1484 TIME-FREQUENCY SIGNAL ANALYSIS TheTwo … · 1484 TIME-FREQUENCY SIGNAL ANALYSIS TheTwo-Dimensional Hermite S-methodforHighResolutionISAR ImagingApplications Srdjan Stankovi´c,

1484 TIME-FREQUENCY SIGNAL ANALYSIS

The Two-Dimensional HermiteS-method for High Resolution ISAR

Imaging ApplicationsSrdjan Stankovic, Irena Orovic, and Andrey Krylov

Abstract–The two-dimensional multiwindowS-method for radar imaging applications isproposed. It represents a combined tech-nique that uses the standard S-method andthe multiple windows approach based on thetwo-dimensional Hermite functions. The pro-posed method provides significant improve-ment of radar image concentration in compari-son with the standard S-method. Also, it doesnot require an additional post-processing algo-rithm. The efficiency of the proposed methodis demonstrated on various examples.

I. INTRODUCTION

The Fourier transform is one of the most fre-quently used signal analysis tool that providescharacterization and interpretation of signalsfor various applications. Hence, in radar imag-ing, the two-dimensional Fourier transform hasbeen frequently employed. Namely, by ap-plying the two-dimensional Fourier transformto the coherently processed radar echoes, thecorresponding radar images are obtained [1]-[4]. It is used for both the synthetic apertureradar (SAR) and the inverse synthetic aper-ture radar (ISAR) systems. However, the highimage resolution can be achieved only in thecase of stationary targets in SAR and constantrotation in ISAR applications. Otherwise, theDoppler shift and the Doppler spread in thereceived signal will cause blurred and smearedimages. Therefore, in order to improve the res-olution in these cases, various time-frequencyapproaches [5]-[8] and/or motion compensa-tion techniques have been employed [9], [10].The motion compensation techniques providegood results usually by estimating target mo-tion parameters, but at the expense of highcomputational load. Concerning the time-frequency representations, some efficient so-

IET Signal Processing, Vol. 4, No. 4, Jan. 2010.

lutions have been provided by using the S-method [5], [6], [8]. It provides highly con-centrated representation as in the case of theWigner distribution, but with reduced numberof cross-terms [11]. The S-method is compu-tationally very efficient technique that startsfrom the two-dimensional Fourier transformand improves its concentration by performinga simple convolution within the frequency do-main window. The appropriate selection ofwindow width is crucial for image resolutionand readability. The narrow window may pro-duce the blurred radar image, while in the caseof wide window, the cross-terms appear. Moredetails about the choice of window width canbe found in [7], [11]. To provide satisfactory re-sults, in some cases the S-method is combinedwith an additional post-processing algorithm[8].

In order to improve performances of theFourier transform and the standard S-methodin radar imaging, the two-dimensional mul-tiwindow S-method is introduced in this pa-per. It can be considered as an extensionof the method introduced in [8]. The pro-posed multiwindow approach is based on theusage of two-dimensional Hermite functions.These functions exhibit some desirable proper-ties such as good time-frequency localization:they are optimally concentrated in the circu-lar regionR = (t, f) : t2 + f2 ≤ C, with Ca constant. A signal expansion into series ofHermite functions provides, at the same time,the insight to its Fourier transform, since theyare the eigenfunctions of the Fourier trans-form. Therefore, the two-dimensional Hermitefunctions are combined with the advantagesof the S-method, providing an enhanced res-olution in radar imaging. Our approach uses

THE TWO-DIMENSIONAL HERMITE S-METHOD FOR HIGH RESOLUTION ISAR... 1485

only a few low order functions, and hence thealgorithm complexity is slightly increased incomparison with the standard S-method. Un-like some common time-frequency techniques,that split the ISAR image into a time-seriesof ISAR images, the multiwindow S-methoduses the whole set of data, in the same wayas the standard S-method does [7], [12]. Itstarts with an already obtained radar image,based on the multiwindow Fourier transform,and improves its concentration by additionalmatrix calculations.

The paper is organized as follows. Brief re-view of the basic radar signal model and ISARmodel is given in Section II. In Section III, themultiwindow Hermite S-method is proposed.The experimental results including the ISARapplication are presented in Section IV. Con-cluding remarks are given in Section V.

