Statistical Properties of FMCWRadar Altimeter SignalsScattered From a RoughCylindrical Surface

ANDREI MONAKOV, Member, IEEESaint Petersburg State University of Aerospace Instrumentation, SaintPetersburg, Russia

MIKHAIL NESTEROV, Member, IEEEJSC “Detal,” Kamensk-Uralskii, Russia

Statistical parameters of the beat signal of the frequency-modulated continuous-wave (FMCW) radar altimeter in case of scat-tering from cylindrical rough surface are determined in the Kirchhoffapproximation. A concept of the double dimensionality of the beat sig-nal is formulated in this paper. Results of this paper can be used toconstruct a simple simulation model to assess signal processing algo-rithms implemented in an FMCW radar altimeter.

Manuscript received March 21, 2015; revised February 16, 2016 and Au-gust 4, 2016; released for publication October 4, 2016. Date of publicationJanuary 9, 2017; date of current version April 17, 2017.

DOI. No. 10.1109/TAES.2017.2650498

Refereeing of this contribution was handled by R. Adve.

Authors’ addresses: A. Monakov is with the Saint Petersburg State Uni-versity of Aerospace Instrumentation, Saint Petersburg 190000, Russia,E-mail: (a_monakov@mail.ru); M. Nesterov is with the JSC “Detal,”Kamensk-Uralskii 623409, Russia, E-mail: (mn@list.ru).

0018-9251/16 C© 2017 IEEE

I. INTRODUCTION

Invented in 1937 by Espenschied and Newhouse [1], [2],a linear frequency-modulated continuous-wave (FMCW)radar altimeter is used to traditionally measure the altitudeof an aircraft above terrain. Nowadays, radar altimeters havealso become excellent tools for remote sensing because theirecho signals along with a measured radar height providevaluable information on the underlying earth surface [3].

Initially, the theoretical background of the FMCW radaraltimetery was based on a representation of the signal scat-tered from the surface immediately below an aircraft as awaveform emitted by a virtual point source that is placedunderneath the surface at the range twice as large as the air-craft height H . This supposition is true for a perfectly flatsurface and it permits to determine principal concepts of theconventional signal processing in FMCW radar altimeters,which are discussed in numerous papers, e.g., [4], [5], andreferences therein. The received signal is multiplied witha replica of the transmitted signal and this product is low-pass filtered. The beat signal is a result of this “deramp”strategy. Its mean frequency is proportional to the radar al-titude. The range compression of the beat signal via the fastFourier transform (FFT) permits to determine position ofthe beat signal spectrum in frequency domain and, hence,to estimate the altitude.

Actually, the earth surface is not flat and its roughnessmakes the aforesaid representation invalid. In the case ofa rough surface, the received signal is a superposition ofwaveforms produced by a number of point reflectors dis-tributed randomly over the illuminated surface area [6],[7]. Each of these waveforms has its own time delay andthe Doppler shift. As a result, measurement errors are intro-duced into the altitude estimates. These errors are causedby the spatial extend of the antenna footprint and can beattributed to the range noise (glint) [8], [9].

In their seminal paper [10], Moore and Williams pro-posed a phenomenological model for the surface scattering.They represented the received signal as a result of incoher-ent summation of a large number of statistically indepen-dent random waveforms from point reflectors spread outrandomly over the reflecting area. Using this model, theyderived the mean returned power from the illuminated areain case the radar is motionless over the surface and showedthat the mean returned power could be expressed as the“convolution of the transmitted pulse form in power unitswith a function that includes effects of antenna pattern,ground properties, and distance.” Further this result waselaborated in a number of publications by the authors andtheir associates and colleagues [11]–[14].

The Moore–Williams phenomenological model wasused in [15], where the spectrum of the altimeter beat signalwas derived. The beat signal was considered as a randomprocess. It was showed that its spectrum had a steep ris-ing edge followed by a gradually descending “tail,” whoselength depended on the radar carrier frequency and param-eters of the illuminated surface area. This result proved thatthe accuracy of FMCW radar altimeters deteriorates due to

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 53, NO. 1 FEBRUARY 2017 323

the increase of the illuminated surface area and its rough-ness. The Doppler effect was not considered in this paper.Analysis of the range noise that worsens the accuracy ofa CW radar frequency modulated by a sinusoid was per-formed by Tartakovskiy [16]. In [17], Axelsson extendedthis analysis to the sawtooth modulation. Both the rangeand the Doppler spread of the beat signal were consideredin this paper.

Introduction of the average impulse response of a roughsurface by Brown [18], where the average impulse responsewas derived as the convolution of three terms: the probabil-ity density function of the heights of the surface specularpoints, the flat surface impulse response (FSIR), and theradar point target response (PTR), enlarged the applica-bility of the Moore–Williams model. It became possibleto represent the received signal for different statistics ofthe surface roughness and transmitted waveforms. Subse-quently, the scope of the Brown model was extended in anumber of publications, e.g., [19]–[21]. But the suppositionthat the received signal is a sum of statistically independentreturns from the surface scattering centers was used in allthese publications.

The Moore–Williams–Brown model permitted to for-mulate the concept of delay/Doppler altimetry, first sug-gested in [22] by Raney, which is a natural application ofsynthetic aperture radar (SAR) principles in radar altime-tery (see, e.g., [23]). A burst of coherent pulses after thepulse compression and the along-track FFT renders the re-flected signal power distribution upon the two-dimensional(2-D) delay/Doppler map (DDM). Slant range correctionof the DDM and subsequent multilook signal processingreduce the measurement noise and increase the radar reso-lution in comparison to the conventional radar altimetry.

