Optimizing Single Sweep Range and Doppler Processing for FMCW Radar using Inverse Filtering

Transcription

1 Optiizing Single Sweep and Doppler Proessing for FMCW Radar using Inverse Filtering AJ de Jong and Ph van Dorp Oude Waalsdorperweg AK, Den Haag The Netherlands ABSTRACT We disuss range and Doppler proessing for FMCW radar using only a single pulse or frequeny sweep The first step is orrelation proessing, for whih the range and Doppler resolution are liited by the abiguity funtion We show that this resolution an be optiized with an additional inverse filtering step The ethod is deonstrated for sinusoidal FMCW radar easureents Several regularized inverse filters were opared and the non-adaptive pseudo inverse filter gave the best results 1 INTRODUCTION In this paper we disuss a tehnique for siultaneously obtaining range and Doppler inforation with FMCW radar, using only a single pulse or frequeny sweep The single sweep approah ontrasts with the ulti sweep approah, where the range info is obtained fro the pulse delay and the Doppler info fro the phase hanges in the range ells fro pulse to pulse An advantage of single sweep proessing is that for a given observation tie T the unabiguous range is onsiderably larger than for the ulti sweep approah The reason is that for ulti sweep proessing, N sweeps of duration T/N are required to obtain a Doppler axis with N different Doppler ells A disadvantage of single sweep proessing is that the range and Doppler resolutions are generally worse, as we disuss in hapter 2 In this paper we fous on the range and Doppler resolutions for single sweep radar, and in hapter 3 we show how these resolutions an be optiized using inverse filtering The ethod is applied to atual easureents of a sinusoidal FMCW radar, as disussed in hapter 4 2 CORRELATION PROCESSING 21 Transitted and reeived signals The transitted signal is an FM signal around a arrier frequeny f The phase of the transitted signal is given by ϕ ( t) = ϕ0 + 2πf t + ϕ ( t), and the instantaneous frequeny is found as 1 dϕ 1 dϕ f ( t) = f + f + f ( t) 2π dt 2π dt Paper presented at the RTO SET Syposiu on Target Identifiation and Reognition Using RF Systes, held in Oslo, Norway, Otober 2004, and published in RTO-MP-SET-080 RTO-MP-SET

2 Optiizing Single Sweep and Doppler The boundaries of f (t) deterine the bandwidth B of the odulation Ignoring the initial phase ϕ 0, the oplex transitted signal reads: ψ ( t) exp[ iϕ( t)] = exp[ iϕ ( t)]exp( i2πf t) µ ( t) exp( i2πf t) T It follows that the transitted signal an be written as a CW signal ties a oplex envelope µ(t) We also need an expression for the reeived signal of a nuber of targets at ranges R and with radial veloities u r We will use the following approxiation, holding for ost pratial ases [1]: ψ R ( t ) = V µ ( t τ ) exp[ i2π( f + ν ) t], where τ = 2R / is the delay and ν = 2f u r / is the Doppler frequeny The oplex fator V ontains the attenuation and a possible phase shift 22 The abiguity funtion The objetive is to extrat the aplitude V, the delay τ and the Doppler frequeny ν for eah target fro the reeived signal ψ R (t) An ideal approah would be to perfor suh an operation on ψ R (t) that we arrive at a funtion of the following for: Z τ, = V δ( τ τ, ν ν ) ( This would be ideal beause this funtion Z(τ, shows sharp peas at τ = τ and ν = ν with aplitude V for all targets An attept to arrive at this ideal Z(τ, is to orrelate ψ R (t) with the following referene signal [1]: ψ ( t, = µ ( t) exp[ i2π( f + t] F This results in Z ( τ, ψ ( t) ψ ( t τ, dt = V χ( τ τ, ν ν ), t R * F where the funtion χ(τ, is defined as χ( τ, = t * µ ( t ) µ ( t τ) exp( i2πνt) dt The funtion χ(τ,, whih is opletely deterined by the odulation µ(t), is nown as the abiguity funtion [3] In ase χ(τ, = δ(τ,, orrelation proessing would be perfet Indeed for a (quasi) rando odulation, the expetation of the abiguity funtion is a delta funtion [1] This for instane eans that orrelation proessing is ideal for FM radio signals More generally, the response Z(τ, to a olletion of point targets ontains a shifted opy χ(τ τ,ν ν ) for every target The range and Doppler resolutions of the targets is desribed by the sharpness of the entral pea at χ(0,0) 34-2 RTO-MP-SET-080

