receiver noise is complex additive-white-Gaussian-noise (AWGN), the logarithmic likelihood functionof ξ is

L (r(l); ξ) = C − 1σ2

n

L−1∑l=0

∣∣∣∣∣r(l) −P∑

i=1

αi(l)bD (θi) � bR (ri)

∣∣∣∣∣2

,

(49)where C is a constant and σ2

n is the receiver noise power.Then the MSE matrix of ξ (ξ is an unbiased estimation of ξ)

satisfies the information inequality [28]

E{

(ξ − ξ)(ξ − ξ)H}� J−1

ξ , (50)

where Jξ is the Fisher information matrix (FIM) of ξ.With the results in [29], the FIM can be calculated by

Jξ =2L

σ2n

�{C �

(ST ⊗

[1 11 1

])}, (51)

where S is a P × P matrix, whose i, jth entry is

[S]ij =1L

L∑l=1

αi(l)α∗j (l), (52)


and

C = DH P⊥AD = (P⊥

AD)H P⊥AD. (53)

In (53),

D =[∂b(θ1 , r1)

∂θ1,∂b(θ1 , r1)

∂r1,∂b(θ2 , r2)

∂θ2,∂b(θ2 , r2)

∂r2,

. . . ,∂b(θP , rP )

∂θP,∂b(θP , rP )

∂rP

], (54)

and A is an N × P matrix, given by

A = [b(θ1 , r1),b(θ2 , r2), . . . ,b(θP , rP )], (55)

where P⊥A is the projection matrix onto the orthogonal comple-

ment of A’s column space:

P⊥A = I − A(AH A)−1AH , (56)

and I is the identity matrix.According to the above definitions, Jξ is a block matrix with

P × P blocks, and

Jξ =2L

σ2n

·�

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎣

[S]11 G11 [S]12 G12 · · · [S]1P G1P

[S]21 G21 [S]22 G22 · · · [S]2P G2P

......

. . ....

[S]P 1 GP 1 [S]P 2 GP 2 · · · [S]PP GPP

⎤⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

.

(57)

In (57), each Gij is a 2 × 2 block, and its entries can be calcu-lated through

[Gij ]11 = pHθipθj

,

[Gij ]12 = pHθiprj

,

[Gij ]21 = pHripθj

,

[Gij ]22 = pHriprj

, (58)

where

pθi= P⊥

A∂b(θi, ri)

∂θi= −j

4πfcd cos θ

cP⊥

A [n � b(θi, ri)],

and

pri= P⊥

A∂b(θi, ri)

∂ri= −j

4πΔf

cP⊥

A [m � b(θi, ri)].

In the above two equations, n = [−(N − 1)(1 + δm0)/2,−(N − 3)(1 + δm1)/2 . . . , (N − 1)(1 + δmN −1)/2]T .

The CRB can be calculated by directly inverting the Jξ in(57). However, if the reflection amplitudes of different targetsare statistically un-correlated, we can get a more explicit andintuitive expression of the CRB.

For targets with un-correlated amplitudes, if the snapshotnumber L is sufficiently large,

[S]ij =1L

L−1∑l=0

ai(l)a∗j (l) ≈ 0, where i �= j. (59)

Then the off-diagonal blocks in (57) are zero, and the FIM isreduced to a diagonal block matrix. The inverse of Jξ is equalto the inversion of each diagonal block. In this case, the CRBsof the ith target’s direction and range estimates are

CRBθi=

σ2nc2

2L[S]ii(4πfcd cos θ)2γ|P⊥

A (m � b(θi, ri))|2 ,(60)

and

CRBri=

σ2nc2

2L[S]ii(4πΔf)2γ|P⊥

A (n � b(θi, ri))|2 , (61)

where

γ = |P⊥A (m � b(θi, ri))|2 |P⊥

A (n � b(θi, ri))|2

− 12|(m � b(θi, ri))H P⊥

A (n � b(θi, ri))|2

− 12�{[

(m � b(θi, ri))H P⊥A (n � b(θi, ri))

]2}.

