Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 DOI 10.1186/s13634-015-0217-y

RESEARCH Open Access

Impacts of frequency increment errors onfrequency diverse array beampatternKuandong Gao*, Hui Chen†, Huaizong Shao, Jingye Cai and Wen-Qin Wang†

Abstract

Different from conventional phased array, which provides only angle-dependent beampattern, frequency diversearray (FDA) employs a small frequency increment across the antenna elements and thus results in a rangeangle-dependent beampattern. However, due to imperfect electronic devices, it is difficult to ensure accuratefrequency increments, and consequently, the array performance will be degraded by unavoidable frequencyincrement errors. In this paper, we investigate the impacts of frequency increment errors on FDA beampattern.We derive the beampattern errors caused by deterministic frequency increment errors. For stochastic frequencyincrement errors, the corresponding upper and lower bounds of FDA beampattern error are derived. They are verifiedby numerical results. Furthermore, the statistical characteristics of FDA beampattern with random frequencyincrement errors, which obey Gaussian distribution and uniform distribution, are also investigated.

Keywords: Frequency diverse array (FDA); Frequency increment errors; Beampattern error; FDA beampattern;FDA radar

1 IntroductionBeampattern is widely used to assess the performance ofphased arrays [1]. However, a limitation of phased arrayis that the beam steering is fixed at one angle for allthe ranges [2]. Recently, a flexible array called frequencydiverse array (FDA) has been proposed [3-5]. Differentfrom phased array, a small frequency increment, as com-pared to the carrier frequency, is applied across the arrayelements [6]. This small frequency increment results in arange angle-dependent beampattern [7,8]. Several inves-tigations have been carried on FDA radars. The time andangle periodicity of FDA beampattern was analyzed in [9].A linear FDA was proposed in [10] for forward-lookingradar ground moving target indication. The multipathcharacteristics of FDA radar over a ground plane wereinvestigated and compared with phased array in [11].FDA radar full-wave simulation and implementation withlinear frequency-modulated continuous waveform werepresented in [12,13]. In [14], we have investigated the FDA

*Correspondence: kuandonggao@gmail.com†Equal contributorsSchool of Communication and Information Engineering, University ofElectronic Science and Technology of China, 2008, Road XiYuan, Chengdu,611731 Sichuan, China

Cramér-Rao lower bounds for estimating direction, range,and velocity. More recent work about the applications ofFDA in MIMO radars can be found in [15-19].

Perfect frequency increments are often assumed inexisting literatures [20]. However, in an actual arraysystem, there will have imperfect errors including ele-ment position errors, mutual coupling, phase errors, andfrequency increment errors [21-24]. Some results havebeen reported about the impacts of element positionerror, mutual coupling and phase error on beampattern,and direction-of-arrival (DOA) estimation performances.Since FDA beampattern has similar properties with con-ventional phased array for the impacts of element positionerrors, mutual coupling, and phase errors, this paper con-siders only the impacts of frequency increment errors onFDA beampattern. Since FDA beampattern is dependenton the angle and range, it has a potential for target local-ization , which is different from traditional time-of-arrival(TOA) and angle-of-arrival (AOA)-based localization[25-28]. The contributions can be summarized as follows.(i) More tighter bounds of FDA beampattern deviationare derived. (ii) Statistical analysis of FDA beampatternin terms of expectation value, variance, and probabilitydensity function (PDF) are provided.

© 2015 Gao et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly credited.

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 2 of 12

The rest of this paper is organized as follows. Section 2formulates the data models of FDA radar without andwith frequency increment errors, respectively. Section 3analyzes the impacts of deterministic frequency incre-ment errors on FDA beampattern. Thereafter, the impactsof random frequency increment errors are investigatedin Section 4. Finally, simulation results are provided inSection 5, and conclusions are drawn in Section 6.

2 FDA beampattern without frequency incrementerrors

Suppose an N-element uniform linear FDA with inter-element spacing denoted as d. The radiated frequencyfrom the nth element is as follows:

fn = f0 + n�f , n = 0, 1, 2, . . . , N − 1 (1)

where f0 and �f are the carrier frequency and frequencyincrement, respectively. Taking the first element as thereference for the array, under far-field condition, onemight express the direct wave component of the electricfield emitted from the FDA at the observation point (θ , r)as [17]:

A(θ , r, t) =N−1∑n=0

anςn (θ |wn )ejwn

(t− r+dn sin θ

c0

)

r(2)

where N is the number of FDA elements, an representsthe complex excitation coefficient for the nth element,ςn (θ |wn ) stands for the far-field vector radiation patternfor the nth element at range r and angular frequency wn =2π fn, c0 is the light speed, dn is the element position ofthe nth element reference relative to the first element, andt is the time parameter. In accordance with the far-field

assumption, ejwn(

t− r+dn sin θc0

)/r corresponds to the delayed

carrier with free space loss.To interpret the effect of frequency diversity within the

scope of an array factor, we should factor the vector ele-ment pattern out of Equation 2. This can indeed be doneunder certain conditions. Assuming all elements in theFDA are identical, we can eliminate the frequency depen-dence in the element factor. Since r � dn, we have:

ςn(θ |wn) ≈ ς(θ |w0). (3)where w0 is the carrier angular frequency. So Equation 2can be rewritten as:

A(θ , r, t) = ς(θ |w0 )

N−1∑n=0

anejwn

(t−(

r+dn sin θc0

))

r. (4)

Further simplification becomes possible by consideringparticular FDA arrangements that are simple to handleand yet able to provide valuable insight. A uniform lin-ear FDA utilizing discrete, linear frequency increments issuch a practical configuration, and it is examined in this

section as a special case. By definition, the elements areexcited with uniform amplitude, but they are allowed tohave a phase progression across the array. These specifica-tions translate to the following expressions for dn and an:

dn = nd and an = e−jnφa (5)

where φa stands for the phase progression. SubmittingEquations 1 and 5 into Equation 4 yields:

A(θ , r, t) = ς (θ |w0 )

N−1∑n=0

e−jnφa ej2π(f0+n�f )(

t−(

r+nd sin θc0

))

r

= ς (θ |w0 )

rejϕ0

N−1∑n=0

e−jnφa ej2πn�f(

t−(

r+nd sin θc0

))−j2π f0 nd sin θ

c0

(6)

where ϕ0 = 2π f0t − 2π f0 rc0

. For notation convenience, wedefine ς = ς (θ |w0 ) ej2π f0t−j2π f0 r

c0 , Equation 6 can thenbe rewritten as:

A(θ , r, t) = ς

r

N−1∑n=0

e−jnφa ej2πn(

t�f − r+nd sin θc0

�f −f0 d sin θc0

)

(7)

Since ndsinθ � r, Equation 7 can be reformulated as:

A(θ , r, t) ≈ ς

r

N−1∑n=0

e−jnφa ej2πn(

t�f − rc0

�f −f0 d sin θc0

)

= ς

rwH v(θ , r, t)

(8)

where

w = [ 1 ejφa . . . ej(N−1)φa]T (9)

and

v(θ , r, t) =[

1 ej2π(

t�f − rc0

�f −f0 d sin θc0

). . .

ej2π(N−1)(

t�f − rc0

�f −f0 d sin θc0

) ]T.

(10)

with [ ·]T and [ ·]H being the transpose operator andhermitian transpose operator, respectively. For simplicity,ς = 1 is assumed in the following discussions. Figure 1shows the ideal linear uniform FDA beampattern witht = 1/�f .

2.1 FDA beampattern with frequency increment errorsSuppose the frequency increment errors being ρ =[ 0 ρ1 . . . ρN−1 ], Equation 8 can be rewritten as:

A(θ , r) = wH v(θ , r, t)r

(11)

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 3 of 12

Figure 1 Ideal two-dimension FDA beampattern with N = 16,�f = 30 KHz, t = 1/(3 × 104), φa = 0, and w = 1N×1.

where

v(θ , r, t) =[

1 ej2π(

t(�f +ρ1)− rc0 (�f +ρ1)−f0 d sin θ

c0

). . .

ej2π(N−1)(

t(�f +ρN−1)− rc0 (�f +ρN−1)−f0 d sin θ

c0

)]T

=[

1 ej2π(

t�f − rc0

�f −f0 d sin θc0

)ej2π

(tρ1− r

c0ρ1). . .

ej2π(N−1)(

t�f − rc0

�f −f0 d sin θc0

)ej2π

(N−1)(tρN−1− r

c0ρN−1

)]T.

(12)

Then Equation 11 can be reformulated as:

A(θ , r) = wH v(θ , r, t)r

= wH(v(θ , r, t) � �v(r, t))r

= wHdiag(�v(r, t))v(θ , r, t)r

= wH C(r, t)v(θ , r, t)r

(13)

where

�v(r, t) =[

1 ej2π(

tρ− rc0

ρ)

. . . ej2π(N−1)(

tρ− rc0

ρ)]T

and

C = diag (�v(r, t))

and � denotes Hadamard product. Note that diag(�v(r, t))denotes a diagonal matrix with �v(r, t) being its diagonalelements. For small frequency increment errors, Taylorseries expansion about C is performed as follows:

C(r, t) = I + C+(r, t). (14)

According to Equation 13, it can be known that:

C+(r, t) = diag(

j2π

(− (N � ρ)r

c0+ t(N � ρ)

)

�(

ej2π(− (N�ρ)r

c0+t(N�ρ)

)))+ O(ρ)

≈ diag(

j2π

(− (N � ρ)r

c0+ t(N � ρ)

)

�(

ej2π(− (N�ρ)r

c0+t(N�ρ)

))).