The DDM can be viewed as an average impulse re-sponse of the rough surface just as in the case of theMoore–Williams–Brown model, but the FSIR and the PTRare 2-D functions of the range and the Doppler frequency.So there are several methods to describe power distribu-tion of the received echo on the DDM based on differentnotations for the 2-D FSIR. Halimi et al. [24], [25] pro-posed analytical model based on geometrical simplificationof the range-Doppler grid. Another approach proposed byMarchan-Hernandez et al. in [26] utilizes the change ofvariables between the space and delay-Doppler domainsfor the surface integral representation of the FSIR [27].The DDMs were also used to retrieve surface wind speeddata in the global navigation satellite system reflectometryairborne experiment [28].

In his paper [17], which has already been cited, Axels-son formulated the following statement of the crucial im-portance: “In all [. . .] applications, an improved knowledgeof the basic relationships between the scattering surface andthe output signal is necessary both for the interpretation ofthe data and for the optimum design of the radar.” As it canbe easily followed, all aforesaid papers on the FMCW radaraltimeter use the supposition that goes back to the Moore–Williams model: Specular points, whose waveforms givebirth to the received signal, are statistically independent.

This supposition is true only when the standard deviationof their heights is large and the correlation length of thesurface is comparable to the radar wavelength. In all othersituations, it is not valid. Hence, to comply with the citedstatement, it is necessary to design a new signal modelthat takes into account a finer mechanism of interaction ofthe transmitted signal and the underlying surface. In otherwords, the theory of electromagnetic wave scattering fromrough surfaces has to be applied to design the desired model.

Scattering of electromagnetic waves from rough sur-faces is one of the main issues in microwave remote sensingtheory and practice. There are a large number of publica-tions devoted to the problem starting from the first classicalmonographs [29]–[31]. Current state of the art in the fieldcan be found in the critical survey [32].

Exact numerical computations within the bounds of 2-D and 3-D scattering problems are able to overwhelm evenmodern computer facilities for most conditions encounteredin practice. Hence, one has to resort to some approximationsthat use the electrodynamics equations with different levelsof mathematical rigour. The Kirchhoff’s approximation, thesmall perturbation method, the two-scale model, and varietyof unifying theories are widely used for this purpose [29]–[31], [33]–[36].

The aim of this paper is to create a new statistical modelof the beat signal of the FMCW radar altimeter, whichcan be easily realized on modern computers and has a rig-orous analytical and physical background. To fulfill theaim, it is necessary to derive analytical expressions for sta-tistical characteristics of signals of the FMCW radar al-timeter scattered from a rough surface. Calculations areperformed using the most popular method to solve wavescattering problems—the Kirchhoff’s approximation. Thismethod was first applied to analyze the FMCW radar al-timeter signals by Russian specialists Zhukovskiy et al. intheir monograph [37]. Unfortunately, this textbook was nottranslated into English and is not known outside Russia.Besides, the beat signal in this monograph is considered asa scalar nonstationary random process. This supposition, asit will be shown in this paper, should be reconsidered.

The cylindrical (1-D) rough surface is used as a math-ematical model of the surface. This model is of specialinterest because analytical calculations in this case can bevalidated via direct comparison of simulations results withanalytical calculations. This model can be realized evenwith conventional desktop systems using robust computa-tions [38]–[40]. It is anticipated that fundamental propertiesof the model will be identical to the more numerically de-manded 2-D problem and the analytical methods used forthe 1-D rough surface can be extended to the 2-D situationin future works.

This paper is organized as follows. Representation of thesignal scattered from a rough 1-D surface in the Kirchhoff’sapproximation is derived in Section II. Scattering of a har-monic signal is considered in Section III, where the coher-ent component of the received signal and its two-frequencycorrelation function (CF) are derived. These functions areused in Section IV to perform the statistical analysis of the

324 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 53, NO. 1 FEBRUARY 2017

Fig. 1. Geometry of surface scattering.

altimeter beat signal. Simulation results and their compari-son with the theoretical calculations of the previous sectionsare described in Section V. Finally, this paper ends with aconclusion in Section VI.

II. SCATTERED SIGNAL

Let a rough, perfectly conducting, flat in the mean sur-face, whose average elevation is the horizontal plane z = 0,be illuminated with a monochromatic wave of the frequencyω that is transmitted by a point source. All calculations weconduct in the coordinate system XYZ fixed relatively to thesource (see Fig. 1). The origin 0 is situated on the horizon-tal plane, in a point corresponding to the source position atthe initial moment of observation t0 = 0. The source movesuniformly at a constant height H along the X-axis and itsspeed is V .

The received signal, according to the Kirchhoff’smethod of calculation of the field backscattered from arough surface [30] is

E(t, ω) = πkC

j (2π)3

∫S

F 2(R)exp(−j2kR)

R2dS (1)

where k = ω/c is the wavenumber, C is a constant param-eter depending on the amplitude of the incident sphericalwave, F (R) is the antenna pattern in the direction of a sur-face point with the radii-vector R = (x, y, ξ (x + V τ, y)),R = R/R is the unit vector corresponding to R, R = |R|is the point range, ξ (x, y) is a random surface, S is thehorizontal plane XOY, and τ = t − t0. Let us suppose thatrandom surface heights are much smaller than H . Then, itis possible to use the following expansion:

R =√

x2 + y2 + (H − ξ (x + V τ, y))2

≈ r − H

rξ (x + V τ, y) = r − ξ (x + V τ, y) cos θ (2)

where r =√

x2 + y2 + H 2 is the length of the radii-vectorr = (x, y, H ). Substitution of (2) into (1) yields

E(t, ω) = πkC

j (2π)3

∫S

F 2(r)exp(−j2kr)

r2

× exp[j2kξ (x + V τ, y) cos θ] dS (3)

where r is the unit vector in the direction of the vector r.Equation (3) is a starting point for the statistical analysis

of the received signal. We conduct this analysis for a roughcylindrical surface z = ξ (x) with corrugations extendinguniformly in the Y -direction.