3 Optiizing Single Sweep and Doppler 3 OPTIMIZING CORRELATION PROCESSING 31 Inverse filtering The previous hapter ay suggest that the abiguity funtion χ(τ, gives the ultiate range and Doppler resolution for a given odulation µ(t) However, orrelation proessing as desribed in hapter 2 is only optial for (quasi) rando signals In this hapter we show how the resolution and the signal to noise ratio of the orrelation result Z(τ, an be iproved using the well-nown tehnique of inverse filtering [4] We reall the orrelation result: Z ( τ, = χ( τ τ, ν ν ) V We now interpret χ(τ, as a point spread or blurring funtion Beause χ(τ, is exatly nown fro µ(t), the blurred shape an be foused to a point with an inverse filter derived fro χ(τ, The inverse filter is ost onveniently desribed in the Fourier doain, and therefore we introdue the following Fourier transforations: ζ( = Fpq { Z( τ, }, ξ( p, = F { χ( τ, } pq In the Fourier doain the orrelation is given by ζ( p, = V exp( ipτ ) exp( iqν ) ξ( Note that in this representation the blurring funtion ξ( an be taen out of the suation This suggests eliination of the blurring with the following filtered version of ζ(: 1 G( H ( ζ( ζ( = V exp( ipτ ) exp( iqν ) ξ( Here, H( is the inverse filter, whih is the inverse of the Fourier transfor of the abiguity funtion Via an inverse Fourier transfor we obtain: 1 τν g ( τ, = F { G( } = δ( τ τ, ν ν ) Thus we eventually arrive at the ideal point target response V 32 Pseudo inverse filtering For the appliation of inverse filtering in pratie, regularization of the inverse filter is neessary One possibility is the pseudo inverse filter [4]: 1 H ( P ξ( for ζ( > Kσ N, H P ( = 0, for ζ( < Kσ N RTO-MP-SET

4 Optiizing Single Sweep and Doppler Here σ N is the noise level and K > 1 is a threshold paraeter We see that for ( values where ζ( does not suffiiently doinate the noise, H P ( is leared This eans that for these ( values, ζ( does not ontribute to the resulting G( and g(τ, For noisy signals, noise outliers will be ore abundant, and this is partiularly devastating at ( values where ξ( is very sall The influene of suh outliers an be largely suppressed with a non-adaptive approah, where the filter is leared where ξ( is sall rather than ζ( This leads to the nonadaptive pseudo inverse filter: 1 H ( PN ξ( for ξ( > K', H PN ( = 0, for ξ( < K' Here K' is a new threshold paraeter Possibly, one an avoid the disontinuities at the edges of the regions where the filter is leared with the following odifiation: * ξ ( q H PN ( 2 ξ( + ) = 2 K' This an be regarded as a soothed version of the non-adaptive pseudo inverse filter 33 The Wiener filter Siilar to the pseudo inverse filter is the Wiener filter [4], whih is the optial filter in the sense of iniizing the expeted least square error in the resulting g(τ,: H W ξ ( ( = 2 2 ξ( + σ / ζ( * 2 N The Wiener filter is also an adaptive filter requiring the estiation of σ N One an introdue an artifiial threshold paraeter by ultiplying σ N with a fator K, leading to a soothed version of the pseudo inverse filter H P ( 4 APPLICATION TO SINUSOIDAL FMCW RADAR MEASUREMENTS 41 Sinusoidal FMCW radar We have applied orrelation proessing and subsequent inverse filtering on atual easureents of a sinusoidal FMCW radar For this odulation, we have the following transitted signal [2]: ψ ( t) = exp( i2πf t + ia os 2πf t) = exp( ia os 2πf t) exp( i2πf t), T with the following instantaneous frequeny: 1 dϕ f ( t) = = f af sin(2πf t) 2π dt 34-4 RTO-MP-SET-080

5 Optiizing Single Sweep and Doppler We see that the instantaneous frequeny variation of the transitted signal is sinusoidal with bandwidth B = 2af The abiguity funtion is found as χ( τ, = exp[ ia (1 os 2πf τ) os 2πf t ia sin 2πf τsin 2πf t]exp( i2πνt) dt < T > Here the integral is only arried out over one odulation period T = 1/f Norally the delay τ will be sall opared to the odulation period T, so that the latter expression an be siplified by a first order approxiation of sin(2πf τ) and os(2πf τ) This leads to χ( τ, exp( ia 2πf τ sin 2πf t) exp( i2πνt) dt < T > We now borrow the following expression fro the theory of Bessel funtions: s= exp( iz sin( θ)) = J ( z) exp( isθ), s where s runs over all integers Substitution in the previous expression gives χ( τ, = J ( a2πf τ) exp[ i2π( sf t] dt s= s < T > For the oputation of Z(τ, one an freely hoose a set of ν values Choosing ν = nf with integer n, the integral in the latter expression siplifies to δ ns and we get n n n χ( τ, nf ) = J ( πbτ) = ( 1) J ( πbτ) This abiguity funtion is shown in figure 1 The range, whih is proportional to τ runs vertially In this and all following iage plots, darer pixels have higher values The ross setion χ(τ,0) is desribed by J 0 (πbτ), with its global axiu at τ = 0 and its osillations along the τ axis Cross setions for different ν = nf are desribed by n th order Bessel funtions The general behaviour of J n (x) for n 0 is that J n (0) = 0, and that J n (x) slowly inreases when x approahes n Around n, it subsequently arrives at its global axiu and first non-trivial zero and then starts osillating This explains the bowtie shape of figure 1 The Fourier transfor ξ( of the abiguity funtion χ(τ, is shown in figure 2 The non-adaptive pseudo inverse filter H PN ( following fro it is shown in figure 3 Note the interhange of darer and lighter regions between figures 2 and 3, illustrating that figure 3 is the inverse of figure 2 Also note the white regions with sharp edges where H PN ( is leared beause ξ( < K' Figure 4 shows the inverse filter h PN (τ, in the range Doppler doain 42 Inverse filtering for atual easureents We now disuss the appliation of inverse filtering on easureents of a ground based sinusoidal FMCW radar observing airborne targets The range and Doppler resolutions are about 1 and 200Hz respetively Inverse filtering was arried out with Z(τ, and χ(τ,, whih gave better results than when we wored with Z(τ, and χ(τ, We tried the three different regularized filters disussed in hapter 3 and we found that all of the wored very well for targets doinating the noise Figure 5 shows RTO-MP-SET