(15)

where N =[ 0, 1, 2, . . . , N − 1] denotes element number ofFDA radar, and O(ρ) represents for the high order termsabout ρ, which can be ignored for Equation 15. UsingEquation 15, we can rewrite the beampattern (13) as:

A(θ , r, t) = wH C(r, t)v(θ , r, t)r

= wH v(θ , r, t) + wH C+(r, t)v(θ , r, t)r

= A(θ , r, t) + wH C+(r, t)v(θ , r, t)r

= A(θ , r, t) + �A(r, t)

(16)

where �A(r, t) = wH C+(r,t)v(θ ,r,t)r denotes the FDA beam-

pattern deviation. Figure 2 shows the FDA beampatternwith frequency increment errors. The array parametersare the same as that for Figure 1. Due to the frequencyincrement errors, the FDA beampattern sidelobes are dif-ferent from Figure 1. Stronger sidelobe peaks will makethe FDA energy scattering and consequently degrade thearray performance.

3 Impacts of deterministic frequency incrementerrors

Firstly, we consider uniform frequency increment errors,i.e., ρ = [0 ρ1 . . . ρN−1] = [0 ρ . . . ρ]. In this case,Equation 13 can be rewritten as:

A(θ , r)= wH diag (�v(r, t)) v(θ , r, t)r

=N∑

n=1

ej2π(n−1)(

t(�f +ρ)− rc0 (�f +ρ)−f0 d sin θ

c0

)

r

= kr

sin(

πN2

(t(�f + ρ

)− rc0

(�f + ρ) − f0 d sin θc0

))sin(

πt(�f +ρ)− r

c0 (�f +ρ)−f0 d sin θc0

2

)

= kr

sin(

πN2

(�f(

t+ ρ�f t)

− �fc0

(r+ ρ

�f r)

− f0 d sin θc0

))

sin(

π�f(

t+ ρ�f t)− �f

c0

(r+ ρ

�f r)−f0 d sin θ

c02

)

(17)

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 4 of 12

Figure 2 Two-dimension FDA beampattern with frequency increment error ρn and ρn ∼ N(0, ρ2) with ρ = 500.

where k = ejπN(

t(�f +ρ)− rc0

(�f +ρ)−f0 d sin θc0

)and w = 1N×1.

Equation 17 arrives the maximum value when:

�f(

t + ρ

�ft)

− �fc0

(r + ρ

�fr)

−(

f0d sin θ

c0

)= 2m

m = 0, 1, 2, . . . .(18)

If no frequency increment error exists, the FDA main-lobe will pass the location

(0◦, c0

�f

)at t = 1

�f . However,when there are frequency increment errors, the location(

0◦, c0�f

)may be not at the beampattern maximum point.

According to Equation 18, the corresponding range erroris as follows:

�r = ρ

�fr. (19)

Since the phase error caused by time error can be equiv-alently regarded as range error and angle error, Equation18 can be reformulated as:

�f(

t + ρ

�ft)

− �fc0

(r + ρ

�fr)

−(

f0d sin θ

c0

)

= �ft + ρ

�ft − �f

c0

(r + ρ

�fr)

−(

f0d sin θ

c0

)

= �ft − �fc0

(r + ρ

�fr − ρc0

2�f 2 t)

− f0dc0

(sin θ − ρc0

2�f f0dt)

= 2m, m = 0, 1, 2, . . .(20)

Therefore, the corresponding range error is as follows:

�r = ρ

�fr − ρc0

2�f 2 t (21)

and the angle error is as follows:

�θ = arcsin(

ρc02�f f0d

t)

. (22)

If the range error and angle error are required to besmaller than 50 m and 0.5◦, respectively, the frequencyincrement error should be smaller than 150 Hz.

Figure 3a,b shows the FDA beampattern with uniformfrequency increment errors ρ, frequency increment �f =30 KHz, and t = 1/�f . It can be noticed that the phaseerror for θ = 0.5◦ is ρ/�f = 5 × 10−3. Note that sincethe range attenuation exists, the range error has a smalldeviation from the calculation of Equation 21.

4 Impacts of stochastic frequency incrementerrors

In this section, we investigate the statistical propertiesof FDA beampattern with random frequency incrementerrors.