III. CHARACTERISTICS OF A HARMONIC SIGNALSCATTERED FROM THE SURFACE

Since the surface ξ = ξ (x) does not depend on y, itis possible to integrate in (3) over this coordinate assum-ing that the antenna pattern is axisymmetric F (r) = F (θ).Using the stationary phase method of integration, we have

E(t, ω) = −√

kC

8π3/2ejπ/4

∫ ∞

−∞F 2(θ)

exp(−j2kρ)

ρ3/2

× exp[j2kξ (x + V τ ) cos θ] dx (4)

where ρ = √x2 + H 2, cos θ = H/ρ, and sin θ = x/ρ.

Let us assume that the antenna pattern is Gaussian

F (θ) = exp(−α2 tan2 θ) (5)

where α2 = 12 ln 2/ tan2 θa

2 , and θa is the antennabeamwidth.

Now it is possible to determine the coherent componentand statistical characteristics of the incoherent componentof the received signal assuming that the surface ξ = ξ (x) isa Gaussian random process, in which the standard deviationis s and the correlation coefficient is

r(x) = exp[−(x/L)2]

where L is the surface correlation length. These calculationsare performed in Appendix A.

A. Coherent Component

In Appendix A, it is shown that the coherent componentof the received signal is

E(t, ω) ≈ −√

P0e−p2−j2kH = E0e

−p2(6)

where P0 = ( C8πH

)2 is the power of the signal E0 =−√

P0e−j2kH that is reflected from a perfectly flat surface

and p = √2ks is the roughness parameter. Equation (6) is

well known in the theory of wave scattering from roughsurfaces [29], [30].

B. Incoherent Component

The incoherent component is a purely random processwithin the received signal [29], [30] and it can be fullycharacterized with the two-frequency CF [31]

R(t1, t2, ω1, ω2) = 〈[E(t1, ω1) − E(t1, ω1)]

× [E∗(t2, ω2) − E∗(t2, ω2)]〉 (7)

MONAKOV AND NESTEROV: STATISTICAL PROPERTIES OF FMCW RADAR ALTIMETER SIGNALS SCATTERED 325

where the angular brackets designate the ensemble averag-ing. This function is derived in Appendix A as

R(t1, t2, ω1, ω2) = P0H√

k1k2e−(p2

1+p22)e−j2(1−2)

×∞∑

m=1

(2p1p2)m

m!

√b1mb∗

2m

(1 + mH 2

L2

(1

b1m

+ 1

b∗2m

))

× exp

[−mV 2τ 2

L2

(1 + mH 2

L2

(1

b1m

+ 1

b∗2m

))−1]

(8)

where τ = t2 − t1, p1,2 = √2k1,2s, 1,2 = k1,2H , and

b(1,2)m = (2m − 1)/4 + 2α2 − p21,2 + j1,2.

IV. BEAT SIGNAL OF THE FMCW RADAR ALTIMETER

The beat signal (BS) is a process containing all avail-able information on the reflecting surface obtained in theFMCW radar altimeter. Statistical analysis of the BS ismore difficult than that of the harmonic signal because theBS is a nonstationary process, the BS is a result of signalprocessing of a wideband waveform scattered from the sur-face, and it is impossible to neglect for the radar spatialdisplacement during the signal modulation period.

Let us suppose the transmitted signal st (t) is a continu-ous wideband waveform with a periodic frequency modu-lation (FM) of the period Tr . Then, the signal can be repre-sented as the following Fourier series:

st (t) =∑

q

cq exp[jωqt]

where ωq = qr is a frequency of the qth signal harmonicof the modulation frequency r = 2π/Tr . Then, using (4)the received signal can be written as

sr (t) = −∑

q

cqejωq t

√kqC

8π3/2ejπ/4

∫ ∞

−∞F 2(θ)

× exp[−j2kqρ]

ρ3/2exp[j2kqξ (x + V t) cos θ] dx

(9)

where kq is the wavenumber corresponding to the qth har-monic.

Assuming that within the bandwidth of the transmittedsignal, the value of

√kq changes little and performing the

substitution√

kq ≈ √k1, where k1 = ω1/c is the wavenum-

ber corresponding to the instantaneous carrier frequency ω1,(9) can be rewritten as

sr (t) = −√

k1C

8π3/2ejπ/4

∫ ∞

−∞F 2(θ)

∑q

cq

× exp

[jωq

(t − 2ρ − 2ξ (x + V t) cos θ

c

)]dx

ρ3/2

(10)

Then, the BS at the moment t = t1 is

e(t1) = sr (t1)s∗t (t1) = −

√k1C

8π3/2ejπ/4

∫ ∞

−∞F 2(θ)

× st

(t1 − 2ρ − 2ξ (x + V t1) cos θ

c

)s∗t (t1)

dx

ρ3/2.

