Radio Science Bulletin 313 - June 2005 - URSI

whereH[ P ] [ U][ ][ V] ,N= Σ (27)[] Σ = Diag ⎡s() s( )! s() r e( r + )!e( N )with⎣1, 2, , 1, , ⎤⎦,() ( ) () ( ) ( )s 1 ≥ s 2 ≥…≥ s r ≥ e r + 1 ≥…≥e N ≥0.As noted earlier, the singular values { e( r + 1 ),!,e( N)}should be zero if the matrix [ P N ] has rank r, and the rankof [ P ] is approximately r when the remaining singularNvalues are very small. Definewhere [ ] j[ ]1 H( ) 4 1[] 4 [ ][ ] [ ][ ]X ≡ U Σ = Σ Ur≡ ⎡ , , ⎤,1 ⎢ ⎣⎥ ⎦2 r 1X ≡ ⎡ X , , X + ⎤,2 ⎢ ⎣⎥ ⎦1[ X] [ X] " [ X][ ] [ ] " [ ]H,(28)X is the jth column block of the matrixX truncated to the first r elements. Although the form of(27) is the same as (5) and (16), the decomposition in (28)differs from (18).In this case, the reduced state matrices [ A r], [ r]and [ C r]are given by[ A ] = [ X ][ X ]+ if and only if[ r][ ]r(1)2 1[ B ] [ X] 1 ,r1 2B ,A X = [ X] ,(29)= (30)−[ U][ ] 1/4 ,⎡C ⎣⎤=⎦Σ(31)[ C ]r(1),= ⎡C ⎣⎤(32)⎦where ⎡C ⎣⎤⎦is the ith block row of matrix [ C ] truncatedto the first r elements. If the reduced state-spacematrices[ A r], [ B r], and [ C r]are known, then the poles ofthe system can be extracted using the outlined procedure forSS1.5. Computation of Poles forTemporal Scattered FieldIn this study, the temporal back-scatteredelectromagnetic fields from five objects are used as impulseresponsesequences for the three direct-data-domain methods(MPM, SS1, SS2). To obtain the simulated scattered fieldsin the time domain, the frequency-domain back-scatteredfields are first calculated with WIPL-D [15], a method-ofmomentscode. WIPL-D uses an electric field integralequation (EFIE) for conducting structures and the PMCHW(Poggio-Miller-Chang-Harrington-Wu) Equation forcomposite structures. The time-domain data is calculatedby computing the inverse Fourier transform of the frequencydomainresult. Before taking the inverse Fourier transformof the frequency domain data, a Gaussian window is appliedto limit the bandwidth of the simulations.The advantages of the Gaussian window are two fold:(1) it provides a smooth roll off in time and frequency; and(2) it is well suited for numerical computation, as it generatesessentially band-limited frequency-domain data. Theexpression for this window in the time domain is [16]with()g t=σ1e2−a( − ),(33)ct ctγ = 0 .(34)σThe parameter c is the speed of light in free space, t 0 is atime delay that represents the peak time shift from theorigin, and σ is the pulse width that can be defined so thatthe peak value has dropped to about 2% at time t forct -ct0=± 2σ. Figure 1(a) shows an example of thetemporal Gaussian pulse, where σ = 1.0 and lm. The unitlm denotes a light meter, which is the length of time for anelectromagnetic wave to travel 1m. When the medium isfree space, 1lm = 3.33564ns. Since the Fourier transform ofa Gaussian function is Gaussian, the Fourier transform of(33) is( )πσ f 21 −( )c − j2πf t=0 (35)G f e ecand the frequency band corresponding to the pulse width isplotted in Figure 1(b).To illustrate the three fundamental data types (smooth,rapidly fluctuating, noise-like), five geometrically simpleobjects are analyzed: a thin conducting wire, a perfectlyconducting sphere, a perfectly conducting finite closedcylinder, a dielectric sphere, and a composite metallicdielectricsphere.5.1 Thin Conducting WireThe first example is a thin-wire scattering element of,TheRadio Science Bulletin No 313 (June, 2005) 31