4.1 Error boundary of FDA beampatternSuppose the frequency increment errors are random. Inthis case, it is not possible to derive a deterministic beam-pattern expression like Equation 17. Here, we are inter-ested in the maximum beampattern deviation, namely:

maxt,r,θ

|�A(t, r, θ)| . (23)

According to Equation 16 and using the Cauchy-Schwarz inequality, we have:

maxt,r,θ

|�A(θ , r, t)| ≤ 1r‖wmaxC+‖2‖v(θ , r)‖2

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 5 of 12

−50 0 50−40

−35

−30

−25

−20

−15

−10

−5

0

Angle (Degree)

Bea

mpa

ttern

(dB

)

idealρ / Δ f=5e−3ρ / Δ f=2e−3ρ / Δ f=1e−3

−0.5 0 0.5 1−1.7

−1.65

−1.6

−1.55

−1.5

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

−35

−30

−25

−20

−15

−10

−5

0

Range(m)

Bea

mpa

ttern

(dB

)

idealρ / Δ f=5e−3ρ / Δ f=2e−3ρ / Δ f=1e−3

9940 9960 9980 10000 10020−1.65

−1.6

−1.55

−1.5

−1.45

a

b

Figure 3 FDA beampattern profiles with frequency increment errors.(a) In angle dimension. (b) In range dimension.

= 1r‖wmaxC+‖2

√N

≤ 1r|wmax|‖C+‖2

√N

= �Amax

(24)

where |·| and ‖·‖2 denote the absolute value and Euclideannorm, respectively, and wmax denotes the maximum ele-ment of w. Note that compared with [29], |wmax| ≤|w|2, the bound expressed in Equation 24 is much tighter.According to Equation 16, C+ can be rewritten as:

‖C+‖2 =∥∥∥∥diag

((j2π

(− (N � ρ)r

c0+ t(N � ρ)

))

�(

ej2π(− (N�ρ)rc0

+t(N�ρ))))∥∥∥∥

2

=∥∥∥∥diag

(j2π

(− r

c0+ t)

(N � ρ)

)∥∥∥∥2

= 2π

∣∣∣∣− rc0

+ t∣∣∣∣ · ∥∥diag(N � ρ)

∥∥2.

(25)

The maximum deviation, in a linear scale, is the same overthe whole beampattern. When only C+ (and thus C) isknown, or when the influence of unknown factors on C+,e.g., aging or temperature, comes into play, it makes senseto consider as random matrix, with some statistical modelfor the entries of C+. We then have to calculate:

max ‖C+‖2 (26)

for the random matrix C+ to find an upper bound.

max ‖C+‖2 = max 2π

(− r

c0+ t)∥∥diag(N � ρ)

∥∥2

= max 2π

(− r

c0+ t)

‖N � ρ‖2

= 2π

∣∣∣∣− rc0

+ t∣∣∣∣ · ‖N‖2 |ρmax| .

(27)

When applying this result to an array with uniformweighting wmax = 1, Equation 24 leads to:

�Amax = 2π

r

∣∣∣∣− rc0

+ t∣∣∣∣ · ‖N‖2 |ρmax|

√N . (28)

For a constant time t = 1/�f , the time error caused byfrequency increment error will become amplitude error ofFDA beampattern, as shown follows:

�Amax = 2π

r

∣∣∣∣− rc0

+ 1�f

∣∣∣∣ · ‖N‖2 |ρmax|√

N . (29)

It can be noticed from Equation 29 that the errorscaused by time and by range counterbalance the totalbeampattern error. Therefore, we delete 1/�f . Then,Equation 29 can be reformulated as:

�Amax ≤ 2π

r

∣∣∣∣− rc0

∣∣∣∣ ‖N‖2 |ρmax|√

N

= 2π

c0‖N‖2 |ρmax|

√N .

(30)

Equation 30 gives FDA beampattern error bound whichindicates its worst case. Since it is caused by the frequencyincrement errors, the maximum device errors which pro-duce the FDA frequency increment are regulated by thebound. This gives guideline about device selections andpredicts the possible FDA beampattern derivation bound.

4.2 Statistical properties of FDA beampatternIn most cases, we know the statistical properties of fre-quency increment errors and would like to derive therespective properties of impacted beampattern. In thissubsection, we will give the formulas of expectation value,variance, and PDF about FDA beampattern based ondifferent PDFs of the frequency increment errors.