(11)

Signal st (t1−(2ρ − 2ξ (x + V t1) cos θ)/c) s∗t (t1) is also

the BS but received from a point source, which signal isdelayed for time interval td = (2ρ − 2ξ (x + V t1) cos θ)/crelatively to the transmitted signal. Let us assume that thetransmitted signal has a saw-toothed FM

st (t) = u(t ; Tr ) exp[j

(ω0t + ν

2t2

)], t ∈ [0, Tr ] (12)

where ω0 is the carrier frequency, ν is the FM rate, u(t ; Tr )is the video-pulse envelope, and Tr is the pulse width. Then

st (t1 − td )s∗t (t1) ≈ u(t1; Tr ) exp[−j (ω0 + νt1)td ]

= u(t1; Tr ) exp[−j (ω0td + dt1)] (13)

where d = νtd is a range beat frequency corresponding tothe delay td . In (13), we neglected the phasor exp[jνt2

d /2],which has no appreciable effect on the final result for smalland middle altitudes. If necessary, this phasor can easily betaken into account because it does not depend on the currentfrequency ω1. Then, the BS samples at two moments t1 andt2 are

e(t1,2) =√

k1,2C

8π3/2ejπ/4u(t1,2; Tr )

∫ ∞

−∞F 2(θ)

× exp

[−j2k1,2

(H

cos θ− ξ (x + V t1,2 cos θ)

)]dx

ρ3/2.

(14)

The peculiarity of (14) compared with (4) is that due tothe periodicity of the FM the wavenumbers k1 and k2 mustbe computed according to the following formula:

k1,2 = ω0 + νt1,2

c, (15)

where t1,2 = [t1,2/Tr ], and [·] designates the fractional partof a number. From (15) it follows that e(t1,2) = E(t1,2, ω1,2).Hence, it is possible to use (6) and (8) to calculate the meanvalue and the CF of the BS. But it is necessary to take intoaccount the connection between the moments t1 and t2 andthe carrier frequency samples ω1 and ω2 when calculatingthe wavenumbers (15).

Consequently, the expectation and the CF of the BS,which is derived in Appendix B, can be written in the fol-lowing form:

e(t) = e0(t)e−p2,

R(t1, t2) = P0e−(p2

1+p22)ejH τ u(t1; Tr )u(t2; Tr )

×∞∑

m=1

2p1p2

m!

exp

[−mV 2 τ 2

L2

(1 − j mH τ

2(k0L)2

)−1]

√1 − j mH τ

2(k0L)2

(16)

326 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 53, NO. 1 FEBRUARY 2017

where H = ντH is the range beat frequency correspond-ing to the delay τH = 2H/c, τ = t2 − t1, and τ = t2 − t1.

The CF (16) is a product of two terms. The term

R1(t1, t2) = P0u(t1; Tr )u(t2; Tr )ejH (t1−t2)

corresponds to the deterministic component of the CF,which is a periodic sequence of radiopulses of the carrierfrequency H . This component has a line energy spectrum,concentrated at harmonics of the modulation frequencyr = 2π/Tr . Maximum of the spectrum envelope falls onthe range beat frequency H . This component is a proofthat the BS is a nonstationary random process. In order toovercome this difficulty in further spectral analysis, let usaverage this component

R1(τ ) = P0

2Tr

∫ Tr

−Tr

u(t1; Tr )u(t1 + τ ; Tr )ejH τ dt1

= P0ejH τ(τ ; Tr ) (17)

where (τ ; Tr ) =[1 − |τ |

Tr

(1 − e−jH Tr sign(τ )

)].

The second term

R2(τ , τ ) = e−(p21+p2

2)

×∞∑

m=1

2p1p2

m!

exp

[−mV 2 τ 2

L2

(1 − j mH τ

2(k0L)2

)−1]

√1 − j mH τ

2(k0L)2

(18)

is the CF of a multiplicative noise that distorts the pe-riodic sequence of radiopulses. This CF is also periodicwith respect to the argument τ of the period Tr and ape-riodic with respect to the argument τ . It is obvious thatthis CF characterizes the spreading, which affect the BSspectral lines because of the range glint and the Dopplerscintillation [8].

Equation (18) is general for the spectral analysis of theBS. However, the series in it converges very slowly for largep. Thus, further transformations are necessary.

Let V = 0 and the signal source does not move. In thiscase, the CF of the multiplicative noise is a strictly peri-odic function, and the series in (18) can be approximatelysummed

R2(τ , τ ) =(

1 − e−2p2) exp

[−(

√2νsτ )2/c2

]√

1 − j H τ

2(k0L)2

(19)

This CF describes the SB spectrum spreading becauseof the interference of reflections from the surface specularpoints in the antenna footprint. Therefore, it corresponds tothe range glint. As it follows from (19), the CF is a com-plex periodic function of the argument τ ∈ [−Tr/2, Tr/2].The corresponding PSD is situated in the vicinity of therange beat frequency H and has an asymmetry. Fur-ther discussion of the received result we postpone toSection V, where a comparison of analytical results andresults of mathematical simulation is performed.

Let us return to the general situation when the signalsource moves. Two variables τ and τ are introduced in (18).