(62)

Remarks: There are intuitive explanations of the results in(60-62). Due to the Cauchy-Schwartz inequality,

|P⊥A (m � b(θi, ri))|2 |P⊥

A (n � b(θi, ri))|2

≥ |(m � b(θi, ri))H P⊥A (n � b(θi, ri))|2 , (63)

and

|(m � b(θi, ri))H P⊥A (n � b(θi, ri))|2

≥ �{[

(m � b(θi, ri))H P⊥A (n � b(θi, ri))

]2}. (64)

Thus γ is no smaller than 0. Furthermore, the equalities in(63) and (64) will be valid when m = n. In the LFDA, thetransmitting frequency shifts linearly among successive arrayelements. If N 2δ is small, m and n will be approximately equal,which makes γ ≈ 0 and the CRBs approach infinity. However,in the RFDA, m is a random vector. Hence γ �= 0, and the CRBsare limited. This result explains in another perspective, why theestimation of direction and range in an RFDA are un-aliased.

B. Mutual Coherence Based Performance Analysis

As shown in Subsection IV-B, sparse recovery algorithms forcompressive sensing are effective in target location and sidelobeelimination in the RFDA. In this subsection, a mutual coherencebased guarantee will be provided for exact recovery in noiselesscases, and for the reconstruction error in noisy cases.

In compressive sensing, the restricted isometry coefficient(RIC) is an effective metric to evaluate the recovery perfor-mance. However, finding the RIC of an observing matrix isexhaustive. In this paper, we adopt mutual coherence, whichis much easier to compute, as an acceptable metric [16] forderiving the recovery performance guarantee.

As defined in (48), the calculation of mutual coherence isbased on the inner products between the columns of the ob-serving matrix. According to (6), these columns of the RFDAare determined by the directions and ranges of potential targets.However, it is quite difficult to give a closed form of mutual


coherence for arbitrary directions or ranges. In this work, wewill give the performance guarantee when potential targets arelocated at grid intersections, which we will call simply “grids”.

Definition 2: The grids are directions and ranges satisfyingthe following two equalities:

SNa

(2fcd sin θ

c

)= 0 or N,

and

Φ(

2Δfr

c

)= 0 or 1.

According to Definition 2, there are N direction grids from−π/2 to π/2, expressed as

θk = arcsinck

2fcdN, k = 0, 1, . . . , N − 1.

However, the range grids depend on the distribution of mn . Forthe discrete uniform distribution, there are M direction gridsfrom 0 to cΔf/2, expressed as

ri =cΔfi

2M, i = 0, 1, . . . ,M − 1.

Then if the 1st-order-AC is satisfied, there are M × N grids inan RFDA. Comparing (48) with (10), the mutual coherence ofthe RFDA is equal to the the beampattern’s highest sidelobe,where the directions and ranges are at grids.

With Theorem 1, we can find that if the 1st-order-AC is satis-fied, the real and imaginary parts of the beampattern are jointlyGaussian distributed. When it comes to mutual coherence, wehave the following lemma.

Lemma 1: Under the 1st-order-AC, if the carrier frequenciesare discrete uniformly distributed, and all the potential targetsare located at grids, the cumulative distribution of the mutualcoherence of Φ satisfies the following inequality:

Pr{μ < r} ≥ 1 − (M − 1)Ne−N r 2. (65)

Proof: With Definition 2 and Theorem 1, it can be seen thatthe real and imaginary parts of sidelobes at grids (except whereΦ(p) = 1, because according to (13) and (22), these sidelobesare deterministic and equal to zero) are zero-mean, i.i.d. Gaus-sian. Their variances are

σ2r = σ2

i =1

2N. (66)

Then the magnitudes of these sidelobes are Rayleigh dis-tributed [19], where

Pr{|β(q, p)| > r} ≤ e−N r 2. (67)

The number of grids where Φ(p) �= 1 is (M − 1)N , whichmeans the probability that the highest sidelobe among the (M −1)N grids is less than r is

Pr{μ < r} ≥ (1 − e−N r 2)(M −1)N , (68)

when e−N r 2 � 1. Furthermore, the right side of (68) can beapproximated as

(1 − e−N r 2)(M −1)N ≈ 1 − (M − 1)Ne−N r 2

. (69)

