radar phase noise modeling and effects-part i : mti filtersdownload.xuebalib.com/5ukvphxlt9w.pdf ·...

Radar Phase Noise Modelingand Effects–Part I: MTI Filters

AUGUSTO AUBRY, Member, IEEEConsorzio Nazionale Interuniversitario per le TelecomunicazioniParma, Italy

ANTONIO DE MAIO, Fellow, IEEEVINCENZO CAROTENUTO, Member, IEEEUniversita degli Studi di Napoli Federico IINapoli, Italy

ALFONSO FARINA, Fellow, IEEEAlenia Marconi Systems S.p.A.Rome, Italy

Random and unwanted fluctuations, which perturb the phase ofan ideal reference sinusoidal signal, may cause significantperformance degradation in radar systems exploiting coherentintegration techniques. To quantify the resulting performance loss,we develop a fast-time/slow-time data matrix radar signalrepresentation, modeling the undesired phase fluctuations viamultivariate circular distributions and describing the phase noisepower spectral density (PSD) through a composite power-law model.Hence, we accurately predict the performance degradationexperienced by moving target indication (MTI) algorithms forclutter cancellation, providing a closed form expression for theimprovement factor I. The subsequent analysis shows that phasenoise affects I directly through its characteristic function (CF).Additionally, I shares a robust behavior with respect to the actualphase noise multivariate circular distribution, as long as the phasenoise PSD correctly represents the available measurements.

Manuscript received July 24, 2014; revised May 4, 2015, June 10, 2015;released for publication October 7, 2015.

DOI. No. 10.1109/TAES.2015.140549.

Refereeing of this contribution was handled by S. Watts.

The work of A. Aubry and A. De Maio has been supported by Selex ES(COLB/CTR/2011/24/A). The work of V. Carotenuto has been supportedin part by “Progetto TELEMACO” (PON03PE_00112_1/F1).

Authors’ addresses: A. Aubry, Consorzio Nazionale Interuniversitarioper le Telecomunicazioni, Unita di Ricerca DIETI, Via Usberti 181/A,I-43124 Parma, Italy; A. De Maio, V. Carotenuto, Universita degli Studidi Napoli Federico II, Dipartimento di Ingegneria Elettrica e delleTecnologie dell’Informazione, Via Claudio 21, I-80125 Napoli, Italy;A. Farina, Selex ES, via Tiburtina Km. 12.4, I-00131 Roma, Italy.Corresponding author is A. De Maio, E-mail: ([email protected]).

0018-9251/16/$26.00 C© 2016 IEEE

I. INTRODUCTION

Oscillators represent an essential part of radar systemsand are commonly used to perform frequency and timingsynchronization [1–5]. They are exploited in radiofrequency (RF) transmitters and receivers to provide thesignal for frequency conversion (upconversion anddownconversion), in digital systems to generate the clocksignal to synchronize operations, and in analog-to-digitalconverters to provide the necessary reference phase (clocksampling and synthesis). Unfortunately, the outputs of theoscillators are not perfectly periodic and suffer from manyimperfections, making difficult the availability of a precisetime reference. Practical clocks [chapter 12 in 4, 6] areaffected by phase and frequency instabilities producingthe so-called phase noise. It consists of random andunwanted fluctuations that inevitably perturb thetime-domain linearity of the phase of an ideal sinusoidaloscillation. Assessing the effect of noisy oscillators on theperformance of radar systems may allow the developmentof algorithms able to mitigate these unwanted distortions,as well as to quantify the resulting performance loss.

Nowadays, most of the modern radar systems arecoherent, namely, they detect the phase of the receivedsignal, relative to a well-controlled reference, as well asthe time delay and the amplitude. Each slow-time sample[chapter 8, 4] of the received signal is treated as a complexnumber having an amplitude and a phase angle. Thenecessity of coherence depends on the specific applicationof the radar system. However, any system that needs tocancel clutter, to measure the Doppler frequencycharacteristics, or to generate the electromagnetic imageof a target requires phase coherency [chapters 8 and 9 in 1,chapters 2, 4, and 17 in 2, chapter 12 in 4]. Otherwisestated, to appropriately exploit the phase history of thereceived signal, the transmitted waveform must exhibit aknown phase. To guarantee the previous requirement, themost common technique is to synthesize a transmit signalfrom a set of very stable continuously operatingoscillators, i.e., coherent oscillators (COHO) and stablelocal oscillators (STALO). The former operates at theintermediate frequency (IF), while the latter works at afrequency on the order of the desired output RF signal.The integrity of the phase measurement depends on thestability of the oscillators that generate the transmit signaland provide local references for the downconversionprocess. In fact, the phase measurement is performedcomparing the phase of the received signal with that of thereference wave used for generating the transmitted signal.One of the most important consequences of phase noise isthat in a clutter cancellation system, such as a movingtarget indication (MTI) or a pulse-Doppler processor(PDP), some of the clutter signal energy will be spreadthroughout the passband of the processor, limiting theclutter cancellation and thus the target detectability[chapters 8 and 9 in 1, chapter 12 in 4].

Evidently, the effect of phase noise on the radarperformance depends on its statistical characterization,

698 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 2 APRIL 2016

namely, its multivariate probability density function (pdf)and its power spectral density (PSD). In radar literature,some studies have been conducted to quantify the radarperformance degradation due to phase noise, modeling theunwanted fluctuations as a discrete white process andassuming stationary (zero-Doppler) clutter [7–9]. Clearly,such a statistical model cannot be completely adequate; infact, experimental measurements have highlighted thatphase noise is a correlated random process [chapter 12 in4, 10], possibly non-Gaussian distributed [8, 9], and theclutter usually shares a nonzero Doppler bandwidth [11].Finally, in [12–14], interesting studies about phase noiseimpairments in bistatic synthetic aperture radar (SAR)applications can be found. Notice that the monostaticconfiguration benefits of the self-coherence effect [chapter12, 4] that may significantly reduce the amount of theeffective phase noise power.

Based on the previous observations, in this two-partstudy, we formalize the radar signal model in the presenceof phase noise. Specifically, we develop a fast-time/slow-time data matrix signal representation [chapters 15and 17, 4], where the undesired phase fluctuations,affecting the reference signal produced by real radaroscillators, are modeled via multivariate circulardistributions [15]. Furthermore, we describe the phasenoise PSD through a composite power-law model andshow the ability of the proposed parametric model to fitthe available real phase noise PSD measurements. Owingto the proposed framework, we assess the performance ofcoherent integration techniques when phase noise ispresent, providing an analytic expression to predict theperformance degradations experienced by MTI algorithms(operating in the presence of phase noise), in terms ofimprovement factor I [chapter 17, 4]. The study of PDPalgorithms and sidelobe blanking architectures isaddressed in Part II of this two-part manuscript. Withreference to MTI algorithms, the proposed frameworkreveals that phase noise affects I directly through itscharacteristic function (CF). Additionally, I is robust withrespect to the actual phase noise multivariate circulardistribution, as long as the phase noise PSD modelcorrectly represents the available measurements.

The remainder of Part I is organized as follows.Section II is devoted to the description of the systemmodel. Section III focuses on phase noise modeling, whileSection IV provides the performance analysis of coherentintegration techniques, with emphasis on MTI algorithms.Section V contains some concluding remarks.

