Abstract—An elementary analysis of X-band SAR images of the flashes from the rotating blades of a hovering helicopter is presented. The analysis is supplemented by simulations based on a physical optics approximation. It is shown that the SAR image consists of a set of ghost images. The ghosts have the form of a ‘comb’ of periodically repeated point spread functions. Each one corresponds to the point spread function of the unmodulated SAR for the leading and trailing blade edges weighted by an envelope consisting of a much broader ‘sinc’ function. Due to interference effects the blade scattering is effectively confined to a small length of the blade. Some effects of varying the rotation rate are presented.

Index Terms— SAR, helicopters, micro-Doppler.

I. INTRODUCTION Vibrating or periodically interrupted targets modulate

scattered coherent real aperture radar signals to produce spectral structure which can be analysed to give information additional to that obtained from a static target. This is known as the ‘micro-Doppler effect’ [1,2]. An interesting example is shown in figure 1.

Figure 1 – Schematic of blade backscattering from a hovering helicopter. Figure 1 shows radar reflections from the main and tail rotor blades of a hovering helicopter. Helicopters have either metallic or composite blades. Both types produce strong scattering, the composite blades often having a metal leading edge. Modelling of the blade scattering [3] and experimental data [1,4] shows that the blades produce ‘flashes’ when they are approximately broadside on to the radar beam. These flashes last for a time of the order of a millisecond. In general

This work was funded by the UK Ministry of Defence Research Acquisitions Office (ISTAR).

blade flashes are produced when the blades are both moving towards and away from the radar, although there is asymmetry leading to different flash intensities for leading and trailing blade edges. When the rotor contains an even number of blades a flash is produced simultaneously by both advancing and retreating blades. When the number of blades is odd the flashes alternate between advancing and retreating blades. Similar arguments also hold for the tail rotor. Reflections can also come from the airframe structure behind the blades and are chopped by the rotor blades. However, it can be shown that this case is included in an analysis of the case shown in figure 1 and so the more general case is discussed here. This type of target produces both amplitude and phase modulation of the scattered signals. The amplitude modulation is very strong because the blades effectively chop the radar signal resulting in a periodic ‘pulse’ modulation with extensive harmonics. The phase modulation is also pronounced because the blades are rotating and the total path length of a wave incident on, and reflected from, a particular blade varies while the blade is illuminated. However, because of the radar pulsing there is a stroboscopic effect and when there is an exact number of pulses between each blade ‘broadside on’ event the blades appear stationary to the radar. In general the blades appear to be rotating at a much smaller rate than the actual rate and Doppler effects are therefore much reduced. Nevertheless there is in general a small effective signal Doppler shift and this varies continuously from a maximum (minimum) at the ends of the blades to near zero at the blade roots. Note that, the range to the radar is usually very large so, in principle, the blade flash occurs all the way along the blade.

The micro-Doppler effect usually relates to real aperture radars but it is of interest to find out if vibrating and periodically interrupted targets produce visible effects in SAR images. An elementary (and highly simplified!) analysis of the SAR image of rotating helicopter blades is presented in this paper. It is shown that, to first order, the result of chopping the radar signals is to introduce ghost images. The ghosts have the form of a ‘comb’ of periodically repeated ‘sinc’ functions (each corresponding to the point spread function of the unchopped SAR) weighted by an envelope consisting of a much broader ‘sinc’ function.