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 6 of 12

4.2.1 Expected valueUsing Equations 8 and 13, it is straightforward to showthat:

E {�A(θ , r, t)} = E{

wH C(r, t)v(θ , r, t)}− wH v(θ , r, t)

= wH E {C(r, t)} v(θ , r, t) − wH v(θ , r, t)(31)

where E{·} denotes the expected value.Assume that the frequency increment error ρn of the

nth element satisfies the Gaussian statistical model ρn ∼N(0, σ 2) and the PDF is as follows:

fn(ρ) = 1√2πσ

e− ρ22σ2 . (32)

Therefore, the expected value of the nth element indiag(C) can be calculated as [30]:

E {Cnn(r, t)} = E{

ej2π(n−1)(

tρ− ρrc0

)}

= 1√2πσ

+∞∫−∞

ej2π(n−1)(

tρ− ρrc0

)e− ρ2

2σ2 dρ

= e−2π2(n−1)2(

t− rc0

)2σ 2

.(33)

where Cnn denotes the nth element of diag(C). Whenapplying Equation 33 onto 31, we can get the expectedvalue of FDA beampattern deviation with Gaussian distri-bution frequency increment errors as follows:

E {�A(θ , r, t)} = wH E {C(r, t)} v(θ , r, t) − wH v(θ , r, t)= wH (diag

(bg)− I

)v(θ , r, t)

(34)

where

bg =[

1, e−2π2(

t− rc0

)2σ 2

, . . . , e−2π2(N−1)2(

t− rc0

)2σ 2]

.

(35)

and I denotes N ×N unit matrix. In this case, the expectedvalue of the FDA beampattern deviation is dependent onthe variance of frequency increment errors and is not 0 forthe Gaussian distribution model.

For another case, we assume that ρn is uniform dis-tribution, i.e., ρn ∼ u(−ρmax, ρmax) with minimum

and maximum values −ρmax and ρmax. We can get theexpected value of nth element in diag(C) as follows:

E {Cnn(r, t)} = E{

ej2π(n−1)(

tρ− ρrc0

)}

=ρmax∫

−ρmax

ej2π(n−1)(

t− rc0

)ρ 1

2ρmaxdρ

=sin(2π(n − 1)

(t − r

c0

)ρmax)

2π(n − 1)(

t − rc0

)ρmax

.

(36)

By using Equation 36 onto 31, the expected value ofFDA beampattern deviation with uniform distributionfrequency increment error can be reformulated as:

E {�A(θ , r, t)} = wH E {C(r, t)} v(θ , r, t) − wH v(θ , r, t)= wH (diag (bu) − I

)v(θ , r, t)

(37)

where

bu =⎡⎣1,

sin(

2π(

t − rc0

)ρmax

)2π(

t − rc0

)ρmax

, . . . ,

sin(

2π(N − 1)(

t − rc0

)ρmax

)2π(N − 1)

(t − r

c0

)ρmax

⎤⎦ .

(38)

Similarly, the expected value of the FDA beampatterndeviation is dependent on the maximum value of fre-quency increment error and not 0 for the uniform distri-bution model. From Equations 34 and 37, the expectedvalues of FDA beampattern have range offset and angleoffset, which is caused by time dependent error on the twodistributions.

4.2.2 VarianceThe beampattern deviation variance var {�A(θ , r, t)}equals to the beampattern variance:

var {�A(θ , r, t)} = var {A(θ , r, t)}= var

{wH C(r, t)v(θ , r, t)

}.

(39)

As we deal with vectors and matrices, we utilize thecovariance matrix cov{·}, and var{·} = cov{·} for anyscalar. We thus have:

var {�A(θ , r, t)} = cov{

wH C(r, t)v(θ , r, t)}

(40)

Using the Kronecker product ⊗, the vectorization trans-formation vec(·) and the identity:

vec {ABC} = CT ⊗ Avec {B} . (41)

Since

wH C(r, t)v(θ , r, t) = vT (θ , r, t) ⊗ wHvec {C(r, t)} (42)

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 7 of 12

we use c = vec {C(r, t)} and the vector t(θ , r, t) =vT (θ , r, t) ⊗ wH , which utilizing:

cov {AX} = Acov {X} AH (43)

to yield:

var {�A(θ , r, t)} = t(θ , r, t)cov {c(r, t)} tH(θ , r, t). (44)

Since C is a diagonal matrix and its entries are indepen-dent random variables, cov(c) is a diagonal matrix and hasnon-zero value with the nNth entry. We then have:

var {�A(θ , r, t)} =∑

ktk(θ , r, t)tH

k (θ , r, t)cov{c(r, t)}kk

(45)

where k = (n − 1)N + 1 denotes the kth entry of vec-tor. So calculating the var {�A(θ , r)} can be equivalentlyobtained by calculating cov{c(r)}kk . This again is a verygeneral result that we can evaluate and review for differentstatistical models for cov{c(r)}kk .

cov{c(r, t)}kk =+∞∫

−∞

[c(r, t)kk − E(c(r, t)kk)

]2f (ρ)dρ

=+∞∫

−∞

[C(r, t)nn − E(C(r, t)nn)

]2f (ρ)dρ

(46)