II. T B — S

M ISAR M

The commonly used techniques for radarimaging are SAR and ISAR. These methodsare designed to provide high resolution imagesof stationary and moving targets. The SARmethod is used for stationary targets whilethe radar is moving at a uniform speed andat a constant altitude [3]. On the other hand,the ISAR method is designed for moving tar-get while the radar is fixed. In both systems,radar transmits a signal usually in the form ofM coherent chirps [3], [6]:

xTx(t) = e−jω0tM−1∑

m=0

xa(t−mTr), (1)

where:

xa(t) =

ejπBfrt

2

for 0 ≤ t ≤ Tr0 otherwise.

(2)

The model parameters are:1. t : Fast time,2. ω0: Radar operating frequency,3. Tr: Pulse repetition rate4. mTr: Indexes the slow time5. MTr: Total signal duration6. fr = 1/Tr: Repetition frequency7. B: Bandwidth of the transmitted pulse.

Assuming that the distance between theradar and the target is d (also known as therange), the received signal is delayed with re-spect to the transmitted signal by 2d/c, as fol-lows:

xRx(t) = σe−jω0(t−2d/c)

M−1∑

m=0

xa(t−2d

c−mTr),

(3)where σ is the reflection coefficient, while c isthe speed of light. The received baseband sig-nal produced by the m-th chirp and returnedfrom the single point scatterer can be writtenas [6], [7]:

xRx(m, t) = σejω02d/ce−j2πBfr(t−mTr)2d/c.

(4)Using the matched filtering of the m-th re-ceived pulse and the m-th transmitted pulse(the received signal is multiplied with the com-plex conjugate of the transmitted signal), pro-duces:

x(m, t) = σejω02d/cej2πBfr(t−mTr)ejπBfr(2d

c )2

.(5)

Since B ω0, the constant phase shift

ejπBfr4d2

c2 can be neglected with respect toother two components [9]. Thus, by substi-tuting t−mTr = nTs into (5), where Ts is thesampling interval within the chirp, the two-dimensional discrete mixed signal can be ob-tained in the form [6], [7]:

x(m,n) = σejω02d/ce−jθn, (6)

where θ = 2πBfr2d/cTs is the angular fre-quency. The Fourier transform will result in apulse at the angular frequency θ, proportionalto the range d. Note that, in the case of targetthat is moving with respect to radar, the rangeis time dependent: d(t) = d(nTs +mTr).

ISAR model : Let us assume that a loca-tion of the p-th target point scatterer is (xp,yp). The range coordinate (in direction of theradar-target line) is denoted by xp, while ypdenotes the cross-range coordinate (in the di-rection normal to the radar-target line). Thetotal range of the p-th scatterer is then definedas [7]:

d(t) = R(t), R = R0,


R(t) = R0 + xp cos(ωRt) + yp sin(ωRt)

limωR→0

[R(t)] = R0 + xp + ypωRt, (7)

where ωRt is the rotation angle of point scat-terer. The Doppler shift is obtained as [7]:

ωd =2ω0c

d

dt[d(t)] =

=2ω0c

d

dt[R0+xp cos(ωRt)+yp sin(ωRt)], (8)

For ωRt → 0, ωd =2ω0c ypωR ∼ yp holds, as-

suming that the translation is compensated.According to these conditions, d(t) ≈ xp, whilethe velocity is v(t) ≈ ypωR = v.Under previous assumptions, the two- dimen-sional Fourier transform of the signal corre-sponding to the p-th scatterer can be approxi-mated by [9]: 2πσpδ(ωt−c1yp)δ(ωm−c2mxp),where c1 = 2ωRω0/cand c2 = 4πB/c. In otherwords, it produces a peak on the proper scat-terer position.