Substituting (69) into (68), Lemma 1 is proven. �With the above lemma, we have the following guarantee for

exact recovery in the noiseless case.Theorem 2: If the carrier frequencies are discrete uniformly

distributed, and all the potential targets are located at grids,by using BP, the RFDA which satisfies the 1st-order-AC canexactly reconstruct K targets with a probability higher than1 − ε (ε ≤ 1) in the noiseless case, where

K ≤ 12

(1 +

√N

ln(MN − N) − ln ε

). (70)

Proof: According to the corollaries in [30], if the mutualcoherence

μ ≤ 12K − 1

, (71)

the K non-zero entries of x can be exactly recovered in thenoiseless case. Then with Lemma 1, and by substituting r =1/(2K − 1) into (65), we have that the probability of exactrecovery is

Pr(x = x) ≥ 1 − (M − 1)Ne−N ( 12 K −1 )2

. (72)

Thus the probability of exact recovery is larger than 1 − ε, if

1 − (M − 1)Ne−N ( 12 K −1 )2

≥ 1 − ε. (73)

Theorem 2 is proven, because it is easy to verify that (73) isequivalent to (70). �

Corollary 2: Under the 1st-order-AC, if the target number,K, satisfies (70), then in a noisy case, where the noise powerof each array element is σ2

n , the reconstruction error of QCBPsatisfies the following constraint, with a probability higher than1 − ε:

‖x − x‖2 ≤√

3(1 + η)1 − (2K − 1)η

(σl + σn ), (74)

where η =√

(ln N + ln(M − 1) − ln ε)/N .Proof: Corollary 2 can be proven by directly combining The-

orem 2 in this paper and Theorem 2.1 in [31]. �Theorem 2 gives a sufficient condition for exact recovery

in noiseless cases, and Corollary 2 gives an upper bound ofreconstruction error in noisy cases. We noted that these twobounds are quite loose in practice, and are seeking tighter results.

VI. NUMERICAL RESULTS

Numerical simulations verify the results achieved in thiswork, and demonstrate the merit of the new proposed FDAstructure. In all the simulations, a linear array with N = 128elements was chosen as the archetype of the RFDA. The cen-ter carrier frequency fc was 3 GHz, and the frequency incre-ment Δf = 1 MHz. The inter-element distance d = 0.025 m,which equals the quarter wavelength. For carrier frequency dis-tributions that are discrete and continuous uniform, the parame-ter M = 64. For Gaussian distributions, the standard deviationσ = 18.5, where the root-mean-square bandwidth [28] is equalto those of the previous uniform distributions.


Fig. 6. The differences between the variances of the real and imaginary partsof the beampatterns (in decibel). The right column shows the theoretical valuesprovided in (83) (by the 2nd order Taylor approximation). The upper, middle,and lower rows are the results for Gaussian, discrete, and continuous uniformlydistributed carrier frequencies, respectively.

In the results, the expectations w.r.t. the random vector mare all calculated via 10,000 Monte-Carlo trials with differentsample tracks of m, unless otherwise specified. However, con-sidering the massive computation time, in Subsection VI-B andSubsection VI-C we conducted just 1,000 trials, with differentnoise but the same m, to find the averaged successful recoveryrate and MSE for each SNR point.

With the above setup, simulations of the beampatterns’asymptotic distribution, target detection performance, andCRB/MSE of direction/range estimation were conducted. Re-sults are provided in the following subsections.

A. Asymptotic Distribution

Because of the difficulties in direct verification of Theo-rem 1, we decided to verify that the stochastic characteristicsof ρ(q, p) = e−jαβ(q, p) match the results given in (78), (82),and (83). We also verified that the real and imaginary parts ofρ(q, p) are Gaussian distributed.

The verifications of (78) and (82) are similar to the simula-tions of the mean and variance of the beampattern. The resultsare omitted here because they are almost the same as those wereshown in Fig. 3 and Fig. 4.