Assume that all the random frequency increment errorshave the same distribution. For the first case, frequencyincrement error ρn of the nth element satisfies the Gaus-sian statistical model ρn ∼ N(0, σ 2). According to theEquations 13 and 36, 46 can be rewritten as:

cov{c(r, t)}kk =+∞∫

−∞

[C(r, t)nn − E(C(θ , r, t)nn)

]

×[

C(r, t)nn − E(C(r, t)nn)]∗

f (ρ)dρ

=+∞∫

−∞

[C(r, t)nnC(r, t)∗nn − E(C(r, t)nn)C(r, t)∗nn

− C(r, t)nnE(C(r, t)∗nn) +E(C(r, t)nn)E(C(r, t))∗nn]

f (ρ)dρ

=+∞∫

−∞C(r, t)nnC(r, t)∗nnf (ρ)dρ

−+∞∫

−∞E(C(r, t))nnC((r, t)∗nn)f (ρ)dρ

−+∞∫

−∞C(r, t)nnE(C(r, t))∗nnf (ρ)dρ

++∞∫

−∞E(C(r, t))nnE(C(r, t))∗nnf (ρ)dρ

=+∞∫

−∞C(r, t)nnC(r, t)∗nnf (ρ)dρ − E(C(r, t))2

nn

(47)

where [ ·]∗ denotes the conjugate operator. Here, the

E (C(r, t)nn) = e−2π2(n−1)2(

t− rc0

)2σ 2 = E

(C(r, t)∗nn

)is uti-

lized. Applying Equation 32, the first term of Equation 47can be rewritten as:

+∞∫−∞

C(r, t)nnC(r, t)∗nnf (ρ)dρ = 1√2πσ

+∞∫−∞

ej2π(n−1)(

tρ− ρrc0

)e−j2π(n−1)

(tρ− ρr

c0

)e− ρ2

2σ2 dρ = 1

(48)

Applying it to Equation 47 yields:

cov{c(r, t)}kk = 1 − E(C(r, t))2nn

= 1 − e−4π2(n−1)2(

t− rc0

)2σ 2 (49)

where k is the kth element of c(r, t), which is used todistinguish from n. Consequently, we can get the FDAbeampattern deviation variance:

var {�A(θ , r, t)} =∑

ktk(θ , r, t)tH

k (θ , r, t)

×(

1 − e−4π2(n−1)2(

t− rc0

)2σ 2)

.(50)

Similarly, we can get the FDA beampattern variancewith the random frequency increment errors, which areuniformly distributed as aforementioned:

var {�A(θ , r, t)} =∑

ktk(θ , r, t)tH

k (θ , r, t)

×⎛⎜⎝1 −

⎛⎝ sin(2π(n − 1)

(t − r

c0

)ρmax)

2π(n − 1)(

t − rc0

)ρmax

⎞⎠

2⎞⎟⎠.

(51)

Compared with Equations 50 and 51, we can see thatthe FDA beampattern variances at both kinds of randomfrequency increment error distributions are dependent onthe statistics properties. Given the same frequency incre-ment errors, the variances will decrease when the numberof elements is increased.

4.2.3 PDFWe have shown how to calculate the expectation valueand the variance of the beampattern deviation and thebeampattern itself. A remained question is: What formwill their respective PDF have? In this subsection, we will

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 8 of 12

derive the PDF of beampattern based on the PDF of thefrequency increment errors.

For the first case, we investigate the PDF of beampatternwhen the frequency increment errors obey the Gaussiandistribution as aforementioned. According to Equation 13,we derive the PDF of C. It is known that:

Cnn = e−j2π(n−1)(

t− rc0

)ρ

= cos(

2π(n − 1)

(t − r

c0

)ρ

)

− j sin(

2π(n − 1)

(t − r

c0

)ρ

)= real(Cnn) − j

(imag(Cnn)

)(52)

where real(Cnn) = cos(

2π(n − 1)(

t − rc0

)ρ)

and

imag(Cnn) = sin(

2π(n − 1)(

t − rc0

)ρ)

denote the realpart and imaginary part of Cnn, respectively. Conse-quently, we have:

ρ = arcsin(imag(Cnn))

2π(n − 1)(

t − rc0

)= arcsin(gn)

2π(n − 1)(

t − rc0

)= h(gn)

(53)

where gn = imag(Cnn) denotes the imaginary part of Cnn.As we known that if a variable x has PDF fx(x), the PDF ofy = hg(x) is as follows:

fy(y) = fx[ h(y)]∣∣h′(y)

∣∣ (54)

where x = h(y) is the inverse function of hg(x), fy(·)denotes the PDF responding to y, and [ ·]′ denotes thederivation operation. We have:

fgn(gn) = fρ[ h(gn)]∣∣h′(gn)

∣∣= 1√

2πσe− h2(gn)