Non-uniform motion: When the target ro-tates with constant angular velocity ωR, fora longer period of time, the position of p-thscatterer is given by: yp = rp sin(ωRt), xp =rp cos(ωRt),where rp =

√xp + yp. Hence, for

the time interval [t, t+∆t], the cross-range co-ordinate changes within the interval [7]:

[yp, yp +∆yp] =

= [rp sin(ωRt), xp sin(ωR∆t) + yp cos(ωR∆t)].(9)

The line-of-sight projection of the point scat-terer velocity changes within:

[vs(t), vs(t+∆t)] =

= [rpωR sin(ωRt), rpωR sin(ωR(t+∆t)] =

= [ypωR, xpωR sin(ωR∆t) + ypωR cos(ωR∆t)].(10)

Furthermore, for small ωR∆t, (10) can be writ-ten as [7]:

[vs(t), vs(t+∆t)] = [ypωR, ypωR + xpωR∆t].(11)

Thus, at the frequency that correspondstoypωR, a function corresponding to the linearfrequency modulated signal is obtained insteadof delta pulse [7]. The same holds for the range

coordinate xp. If ωR∆tis not small enough,or if ωRis not constant, the instantaneous fre-quency can be highly non-linear, resulting in aspread two-dimensional Fourier transform andblurred image.

III. T-F ! A!"" ISAR:

T$-D" M$$

S-

A. The Two-dimensional Fourier Transformand the S-method

The two-dimensional Fourier transform hasbeen widely used in radar imaging. For thereceived signal x(m,n), it is given in the form:

X(p, q) =M−1∑

m=0

N−1∑

n=0

x(m,n)e−j(2πmp/M+2πnq/N).

(12)The ISAR image is obtained by using the pe-

riodogram, defined as Per(p, q) = |X(p, q)|2 ,which is the square modulus of the Fouriertransform. Due to the nonuniform target mo-tions, the Doppler shift corresponding to thecross-range of each scatterer, is time-varying.The received radar signal will be frequencymodulated. Consequently, its two-dimensionalFourier transform will be spread, resulting inthe blurred or smeared image.

In order to improve the concentration ofthe periodogram, the S-method has been used.The S-method for a fixed range cell is definedas [6], [11]:

SMR(p, q) =L∑

i=−L

P (i)X(p+ i, q)X∗(p− i, q),

(13)where P(i) is a frequency window which iscommonly of rectangular form. Similarly, fora fixed cross-range cell, the S-method can bewritten as:

SMCR(p, q) =L∑

j=−L

P (j)X(p, q+j)X∗(p, q−j).

(14)Consequently, the two-dimensional S-methodthat improves the concentration of radar im-ages along both coordinates (range and cross-range) is defined as [6], [13]:

SM(p, q) =


=L∑

i=−L

L∑

j=−L

P (i, j)X(p+i, q+j)X∗(p−i, q−j)

= Per(p, q)+

+2Re

L∑

i=1

L∑

j=1

P (i, j)X(p+ i, q + j)×

×X∗(p− i, q − j) . (15)

Note that, in some applications, the one-dimensional S-method applied to the cross-range coordinate could provide satisfactory re-sults [8]. Namely, the target velocities in-duce spreading in the cross-range and focusingalong this axis is needed. However, as a resultof nonuniform rotation (or fast movements)performed by the target and radar setup, thespreading may appear along the range axisas well. In this case the two-dimensional S-method can be used [8].

The S-method is very suitable for hardwarerealization and some efficient solutions have al-ready been proposed and analyzed in [14], [15].

B. The multiwindow S-method based on theHermite functions

Spatial function in the Sturm-Liouvilleboundary value problem, which appears, forexample, in the treatment of the harmonic os-cillator in quantum mechanics, is called Her-mite function [16]. The Hermite functions arethe eigenfunctions of the Fourier transform:

FT (Ψk(x)) = ikΨk(x), (16)

where FT (.) denotes the Fourier transform.Thus, an expansion of signal into a series ofHermite functions enables one to analyze ofthe signal and its Fourier transform simulta-neously. Furthermore, the Hermite functionsare orthogonal and well localized in both timeand frequency domains. Hence, the set of Her-mite functions could be employed for efficienttime-frequency analysis. The two-dimensionalHermite functions are defined as follows [16],[17]:

Ψkl(x, y) =

=(−1)k+lex2/2+y2/2√

2k+lk!l!π

dk(e−x2

)

dxkdl(e−y

2

)

dyl. (17)

Some examples of two-dimensional Hermitefunctions are plotted in Fig 1. It is interest-ing to note that a two-dimensional Hermitefunction can be obtained by calculating one-dimensional Hermite functions for each coor-dinate separately:

Ψkl(x, y) = Ψk(x)Ψl(y) =

=(−1)kex2/2√2kk!