A. Notation

We adopt the notation of using boldface for vectors a(lowercase), and matrices A (uppercase). The (nth, mth)entry of A and the nth element of a are denoted by An,m

and an, respectively. The transpose, the conjugate, and theconjugate transpose operators are denoted by the symbols(·)T, (·)*, and (·)†, respectively. tr(·) is the trace of thesquare matrix argument. I and 0 denote, respectively, the

Fig. 1. Block diagram for radar processing scheme.

identity matrix and the matrix with zero entries (their sizeis determined from the context). R

N, CN, and H

N are,respectively, the sets of N-dimensional vectors of realnumbers, of N-dimensional vectors of complex numbers,and of N × N Hermitian matrices. For a given vectora ∈ C

N, diag(a) indicates the N-dimensional diagonalmatrix whose ith diagonal element is ai, for i = 1, . . ., N,whereas its Euclidean norm is denoted by ‖a‖. The curledinequality symbol � (and its strict form �) is used todenote generalized matrix inequality: for any A ∈ H

N ,A � 0 means that A is a positive semidefinite matrix(A � 0 for positive definiteness). The letter j representsthe imaginary unit (i.e., j = √−1), while the letter i oftenserves as an index. For any complex number x, |x| and arg(x) denote the modulus and the argument of x,respectively. For any real number x, �x represents thenearest integer lower than or equal to x. In addition, ⊗ and� denote the convolution operator and Hadamard(element wise) product, respectively. Finally, E[·]denotesstatistical expectation.

II. SYSTEM MODEL

We consider a monostatic radar that transmits a burstof N pulses. In Fig. 1, we show the block scheme for theradar transmit and receive chain [chapter 1 in 2, chapter 2in 16]. Specifically,

• s(t) = AN∑

n=1p(t − (n − 1)T ), t ∈ [0, NT ) denotes

the baseband equivalent of the radar transmitted train,where A > 0 is an amplitude factor related to thetransmitted power, p (t) is a unit energy rectangular pulsewaveform of duration Tp, with an effective (peak-to-null)single-side bandwidth Wp = 1/Tp, T is the pulse repetitioninterval (PRI), and 1

Tis the pulse repetition frequency

(PRF);• exp[j(2π fct + φ (t))] and exp[–j(2π fct + φ (t))]

account for the upconversion and downconversionprocess, respectively, where fc is the carrier frequency andφ (t) is the real valued passband random process modelingthe oscillator phase noise [chapter 2 in 1, chapter 12 in 4,10], whose statistical characterization is addressed inSection III;

• the block “channel” models the backscattering fromthe environment (encompassing the clutter signal cRF (t),as well as a prospective target return sRF (t)) and the RFwhite noise wRF (t); its output is the radar received signalrRF (t) = sRF (t) + cRF (t) + wRF (t);

AUBRY ET AL.: PHASE NOISE MODELING AND ITS EFFECTS ON MTI FILTERS 699

• the block “matched filtering” represents the matchedfilter to the radar pulse p(t); its output is the possibly rangecompressed signal y(t) = r(t) ⊗ p∗(−t);

• rk,n = y(kTp + (n–1)T), k = kmin, kmin + 1, . . . ,

� TTp

, n = 1, 2, . . ., N, denotes the fast-time/slow-time datamatrix, with kmin Tp > 0 the initial time of the fast-timesampling process, accounting for the radar eclipsing.

As a first step toward the derivation of thefast-time/slow-time data matrix signal model, let usevaluate the output to the matched filter due to anonambiguous pointlike moving target with complexbackscattering amplitude α1, Doppler frequency fd, andround-trip delay τ = 2R

c, where R is the target range and c

the propagation velocity in the medium. In this case, thedownconverted received signal is

r(t) = sRF (t) exp [−j (2πfct + φ(t))]

= A1α1

N∑n=1

Ap(t − (n − 1)T − τ )

× exp (j (2πfc(t − τ ) + φ(t − τ )))

× exp (j (2πfd (t − τ ))) exp (−j (2πfct + φ(t)))

= A1 α1 A

N∑n=1

p(t − (n − 1)T − τ ) exp [j (φ(t − τ )

− φ(t) + 2πfd (t − τ ) − 2πfcτ )] , (1)

with A1 accounting for the effects of the transmit-receiveantenna gains, the two-way path loss, and other factorsinvolved in the radar equation. Hence, following the stepsin [chapter 2 in 16, chapter 2 in 17], we get the signal atthe output of the matched filter given by

y(t) = α

N∑n=1

χp (t − (n − 1)T − τ, fd, n, τ )

× exp [j2π(n − 1)fdT ] , (2)

where α = A1α1A exp(−j2πfcτ ) ∈ C and

χp (t2, f, n, τ ) =∫ +∞

−∞p (t1) p∗ (t1 − t2)

× exp [j (δφ (t1 + τ + (n − 1)T ))] exp (j2πf t1) dt1

indicate a generalized ambiguity function of the pulsewaveform p(t) accounting for the phase noise, with δφ(t)= φ (t – τ ) – φ (t).1 Notice that χp(t2, f, n, τ ), n = 1, . . .,N, is a random variable that depends on the randomprocess δφ(t). Also, the actual integration limits arebounded due to the finite duration of the pulse p(t). InAppendix A, it is shown that under some mild technicalconditions, the phase noise process involved in thenth PRI, n = 1, . . ., N, i.e., δφ (t1 + τ + (n–1)T),t1 ∈ [0, Tp], can be approximated as a random variableδφn = φ((n − 1)T ) − φ(kTp + (n − 1)T ), where k is therange bin associated with the round-trip delay τ , i.e.,|kTp − τ | ≤ Tp

2 . Specifically, denoting by Bφ the

1 For notational simplicity, we omit the explicit dependence on τ of δφ(t).

single-side bandwidth where the phase noise PSD isgreater than or equal to2 10−4

Wp, with 2Wp the single-side

radar bandwidth around the carrier frequency, and by Sφ

the phase noise power value associated with the frequencyinterval [−Bφ, Bφ], then ∀t1 ∈ [0, Tp]

E[|δφ (t1 + τ + (n − 1)T ) − δφn|2

] � 1, n = 1, . . . , N,

as long as (πTpBφ)2Sφ � 1. Under the previouslymentioned assumptions, exp[j (δφ (t1 + τ + (n – 1)T) –δφn)] � 1 and consequently,

χp(t2, f, n, τ )

� exp(jδφn)

(∫ +∞

−∞p(t1)p∗(t1 − t2) exp(j2πf t1)dt1

)

� exp(jδφn)

(∫ +∞

−∞p(t1)p∗(t1 − t2)dt1

)= exp(jδφn)Rp(t2), n = 1, . . . , N, (3)

where the last approximation assumes a Doppler-tolerantpulse p(t), with Rp(t2), t2 ∈ R,the autocorrelation functionof p(t). Interestingly, the required conditions hold true forphase noise PSDs of practical interest (please see realphase noise data analysis in Subsection III.B). Based on(3), the signal received from a pointlike target afterbaseband conversion and filtering can be expressed as

y(t) = α

N∑n=1

Rp(t − (n − 1)T − τ )

× exp(jδφn) exp (j2π(n − 1)fdT ) . (4)

Let us now focus on the output of the matched filterdue to the clutter and observe that the clutter signal can bemodeled as the superposition of the returns fromindependent pointlike scatterers [p. 22, 16]. Hence, owingto the linearity of the matched filter, we can compute theclutter output applying the developed approximations (4)to each pointlike source and superimposing the signalsbelonging to the same range cell.