The variable τ ∈ (−∞, ∞) appears only in conjunctionwith the speed V and characterizes the Doppler spreadingof the spectrum. The variable τ ∈ [−Tr/2, Tr/2] is presentin the expression also in the case when V = 0. Therefore, itdescribes only the range glint. The Doppler spreading of theBS spectrum superimposes on the spectrum changes causedby the range glint. However, the nature of these phenom-ena is fundamentally different. Therefore, it is convenientto introduce the concept of the “fast” τ ∈ [−Tr/2, Tr/2]and “slow” τ ∈ (−∞, ∞) times to interpret (18) . Theseconcepts are borrowed from the theory of SAR [23]. Signaltime delay is estimated in the “fast” time, the “fast” timedomain belongs to the interval that is equal to the periodof modulation Tr , its quantum in digital signal processingis the sampling period of the BS Ts . The source move-ment takes place in the “slow” time, the Doppler frequencyshift of the received signal manifests itself in this time, itsquantum is the modulation period Tr .

This interpretation permits to consider the BS as a 2-Drandom process and to introduce an appropriate 2-D PSD

S(, ) =∫∫

R(τ , τ ) exp[−j (τ + τ )] dτdτ

where R(τ , τ ) = R1(τ )R2(τ , τ ), and are the “slow”and “fast” frequencies, respectively.

In Appendix C, the following equation for the 2-D spec-trum is derived:

S(, ) = P0

√πL

V

(1 − e−2p2)3/2

√2p

exp

[−

(L

4βV

)2]

×∫ Tr

−Tr

e−γ 2 τ 2(τ ; Tr )

× exp[−j

( − apx()

)τ]

dτ (20)

where γ = √2νs/c, β2 = p2/2(1 − e−2p2

). The last inte-gral can be easily computed numerically. Therefore, thisexpression can be considered final and should be analyzed.As it follows from (20), the 2-D PSD is located on the plane(, ) in the form of a “ridge,” which apex lies along theline

apx() = H√1 − (

/max)2

(21)

where max = 2π · 2V/λ is the maximum Doppler fre-quency of the received signal.

This bearing is fully consistent with the physics of thespectrum spreading because the range beat frequency andthe Doppler shift of the BS corresponding to a specular pointon the surface are unambiguously related, and this relation,as it follows from Fig. 1, is expressed with (20) precisely.Further, due to the roughness of the surface the 2-D PSDis spreading over the “slow” (Doppler) and “fast” (beatrange) frequencies. However, if the spreading of the PSDover the “fast” frequency depends only on the standarddeviation of the surface roughness s, the spreading overthe “slow” frequency is more complicated. For smallvalues of the roughness parameter (p 1), the Doppler

MONAKOV AND NESTEROV: STATISTICAL PROPERTIES OF FMCW RADAR ALTIMETER SIGNALS SCATTERED 327

Fig. 2. PSD of the beat signal for V = 0. (a) p = 1 (b) p = 16.

Fig. 3. 2-D spectrum for V = 50 m/s and p = 1. (a) Estimation. (b) Calculation.

spreading depends only on the correlation radius of thesurface L since in this case β ≈ 1/2. For large values of theroughness parameter β ≈ p/

√2, and the spreading begins

to depend on the ratio s/L, i.e., on the mean slope of thesurface.

V. SIMULATION AND COMPARISON OF RESULTS

To verify the derived results computer simulation wasperformed under the following conditions:

1) Radar wave length λ = 5 cm;2) Frequency deviation �F = 150 MHz;3) Antenna beamwidth θa = 30◦;4) Modulation period Tr = 1 ms;5) Source height H = 100 m;6) Source speed V = 50 m/s;7) Surface correlation length L = 1 m.

The surface profile z = ξ (x) is a Gaussian random pro-cess, whose correlation coefficient is r(x) = exp[−(x/L)2].The received signal was simulated using (4) via numericalintegration. The function z = ξ (x) was represented withits samples taken uniformly with the period λ/8 (eightsamples per wavelength). This sampling period, on one

hand, is necessary for accurate calculation of the integralin (4); on the other hand, it increases the simulation timesignificantly, because the number of points on the surfacebecomes extremely large. The latter circumstance explainsthe small number of statistical runs that were used in thesimulation.

The simulation results and analytical calculations areshown in Figs. 2 and 4. The power spectra of the BS com-puted in accordance to (19) are presented in Fig. 2 fordifferent values of p and the motionless (V = 0) signalsource. Estimates of the PSD, averaged over K = 100 in-dependent signal realizations, are also shown in the figures.As it can be seen from the figures, the power spectra ex-pand toward higher frequencies and acquire a right asym-metry with increase of the surface roughness. This trans-formation is known in the theory of altimetry [37] and is aconsequence of the extension of the surface area compos-ing the received signal. Comparison of the estimates andtheoretical calculations indicates correctness of the latter.

Figs. 3 and 4 present the double-dimension spectraldensities of the BS obtained in the simulation and computedusing (20) for p = 1 and p = 16, respectively. The numberof independent statistical trials was equal to K = 40. The

328 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 53, NO. 1 FEBRUARY 2017

Fig. 4. 2-D spectrum for V = 50 m/s and p = 16. (a) Estimation. (b) Calculation.

dwell time was equal to 0.5 s. The dashed curves in thefigures correspond to (21).

In general, there is a good agreement between the ana-lytical calculations and the simulation results. Estimates ofthe PSD behave as it was predicted in the analysis of (20):They line up along the curve (21), and their expansion overthe Doppler and range beat frequencies with the increaseof the roughness parameter p corresponds to the theory.The analytical and simulation results slightly differ in twoaspects: The principal peak is wider and the side-lobe levelis larger for the analytically calculated PSD. Thus, it can beargued that the simulation confirmed the correctness of thetheoretical calculations.