√π

dk(e−x2

)

dxk(−1)ley2/2√2ll!√π

dl(e−y2

)

dyl.

(18)Although they seem to be complex for real-

ization, the Hermite functions could be easilycalculated by using the recursive realization asfollows [18]:

Ψ0(x) =14√πe−x

2/2,

Ψ1(x) =

√2x4√π

e−x2/2,

Ψk(x) = x

√2

kΨk−1(x)−

−√

k − 1k

Ψk−2(x), ∀k ≥ 2. (19)

In order to improve the concentration in thetime-frequency domain, the Hermite functionshave been used in the definition of multiwin-dow spectrograms, denoted as HSpec(t,ω) [19]-[21], which could be written in the discreteform as:

HSpec(n, ω) =

=K−1∑

k=0

(√

dk

∣∣∣∣∣

∑

m

x(m)Ψk(m− n)e−jωm

∣∣∣∣∣

)2,

(20)where dk are the weighting coefficients. Ithas been shown that the multiwindow spectro-gram outperforms the standard one and canprovide highly concentrated time-frequencyrepresentation [20], [21]. Similarly, to im-prove the concentration of radar images, themultiple windows approach based on theHermite functions can be adapted for two-dimensional case and the existing processingtools: the two-dimensional Fourier transformand the S-method (the one-dimensional andtwo-dimensional cases).


Fig. 1. Illustrations of some two-dimensional Hermite functions: a) Ψ00(x, y), b)Ψ24(x, y), c)Ψ44(x, y)

Analogous to the standard S-method, thetwo-dimensional multiwindow S-method canbe defined as a convolution of Hermite basedFourier transforms, as follows:

HSM(p, q) =

=L∑

i=−L

L∑

j=−L

P (i, j)HXK(p+ i, q + j)×

×HXK∗(p− i, q − j), (21)

where HXK(p, q)denotes the multiwindowtwo-dimensional Fourier transform. Itis obtained as a sum of two-dimensionalFourier transforms calculated by using two-dimensional Hermite functions of various or-ders, as follows:

HXK(p, q) =K−1∑

k=0

K−1∑

l=0

HXkl(p, q) =

=K−1∑

k=0

K−1∑

l=0

M−1∑

m=0

N−1∑

n=0

dklΨkl(m,n)

×x(m,n)e−j(2πmp/M+2πnq/N). (22)

where dkl are the weighting coefficients, whileK is the number of functions used along bothrange and cross-range direction (axis). Fur-thermore, in order to simplify the realization,HXkl(p, q) is obtained as a composition ofone-dimensional cases, as follows:

HXkl(p, q) =N−1∑

n=0

[√dlΨl(n)e

−j2πnp/N

×M−1∑

m=0

√dkΨk(m) x(m,n)e−j2πmq/M

]. (23)

The one-dimensional weighting coefficientsdk and dl are determined according to the pro-cedure described in [21], while dkl=

√dkdl =


dlk holds. The different order Hermite func-tions act as multiple windows that improve theconcentration in both directions.

The two-dimensional multiwindow peri-odogram is obtained as: HPer(p, q) =∑K−1

k=0

K−1∑

l=0

∣∣HXkl(p, q)∣∣2 .

Consequently, the multiwindow Hermite S-method can be written in the form:

HSM(p, q) = HPer(p, q)+

+2Re

L∑

i=1

L∑

j=1

P (i, j)HXK(p+ i, q + j)

×HXK ∗(p− i, q − j)

(24)

Observe that in case of the standardS-method given by (15), the terms 2Re∑L

i=1

∑Lj=1 P (i, j)X(p+ i, q + j)

×X∗(p− i, q − j) are used to improve the pe-riodogram concentration (all even phase deriv-atives are removed from the spread factor) [6],[8]. Since the same idea is used for multi-window approach, the multiwindow S-methodimproves the resolution by eliminating evenphase derivatives that remain after the multi-window periodogram removed first K -1 deriv-atives [21].

In some applications, it might be sufficientto calculate the Hermite S-method along onlyone axis. Thus, the Hermite S-method inrange and cross-range direction could be, re-spectively, defined as:

HSMR(p, q) =

=L∑

i=−L

L∑

j=−L

P (i, j)HXK(p+i, q)HXK∗(p−i, q),

HSMCR(p, q) =

=L∑

i=−L

L∑

j=−L

P (i, j)HXK(p, q+j)HXK∗(p, q−j).