As to the output of the matched filter due to the RFnoise, it is given by

y(t) =∫ Tp

0p∗ (t1) w (t1 + t) exp (−jφ (t1 + t)) dt1, (5)

with w(t) a complex, zero-mean, circularly symmetricGaussian random process with constant PSD σ 2 within thereceiver bandwidth [–Wp, Wp]. Following the same stepsas in Appendix A, it is not difficult to show that

E[|φ (t1 + t) − φ(t)|2] � 1, ∀t1 ∈ [

0, Tp

]

2 Notice that the factor 10−4 can be replaced by any positive real numbermuch lower than “1” without affecting the theoretical results obtained inthe paper. Nevertheless, from a practical point of view, it is better tospecify a value, and to this end, we have chosen 10−4.


if (πTpBφ)2Sφ � 1. Hence, exp[–j(φ (t1 + t) – φ (t))] � 1,t1 ∈ [0, Tp], and

y(t) �(∫ Tp

0p∗ (t1) w (t1 + t) dt1

)exp(−jφ(t)). (6)

Summarizing, the N-dimensional vectorr = [rk,1, rk,2, . . . , rk,N ]T ∈ C

N of the received waveformsamples obtained after baseband conversion, filtering, andsampling at the range bin of interest k, can be expressed as

rk,n = αpn exp (jδφn) + cn exp (jδφn) + wn,

n = 1, . . . , N, (7)

where

• p = [p1, . . . , pN ]T ∈ CN with pi = 1√

N

exp(j2π(i − 1)νd ), i = 1, . . ., N, denote the unit normtarget steering vector, where νd = fdT is the targetnormalized Doppler frequency, with fd in Hz the actualtarget Doppler frequency;

• α ∈ C accounts for target reflectivity, potentialstraddle loss, channel propagation effects, and other termsinvolved into the radar range equation [chapter 1, 2]. Thephase of α is modeled as a uniform random variable over]0, 2π], while its amplitude is assumed deterministic andunknown;

• y = [δφ1, . . ., δφN]T ∈ RN represents the oscillator

phase noise perturbation, which is statisticallyindependent of c, α, and w;

• c = [c1,. . ., cN]T ∈ CN is the vector of the clutter

samples, modeled as a complex, zero-mean, circularly

symmetric Gaussian random vector, with covariancematrix E[cc†] = M;

• w = [w1, . . . , wN ]T ∈ CN is the vector of the white

noise samples, whose components are independent andidentically distributed (iid) complex, zero-mean, circularlysymmetric Gaussian random variables, namely,E[ww

†] = σ 2

w I .

Exploiting the previously mentioned definitions, thereceived signal (7) can be expressed in a vectorial compactform as

r = α y � p + y � c + w, (8)

with y = [exp(jδφ1), . . . , exp(jδφN )]T . Beforeconcluding this section, it is worth pointing out that thevector y accounts for both the phase noise perturbationintroduced at the transmitter side (upconversion), as wellas the phase noise contribution produced at the receiver

side (downconversion). The next section is devoted to thestatistical characterization of the overall phase noisecontribution.

III. PHASE NOISE MODELING

According to the developed signal model, phase noiseobservations φ(t) can be regarded as circular samples, i.e.,points on the unit circle. Otherwise stated, the process (t) = exp(jφ(t)) belongs to the class of “Circular Data”processes, characterized by circular distributions [15]. InSubsection III.A, we introduce some statistical models formultivariate circular variables (wrapped distribution), andin Subsection III.B, we describe a phase noise spectralmodel.

A. Multivariate Circular Models

In this subsection, we describe the important family ofwrapped distributions. Given a random vector of R

N withan assigned probability distribution, we can wrap eachcomponent around the circumference of the unit radiuscircle so as to produce a wrapped distribution. Otherwisestated, if xi, i = 1, . . . , N , denote random variables onthe line, i.e., x = [x1, x2, . . . , xN ]T represents a randomvector of the Euclidean space R

N , the corresponding(wrapped) random variables xw

i , i = 1, . . . , N, are givenby

xwi = xi(mod 2π), i = 1, . . . , N . (9)

As a consequence, if x1, x2, . . . , xN have joint pdf f(·),then the joint pdf fw(·) of xw

1 , xw2 , . . . , xw

Nis [chapter 3, 15]

fw (θ1, θ2, . . . θN ) =⎧⎨⎩

∞∑k1,k2,...,kN=−∞

f (θ1 + 2πk1, θ2 + 2πk2, . . . , θN + 2πkN ) , 0 ≤ θ1, . . . , θN < 2π,

0 otherwise(10)

The most important property of an N-dimensionalwrapped distribution concerns its CF [chapter 3, 15],which is a powerful tool to analyze and handle circulardata [chapter 3, 15]. It is defined for a circular randomvector [ϑ1, ϑ2, . . . , ϑN ]T ∈ [0, 2π[N , as the N th-ordersequence

φ (p1, p2, . . . , pN ) : p1, p2, . . . , pN = 0, ±1, ±2, . . . ,

given by

φ (p1, p2, . . . , pN )

= E[exp (jp1ϑ1 + jp2ϑ2 + . . . + jpNϑN )

], (11)

i.e., the set of all the joint trigonometric moments of thecircular random variables ϑ1, ϑ2, . . . , ϑN . As to wrappeddistributions, denoting by x (·) the standard CF ofx, it can be shown that the CF of xw

1 , xw2 , . . . , xw

Nis


[chapter 3, 15]

φ (p1, p2, . . . , pN ) = x (p1, p2, . . . , pN ) ,

p1, p2, . . . , pN = 0, ±1, ±2, . . . , (12)

i.e., the samples of the standard CF.Interestingly, it can be proved that any multivariate

circular distribution is determined by its CF, a propertyreferred to as “uniqueness property” [chapter 4, 15];otherwise stated, there exists a one-to-one mappingbetween the pdfs and their joint trigonometric moments, incontrast with classic distributions on the Euclidian spaceand their standard moments [chapter 30, 18]. Based on thepreviously mentioned observation, a meaningful approachto obtain statistical inference on circular data is to estimatetheir joint trigonometric moments. Also, this importantproperty has been exploited in [19] to approximate the vonMises density via a suitable mixture of zero-meanone-dimensional wrapped Gaussian distributions.

We now present some special cases ofmultidimensional wrapped distribution.

Wrapped Gaussian distribution: The wrappedGaussian distribution WN (μ, �) is obtained wrapping theN -dimensional Gaussian distribution N (μ, �), where� � 0, μ = [μ1, μ2, . . . , μN ]T ∈ R

N [20, 21]. From (10),the pdf corresponding to WN (μ, �) is

fw(θ ; μ, �)

= 1√2π det(�)

∞∑k1,k2,...,kN=−∞

h (θ , k1, k2, . . . , kN , μ, �)

with

h (θ , k1, k2, μ, �)

= exp

⎡⎢⎢⎢⎣−1

2

∥∥∥∥∥∥∥∥∥�− 1

2

⎡⎢⎢⎢⎣

θ1 − μ1 + k12π

θ2 − μ2 + k22π...

θN − μN + kN2π

⎤⎥⎥⎥⎦∥∥∥∥∥∥∥∥∥

2⎤⎥⎥⎥⎦ .

Because the CF of x ∼ N (μ, �) is given by x(t) = exp(jμT t − 1

2 tT �t), with t ∈ RN , from (12), we

have

φ (p1, p2, . . . , pN )

= exp

⎡⎣j

⎛⎝ N∑

i=1

μipi

⎞⎠− 1

2

⎛⎝ N∑

i1=1

N∑i2=1

�i1,i2pi1pi2

⎞⎠⎤⎦ .

(13)

Notice that the first-order distributions of xwi ,

i = 1, . . . , N are wrapped Gaussian distributions;specifically, xw

i ∼ WN (μi, �i,i), i = 1, 2, . . . , N,

[chapter 3, 15].Wrapped generalized asymmetric Laplace

distribution: The wrapped generalized asymmetricLaplace (WGAL) distribution WGAL (μ, �, s)is obtained wrapping the N -dimensional generalizedasymmetric Laplace (GAL) distribution GAL(μ, �, s),where s ≥ 1, � � 0, μ = [μ1, μ2, . . . , μN ]T ∈ R

N . From

[22] and (10), the pdf connected to WGAL(μ, �, s) is

fw(θ ; μ, �, s)

= c1

∞∑k1,k2,...,kN=−∞

Ks−1(Q(θ , k1, k2, . . . , kN , �)C(�, μ))

×(

Q (θ , k1, k2, . . . , kN , �)

C (�, μ)

)(s−1)

× exp (G (θ , k1, k2, . . . , kN , �, μ)) , (14)

where Kλ(u) is the modified Bessel function of the thirdkind with index λ > 0 [22] and

c1 = 2

�(s)√

det(�)2π,

G (θ , k1, k2, . . . , kN , �, μ)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−

⎡⎢⎢⎢⎣

μ1

μ2...