VI. CONCLUSION

In this paper, we studied statistical properties of signals(harmonic and beat signal of the FMCW radar altimeter)scattered from a cylindrical rough surface. Statistical analy-sis is based on the Kirchhoff’s approximation. Correctnessof the derived analytical results is confirmed by computersimulation. The major finding of this paper is a thesis ofa double dimensionality of the altimeter beat signal. Re-flected from the surface this signal acquires the propertiesof a random field that depends on the “fast” and “slow”times. Signal time delay is estimated in the “fast” time;the “fast” time domain belongs to the interval that is equalto the period of modulation; its quantum in digital signalprocessing is the sampling period of the beat signal. Thesignal source movement takes place in the “slow” time, theDoppler frequency shift of the received signal manifestsitself in this time, its quantum is the modulation period.As a consequence of the 2-D nature of the beat signal, itsspectrum must also be considered as a 2-D function of therange beat frequency and the Doppler frequency. In case ofthe cylindrical surface, the spectrum of the beat signal lookslike a “ridge” in the plane “range beat frequency—Dopplerfrequency,” located along the line connecting the Dopplerfrequencies and range beat frequencies of specular pointson the surface. Spreading of the beat signal spectrum due tothe surface roughness occurs on both frequencies. However,

if the spectrum spreading over the “fast” frequency dependsonly on the standard deviation of the surface roughness, thespreading over the “slow” frequency depends on the surfacecorrelation length in case of small roughness parameters oron the mean slope of the surface in case of large roughnessparameters.

The concept of the double dimensionality of the beatsignal spectrum allows for a new interpretation of trans-formation of the single-dimensional beat signal spec-trum, which is used in modern FMCW radar altime-ters. Obviously, this spectrum is a projection of thedouble-dimensional spectrum on the plane, in which thetrace corresponds to the range beat frequency. Therefore,the spreading of the 1-D spectrum due to the signal sourcemovement, which introduces an additional error in altitudemeasurements, is the result of the “ridged” nature of thedouble dimensionality of the beat signal.

The suggested concept can be used in radar altimetersbecause it allows to establish physical phenomena occurringin the scattering of frequency-modulated continuous signalson rough surfaces more accurately. Analytical results of thispaper can be used to construct an efficient simulation modelof the beat signal to assess the quality of signal processingalgorithms to be implemented in FMCW radar altimeters.

APPENDIX A COHERENT AND INCOHERENTCOMPONENT ANALYSIS

A. Coherent Component

Calculating the mean value of (4) we derive the coherentcomponent as (see [29]–[36])

E(t, ω) = −√

kC

8π3/2ejπ/4

∫ ∞

−∞exp[−2α2 tan2 θ]

× exp[−j2kρ]

ρ3/2exp[−(

√2ks cos θ)2] dx.

(22)

Intoduction of the roughness parameter p = √2ks and

substitution of x = H tan θ and = kH into the integral

MONAKOV AND NESTEROV: STATISTICAL PROPERTIES OF FMCW RADAR ALTIMETER SIGNALS SCATTERED 329

permits to rewrite (22) as

E(t, ω) = − k1/2C

8(π3H )1/2ejπ/4

∫ π2

− π2

exp[−2α2 tan2 θ]

× exp[−j2 cos−1 θ] exp[−p2 cos2 θ]dθ√cos θ

.

(23)

Since � 1 the main contribution to the integral comesfrom the vicinity of the point θ = 0 where the functioncos−1 θ is stationary. Hence, it is possible to enlarge thelimits of integration to the interval (−∞, ∞). Besides, letus represent the trigonometric functions in arguments of theexponents in (23) with the Taylor expansions confined toinfinitesimals of the second order

E(t, ω) = −k1/2Cejπ/4

8(π3H )1/2e−p2−j2

×∫ ∞

−∞exp{[−(2α2 − p2) − j]θ2} dθ

= −k1/2Cejπ/4

8(π3H )1/2e−p2−j2

√π

(2α2 − p2) + j.

(24)

As � |2α2 − p2|, we obtain

E(t, ω) ≈ −√

P0e−p2−j2kH = E0e

−p2(25)

where P0 = (C

8πH

)2is the power of the signal E0 =

−√P0e

−j2kH that is reflected from a perfectly flat surface.

B. Incoherent Component

Substituting (4) and (23) into (7), we have

R(t1, t2, ω1, ω2) =√

k1k2C2

64π3H

×∫ π

2

− π2

∫ π2

− π2

dθ1dθ2√cos θ1 cos θ2

exp

[−j

(21

cos θ1− 22

cos θ2

)]

× exp[−2α2(tan2 θ1 + tan2 θ2)]

× exp[−(p21 cos2 θ1 + p2

2 cos2 θ2)]

× {exp[2rp1p2 cos θ1 cos θ2] − 1} (26)

where 1,2 = k1,2H , p1,2 = √2k1,2s, k1,2 = ω1,2/c, r =

r(H (tan θ2 − tan θ1) + V (t2 − t1)) is the surface correlationcoefficient.