(25)The proposed approach can be generalized

by using other functions instead of the Hermitefunctions. For example, the Laguerre func-tions could be interpreted as the Wigner dis-tribution of Hermite functions (k-th order La-guerre function is the Wigner distribution of

the k-th order Hermite function), and hence,could be used to generate optimal kernels [20].Furthermore, the windows provided by splinefunctions, e.g. B-spline functions that aretime-limited piecewise polynomials [22], couldbe considered, as well.

IV. E'(

Example 1 : In order to demonstrate how theS-method improves the concentration in thetime-frequency domain, let us observe the sim-ple simulated signal in the form:

x(n) = x1(n) + x2(n) + x3(n) + x4(n),

where,

x1(n) = A(n)ej0.2

256πn2ej

π

2n,

x2(n) = A(n)ej0.24

256πn2ej

π

8n,

x3(n) = A(n)e−j0.18

256πn2e−j

π

4n,

x4(n) = A(n)e−j0.12

256πn2e−j

π

2n.

The time instant n=0 is considered. Thus,the signal consists of four components thatmay correspond to the returned radar sig-nals from four point targets that accelerate.The amplitude is slow-varying: A(n) = 1

2 +12 cos

π256n, −128 ≤ n ≤ 127.The discrete in-

stantaneous frequencies of signal componentsare: ω1(0) =

π2 ,ω2(0) =

π8 ,ω3(0) = −π

4 andω4(0) = −π

2 . The Hermite S-method is cal-culated by using different values of parameterL (L=0, 4, 16, 32), and the results are shownin Fig 2. Here, the same number of Hermitefunctions: K=3 is used for each L. One mightobserve that the concentration of signal com-ponents increases by increasing L, i.e. by in-creasing the number of convolution terms inthe Hermite S-method. However, for high val-ues of L (L=16 and L=32 in Fig 2), the unde-sired cross-terms appear between components.Generally, in most applications, 3 ≤ L ≤ 10should be used [7], [11].

Furthermore, the concentration can be alsoimproved by increasing the number of Hermitefunctions K. The Hermite S-method, imple-mented by using different number of Hermitefunctions (K=1, 2, 3, and 4) is illustrated in


Fig. 2. Concentration improvement by increasing L:a) L=0, b) L=4, c) L=16, d) L=32

Fig 3. Note that the same value of parameterL (L=4) is used for all values of K.

Example 2 : In order to prove efficiency ofthe Hermite S-method in target points de-tection, the simulated range/cross-range dataare used. The radar setup assumes: highresolution radar operating at the frequencyf 0=10.1GHz, bandwidth of the transmittedsignal B=600MHz, the coherent integrationtime T=2s, 64 pulses in one revisit and 256samples within one pulse. The target is at2 km distance from the radar and rotates atωR = 2.10/ sec. The nonlinear rotation withfrequency Ω = 0.5Hz is superimposed:ωR(t) =ωR +A sin(2πΩt), where A = 1.050/ sec. Thetranslation is not assumed within the experi-ment, so there is no need for translation com-pensation. The target is simulated as six pointscatterers in (x, y) plane, where the posi-tions of scatterers at t=0 are: (x1, y1)=(-2.1, 1.44), (x2,y2)=(0, 1.44), (x3,y3)=(2.1,1.44), (x4,y4)=(-1.08, -0.72), (x5,y5)=(1.08, -

Fig. 3. Concentration improvement by increasing K:a) K =1, b) K=2, c) K=3, d) K=4

0.72), (x6,y6)=(0, -2.28). All coordinates arein meters. Radar images obtained by using:the two-dimensional Fourier transform (peri-odogram), the standard S-method and the pro-posed Hermite S-method are given in Fig 4.The parameter L=3 is used for the calcula-tion of both the S-method and the HermiteS-method. The number of Hermite functionsused in the Hermite S-method is K=3.