μN

⎤⎥⎥⎥⎦

T

�−1

⎡⎢⎢⎢⎣

θ1 + k12π

θ2 + k22π...

θN + kN2π

⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭

,

(15)

Q (θ , k1, k2, . . . , kN , �) =

∥∥∥∥∥∥∥∥∥�− 1

2

⎡⎢⎢⎢⎣

θ1 + k12π

θ2 + k22π...

θN + kN2π

⎤⎥⎥⎥⎦∥∥∥∥∥∥∥∥∥

,

C(�, μ) =√

2 + μT �−1μ.

Notice that the CF of x ∼ GAL(μ, �, s) is given by x(t) = (1 + tT � t

2 − jμT t)−s ; as a consequence, from(12),

φ (p1, p2, . . . , pN )

=

⎡⎢⎢⎢⎢⎣

1

1 −(

N∑i=1

μipi

)+ 1

2

(N∑

i1=1

N∑i2=1

�i1,i2pi1pi2

)⎤⎥⎥⎥⎥⎦

s

.

Interestingly, a random vector x ∼ GAL(μ, �, s) hasmean vector sμ and covariance matrix s� + sμμT.Additionally, it can be generated via [22]

x = μz + √z y, (16)

where y ∼ N (0, �) and z is a standard gammadistribution with shape parameter s. Consequently, thefirst-order distributions of xw

i , i = 1, . . . , N areWGAL(μi, �i,i , s), i = 1, 2, . . ., N .

B. Phase Noise Spectral Models

Empirical models based on measurements suggest thatthe phase noise PSD Sφ ( f ) can be described as apower-law function [10, 23], i.e.,

Sφ(f ) = Kα|f |−α, α ∈ {0, 1, 2, 3, 4}, Kα ∈ R,

(17)or through their suitable combinations, where f denotes theoffset frequency from the actual carrier frequency [chapter


TABLE IPhase Noise Models

|f |−α Phase noise Color of � (t)

|f |0 White phase noise Purple|f |−1 Flicker phase noise Blue|f |−2 White FM phase noise White|f |−3 Flicker FM phase noise Pink|f |−4 Random walk FM phase noise Brown

12, 4]. The characterization of the oscillator output PSDbased on the knowledge of the phase noise spectralcharacteristics is a long-standing issue and has beenaddressed in many papers. Among them, the Leeson model[24] remains probably the most widely used. Moreover, inrecent work [10], the authors applied correlation theorymethods to obtain a model for the near-carrier oscillatorPSD. Further interesting results can be found in [25].

The phase noise PSD is related to frequencyfluctuations PSD S�( f ) via the spectral transformation[10, 26]

Sφ(f ) = S�(f )

f 2. (18)

Notice that the different noise sources are usuallyreferred to as “colored” [10]. Specifically, thecorrespondence between the terminology for frequencynoise sources and the resulting phase noise is summarizedin Table I. It is worth pointing out that this representationdoes not correspond to the actual phase noise generationmechanisms, but it is useful as a mathematical abstraction.

In the following, we focus on a simple mathematicalmodel often used in practice [chapter 12 in 4, 10], whichapproximates the phase noise PSD as

Sφ(f ) = K0 + K1

|f | + K2

|f |2 + K3

|f |3 + K4

|f |4 , (19)

where the weights Ki, i = 0, 1, 2, 3, 4 allow to suitablycombine the power-law processes. Notice that K0 definesthe minimum phase noise level at high offset frequencies.

It is worth highlighting that both the combined model(19) and the power-law spectra (17) with α ≥ 1 produce aninfinite value at 0 Hz, so the use of these models is usuallysubject to some maximum phase noise PSD, whichgenerally occurs around the 5- to 10-Hz offset frequency[chapter 12, 4]. As already pointed out, the question ofnear-carrier oscillator PSD has been successfullyaddressed in open literature for the power-law phase noiseprocesses described in Table I [10]. As a matter of fact, theeffect of slowly varying frequency fluctuations inoscillators is characterized through their long-termstability and can manifest itself through non-negligibledrifts in the oscillation frequency. Nevertheless, it is not ofprimary importance in a radar application, as long as, overthe coherent processing interval (CPI), the inducedfrequency drift remains constant or can be compensated.

Before concluding this subsection, we present somereal phase noise PSD measurements and their adherence

TABLE IIPhase Noise Data Set Characteristics

File Number Carrier Frequency (MHz) Carrier Power (dBm)

file1 1000 9.6file2 1610 12file3 2000 17.3file4 2140 10.4file5 2390 7.6file6 390 −1.9file7 500 17.6file8 500 17.8

Fig. 2. (a) Single-sideband phase noise PSD in dBc/Hz versusfrequency offset in Hz: 1000-MHz oscillator carrier frequency availablemeasurements, file1 (solid line) and optimized fitting, i.e., our proposed

model (dashed line). (b) Single-sideband phase noise PSD in dBc/Hzversus frequency offset in Hz: 1610-MHz oscillator carrier frequency

available measurements, file2 (solid line) and optimized fitting, i.e., ourproposed model (dashed line).




with properly tuned composite power-law models. Thedata have been generated from a stable oven-controlledcrystal oscillator (OCXO), operating at 100 MHz, and adirect digital synthesizer (DDS). The availablemeasurements refer to the single-side PSD in dBc/Hz[chapter 12 in 4], for different values of the oscillatoroperating carrier frequency and carrier power. Table IIsummarizes the characteristics of the available data set.Figs. 2–5 show the single-side phase noise PSD in dBc/Hz



model (dashed line). (b) Single-sideband phase noise PSD in dBc/Hzversus frequency offset in Hz: 390 MHz oscillator carrier frequency





versus the frequency offset in Hz, for the availablemeasurements at different oscillator carrier frequency(solid lines) and the optimized fitting (dashed lines); inAppendix B, the description of the procedure adopted forthe estimation of the parameters is explained with somedetails. These plots clearly highlight the capability of theproposed model to describe the phase noise PSD for allthe considered oscillator carrier frequencies. In theperformed analysis, the parameters involved in thecomputation of the estimate (see Appendix B) are equal toM = 101, f1 = 10 Hz, f2 = 39.5275 Hz, f3 = 1000 Hz,and f4 = 10000 Hz for each data set.

IV. COHERENT INTEGRATION TECHNIQUES

Coherent integration techniques [chapters 2 and 4 in 2,chapters 14 and 17 in 4, chapters 10 and 11 in 27] performa weighted coherent sum of the received samples (7) fromthe cell under test, producing the statistic

z =N∑

n=1

h∗nrn = h†r, (20)

where h = [h1, . . . , hN ]T ∈ CN are known deterministic

weights. Equation (20) encompasses a wide class of signalprocessing algorithms, such as coherent detection (forinstance, PDP), clutter cancellation (e.g., MTI), linearfiltering, and spectral analysis (discrete Fourier transformcomputation). Interestingly, the statistic z can be expressedas the sum of two contributions zT and zd, with

• zT = αN∑

n=1h∗

npn exp(jδφn) = αh†diag( y) p,

denoting the useful term;

• zd =N∑

n=1h∗

n(cn exp(jδφn) + wn) = h†diag( y)c

+ h†w, representing the interference term.