In order to define terms and provide a reference for the subsequent analysis consider briefly the image of a single coherent stationary point. The range from the focal point of the radar antenna to a scattering point can be expanded as a time series in t:

Imaging the rotor blades of hovering helicopters with SAR

B. C. Barber Defence Science and Technology Laboratory

Dstl Farnborough, Ively Road, Farnborough, GU14 0LX, UK phone: + 44(0)1252455093, fax: + 44(0)1252455765, email: bcbarber@dstl.gov.uk

1-4244-1539-X/08/$25.00 ©2008 IEEE

( ) ( )20 1 2 1a t a a t a t= + + +…

The azimuth focussing process is carried out by means of a

correlation. It is supposed that the phase of a wave scattered from a point is ( )4 /a tπ λ where λ is the radar wavelength. The factor of 4 is a consequence of the double path from the radar to the point scatterer. A plane wave scattered from a point can then be represented by an exponential of the phase function. The Doppler bandwidth AF∆ is 24A AF a T λ∆ = where AT is the synthetic aperture time interval and there is an offset Doppler frequency 12Af a λ= . For simplicity without loss of generality it is assumed here that Af is mixed to zero. The number of pulses in the aperture is 1PN + and the pulse recurrence frequency is υ . Let τ∆ be the time between the start of a pulse and the start of the next pulse so that 1τ υ∆ = . Again, for simplicity it is assumed that the Doppler band is sampled at the Nyquist rate so that

1AFτ∆ ∆ = , although this limitation can be easily removed. If terms up to second order in the range polynomial are included the discrete azimuth correlation can then be shown to be (see [5] appendix B):

( ) ( )

( ){ }( )

( ) ( )( )( )

0

22

0

20

exp 4 2

exp exp

exp 4 exp

sin 1exp

sin

P

A

N

P Pn

P

P P

P

ag i

ni n iN N

ai i

N

N Ni

N

ζ ξ πλ

π ζ ξ π

ππ ζ ξλ

π ζ ξπ ζ ξ

π ζ ξ

=

− =

⋅ − − −

= −

+ −

⋅ − − ⋅−

∑

where the point spread function is located at ξ and the aximuth correlation variable is ζ . Both ξ and ζ are expressed in units of τ∆ . The correlation is performed against a sampled replica of the expected scattered wave (of unit amplitude) and this gives the usual periodic point spread function in the azimuth direction with a period equal to the synthetic aperture interval. If PN is large the point spread function approximates to the usual ‘sinc’ function .

II. THE SAR IMAGE OF ROTATING HELICOPTER BLADES In this section an elementary analysis is constructed in

order to gain insight into the basic properties of SAR images of blade flashes. Some more detailed properties are obtained in section III from a simulation model. In order to construct an elementary analysis simplifications are made here which nevertheless retain sufficient physics of the problem to provide a basic description of the imaging process. The discussion here centres on the main rotor blades but is also applicable to the tail rotor. The tail rotor is somewhat less important because in many helicopters it is masked by the fuselage. On the other hand, because the plane of the tail rotor

is at right angles to that of the main rotor it has a different polarimetric response. Also, there can be interference effects between the blades and other reflections but these are ignored here.

Experimental data (see, for example, measurements for a real aperture S band radar in [4]) gives timescales of a fraction of a millisecond for the blade flash. One might imagine that at X band the flash duration would be proportionately less due to narrower diffraction lobes. However, for a SAR the effective flash time may be well over a millisecond even at X band. For example a high resolution SAR images only a small fraction of the blade length in each resolution cell and this results in wider effective diffraction lobes. In addition, in the formation of the synthetic aperture there are interference effects between length elements of an individual blade for different pulses which result in most of the scattering effectively resulting from a small part of the blade. This is shown below in the simulations.