We have calculated the number of “cor-rect points” by finding the six largest valueswithin the periodogram, the S-method andthe Hermite S-method (Fig 5), and checkingif they correspond to the true targets posi-tions. Other points are set to zero. The co-ordinates of the six points with largest energyare given in Table I. Note that in the caseof periodogram, the selected points are quitedislocated from the true target points. Fourout of six points could be considered as “cor-rect points”, if the S-method is used, while allselected points in the Hermite S-method are“correct points” for this signal.


Fig. 4. ISAR images of simulated range/cross-range data: a) periodogram, b) standard S-method, c) HermiteS-method

Fig. 5. Six largest values (points) within: a) periodogram, b) standard S-method, c) Hermite S-method

TABLE I

T " ) "' " ("

(x,y) i=1 i=2 i=3 i=4 i=5 i=6True points (-2.1, 1.44) (0, 1.44) (2.1, 1.44) (-1.08, -0.72) (1.08, -0.72) (0, -2.28)Periodogram (-2.25, 1.82) (0, 1) (2.25, 0.8) (-1, -1) (1, -0.4) (0, 1.4)S-method (-2.2, 1.44) (0, 1.44) (2.2, 1.44) (0, 2) (0,2.1) (0, 2.2)

Hermite S-method (-2.2, 1.44) (0, 1.44) (2.2, 1.44) (-1,-0.8) (1, -0.65) (0, -2.28)

Furthermore, all points whose energy isabove the minimal energy of true points areconsidered as “false alarms”. In Fig 6, we haveillustrated the true points and “false alarms”.The number of “false alarms” in the case of pe-riodogram and S-method is 8 (in total 14 pointare presented). Note that within the HermiteS-method there are no “false alarms” in thisexample.

Example 3: The MIG target model isused in this example. The ISAR imagesof MIG25, obtained by using standard peri-

odogram and the standard S-method (with-out post-processing algorithms and by usingthe rectangular window), are shown in Fig 7.aand b, respectively. The value of parameterL=3 is used in the calculation of the stan-dard S-method. Note that some parts of theISAR images are blurred due to the maneuversthat target performs (for example the pointson the nose, wings, etc). The standard S-method cannot provide notable improvementof the concentration, since the starting two-dimensional Fourier transform produces quite


Fig. 6. True target points and “false alarms”: a) periodogram, b) standard S-method, c) Hermite S-method

unfocused image.In Fig 7.c the two-dimensional Hermite pe-

riodogram is shown. It is obtained by us-ing the two-dimensional Hermite functions forK=3 (three Hermite functions are used inboth range and cross-range direction). Itsconcentration can be further improved by us-ing the Hermite S-method. Hence, the im-ages obtained by using the Hermite S-methodfor K=1 (one window function in both rangeand cross-range directions) and K=3 (threewindow functions are used in both range andcross-range directions) are shown in Figs 7.dand 7.e, respectively. As in the case of stan-dard S-method, the parameter L=3 is usedfor the Hermite S-method calculation. Ob-serve that the image in Fig 7.e (the HermiteS-method for K=3 and L=3) is quite focusedand improved comparing to other presentedcases (including the case for K=1).

In this example, the one-dimensional formof Hermite S-method applied to cross-range di-rection is sufficient to provide good results andlow computational costs. Namely, the imagesare sharp enough in the range direction, so theconvolution within the standard S-method andthe Hermite S-method can be applied only tothe cross-range direction. Since a small num-ber of Hermite functions (K=3) is used, thisapproach is still efficient regarding the compu-tational complexity.

The measure of image sharpness [23] is usedin order to quantify the quality of re-focusedimages. This simple quality metric is definedas the intensity normalized sum of the absolute

value of pixel photocurrent convolved with aspatial high pass filter [23]:

IQM = 1E

∑

i

∑

j|Ii,j ∗K| ,

K =

0 −1 0−1 4 −10 −1 0

,

where Ii,j is the intensity at the (i,j )th positionand E =

∑

i

∑

jIi,j is the total intensity of the

image. As an additional objective measure,the image contrast is calculated as follows [24]:

C =

√∑

i

∑

j|Ii,j |4

∑

i

∑j |Ii,j |

2 .

The quality measures, IQM and C, for imagespresented in Fig 7, are given in Table II. Notethat in the case of image obtained by using theHermite S-method, the measures IQM and Chave the highest values, indicating the highestquality.