Conditioned on the random phase vector

ϕ = [φ(0), φ

(kTp

), φ(T ), φ

(T + kTp

), . . . ,

φ((N − 1)T ), φ((N − 1)T + kTp

)]T, (21)

zT is a complex random variable with uniform phase anddeterministic amplitude |α||h†diag( y) p| (nonfluctuatingtarget assumption), and zd is a complex, zero-mean,circularly symmetric Gaussian random variable with

variance (‖M12 diag( y∗)h‖2 + σ 2

w ‖ h‖2). As aconsequence, in the presence of phase noise, the usefuland the interference signals are no longer statisticallyindependent (i.e., only conditional independence holdstrue).

Let us now analyze the second-order statisticalcharacterization of zT and zd that will be useful forsubsequent mathematical derivations. Due to the linearityof the mean operator and the statistical independence of α,ϕ, w, and c, the mean value of zT and zd are, respectively,given by

E [zT ] = E[α]N∑

n=1

h∗npnE

[exp (jδφn)

] = 0;

E [zd ] =N∑

n=1

h∗n (E [cn] E[exp (jδφn)] + E [wn]) = 0.

Also, zT and zd are uncorrelated random variables; in fact,

E [zT zd ] = E[α]N∑

n=1

N∑m=1

h∗npnh

∗m

× (E [cm] E[exp (jδφn) exp (jδφm)]

+ E[exp (jδφn)]E [wm]) = 0;

E[zT z∗

d

] = E[α]N∑

n=1

N∑m=1

h∗npnhm

× (E[c∗m

]E[exp (jδφn) exp (−jδφm)]

+ E[exp (jδφn)

]E[w∗

m

]) = 0.


Finally, the mean power of zT and zd are, respectively,

E[|zT |2] = E

[|α|2]E

[(N∑

n=1

h∗npn exp (jδφn)

)

×(

N∑m=1

hmp∗m exp (−jδφm)

)]

= |α|2N∑

n=1

N∑m=1

h∗npnhmp∗

m

× E[exp (jδφn) exp (−jδφm)

]= |α|2h†diag( p)A diag

(p∗) h, (22)

and

E[|zd |2

] = E

[(N∑

n=1

h∗n (cn exp (jδφn) + wn)

)

×(

N∑m=1

hm

(c∗m exp (−jδφm) + w∗

m

))]

=N∑

n=1

N∑m=1

h∗nhmE[cnc

∗mE[exp(jδφn)

× exp(−jδφm)]] + σ 2w

N∑n=1

|hn|2

= h†E[diag(c)E

[y y†

]diag

(c∗)] h + σ 2

w‖h‖2

= h† (A � E[cc†

])h + σ 2

w‖h‖2

= h† (A � M) h + σ 2w‖h‖2, (23)

with

An,m = E[exp (j (δφn − δφm))

], (n, m) ∈ {1, 2, . . . , N}2.

(24)The term σ 2

w‖h‖2 in (23) accounts for the thermal noisepower, whereas h†(A � M)h is related to the clutterpower. Interestingly, the clutter PSD spreading effect[chapter 8 in 1, chapter 12 in 4], due to phase noise,analytically stems from the Hadamard product betweenthe matrices A and M. Specifically,

1) the minimum eigenvalue of A � M, i.e., thecovariance matrix of the clutter affected by phase noise, isgreater than or equal to the minimum eigenvalue of M,λmin(M). In fact, M � λmin(M)I , which impliesA � M � A � λmin(M)I = λmin(M)I, becauseA � λmin(M)I = λmin(M)I . As a consequence, themaximum achievable improvement factor reduces [28]with respect to the ideal condition;

2) the maximum eigenvalue of A � M is lower thanor equal to that of M [29]. Hence, in conjunction withproperty 1, a lower eigenvalue spread than the ideal case isexperienced, leading to a flatter spectrum.

Finally, note that ∀(n, m) ∈ {1, 2, . . . , N2}An,m � 1 + jE [δφm] − jE [δφn] + E [δφnδφm] , (25)

as long as E[δφ2n] � 1, n ∈ {1, 2, . . ., N}, i.e., in this

regimen A only depends on the second-order momentsof y.

Interestingly, E[|zT |2] and E[|zd |2] are functionallydependent on the phase noise only through the matrix A.Furthermore, indicating the CF of the phase vector ϕ (seeSubsection III.A) as φϕ(i), i = [i1, i2,. . .,i2N]T ∈ {0, ± 1,± 2, . . .,}2N, for all (n, m) ∈ {1, 2, . . ., N}2, it holds

An,m ={

1 n = m

E[exp (j (δφn − δφm))

] = φϕ(i (n,m)

)n �= m

(26)with i (n,m) ∈ {0, ± 1, ± 2, . . .,}2N given by

i (n,m)k =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1, k = 2(n − 1) + 1

−1, k = 2(n − 1) + 2

1, k = 2(m − 1) + 2

−1, k = 2(m − 1) + 1

0, otherwise

, k = 1, 2, . . . , 2N .

(27)The previous equations imply that the phase noiseinfluences the second-order moments of zT and zd directlythrough its CF. It is worth highlighting that the conductedanalysis fully agrees with the study addressed in [7],where the case of iid Gaussian random phase noisesamples is considered.

A. MTI Processing

In this subsection, we exploit the previously derivedgeneral results to characterize the performance of MTIsystems. In this context, the figures of merit are given by[chapter 17, 4]:

• clutter attenuation (CA) that measures the reductionin clutter power at the output of the MTI filter comparedwith the clutter power at the input;

• improvement factor (I) that quantifies the increase insignal-to-clutter power ratio (SCR) due to MTI filtering;as a consequence, it accounts for the effect of the filterboth on the target and the clutter.

Clutter attenuation: CA directly evaluates theeffectiveness of the MTI filter h to suppress the clutterenergy. It is simply the ratio between the clutter power Cin

at the input and the clutter power Cout at the output of thefilter:

CA = Cin

Cout

=1N

tr(M)

h† (A � M) h. (28)

Improvement factor: I is formally defined as the SCRat the output of the MTI filter divided by the SCR at theinput, averaged over all target radial velocities of interest.As a first step toward the evaluation of I, let us consider aspecific normalized target Doppler shift νd; hence,exploiting (22) and (23), the improvement factor can be


factored into the form

I (νd ) = SCRout

SCRin

= Sout

Sin

Cin

Cout

= N |α|2h†diag( p)Adiag ( p∗) h|α|2

1N

tr(M)

h† (A � M) h.

(29)

Finally, averaging over the normalized target Dopplerinterval [ν1, ν2](− 1

2 ≤ ν1 ≤ ν2 ≤ 12 ), we obtain

I ν2ν1

= 1

ν2 − ν1

∫ v2

v1

I (νd ) dνd = Nh† (A � E[

p p†]) h

×1N

tr(M)

h† (A � M) h= h† (A � �ν2

ν1

)h

h†(

A � M)

h, (30)

with M = M1N

tr(M)the normalized clutter covariance matrix

and

�ν2ν1

(l, m) ={

1 if l = m

e(jπ (ν1+ν2)(l−m)) sin[π (ν2−ν1)(l−m)][π (ν2−ν1)(l−m)] if l �= m

.

(31)As ν1 = − 1

2 and ν2 = 12 , (30) reduces to the conventional

average improvement factor

I =∫ 1

2

− 12

I (νd ) dνd = ‖h‖2

h†(

A � M)

h. (32)

Interestingly, when the phase noise is absent, An,m = 1,∀ (n, m) ∈ {1, 2, . . ., N}2, leading to the ideal

improvement factor Iideal = ‖h‖2

h† Mh, [11, 28]. As a

consequence, the improvement factor loss L, due to thephase noise, is given by

L = Iideal

I= h† (A � M

)h

h†Mh, (33)

which allows to quantify the impairment effect on theperformance of the MTI algorithms. The previousequation highlights that the loss factor L depends on theemployed MTI filter, the CPI, the clutter correlation, andthe phase noise CF, but it is functionally independent ofthe clutter power.