The Taylor expansion of the exponent in the braces in(26) yields

R(t1, t2, ω1, ω2) =√

k1k2C2

64π3H

×∫ π

2

− π2

∫ π2

− π2

dθ1dθ2√cos θ1 cos θ2

× exp

[−j

(21

cos θ1− 22

cos θ2

)]

× exp[−2α2(tan2 θ1 + tan2 θ2)]

× exp[−(p21 cos2 θ1 + p2

2 cos2 θ2)]

×∞∑

m=1

1

m![2rp1p2 cos θ1 cos θ2]m. (27)

Since the right-hand side of (27) depends only on thedifference τ = t2 − t1, the received signal is a weak-sensestationary random process and it can be characterized withits mean power P (ω) and the PSD S(; ω1, ω2).

If r(x) = exp(−x2/L2), where L is the surface correla-tion length, the surface PSD is

G(υ) = √πL exp

[−υ2L2

4

]. (28)

Representing rm(x) via the inverse Fourier transform ofthe PSD (28), we have

R(t1, t2, ω1, ω2) =√

k1k2C2

64π3H

∞∑m=1

(2p1p2)m

m!

×∫

dυ1√m

G

(υ√m

)exp[jυV (t1 − t2)]

× I1(1, υ)I2(−2, −υ). (29)

The auxiliary integrals I1,2 are

I1,2(, υ) =∫ π

2

− π2

dθ cosm−1/2 θ exp

[−j

2

cos θ

]

× exp[−2α2 tan2 θ − p21,2 cos2 θ]×

× exp[−jμ tan θ] (30)

where μ = υH . As it was done in case of the coherent com-ponent, let us take into account that the main contributionto integral (30) comes from the vicinity of the point θ = 0.Using the substitution cosm−1/2 θ ≈ exp[−(2m − 1)θ2/4]and the expansions cos θ ≈ 1 − θ2/2, cos−1 θ ≈ 1 + θ2/2,tan θ ≈ θ , we obtain

I1,2(, υ) ≈ e−p21,2−j2

×∫ π

2

− π2

e−[(2m−1)/4+2α2−p21,2+j]θ2

e−jμθdθ

= e−p2−j2

√π

bm

exp

[− μ2

4b(1,2)m

](31)

where b(1,2)m = (2m − 1)/4 + 2α2 − p21,2 + j1,2. Substi-

tuting (31) into (29), we have

R(t1, t2, ω1, ω2) =√

k1k2C2

64π3H

∞∑m=1

(2p1p2)m

m!

×∫

dυ1√m

G

(υ√m

)exp[jυV (t1 − t2)]

× e−p21−j21

√π

b1m

exp

[− μ2

4b1m

]

× e−p22+j22

√π

b∗2m

exp

[− μ2

4b∗2m

]. (32)

330 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 53, NO. 1 FEBRUARY 2017

After integration over the spacial frequency υ and takinginto account (28), finally we obtain

R(t1, t2, ω1, ω2) = P0H√

k1k2e−(p2

1+p22)e−j2(1−2)

×∞∑

m=1

(2p1p2)m

m!

√b1mb∗

2m

(1 + mH 2

L2

(1

b1m

+ 1

b∗2m

))

× exp

[−mV 2τ 2

L2

(1 + mH 2

L2

(1

b1m

+ 1

b∗2m

))−1]

(33)

where τ = t2 − t1, p1,2 = √2k1,2s, 1,2 = k1,2H .

Assuming ω = ω1 = ω2 and/or τ = 0 and 1,2 =k1,2H � 1 in (8), it is possible to calculate the mean power,the CF, and the PSD of the incoherent component:

P ≈ P0(1 − e−2p2),

R(τ ) ≈ P0e−2p2

{exp

[2p2e

− V 2τ2

L2

]− 1

},

S() ≈ P0

√πL

Ve−2p2

∞∑m=1

(2p2)m

m!√

mexp

[−

(L

2V√

m

)2]

(34)

The first equation in (34) is also well known in the theoryof wave scattering from rough surfaces [29], [30]. The lasttwo equation were used to assess the accuracy of analyticalcalculations. Comparison of the analytical calculations ofS() and corresponding estimates obtained in computersimulation confirmed the correctness of (34).

APPENDIX B CF OF THE BEAT SIGNAL

Substitution of (15) into (8) for the two-frequency CFyields the following equation for the CF of the beat signal:

R(t1, t2) = u(t1; Tr )u(t2; Tr )

×P0H√

k1k2e−(p2

1+p22)e−j2(1−2)

×∞∑

m=1

2p1p2

m!

[b1mb∗

2m + mH 2

L2(b1m + b∗

2m)

]−1/2

× exp

[−mV 2(t1 − t2)2

L2

b1mb∗2m

b1mb∗2m + mH 2

L2 (b1m + b∗2m)

]

(35)

where e0(t) is the BS from a scatterer located in the nadirpoint. In contrast to (8), the CF does not depend any longeronly on the difference τ = t2 − t1. This fact proves thatthe BS is a nonstationary random process. Therefore, inour case it is impossible to use directly spectral methodsdesigned for stationary processes. To have an idea aboutthe spectral composition of the BS, let us analyze (35), forwhich we substitute b1m and b2m into the term

b1mb∗2m + mH 2

L2(b1m + b∗

2m)

= [(2m − 1)/4 + 2α2 − p21 + j1][(2m − 1)/4

+ 2α2 − p22 − j2]

+ mH 2

L2[(2m − 1)/2 + 4α2 − p2

1 − p22 + j1 − j2]

Since 1,2 = k1,2H � 1, it is possible to leave in thelast expression only terms containing 1 and 2, i.e., toput

b1mb∗2m + mH 2

L2(b1m + b∗

2m) ≈ 12 + jmH 2

L2(1 − 2)

= 12

[1 + j

mH 2

L2

1 − 2

12

].