Example 4 : Here, the simulation with thenoisy signal is considered. Namely, the signalfrom the previous example has been corruptedby the Gaussian noise with SNR=3dB. TheISAR images obtained by using the standardperiodogram, the standard S-method, the Her-mite periodogram and the Hermite S-methodare presented in Fig 8.

Note that the images obtained by using thestandard periodogram and the standard S-method are quite blurred and smeared (Fig 8.a


Fig. 7. ISAR image of MIG25 obtained by using: a) two-dimensional Fourier transform, b) standard S-method,c) two-dimensional Hermite periodogram, d) Hermite S-method by using K=1, e) Hermite S-method byusing K=3

TABLE II

I ! "

Image obtained by using: IQM C

Standard periodogram — Fig 7.a 0.76 0.00089Standard S-method (L=3) - Fig 7.b 0.71 0.0011

Hermite periodogram (K=3) - Fig 7.c 0.79 0.014Hermite S-method (K=3,L=3) - Fig 7.e 0.86 0.0019

and Fig 8.b, respectively). The Hermite peri-odogram (Fig 8.c) improves the results, but itis still blurred and affected by the noise. How-ever, the Hermite S-method (Fig 8.d) is morerobust in noisy conditions and provides ratherclear and focused image.

V. C"

A multiwindow Hermite S-method is in-troduced as an efficient tool for high res-olution radar imaging. This multiwindowapproach is based on the usage of two-dimensional Hermite functions. Also, the Her-mite based two-dimensional Fourier transformand Hermite periodogram are defined as in-

termediate processing steps. Furthermore,the one-dimensional and two-dimensional ver-sions of the Hermite S-method are consid-ered. The Hermite S-method provides a sig-nificant resolution improvement comparing totwo-dimensional Fourier transform and thestandard S-method. The enhanced resolutioncould be obtained with only a few Hermitefunctions that could be easily calculated byusing the recursive algorithm. The computa-tional complexity of the proposed method isslightly increased comparing to the standardS-method. Hence, it is amenable for imple-mentation in hardware.


Fig. 8. Noisy ISAR image of MIG25 obtained by using: a) periodogram, b) standard S-method, c) Hermiteperiodogram, d) Hermite S-method by using K=3

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The authors are very thankful to anonymousreviewers for useful comments and suggestionsthat helped to improve the paper.

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[1] Carrara, W. G., Goodman, R. S., and Majewski,R. M.: ‘Spotlight Synthetic Aperture Radar–Signal Processing Algorithms’, Artech House,Norwood, Ohio, USA, 1995.

[2] Wong, S., Riseborough, E., and Duff, G.: ‘Experi-mental investigation on the distortion of ISAR im-ages using different radar waveforms’, TechnicalMemorandum TM 2003-196, Defence Researchand Development Canada, Ottawa, Canada,2003.

[3] Chen, V. C., and Ling, H.: ‘Time-FrequencyTransforms for Radar Imaging and Signal Analy-sis’, Artech House, Boston, Mass, USA, 2002.

[4] DeGraaf, S. R.: ‘SAR imaging via modern 2-Dspectral estimation methods’, IEEE Transactionson Image Processing, 1998, 7, 5, pp. 729—761.

[5] Popovic, V., Dakovic, M., Thayaparan, T., andStankovic, LJ.: ‘SAR Image Improvements by Us-ing the S-method’, in Proc. IEEE ICASSP, vol.3, May 2006, pp.III 177-III 180.

[6] Stankovic, LJ., Thayaparan, T., Dakovic, M., andPopovic, V.: ‘S-method in radar imaging’, in Pro-ceedings of the 14th European Signal ProcessingConference (EUSIPCO ’06), Florence, Italy, Sep-tember 2006.

[7] Thayaparan, T., Stankovic , LJ., Wernik, C., andDakovic , M.: ‘Real-time motion compensation,image formation and image enhancement of mov-ing targets in ISAR and SAR using S-methodbased approach’, IET Signal Processing, 2008, 2,3, pp. 247-264.

[8] Stankovic , LJ., Thayaparan, T., Popovic , V.,Djurovic , I., and Dakovic , M.: ’Adaptive S-method for SAR/ISAR Imaging’, Eurasip Jour-nal on Advanced Signal Processing, 2008, ArticleID 593216, 10 pages.