B. Performance Analysis of MTI Systems in the Presenceof Phase Noise

In this subsection, we analyze the performance of thesingle canceller, double canceller, and third-ordercanceller in the presence of phase noise, clutter, andthermal noise, for a radar employing a rectangular pulse ofduration Tp = 10−1 μs. This is tantamount to consideringas MTI filters the following vectors [chapter 2 in 2,chapter 17 in 4], belonging to the class of binomial filters,

1) hsc = [1, −1, 0, 0, . . . , 0]T ∈ CN, for the single

canceller;2) hdc = [1, −2, 1, 0, . . . , 0]T ∈ C

N, for the doublecanceller;

3) htc = [1, −2, 2, −1, 0, . . . , 0]T ∈ CN, for the

third-order canceller.

In all the numerical examples, we consider aGaussian-shaped clutter autocorrelation function, namely[11],

rc(m) = E[c(n)c(n + m)∗

] = Pc exp

[− (mT )2

2σ 2c

], m ∈ N,

(34)where c(n), n ∈ N denotes the clutter random process; theclutter vector c in (7) is consequently given by c = [c(1),c(2), . . ., c(N)]T; Pc represents the clutter power, andσc = 1/(2 πσf ) with σ f the spectral width of the clutterPSD. Otherwise stated, the (ith, kth) element of the cluttercovariance matrix M is modeled as

M i,k = Pc ρ(i−k)2

T , (35)

with ρT = exp(− T 2

2σ 2c

), the one-lag correlation coefficient.As to the phase noise contribution ϕ in (21), we model

it as a zero-mean random vector, with covariance matrixR. For comparison purposes, we analyze the MTIperformance as ϕ is drawn from both a wrapped Gaussiandistribution and a WGAL distribution (see SubsectionIII.A). To comply with the empirical evidence, we modelthe one-side phase noise PSD via a composite power law,namely,

Sφ(f ) = K0 + K1

f+ K2

f 2+ K3

f 3+ K4

f 4, f ∈ [

f0, Wp

],

(36)where f > 0 denotes the one-side offset frequency (thePSD is a symmetric function) and f0 is the lowest offsetfrequency of interest. Based on Wiener-Khintchinetheorem, the autocorrelation function of the process φ (t)is given by

rφ (τ1) = E [φ(t)φ(t − τ1)] = 2∫ Wp

f0

Sφ(f ) cos (2πf τ1) df.

(37)In Appendix C, it is shown that the autocorrelationfunction of the composite power-law model (36) can beexpressed as

rφ (τ1) = 2K0sin(2πWpτ1) − sin(2πf0τ1)

2πτ1

+ 2K1[ci(2πτ1Wp) − ci(2πτ1f0)]

+4K2πτ1

[cos (2πτ1f0)

2πτ1f0− cos

(2πτ1Wp

)2πτ1Wp

−Si(2πτ1Wp

)+ Si (2πτ1f0)

]

+ 4K3π2τ 2

1

[cos (2πτ1f0)

(2πτ1f0)2 − cos(2πτ1Wp

)(2πτ1Wp)2

+ ci (2πτ1f0) − ci(2πτ1Wp

)+ sin

(2πτ1Wp

)2πτ1Wp

− sin (2πτ1f0)

2πτ1f0

]

+16

3K4π

3τ 31

[cos (2πf0τ1)

(2πf0τ1)3 − cos(2πWpτ1

)(2πWpτ1)3


+ 1

2

sin(2πWpτ1

)(2πWpτ1

)2 − 1

2

sin (2πf0τ1)

(2πf0τ1)2

−1

2

[cos (2πτ1f0)

2πτ1f0− cos

(2πτ1Wp

)2πτ1Wp

− Si(2πτ1Wp

)+ Si (2πτ1f0)

]], (38)

where

ci(x) = −∫ ∞

x

cos(t)

tdt,

is the cosine integral function [chapter 5, p. 231, 30] and

Si(x) =∫ x

0

sin(t)

tdt,

is the sine integral function [chapter 5, p. 231, 30].Note that model (36) does not specify the value Pφ of

the phase noise PSD over the bandwidth [0, f0]. However,as shown in Appendix C, under some mild technicalconditions, the value of Pφ does not affect the radarperformance, as long as 1

f0� NT, where NT represents

the radar CPI. As an example, the performance of a radaremploying N = 16 pulses with a PRI T = 1 ms, mainlydepends on the phase noise spectral components greaterthan or equal to 3 Hz; otherwise stated, the phase noisePSD value Pφ over [0, f0], with f0 = 3 Hz, is “irrelevant.”

As to the parameter values of the composite powermodel, we consider K0 = 10−11, K1 = 0.0000025 · 10−4,K2 = 0.00021 · 10−4, K3 = 0.04 · 10−4, and K4 = 10−5

(approximatively corresponding to the parametersinvolved in the optimized fitting of file2; see SubsectionIII.B), unless otherwise stated.

In the following analysis, we study the improvementfactor loss L, due to the phase noise, as a function of thePRF, phase noise pdf, and clutter spectral variance, fordifferent binomial filters, assuming a clutter power PcdB =20 dB and σ 2dB

w = 0 dB. Moreover, the performance of theradar in terms of I and L does not depend on the number ofpulses N, as long as N is greater than or equal to thenumber of MTI filter taps KT. In fact, if hi = 0,i = KT + 1, . . ., N,

h† (A � M) h = h† ( A � M)

h,

where h = [h1, . . . , hKT]T ∈ C

KT , while Ai,j = Ai,j , andM i,j = M i,j , (i, j) ∈ {1,2, . . ., KT}2.

In Fig. 6, we provide the ideal improvement factor Iideal

(Fig. 6a) and the improvement factor loss L (Figs. 6b–6d)versus PRF, assuming a target range R = 30 km, which istantamount to fixing the range bin of interest k influencingy, and σ f = 50 Hz, for hsc, hdc, and htc.3 Specifically, Fig.6b accounts for wrapped Gaussian phase noise, Fig. 6caddresses WGAL phase noise with s = 1, and Fig. 6drefers to WGAL phase noise with s = 10, to deal with

3 Notice that the one-lag correlation coefficient involved in our analysisis a function of T.

Fig. 6. (a) Ideal improvement factor Iideal versus PRF, assuming R = 30km and σ f = 50 Hz. Single canceller (red ♦-marked solid curve); double

canceller (green ◦-marked solid curve); third-order canceller (blue�-marked solid curve). (b) Improvement factor loss L versus PRF,

assuming R = 30 km, σ f = 50 Hz, and wrapped Gaussian phase noise.Single canceller (red ♦); double canceller (green ◦); third-order canceller

(blue �). Actual values (solid lines), small angle approximationpredictions (dashed lines), and prediction based on [7] (dashed-dotted

lines). (c) Improvement factor loss L versus PRF, assuming R = 30 km,σ f = 50 Hz, and WGAL phase noise with s = 1. Single canceller (red ♦);double canceller (green ◦); third-order canceller (blue �). Actual values(solid lines) and small angle approximation predictions (dashed lines).