2σ2∣∣h′(gn)

∣∣ , −1 < gn < 1(55)

where

h′(gn) = 1

2π(n − 1)√

1 − g2n

(t − r

c0

)denotes the derivation of h(gn). Utilizing gn, Equation 13can be rewritten as:

A(θ , r, t) = wH C(θ , r, t)v(θ , r, t)r

=N∑

n=2

(√1 − g2

n + jgn

)vn(θ , r, t)

r+ v1(θ , r, t)

r

=N∑

n=2

(√1 − g2

n + jgn

)vn + v1

(56)

where vn denotes the nth element of v. Defining vn =vn(θ ,r,t)

r for notation convenience, we can find

√1 − g2

2 + jg2 =(

A(θ , r, t)−N∑

n=3

(√1 − g2

n + jgn

)vn−v1

)/v2.

(57)

Let

a2 = imag(

A(θ , r) −N∑

n=3

(√1 − g2

n + jgn)

vn − v1

)

= imag(A(θ , r)) − imag( N∑

n=3

(√1 − g2

n + jgn)

vn − v1

)

and v2 = cosφ + jsinφ, where φ is the phase of v2, utilizingEquation 57, we can get:

g2 cos ϕ +√

1 − g22 sin ϕ = a2. (58)

Solving Equation 58, it yields:

g2 = a2 cos ϕ + sin ϕ√

1 − a22

= ξ2(Ai)(59)

where Ai = imag(

A(θ , r))

denotes the imaginary part ofA. And we define:

g3 =g3 = ξ3(g3)

. . .

gN =gN = ξN (gN ).(60)

The joint PDF of the variables above is as follows:

fAig3...gN(Aig3 . . . gN )

= fg1g3...gN

[ξ2(

Ai(θ , r, t))

, ξ3(g3), . . . , ξN (gN )]|J|

(61)

According to Equations 59 and 60, we can get that:

J =

∣∣∣∣∣∣∣∣∣∣

∂ξ2∂Ai

∂ξ2∂g3

. . .∂ξ2∂gN

∂ξ3∂Ai

∂ξ3∂g3

. . .∂ξ3∂gN

. . . . . . . . . . . .∂ξN∂Ai

∂ξN∂g3

. . .∂ξN∂gN

∣∣∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣

∂ξ2∂Ai

∂ξ2∂g3

. . .∂ξ2∂gN

0 1 . . . 0. . . . . . . . . . . .

0 0 . . . 1

∣∣∣∣∣∣∣∣= cosφ − sinφ√

1 − a22

(62)

where ∂ξ1∂Ai

= cosφ − sinφ√1−a2

2

. Since the each element of C is

responding to only one frequency increment error whichis random and independent, gn is the independent withother gm with m = 1, 2, . . . , N − 1 and m �= n. Therefore,Equation 61 can be rewritten as:

fg2g3...gN

[ξ2(Ai(θ , r, t)), ξ3(g3), . . . , ξN (gN )

]= fg2

[ξ2(Ai(θ , r, t))

]fg3

[g3]. . . fgN

[gN] (63)

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 9 of 12

Moreover, we can get the PDF of FDA beampattern as:

fAi(Ai) =

1∫−1

. . .

1∫−1

fg2g3...gN

[ξ2(Ai(θ , r, t)), ξ3(g3), . . . , ξN (gN )

]

×⎛⎜⎝cosφ − sinφ√

1 − a22

⎞⎟⎠ dg3 . . . dgN .

(64)

Utilizing the same approach, we can get the PDF of realpart of FDA beampattern. Since the relationship betweenreal part and imaginary part is complex, so we cannot givethe PDF of the whole FDA beampattern.

Similarly, we can derive the FDA beampattern variancewith the random frequency increment errors, which areuniformly distributed as aforementioned. The PDF of gn isas follows:

fgn(gn) = fρ[ h(gn)]∣∣h′(gn)

∣∣= 1

2ρmax

1

2π(n − 1)(

t − rc0

)√1 − g2

n

,

− 1 < gn < 1

(65)

and the PDF of FDA beampattern imaginary part is asfollows:

fAi(Ai) =

1∫−1

. . .

1∫−1

fg2g3...gN

[ξ2(Ai(θ , r, t)), ξ3(g3), . . . , ξN (gN )

]

×⎛⎜⎝cosφ − sinφ√

1 − a22

⎞⎟⎠ dg3 . . . dgN .