[9] Djurovic , I., Thayaparan, T., and Stankovic ,LJ.: ’Adaptive Local Polynomial Fourier trans-form in ISAR’, Eurasip Journal on Applied SignalProcessing, 2006, Article ID 36093, 15 pages.

[10] Haiqing, W., Grenier, D., Delisle, G. Y., and Da-


Gang, F.: ‘Translational motion compensation inISAR image processing’, IEEE Transaction onimage processing, 1995, 4, 11, pp.1561-1571.

[11] Stankovic , LJ.: ‘A method for Time-FrequencySignal Analysis’, IEEE Transaction on SignalProcessing, 1994, 42, 1, pp. 225-229.

[12] Thayaparan, T., Stankovic , L., Dakovic , M.:”S-method based approach for image formation,motion compensation, and image enhancement ofmoving targets in ISAR and SAR”, IEEE In-ternational Symposium on Geoscience and Re-mote Sensing, IGARSS 2008.,vol.2, pp. II-29-II-32, July 2008.

[13] Stankovic , S., Stankovic , LJ., and Uskokovic , Z.:’On the Local Frequency, Group Shift and CrossTerms in Some Multidinesional Time-FrequencyDistributions; A method for MultidimensionalTime-Frequency Analysis’, IEEE Transaction onSignal Processing, 1995, 43, 7, pp. 1719-1724.

[14] Stankovic , S., Stankovic , LJ., Ivanovic , V., andStojanovic , R.: ’An Architecture for the VLSIDesign of Systems for Time-Frequency Analy-sis and Time-Varying Filtering’, Annales desTelecommunications, 2002, 57, (9/10), pp. 974-995.

[15] Stankovic , S., Djurovic , I., and Vukovic , V.:’System Architecture for Space-Frequency ImageAnalysis’, Electronic Letters, 1998, 34, 23, pp.2224-2225.

[16] Najafi, M., Krylov, A., and Kortchagine, D.:‘Image Deblocking With 2-D Hermite Trans-form’, Proceeding Int. Conference Graphicon2003, Moscow 2003, pp. 180-183.

[17] Krylov, A., and Kortchagine, D.: ‘HermiteFoveation’, Proceeding Int. Conference Graphicon2004, pp. 166-169, Moscow, 2004.

[18] Kortchagine, D. N. and Krylov, A. S.: ‘Projec-ton Filtering in image processing’, Proceeding Int.Conference Graphicon 2000, pp.42-45, 2000.

[19] G. Fraser, B. Boashash, “Multiple window spec-trogram and time-frequency distributions,” inProc. IEEE ICASSP, vol. 4, 1994, pp. 293-296.

[20] Bayram, M., and Baraniuk, R. G.: ‘MultipleWindow Time-Frequency Analysis’, Proc. of theIEEE Signal Processing International Sympo-sium on Time-Frequency and Time-Scale Analy-sis, pp.173-176, Jun 1996.

[21] Cakrak, F., and Loughin, P. J.: ‘Multiwin-dow Time-Varying Spectrum with InstantaneousBandwidth and Frequency Constraints’, IEEETransactions on Signal Processing, 2001, 49, 8,pp. 1656-1666.

[22] Toraichi, K., Kamada, M., Itahashi, S. and Mori,R.: ‘Window functions represented by B-splinefunctions’, IEEE Trans. on Acoustics, Speech andSignal Processing, 1989, 37, 1, pp. 145-147.

[23] Cohen, M., Cauwenberghs, G., and Vorontsov,M.: ‘Image Sharpness and Beam Focus VLSI Sen-sors for Adaptive Optics’, IEEE Sensors Journal,2002, 2, 6, pp. 680-690.

[24] Callow, H.J., Saebo, T.O., and Hansen, R. E.:‘Towards Robust Quality Assessment of SAS Im-agery using the DPCA algorithm’, Oceans 2005 —Europe, 2005, 2, pp. 1095-1100.

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1484 TIME-FREQUENCY SIGNAL ANALYSIS TheTwo … · 1484 TIME-FREQUENCY SIGNAL ANALYSIS TheTwo-Dimensional Hermite S-methodforHighResolutionISAR ImagingApplications Srdjan Stankovi´c,