Performing substitutions k1k2 ≈ k20, 12 ≈ k2

0H2 and

1 − 2 = H (k1 − k2), where k0 is the wave number cor-responding to the carrier frequency of the transmitted sig-nal, we have

R(t1, t2) = P0e−(p2

1+p22)ejH τ u(t1; Tr )u(t2; Tr )

×∞∑

m=1

2p1p2

m!

exp

[−mV 2 τ 2

L2

(1 − j mH τ

2(k0L)2

)−1]

√1 − j mH τ

2(k0L)2

(36)

Here, H = ντH is the range beat frequency correspondingto the delay τH = 2H/c, τ = t2 − t1, and τ = t2 − t1.

APPENDIX C PSD OF THE BEAT SIGNAL

To calculate the PSD of the beat signal let us take thedirect Fourier transform of R(τ , τ ) in (18) over the “slow”time ∫

R(τ , τ ) exp[−j (τ )] dτ = P0

√πL

Ve−(p2

1+p22)

× (τ ; Tr ) exp

[jH τ

(1 + 2

22max

)]

×∞∑

m=1

(2p1p2)m

m!√

mexp

[− 1

m

(L

2V

)2]

(37)

where max = 2π · 2V/λ is the maximum Doppler fre-quency of the received signal.

It is possible to obtain an approximate formula for theseries in (37). If a function f (x, y) can be represented as

f (x, y) =∞∑

m=1

xm

m!√

me−y2/m

it is easy to show that its Fourier transform, calculated overthe variable y, is

F (η) = √π{exp[xe−η2/4] − 1}.

Then, the function f (x, y) itself can be found as theinverse Fourier transform

f (x, y) =√

π

2π

∫{exp[xe−η2/4] − 1}ejyη dη.

The function in the braces in the integral is even andinfinitely differentiable, it has maximum equal to ex − 1

MONAKOV AND NESTEROV: STATISTICAL PROPERTIES OF FMCW RADAR ALTIMETER SIGNALS SCATTERED 331

at η = 0 , and tends to zero when η → ±∞. Therefore,with a high degree of accuracy it can be approximated by aGaussian curve

exp[xe−η2/4] − 1 ≈ (ex − 1)e−β2η2.

The parameter β can be found by equating the valueof the second derivatives of the functions in the right- andleft-hand sides of the last equation in the point η = 0:

β2 = x

4

ex

ex − 1.

The following approximations hold true: β2 ≈ 1/4 forsmall values of x, and β2 ≈ x/4 for large ones. Using theGaussian approximation to calculate the inverse Fouriertransform, we have the following approximation of thefunction f (x, y):

f (x, y) ≈ ex − 1

2βexp

[−

(y

2β

)2]

.

Then after replacing the series in (37) we get∫R(τ , τ ) exp[−j (τ )] dτ ≈ P0

√πL

Ve−(p2

1+p22)

× (τ ; Tr ) exp

[jH τ

(1 + 2

22max

)]

× e2p1p2 − 1

2βexp

[−

(L

4βV

)2]

(38)

where β2 = p1p2/2(1 − e−2p1p2 ). Taking the exponente2p1p2 out of the brackets in the fraction numerator andcombining it with the exponent standing in front of thefraction, after the substitution p1p2 = p2 = (

√2k0s)2, we

obtain∫R(τ , τ ) exp[−j (τ )] dτ = P0

√πL

V

× (1 − e−2p2)3/2

√2p

exp

[−

(L

4βV

)2]

e−(p1−p2)2

× (τ ; Tr ) exp

⎡⎣ jH τ√

1 − (/max

)2

⎤⎦ (39)

where β2 = p2/2(1 − e−2p2), β2 ≈ 1/4 for p 1, and

β2 ≈ p2/2 for p � 1. Given that p1 − p2 = √2νsτ /c and

calculating the direct Fourier transform over the “fast” time,we get the following equation for the 2-D spectrum:

S(, ) = P0

√πL

V

× (1 − e−2p2)3/2

√2p

exp

[−

(L

4βV

)2]

×∫ Tr

−Tr

e−γ 2 τ 2(τ ; Tr ) exp

[−j( − apx()

)τ]

dτ (40)

where γ = √2νs/c and apx() = H√

1−(/max)2.

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Andrei Monakov (M’06) graduated from the Leningrad Institute of Aviation Instrument Making, Saint Petersburg,Russia, in 1978. He received the Cand. Sc. degree and the Dr. Sc. degree in radar systems from the Saint Petersburg StateUniversity of Aerospace Instrumentation in 1984 and in 2000, respectively.

He is currently a Professor in the Radio Engineering Department, Saint Petersburg State University of AerospaceInstrumentation, Saint Petersburg, Russia. His research interests include digital signal processing, radar theory, remotesensing, and air traffic control.

Mikhail Nesterov (M’06) received the Diploma degree in physics and applied math from Moscow Institute of Physics andTechnology, Dolgoprudny, Russia, and the Cand. Sc. degree in radar systems from the Saint Petersburg State Universityof Aerospace Instrumentation, Saint Petersburg, Russia, in 1995 and 2006, respectively.

He is currently the Head of the Research Department, JSC “Detal,” Kamensk-Uralskii, Russia. His research interestsinclude remote sensing and radar altimetric systems.

MONAKOV AND NESTEROV: STATISTICAL PROPERTIES OF FMCW RADAR ALTIMETER SIGNALS SCATTERED 333

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