6 Optiizing Single Sweep and Doppler Z(τ, for a strong target and figure 6 shows the resulting g(τ, after suessive inverse filtering We only show one figure, beause the different regularized filters perfored equally well for this ase For noisy targets, we expeted the non-adaptive pseudo inverse filter to wor best This is illustrated by figures 7-10 Figure 7 shows Z(τ, for a weaer target, and figures 8 to 10 show g(τ, for H PN (, H P ( and H W ( respetively For the pseudo inverse filter, the value K = 2 was hosen The noise level σ N was estiated fro ( regions where no target ontribution to ζ( was expeted Clearly, the resulting g(τ, is not extreely good for any of the inverse filters, but H PN ( fouses ost of the energy in a relatively sall region In addition, the baground is soother, ontaining fewer outliers Figure 11 shows an interesting easureent with 2 targets The range runs fro about 0 to 100 and the Doppler frequeny fro about 20Hz to 20Hz Ground lutter is suppressed with a band filter around 0Hz, whih is visible as a slight depression in the entre of the figure The iage shows two lear bowtie shapes with different range and different Doppler frequeny Apparently, one target is an inoing target, while the other is an outgoing target Note that the target on the right hand side of the plot is aopanied by a bowtie at the sae range, but with higher Doppler and with lower intensity This ay well be due to JEM (Jet Engine Modulation) of the ain target Figure 12 shows the opression ahieved by inverse filtering using H PN ( Figures 13 and 14 show the sae inforation in surfae plots 5 CONCLUSIONS In the above hapters we disussed the possibilities of optiizing the range and Doppler resolutions of single sweep FMCW radar proessing by an additional inverse filtering step after orrelation proessing As disussed, the inverse filter an be derived fro the abiguity funtion of the hosen odulation, and we have given a nuber of regularized inverse filters for appliation in pratie We have deonstrated the tehnique for atual easureents of a ground based sinusoidal FMCW radar observing airborne targets We have opared the results of different regularized inverse filters, and found that all filters perfored well for targets doinating the noise For noisy targets the non-adaptive pseudo inverse filter perfored best For future researh it would be interesting to opare the perforane of deterinisti odulations to (quasi) rando odulations For rando odulations, the expetation of the abiguity funtion is already a delta funtion without inverse filtering, but one ay expet ore noise outliers It ight be added that after inverse filtering, the range and Doppler resolutions of single sweep proessing an in priniple be as good as for ulti sweep linear FMCW proessing Then, however, the single sweep radar will need a uh high sapling frequeny, a uh larger data buffer and uh faster data proessing than the ulti sweep FMCW radar Instead, one will rather use single sweep radar for its potential of a large unabiguous range and Doppler frequeny, while the observation tie an be short 6 REFERENCES [1] RL Mithell, Radar Signal Siulation (1976) [2] M Solni, Introdution to Radar Systes (1981) [3] SM Sussan, Least Square Synthesis of Radar Abiguity Funtions, Trans IRE, vol IT-8, (1968) [4] M Shwartz L Shaw, Signal Proessing (1975) 34-6 RTO-MP-SET-080

7 Optiizing Single Sweep and Doppler Reiproal Reiproal Figure 1: Modulus of the abiguity funtion χ(τ, The entre of the plot is χ(0,0) Figure 2: Modulus of ξ( The entre of the plot is ξ(0,0) Reiproal Reiproal Figure 3: Modulus of H PN ( The entre of the plot is H PN (0,0) Figure 4: Modulus of h PN (τ, The entre of the plot is h PN (0,0) Figure 5: Modulus of Z(τ, for a strong target Figure 6: Modulus of g(τ, for the target of figure 5 RTO-MP-SET

8 Optiizing Single Sweep and Doppler Figure 7: Modulus of Z(τ, for a wea target Figure 8: Modulus of g PN (τ, for the target of figure 7 Figure 9: Modulus of g P (τ, for the target of figure 7 Figure 10: Modulus of g W (τ, for the target of figure 7 Figure 11: Modulus of Z(τ, for a ase of ultiple targets Figure 12: Modulus of g PN (τ, orresponding to figure RTO-MP-SET-080

9 Optiizing Single Sweep and Doppler Figure 13: Surfae plot orresponding to figure 11 The range axis points towards the reader Figure 14: Surfae plot orresponding to figure 12 The range axis points towards the reader RTO-MP-SET

10 Optiizing Single Sweep and Doppler RTO-MP-SET-080