It was pointed out in the Introduction that there is a stroboscopic effect due to the pulsing of the radar. In general, in the case of a high resolution wideband SAR with a prf of many kHz there may be many pulses contained within each ‘flash’. It is reasonable to suppose that, while the blade reflection is active at a particular point along the blade, there is a constant rate of change of phase from pulse to pulse due to the change in range due to the blade rotation. In addition there is the quadratic phase due to the motion of the radar on which the synthetic aperture focusing depends and which is removed during the azimuth correlation. Now consider the case when there is an integer number of pulses between each ‘broadside on’ event so that the blade rotation is frozen. In this case the set of pulses illuminating the blade during the flash are symmetrically distributed either side of the pulse which occurs at the exact broadside on time. Their phases will be advanced and retarded symmetrically about the phase of the broadside on pulse, the quadratic phase changes having been removed by the correlation. Hence during the formation of the synthetic aperture the resultant of the vector sum of the signals from all the pulses in each flash scattered from each scattering element on the blade will have the same phase as the signals scattered from each element for the single pulse illuminating the blade exactly broadside on. This argument can obviously be generalized to the case when each inter-flash period does not contain an integer number of pulses. Hence the important blade rotation rate is not the actual rate but the reduced stroboscopic one. Moreover the SAR and processing behave in a sense as if the set of pulses contained in each flash is equivalent to one large pulse and the pulse recurrence frequency is reduced to the flash recurrence frequency. This, of course produces aliassing and this is manifested by an image of the blades which is periodically repeated across the image.

Actually, the above arguments overlook the amplitude of the resultant of the correlation of the pulses within a particular set which illuminate a flash. The pulses before and after the central broadside on pulse add along the arcs of a circle due to the advancing and retarding phase brought about by the blade rotation. In addition, the arc radius depends on the position of the element along the blade. Hence the amplitude of the resultant also depends on the distance along the blade. It turns

out (see the simulations in section III) that the maximum occurs near the centre of the blade for the parameters used in the simulation.

Let the actual blade rotation rate be α revolutions per second. Consider first the special case when α is such that the blade rotation is frozen. If there are BN blades the number of flashes per second is BNα . The frozen rotation case occurs when the number of pulses from the start of a flash to the start of the next flash is an integer η where BNη υ α= . Also, the

total number of flashes in the aperture is ( )intF B AN N Tα=

where the total synthetic aperture time is AT . The integer part of B AN Tα is taken because there may be ‘dead pulses’ at the end of the synthetic aperture which do not illuminate a flash or ‘incomplete flashes’ which do not contain a complete set of pulses. In order to further simplify the problem suppose that the pulse recurrence frequency is high enough so that there are a number of pulses in each flash. This implies a high resolution SAR operating with a recurrence frequency of many kHz. It is possible that the number of pulses in each flash may fluctuate since the pulse repetition rate and the rate of rotation may not be sufficiently synchronised over the aperture time. This effect is ignored here (and in fact for the frozen rotor case considered here the number is constant), and it is assumed that the number of pulses contained within a flash is constant from flash to flash. Hence the blade flash is on for a time t∆ and contains 1∆ + pulses. Also, it is supposed that the amplitude of the flash is constant while switched on and that the rate of rotation is constant over the aperture time.

Consider first the image of the thp flash from a particular element on the blade with a scatter cross section of lσ ∆ where σ is the amplitude cross section per unit length of the blade, l∆ is the length of the element and l is a coordinate measured along the blade:

( ) ( ){ } ( )

( )

( ) ( )

( ) ( )

( )

2

22

2

exp 3

4exp exp2

exp

2exp exp

2sin 1

2sin

p

ppm p

P

P

P P

P

P

g l i mN

li m i m pN

l iN

pi iN N

lN

lN

πζ ξ σ ζ ξ

π π αλυ

πσ ζ ξ

π πζ ξ ζ ξ

ζ ξ ωπλυ

ζ ξ ωπλυ

+∆

=− = ∆ − −

∆ ⋅ − ⋅ − −

= ∆ −

∆⋅ − − − −

− ∆ + − ⋅

− −

∑

where 2ω πα= and is the angular rotation rate.

The phase function 24exp

2li m pπ α

λυ∆ − −

in (3) results

from the change of phase from pulse to pulse due to the change in range caused by blade rotation. This, of course, depends on the distance along the blade l .