(d) Improvement factor loss L versus PRF, assuming R = 30 km, σ f = 50Hz, and WGAL phase noise with s = 10. Single canceller (red ♦); doublecanceller (green ◦); third-order canceller (blue �). Actual values (solid

lines) and small angle approximation predictions (dashed lines).

distributions sharing different spreading levels. Also, in allthe plots, the red curve refers to the single canceller, thegreen curve refers to the double canceller, and the bluecurve refers to the third-order canceller. Moreover, thedashed lines refer to the improvement factor loss predictedvia the small angle approximation method (25), which isusually applied in the radar field. Finally, dashed-dottedlines in Fig. 6b account for the predition of L according to[chapter 17 in 4, 7], which is based on the assumption ofstationary clutter and white Gaussian phase noise.

Inspection of Fig. 6a reveals that in the ideal case,increasing PRF is tantamount to improving the MTI filtereffectiveness. This is a reasonable behavior because highervalues of PRF correspond to highly correlated cluttersamples that can be better cancelled. Also, Figs. 6b–6dshow that the higher PRF, the higher L, highlighting thatthe phase noise impairments grow as the clutter correlationincreases. A possible justification of the previously statedresults is that the phase noise spectral spreading becomesmore and more pronounced as the clutter samples PRI


Fig. 7. Improvement factor loss L versus PRF, assuming R = 30 km,σ f = 50 Hz, and K0 = 10−8, K1 = 0.00025 · 10−4, K2 = 0.021 · 10−4,K3 = 4 · 10−4, and K4 = 10−4. Wrapped Gaussian phase noise (black

dashed-dotted lines), WGAL phase noise with s = 1 (red dashed lines),WGAL phase noise with s = 10 (cyan-dotted lines), and small angle

approximation predictions (green solid lines). Single canceller(♦-marked curves); double canceller (◦-marked curves); third-order

canceller (�-marked curves).

becomes narrower: as a matter of fact, if the clutter PSD isflat,4 the phase noise does not impair MTI performance,whereas if the clutter PSD is a Dirac delta function,5 thephase noise significantly spreads the original clutter PSD,and as a consequence, it affects MTI performance.

Interestingly, the curves in Fig. 6b highlight that thesimpler and approximated method [chapter 17 in 4, 7] maylead to a poor prediction of the phase noise effects.Moreover, the results in Figs. 6b–6d suggest that the smallangle approximation holds true for the available phasenoise measurements. In fact, they show that the actual lossfactor L coincides with that predicted by the small angleapproximation method, regardless of the phase noise pdf.This behavior reflects the fact that the MTI performancedegradation only depends on the second-order phase noisestatistical characterization, as long as the phase noisepower is small (see equation (25)). To further assess thephase noise impairments, in Fig. 7, we consider the samescenario as in Fig. 6, but with a higher phase noise power,assuming K0 = 10−8, K1 = 0.00025 · 10−4, K2 = 0.021 ·10−4, K3 = 4 · 10−4, and K4 = 10−4. The curves show thatnow the statistical distribution influences the behavior ofL. Specifically, the WGAL distribution with s = 1 exhibitsthe lowest loss, while the wrapped Gaussian distributionand the WGAL pdf with s = 10 share a similar behavior.

In Fig. 8, we consider the same analysis as in Fig. 6,but for σ f = 20 Hz. As for Fig. 6, the higher PRF, thehigher L, highlighting that the MTI performancedegradation increases as the clutter spectral width reduces.Interestingly, both I and L are greater than the counterpartsin Fig. 6. This is a reasonable behavior because now theclutter PSD is narrower than Fig. 6; this implies that in the

4 In this case, the cancellers are unable to separate the target from clutteralso in the ideal condition of phase noise absence.5 In the ideal condition of phase noise absence, the clutter can beperfectly filtered out by cancellers.

Fig. 8. (a) Ideal improvement factor Iideal versus PRF, assuming R = 30km and σ f = 20 Hz. Single canceller (red ♦-marked solid curve); double

canceller (green ◦-marked solid curve); third-order canceller (blue�-marked solid curve). (b) Improvement factor loss L versus PRF,

assuming R = 30 km, σ f = 20 Hz, and wrapped Gaussian phase noise.Single canceller (red ♦-marked solid curve); double canceller (green

◦-marked solid curve); third-order canceller (blue �-marked solid curve).(c) Improvement factor loss L versus PRF, assuming R = 30 km, σ f = 20Hz and WGAL phase noise with s = 1. Single canceller (red ♦-markedsolid curve); double canceller (green ◦-marked solid curve); third-ordercanceller (blue �-marked solid curve). (d) Improvement factor loss L

versus PRF, assuming R = 30 km, σ f = 20 Hz, and WGAL phase noisewith s = 10. Single canceller (red ♦-marked solid curve); double

canceller (green ◦-marked solid curve); third-order canceller (blue�-marked solid curve).

TABLE IIIImprovement Factor Loss L (dB), Assuming PRF = 1250 Hz, Phase

Noise PSD as in file2 and R = 30 km

Single Canceller Double Canceller Third-Orderσ f Loss Loss Canceller Loss

50 Hz 0.05 0.83 0.3820 Hz 0.33 9.4 1.5

ideal condition, the cancellers can be more effective, but atthe same time, the phase noise spectral spreading for thescenario of Fig. 8 can be more deleterious. Again, the lossfactor L is the same, regardless of the phase noise pdf,confirming that the small angle approximation holds true.

Table III reports the improvement factor loss for atypical study case with PRF = 1250 Hz, as in aAN/SPS-48E radar [31], when the oscillator phase noise iscompatible with the data in file2, the clutter PSD isGaussian shaped, and the target range is R = 30 km.


V. CONCLUSIONS

In this paper, we have formalized the radar signalmodel when the phase noise affects the reference signalproduced by real radar oscillators. Specifically, we havedeveloped a fast-time/slow-time data matrix signalrepresentation, modeling the undesired phase fluctuationsvia a multivariate circular distribution and describing thephase noise PSD through a composite power-law model.Interestingly, the proposed parametric spectral model wellfits the available real phase noise PSD measurements.

We have assessed the performance of coherentintegration techniques operating in the presence of phasenoise, providing an analytic expression to predict theperformance degradations experienced by MTIalgorithms. The proposed framework has highlighted thatphase noise affects I directly through its CF. Additionally,I is robust with respect to the actual phase noise circularmultivariate distribution, as long as the phase noise PSDcorrectly represents with the available measurements.

In the second part of this two-part paper, we continuethe study of the phase noise effects on radar signalprocessors focusing on the performance of PDPalgorithms and sidelobe blanker.

ACKNOWLEDGMENT

We thank Ing. G. Tonelli (Selex ES) who has kindlyprovided the real data used in this study. The authors thankthe associate editor and the referees for their interestingcomments and careful revision of the paper.

APPENDIX A. AMBIGUITY FUNCTIONAPPROXIMATION

Notice that for all t1 ∈ [0, Tp],

E[|δφ (t1 + τ + (n − 1)T ) − δφn|2

]≤ 2

(E[|φ (t1 + (n − 1)T ) − φ((n − 1)T )|2] (39)

+ E[|φ (t1 + τ + (n − 1)T ) − φ

(kTp + (n − 1)T

) |2]),(40)

where the inequality stems from x2 + y2 ≥ 2xy. Let usanalyze E[|φ(t1 + (n − 1)T ) − φ((n − 1)T )|2]; resortingto the Wiener-Khintchine theorem,

E[|φ (t1 + (n − 1)T ) − φ((n − 1)T )|2]= 4

∫ ∞

−∞sin2 (πf t1) Sφ(f )df

≤ 8∫ Bφ

0f 2π2t2

1 Sφ(f )df + 8∫ Wp

Bφ

Sφ(f )df (41)

≤ 4SφBφ2π2t2

1 + 8 · 10−4 � 1 ∀t1 ∈ [0, Tp

], (42)

where Sφ( f ) is the phase noise PSD, Wp is the radarbaseband single-side bandwidth, Bφ is the single-sidebandwidth, the phase noise PSD is greater than or equal to10−4

Wp, and Sφ is the value of phase noise power level over

the frequency interval [−Bφ, Bφ]. Finally, in (41), we use

sin2(x) ≤ x2 and sin2(x) ≤ 1,

and in (42), we exploit

f 2 ≤ B2φ, f ∈ [

0, Bφ

].