For the verification of (83), we recorded the real and imag-inary parts of ρ(q, p) in each trial. The differences betweenthe variance of �{ρ(q, p)} and �{ρ(q, p)} are shown in theleft column of Fig. 6, and the theoretical values are displayedon the right. The simulated and analytical results match well.Moreover, it can be verified that the cross correlations between�{ρ(q, p)} and �{ρ(q, p)} for different {q, p} pairs also matchΔ�(q, p) very well. Based on these facts, one can conclude that(83) is verified.

Finally, we conducted the Kolmogorov-Smirnov (KS) test[32] for the normalized �{ρ(q, p)} and �{ρ(q, p)}. When the

Fig. 7. Target direction and range location with an RFDA. (a) Beamformingresult, (b) Compressive sensing result.

trial number reached 10,000, all the {q, p} pairs passed the testat a 5% significance level.

With above results, the expression of asymptotic distributionof the beampattern in Theorem 1 can be considered as verifiedin this simulation setup.

B. Target Detection Performance of Compressive Sensing

Simulation results for the target recovery performances ofcompressive sensing are provided in this subsection.

The first simulation gives an example of target indication.The discrete uniform frequency diverse distribution was adoptedhere, and there were three targets at different ranges anddirections ( θ1 = −30o , r1 = 10 m; θ2 = 5o , r2 = 70 m; θ3 =60o , r3 = 120 m.). The amplitudes of Target 1 and Target 2were identical and 20 dB larger than that of Target 3. The SNRof Target 3 was 0 dB (measured at the input of each receiver).Data was collected from only one snapshot. The beamformingresult is shown in Fig. 7(a). The ranges and directions of Tar-gets 1 and 2 are correctly indicated. But compared with thefirst two, Target 3 was too weak, and was masked by sidelobes.However, in the compressive sensing result (shown in Fig. 7(b),the SP algorithm was used with a sparsity of six.), all the threetargets were successfully detected in all of our 1,000 trials, andtheir locations were correctly indicated as well.

The second simulation was performed to evaluate the recov-ery performances of different compressive sensing algorithms.There were two targets with identical reflection amplitudes, butdifferent locations. The input SNR (measured on each arrayelement) varied from −24 dB to 6 dB. In the simulation, therecovered support set was composed of the indices of the twostrongest entries in x. A successful recovery was defined asexact coincidence between the recovered and the true supportsets. The results are shown in Fig. 8. In the comparison betweenthe SMV and MMV scenarios, the recovery performances ofthe MMV are better for both types of algorithms. In SMV sce-narios, an 100% successful recovery can be expected if the


Fig. 8. Successful recovery rates of different compressive sensing algorithms.

Fig. 9. MSEs of direction estimates obtained with ML estimator, comparingwith the corresponding CRBs given by (60).

SNR is larger than −7 dB2. And the SNR requirement is about8 dB lower in MMV scenarios, because more observations canmake the recovery more stable [26]. In the comparison of re-covery algorithm types, FOCUSS and M-FOCUSS outperformtheir subspace pursuit counterparts in both SMV and MMVscenarios.

C. CRB and MSE of Direction/Range Estimation

In Fig. 9 and Fig. 10, the CRBs provided in (60) and (61) arecompared with corresponding MSEs of the direction and rangeestimates obtained via maximum likelihood (ML) estimation,for various SNRs. The amplitudes of the targets were kept con-stant, and noise power was varied to get each of the data points.A single target scenario was considered in this simulation.

The results are averaged w.r.t. different sample tracks of therandom vector m. It can be seen that the theoretical values

2The successful recovery ratio seemed to be even higher than the detectionprobability of matched filtering. An intuitive explanation is that, in this simula-tion, the algorithms had already known the number of targets (sparsity number).Thus the exact recovery of the support set is easier than the correct hypothesisdetection from noisy observations.

Fig. 10. MSEs of range estimates obtained with ML estimator, comparingwith the corresponding CRBs given by (61).

of CRB match the MSEs well in high SNR situations (SNR> 0 dB). This result validates the correctness of the estimationerror bounds provided in Subsection V-A.