Suppose that the start time of the synthetic aperture coincides with the start of the first pulse in the synthetic aperture sequence. The image of a point on the blade is then the sum over all the flashes in the synthetic aperture of the image of a single flash. The sum over all the flashes for the small length l∆ of blade located at l along the blade is then given by :

( ) ( ) ( )

( ) ( )

( ) ( )

( )

( )( )

( )

0

2

4

exp exp

2sin 1

2sin

1sin

sin

FN

Pj

P P

P

P

F

P

P

g l g

i iN N

lN

lN

NN

N

ζ ξ σ ζ ξ

π πζ ξ ζ ξ

ζ ξ ωπλυ

ζ ξ ωπλυ

πη ζ ξ

πη ζ ξ

=− = ∆ −

∆= − − − −

− ∆ + − ⋅

− −

+ −

⋅−

∑

Note that in the summation in (4), j p η= . When PN and FN are large ( )1F PN Nη + ≈ and

P FN Nη ≈ . The last periodic sinc function in (4) then becomes

( ) ( ){ }sin sin FNπ ζ ξ π ζ ξ− −

And this is a sinc function with the full resolution of the SAR but which is periodically repeated whenever

FNζ ξ− = . Hence each point spread function in the periodically repeated sequence (‘comb’) has a width given by

1A AF τ∆ = and this is equal to the resolution which applies when the aperture is continuous and unchopped. The period is given by 1A A FF Nτ∆ = and this is equal to FN point spread function widths, where FN is the number of flashes in the aperture. The first sinc function in (4):

( ) ( )

( )

2sin 1

2sin

P

P

lN

lN

ζ ξ ωπλυ

ζ ξ ωπλυ

− ∆ + −

− −

is more complicated and contains two distinct scales. Close to a blade image, ( ) 2PN lζ ξ ω λυ− << and then the sinc function is peaked whenever 2 lω λυ is near an integer. For the example presented in the next section this turns out to be near the middle of the blade. Hence the blades appear to be much shorter in the images than they really are. This effect is demonstrated in the simulations shown in the next section. The other scale is ( )1PN ∆ + and is much longer and defines the width of the envelope of the periodically repeated blade images. It is essentially the point spread function corresponding to a single flash interval t∆ and has a width given by . 1A A AF t Tτ∆ ∆ = . This is greater than the point spread function width for a continuous unchopped aperture by a factor of AT t∆ resulting in a very wide comb of images which can in principle be many km. The analysis above refers to the ‘frozen’ case where there is an integer number of pulses from flash to flash. In general this will not be the case and the rotor blades will have a small effective (stroboscopic) rotation. The small rotation gives rise to a non-zero radial velocity at the broadside on blade position. This results in azimuth shifts in the blade images. The exact stroboscopic velocity depends on some parameters which are not in general easy to determine. For example, how far away the blade rotation rate is from a frozen case, and this depends on the factors of the pulse recurrence frequency, the blade rotation rate etc. In general the receding and leading edges of the blades are shifted in opposite directions but remain well focussed. A general discussion of this aspect of the problem is not attempted here but examples are given in the simulations presented next.

III. SIMULATIONS The above analysis is incomplete but gives a qualitative

description of the basic aspects of the images of rotating helicopter blades. A more complete and refined simulation is briefly presented in this section. The simulation is based on a completely fictitious, but entirely realistic, satellite radar system. Relevant orbit parameters, radar parameters and helicopter parameters for the simulation are listed in table 1. The helicopter blades were assumed to extend from 1 metre to 6 metres and were divided into 32 elements per metre. The time of the centre of first pulse at the beginning of the synthetic aperture was assumed to occur when a blade was exactly broadside on to the beam. Sets of fixed numbers of pulses were defined (eleven in the simulations presented here) which coincided approximately with subsequent ‘broadside on blade events’. For each pulse in the set the phase of each element was computed on the basis of where it was with respect to a broadside on blade as a suitable reference. The sum of the signals scattered from each element for each pulse then constituted the scattered signal which was then azimuth compressed.