To study E[|φ(t1 + τ + (n− 1)T ) − φ(kTp

+ (n − 1)T )|2], we can apply the previous steps to

E[|φ (t1 + (n − 1)T ) − φ((n − 1)T )|2] ,

with t1 = t1 + τ − kTp, where |t1| ≤ 32Tp because k is the

range bin and consequently, |τ − kTp| ≤ 12Tp.

APPENDIX B. COMPOSITE POWER-LAW MODELPARAMETER ESTIMATION

As a first step toward the estimation of the modelparameters, let us denote by

• fi, i = 1, 2, . . ., N1, the discrete set of offsetfrequencies, where the phase noise PSD has beenmeasured;

• Sm(fi), i = 1, 2, . . ., N1, is the phase noise PSDmeasurement in dBc/Hz obtained in correspondence of theoffset frequency from the carrier fi, i = 1, 2, . . ., N1.

Hence, the estimation of the parameters K0, K1, K2, K3,K4, is obtained via the least-square (LS) fitting of themodel SdB( f ) = 10 log10 (Sφ( f )) with the PSDmeasurements Sm(fi), i = 1, 2, . . ., N1, namely, they aredefined as the optimal solution to

minK0,K1,K2,K3,K4

N1∑i=1

|SdB (fi) − Sm (fi) |2. (43)

Problem (43) is a non-convex optimization problem, andwe can obtain suboptimal solution performing a searchover a discrete grid of points or using a search gridalgorithm. In both cases, we need a good initial point tocompute an optimized solution with a low computationalcomplexity. To this end, we fix a set of four differentfrequencies, f1, f2, f3, f4 and evaluate the mentionedestimate according to the following procedure:

• K0 = 10Nf loor /10, with Nf loor = 1M

N1∑i=N1−M+1

S(fi),

the average value of the phase noise PSD measurements,over the last M offset frequencies;

• K1, K2, K3, K4, the solution to the following systemof linear equations

K11

f1+K2

1

f 21

+K31

f 31

+K41

f 41

= 10Sm(f )1/10−K0, (44)

K11

f2+K2

1

f 22

+K31

f 32

+K41

f 42

= 10Sm(f2)/10−K0, (45)

K11

f3+K2

1

f 23

+K31

f 33

+K41

f 43

= 10Sm(f3)/10−K0, (46)


K11

f4+ K2

1

f 24

+ K31

f 34

+ K41

f 44

= 10Sm(f4)/10 − K0.

(47)

This is tantamount to predicting k0 as the phase noise PSDfloor, evaluated as the average value of the last M PSDvalues and estimate the remaining parameters to match thephase noise PSD measurements in correspondence of thefrequencies f1, f2, f3, f4.

APPENDIX C. AUTOCORRELATION FUNCTION OFPOWER-LAW COMPOSITE PSD

Due to the linearity of the integral operator involved in(37), we can separately analyze the different addends in(36). Specifically, we have

• White phase noise

2∫ Wp

f0

K0 cos (2πf τ1) df

= 2K0sin

(2πWpτ1

)− sin (2πf0τ1)

2πτ1, (48)

• Flicker phase noise

2∫ Wp

f0

K11

fcos (2πf τ1) df = 2

∫ 2πτ1Wp

2πτ1f0

K11

νcos(ν)dν

= 2K1

[∫ ∞

2πτ1f0

1

νcos(ν)dν −

∫ ∞

2πτ1Wp

1

νcos(ν)dν

]

= 2K1[ci(2πτ1Wp

)− ci (2πτ1f0)], (49)

• White FM phase noise

2∫ Wp

f0

K21

f 2cos (2πf τ1) df

= 4πτ1

∫ 2πτ1Wp

2πτ1f0

K21

ν2cos(ν)dν

= 4K2πτ1

([−cos(ν)

ν

]2πτ1Wp

2πτ1f0

−∫ 2πτ1Wp

2πτ1f0

1

νsin(ν)dν

)

= 4K2πτ1

[cos (2πτ1f0)

2πτ1f0− cos

(2πτ1Wp

)2πτ1Wp

−Si(2πτ1Wp

)+ Si(2πτ1f0)

], (50)

• Flicker FM phase noise

2∫ Wp

f0

K31


= 8π2τ 21

∫ 2πτ1Wp

2πτ1f0

K31

ν3cos(ν)dν

= 4K3π2τ 2

1

([−cos(ν)

ν2

]2πτ1Wp

2πτ1f0

−∫ 2πτ1Wp

2πτ1f0

1

ν2sin(ν)dν

)

= 4K3π2τ 2

1

([−cos(ν)

ν2

]2πτ1Wp

2πτ1f0

+[

sin(ν)

ν

]2πτ1Wp

2πτ1f0

−∫ 2πτ1Wp

2πτ1f0

1

νcos(ν)dν

)

= 4K3π2τ 2

1

[cos (2πτ1f0)

(2πτ1f0)2 − cos(2πτ1Wp

)(2πτ1Wp

)2

+ sin(2πτ1Wp

)2πτ1Wp

− sin (2πτ1f0)

2πτ1f0

+ci (2πτ1f0) − ci(2πτ1Wp

) ], (51)

• Random Walk FM phase noise

2∫ Wp

f0

K41


= 16π3τ 31

∫ 2πτ1Wp

2πτ1f0

K41

ν4cos(ν)dν

= 16

3K4π

3τ 31

([−cos(ν)

ν3

]2πτ1Wp

2πτ1f0

−∫ 2πτ1Wp

2πτ1f0

1

ν3sin(ν)dν

)

= 16

3K4π

3τ 31

([−cos(ν)

ν3

]2πτ1Wp

2πτ1f0

+[

1

2

sin(ν)

ν2

]2πτ1Wp

2πτ1f0

−∫ 2πτ1Wp

2πτ1f0

1

2

1

ν2cos(ν)dν

)

= 16

3K4π

3τ 31

[cos (2πf0τ1)

(2πf0τ1)3 − cos(2πWpτ1

)(2πWpτ1

)3

+1

2

sin(2πWpτ1

)(2πWpτ1

)2 − 1

2

sin (2πf0τ1)

(2πf0τ1)2 − cos (2πτ1f0)

4πτ1f0

+cos(2πτ1Wp

)4πτ1Wp

+ Si(2πτ1Wp

)2

− Si (2πτ1f0)

2

].

(52)

Let us now analyze the effect of the value Pφ of the phasenoise PSD in the bandwidth [0, f0] on the systemperformance. To this end, we assume thatφ1(t) = φ(t) ⊗ h1(t) and φ2(t) = φ(t) ⊗ h2(t) arestatistically independent random processes,6 where h1(t)and h2(t) denote, respectively, the ideal low-pass filter withsingle-side bandwidth f0 and the ideal bandpass filter over[f0, Wp]. Hence, we have that φ(t) can be expressed as thesum of the two statistically independent processes φ1(t)and φ2(t), where φ2(t) shares the autocorrelation function(38), while φ1(t) is characterized by the autocorrelationfunction

rφ1 (τ1) = Pφ

sin (2πf0τ1)

(2πf0τ1)� Pφ, |τ1| ≤ CPI. (53)

Equation (53) means that φ1(t) can be modeled as arandom variable. Because exp(jφ1(t)) produces a fixedrotation on the received signal, it does not affect thesystem performance.

6 By doing so, the realization of φ1(t) does not influence the pdfsdescribing the random process φ2(t).


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