VII. CONCLUSION

We proposed a new frequency diverse array structure, namedRFDA, to locate targets’ directions and ranges without coupling.By randomly assigning the carrier frequencies of array elements,the RFDA realizes a random sparse sampling of the targets’information simultaneously in the spatial-frequency domainwith a low system complexity. The beampattern of the RFDAis thumbtack-like, but with random sidelobe bases. Stochas-tic characteristics of the beampattern, the mean, variance, andasymptotic distribution were analytically derived, and two signalprocessing algorithms were introduced. In addition, the Cramer-Rao bounds for direction/range estimation, as well as mutualcoherence based limits for compressive sensing, were providedas performance guarantees of this new array. Numerical simu-lations demonstrated the RFDA’s performance and verified thetheoretical results.

APPENDIX

According to (30), the variance of yn (q, p) is independent ofn, hence the sum of {yn (q, p)}N −1

n=0 is asymptotically complexGaussian distributed [18]. Then the mean of β(q, p) is

Em

{[β1(q, p)β2(q, p)

]}=[�{β(q, p)}�{β(q, p)}

]. (75)

Substituting (13) into (75), and noticing that g(mn ) is even, wehave

�{β(q, p)} =1N

SNa (q)Φ(p) cos α

�{β(q, p)} =1N

SNa (q)Φ(p) sin α. (76)

The expression of Mβ can be derived as follows. De-note zn (q, p) = e−jαyn (q, p) = 1/N · ej2π (n−(N −1)/2)q


ej2πmn [p+δ(n−N −12 )q ] , and then

ρ(q, p) �N −1∑n=0

zn (q, p)

= e−jαβ(q, p)

=1N

N −1∑n=0

ej2π (n−N −12 )q ej2πmn [p+δ(n−N −1

2 )q ].

(77)

Moreover, the mean of ρ(q, p) can be achieved directly by

ρ(q, p) � Em {ρ(q, p)} =1N

SNa (q)Φ(p) + Δρ(q, p), (78)

where

Δρ(q, p) =1N

∞∑k=1

1k!

Φ(k)(p)δkΨNk (q). (79)

By defining σ2r , σ2

i as the variances, and σri as the covarianceof or between the real and imaginary parts of ρ(q, p), we havethat

Em{|ρ(q, p) − ρ(q, p)|2} = σ2

r + σ2i , (80)

and

Em{[ρ(q, p) − ρ(q, p)]2

}= σ2

r − σ2i + j2σri. (81)

Between ρ(q, p) and β(q, p), the only difference is the phasefactor ejα , so they have same variances, given by

Em{|ρ(q, p) − ρ(q, p)|2} = σ2

β (q, p). (82)

In addition, the explicit expression of Em{[ρ(q, p) − ρ(q, p)]2} can be derived by

Em{[ρ(q, p) − Em{ρ(q, p)}]2}

= Em{ρ2(q, p)

}− ρ2(q, p)

=1

N 2 Em

{N −1∑l=0

N −1∑n=0

ej2π (n−N −12 )q ej2πmn [p+δ(n−N −1

2 )q ]

·ej2π (l−N −12 )q ej2πml [p+δ(l−N −1

2 )q ]}− ρ2(q, p)

=1N

ρ(2q, 2p) − 1N 2

N −1∑n=0

ej2π (2q)(n−N −12 )

·Φ2(

p + δ

(n − N − 1

2q

))

=1N

ρ(2q, 2p) − 1N 2

N −1∑n=0

ej2π (2q)(n−N −12 )[Φ2(p)

+∞∑

k=1

1k!