Orbit radius 7168 km Earth radius 6368 km Satellite orbit angular rate 1.0403×10-3 rad/s

Min. antenna to target distance 1768 km

Grazing angle at target 20°

Antenna to target angle 13.40°

Target latitude 50.0°

Antenna latitude 46.53° Antenna to target longitude difference 19.51°

Satellite heading angle 22.10°

Squint angle 0°

Wavelength 0.03m Azimuth and range resolution 1m Aperture time 3.800s

Pulse recurrence frequency 6800.0Hz

No of pulses in aperture 25840

Range polynomial coefficients as defined in equation (1)

1a = −330.5 m/s

2a = 13.421 m/s²

Helicopter blade length 6m

No of blades 4 and 5 Table 1 – Satellite orbit parameters, radar parameters, and helicopter parameters.

The simulation is essentially a physical optics approximation. It is assumed that the elements all have the same scatter cross section, although the cross section presented here is arbitrary. It was explained above that those pulses occurring before and after the central broadside on pulse add along the arcs of a circle due to the advancing and retarding phase brought about by the blade rotation. Hence those pulses which illuminate the blade well away from the broadside on position the circular arc will wind up to give a small resultant. In other words these pulses are attenuated by the effective diffraction lobe. Hence, bearing in mind that the objective here is to determine the structure of the images rather than absolute cross-sections, it is only necessary to consider pulses near to the broadside on position. It was found by trial and error that that five pulses distributed uniformly on each side of the pulse corresponding to the blade position nearest to the broadside on position gave an acceptable approximation to a large number. There were therefore a total of eleven pulses in each set.

As was pointed out in the previous section the ghost images of the rotating blades are spread over many km in azimuth and is one point spread function wide in range. Consequently, the complete image is difficult to display. The images shown in here are small parts of the total and measure just 640m in azimuth. The total number of pulses used was 25840 and this corresponded to 25840m since there was one pulse per metre over the ground giving an azimuth resolution of one metre.

Figure 2 – Ghost images of 4 blades rotating at 272 revs. per minute. Scattering from the whole of each blade.

Figure 2 shows the image of a blade at the centre of the total image at 12920 metres. In addition there are four ‘ghosts’ either side. Notice that each image has a double peak. For the purposes of this simulation it was assumed that the scattering from the front edge of the blade was 6dB higher than that from the rear edge. Hence one of the peaks is 6db higher than the other. Also, the centres of the blade edges are separated by 6m in this (even blade number) case. The rotation rate (272 rpm) results in every blade being exactly broadside on every 375 pulses for a pulse recurrence frequency of 6800Hz. This is therefore a frozen case. This rotation rate is, however, approximately representative of some helicopters. Notice that there is a sinusoidal amplitude modulation of the ghosts. The image width was chosen to cover one complete cycle of the modulation. The modulation is a result of interference effects not included in the approximate analysis in the previous section. Notice also that the width of the peaks is only one point spread function width (1 metre) and not the full width of the blade (5 metres) as a result of interference effects which were included in the analysis.

Figure 3 – Ghost images of 4 blades rotating at 272 revs. per minute.

Scattering from a section of blade from 3.75 to 4.75 metres. Figure 3 shows the same image as figure 2 but instead of

the blade extending fron 1 to 6 metres on each side the blade has been taken to extend from 3.75 metres to 4.75 metres and this represented the maximum for this case. Notice that the peak of the blade image is 32.0 for this image. Compare this with figure 2 where the peak is 29.7. The images are otherwise comparable. Hence reducing the scattering length has increased the image amplitude slightly. This demonstrates that the scattering effectively comes from only about a one metre length of blade roughly in the centre. This is a consequence of the interference effects mentioned above. It is easy to show that for the parameters used in this simulation the pulse to pulse phase shift within a flash is 2π at a distance of 3.60m along the blade. The effective centre of the image of each blade ought to be centred on this distance instead of the 4.25 metres actually measured, an error which has not been resolved yet.