(66)

5 Simulation results5.1 Example 1: FDA beampattern boundConsider a 16-element uniform linear FDA with half ofwavelength λ spacing between neighbor elements. Thecenter frequency f0 is 10 GHz, and the increment fre-quency �f is 30 KHz. The target is located at the range10 km, angle 0◦, and the time is on 1/�f . Figure 4a,bshows the comparisons of ideal FDA beampattern Bideal,FDA beampattern upper and lower bound Bbound, andFDA beampattern with random frequency incrementerrors BRandom. In the FDA beampattern with random fre-quency increment errors, the frequency increment errorsare Gaussian distribution, i.e., ρn ∼ N(0, 902) and therandom FDA beampatterns are based on 50 indepen-dent Monte Carlo simulation runs. It can be shown thatthe present bounds hold for all the simulated beampat-tern realizations for both Figure 4a,b. Simulations resultsshow that the bounds are tighter in range dimension than

−50 0 50−25

−20

−15

−10

−5

0

5

Angle (Degree)

Bea

mpa

ttern

(dB

)

Bideal

BRandom

Bbound

0.8 0.9 1 1.1 1.2

x 104

−25

−20

−15

−10

−5

0

5

Range (m)

Bea

mpa

ttern

(dB

)B

idealB

RandomB

bound

a

b

Figure 4 Profiles for the ideal, the bounds, and the errors FDAbeampattern. (a) In angle dimension. (b) In range dimension.

that in the angle dimension which is caused by rangecounterbalance of the total error in beampattern.

Furthermore, we give two figures about expectation andvariance about FDA beampattern errors caused by fre-quency increment errors, shown as in Figures 5 and 6,respectively. Figure 5 compares the expectation value oftheoretical result and empirical result. The theoreticalresult is based on the method of Section 4.2.1. The empiri-cal result is based on the 10,000 independent Monte Carlosimulation runs. The comparisons show that when theFDA beampattern errors have smaller value, two resultsare close to each other. The variances of theoretical resultand empirical result are shown in Figure 6. Similar withFigure 5, the theoretical result is based on the method ofSection 4.2.2. The empirical result is based on the 1,000independent Monte Carlo simulation runs. Different fromthe expectation, the empirical result fluctuates around thetheoretical result along with increase of σ .

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 10 of 12

10−1

100

101

102

−250

−200

−150

−100

−50

σ

Bea

mpa

ttern

err

or (

dB)

Theoretical Empirical

Figure 5 The comparison of Gaussian random FDA beampattern error expectation value.

5.2 Example 2: FDA beampattern PDFConsider a uniform linear FDA with four elements. Inthis example, FDA beampattern amplitude error does notdivide into the factor r0, and its value is very large com-pared with that of Figures 5 and 6. Other array parametersare the same to that of example 1. Figure 7 shows PDFsof theoretical and empirical results, and Figure 8 showsthe PDFs of imaginary part of random FDA beampat-tern with different σ s. It can be known that lower σ

enjoys the smaller beampattern errors, and PDF curvesare centrosymmetric with the center about Ai = 0. Onemight use this figure and Equation 64 to specify toler-ance requirements of frequency increment errors to fulfilla given beampttern requrement with certain probability.For instance, if probability of the beampttern errors at thedomain (−0.5 to 0.5) is not smaller than 0.95, the standarddeviations of ρ for every elements of FDA array will berequired to be no larger than 60.

10−1

100

101

102

−110

−100

−90

−80

−70

−60

−50

−40

σ

Bea

mpa

ttern

err

or (

dB)

Theoretical Empirical

Figure 6 The comparison of Gaussian random FDA beampattern error variance value.

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 11 of 12

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Ai

PD

F

TheoreticalEmpirical

Figure 7 The PDF of Gaussian random FDA beampattern error.

6 ConclusionsIn this paper, we have investigated the impacts of fre-quency increment errors on FDA beampattern based ondeterministic errors and random errors. For uniform andlinear deterministic frequency increment errors, the spe-cific beampattern error formulations are provided, whichgives guideline for device selection. For the stochastic fre-quency increment errors, we have derived a very tight

worst-case boundary of the FDA beampattern. Simulationresults show that all the random beampatterns are held forthe derived bounds, and the worst-case boundary is help-ful to the FDA system design. At last, we derived the sta-tistical properties of the expectation, variance, and PDF.They can be used to analyze the probability of FDA beam-pattern fluctuations for the given distribution frequencyincrement errors.

−1 −0.5 0 0.5 10

0.5

1

1.5

2

Ai

PD

F

σ=60σ=90σ=150

Figure 8 The PDF of imaginary part of random FDA beampattern.

Gao et al. EURASIP Journal on Advances in Signal Processing (2015) 2015:34 Page 12 of 12

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors formulated and discussed the idea together. Additionally, KG wrotethe paper. All authors read and approved the final manuscript.

AcknowledgementsThis work was supported in part by the Program for New Century ExcellentTalents in University under grant no. NCET-12-0095.

Received: 3 November 2014 Accepted: 12 March 2015

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