δkqk

(n − N − 1

2

)k

Φ2(k)(p)]

=1

N 2 SNa (2q)[Φ(2p) − Φ2(p)] + Δ�(q, p) + jΔ�(q, p),

(83)

where Δ�(q, p) and Δ�(q, p) are defined in (38) and (39), re-spectively. Substituting (82-83) into (80-81), and noting thatSN

a (·) and Φ(·) are real, one finds that

σ2r =

12N

[1 − Φ2(p) − SN

a (2q)N

(Φ2(p) − Φ(2p)

)]

+12[Δσ 2

β(q, p) + Δ�(q, p)

], (84)

σ2i =

12N

[1 − Φ2(p) +

SNa (2q)N

(Φ2(p) − Φ(2p)

)]

+12[Δσ 2

β(q, p) − Δ�(q, p)

]. (85)

andσri =

12Δ�(q, p). (86)

Hence, the covariance matrix of the real and imaginary parts ofρ(q, p) is

Mρ(q, p) =

[σ2

r (q, p) Δ�(q, p)/2Δ�(q, p)/2 σ2

i (q, p)

]. (87)

Because β(q, p) = ejαρ(q, p), we have that

Mβ(q, p)

=[

cos α − sin αsinα cos α

]Mρ(q, p)

[cos α sinα− sin α cos α

]

=[

A BB C

], (88)

where A, B, and C were defined in (33), (34), and (35), respec-tively. Theorem 1 is proven. �

ACKNOWLEDGMENT

The authors would like to thank J. Ballard for the proofread-ing, and anonymous reviewers for the valuable suggestions thathave helped to improve the quality of the manuscript.

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Yimin Liu (M’12) received the B.S. and Ph.D. de-grees (both with honors) in electronics engineeringfrom the Tsinghua University, Beijing, China, in 2004and 2009, respectively.

From 2004, he was with the Intelligence SensingLaboratory, Department of Electronic Engineering,Tsinghua University. He is currently an AssociateProfessor at Tsinghua, where his field of activity isstudy on new concept radar and other microwavesensing technologies. His current research interestsinclude radar theory, statistic signal processing, com-

pressive sensing and their applications in radar, spectrum sensing, and intelligenttransportation systems.

Hang Ruan received the B.E. degree from TsinghuaUniversity, Beijing, China, in 2015. He is currentlyworking toward the Ph.D. degree in the Departmentof Electronic Engineering, Tsinghua University. Hismain research interests include radar spectrum shar-ing and signal processing.

Lei Wang was born in Hebei, China, in 1991. Hereceived the B.S. degree in electronic information sci-ence and technology from Xidian University, Xi’an,China, in 2014. He is currently working toward thePh.D. degree in electronic engineering at TsinghuaUniversity, Beijing, China. His research interests in-clude ISAR and recognition in random stepped fre-quency radar.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion,Haifa, Israel and the Ph.D. degree from StanfordUniversity, CA, USA. He is the Eugene and MarthaLohman Professor of electrical engineering in thePreston M. Green Department of Electrical and Sys-tems Engineerin, Washington University (WUSTL),St. Louis, MO, USA. He served as a Chair of this de-partment from 2006 to 2016. Under his departmentchair leadership, the undergraduate enrollment hasmore than tripled in four years and the master’s en-

rollment grew seven-fold in the same time period. He is also a Professor inthe Department of Biomedical Engineering (by courtesy), a Professor in theDivision of Biology and Biomedical Studies, and the Director of the Centerfor Sensor Signal and Information Processing at WUSTL. Prior to serving atWUSTL, he was a faculty member at Yale University and the University ofIllinois at Chicago.

Dr. Nehorai served as an Editor-in-Chief of the IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 2000 to 2002. From 2003 to 2005, he was the VicePresident (Publications) of the IEEE Signal Processing Society (SPS), the Chairof the Publications Board, and a member of the Executive Committee of thisSociety. He was the Founding Editor of the special columns on LeadershipReflections in the IEEE Signal Processing Magazine from 2003 to 2006. Hereceived the 2006 IEEE SPS Technical Achievement Award and the 2010 IEEESPS Meritorious Service Award. He was elected Distinguished Lecturer of theIEEE SPS for a term lasting from 2004 to 2005. He received several best paperawards in IEEE journals and conferences. In 2001, he was named UniversityScholar of the University of Illinois. He was the Principal Investigator of theMultidisciplinary University Research Initiative project titled Adaptive Wave-form Diversity for Full Spectral Dominance from 2005 to 2010. He is a Fellowof the Royal Statistical Society since 1996 and of AAAS since 2012.

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