Figure 4 shows the effect of varying the blade rotation rate. The blade rotation rate of 272 rpm used in the examples

shown in figures 2 and 3 is shown along line 272 in the centre of figure 4. Notice how changes of only 1 rpm can make large changes to the images. For some of the rotation rates the leading and trailing edge images are completely separated.

Figure 4 – Ghost images of 4 blades for rotation rates from 245 to 275 rpm showing the effect of varying the rotation rate.

Figure 5 – Ghost images of 5 blades for rotation rates from 245 to 275 rpm showing the effect of varying the rotation rate. Figure 5 shows the same case as figure 4 except for 5 blades instead of 4. In figure 4 the frozen rotation rates are 272, 255 and 250 rpm and the frozen rates in figure 5 are 272 and 255 rpm. Notice how the images are very similar for these rotation rates and compare the sections through the images at line 255 in figures 4 and 5 with the sections through line 272 in figures 2 and 3.

IV. DISCUSSION It is plain from the above results that the SAR image of the

rotating blades of a hovering helicopter is very distinctive. Nothing concerning the actual amplitude of the ghosts has been presented here but they have a low level and it is necessary for the background clutter to also have a low level

so that the ghosts can be observed (eg a tarmac runway). The ghosts are unlike any other such images largely because they have roughly constant peak amplitude across a wide area of the radar footprint. On the other hand because they have constant amplitude it is difficult to locate the helicopter in azimuth and necessary to use additional methods to locate it.

In the case of SAR the effective blade flash time is much longer than might be expected on the basis of real aperture radar measurements. This is a consequence of interference effects. One might anticipate that the length of the effective scattering domains on the blades is of the order of the azimuth spatial resolution. However, in addition there appears to be only one such domain and this is located near the centre of the blades.

A study of figures 4 and 5 reveals that the location of the peaks along an azimuth line varies from line to line (ie with the rotation rate). Sometimes the receding blade appears before the advancing blade and sometimes the reverse is true. For the even blade case it is difficult to determine a pattern and if the radar pulse recurrence frequency is varied the situation becomes even more confusing. It is therefore difficult to measure the rotation rate of the blades from the SAR image with any certainty.

The analysis and simulations presented here are a greatly simplified version of a very complicated problem. Nonetheless the results are believed to give a reasonable picture of the SAR imaging process for the rotating blades of a hovering helicopter.

REFERENCES [1] T. Thayaparan, S. Abrol, E. Riseborough, L. Stankovic, D. Lamothe and

G. Duff, “Analysis of radar micro-Doppler signatures from experimental helicopter and human data,” IET Radar, Sonar Navig., vol. 1, pp. 289-299, Aug. 2007.

[2] V. C. Chen, F. Li, S-S. Ho and H. Wechsler, “Micro-Doppler effect in radar: phenomenon, model, and simulation study,” IEEE Trans. Aerospace and Electronic Systems, vol. 42, pp. 2-21, Jan. 2006.

[3] P. Pouliguen, L. Lucas, F. Muller, S. Quete and C. Terret, “Calculation and Analysis of Electromagnetic Scattering by Helicopter Rotating Blades,” IEEE Trans .Antennas and Propagation, vol. 50, pp. 1396-1408, Oct. 2002.

[4] J. Misiurewicz, K. Kulpa, and Z. Czekala, “Analysis of recorded helicopter echo,” In Proceedings of Radar 97 (IEE Conf. Publ. No. 449), pp. 449-453, 14-16 Oct. 1997, Edinburgh.

[5] B. C. Barber, “Analysis of binary quantisation effects in the processing of chirp and synthetic aperture radar signals”, In Mathematics in Signal Processing, Eds. T. S. Durrani, J. B. Abbiss, J. E. Hudson, R. N. Madan, J. G. McWhirter and T. A. Moore. Clarendon Press, Oxford, 1987.

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