Numerically efficient results for short pulse propagation in a complex environment is presented in terms of multiple diffraction. As such, the time domain (TD) diffraction propagators are based on discretization of a generic short-pulse in terms of narrow rectangles, and on closed form representations of diffracted fields when the excitation is a narrow rectangular pulse

Albani, M

Capolino, F

Erricolo, D

Erricolo, Danilo

Capolino, Filippo

Albani, Matteo

eScholarship - University of California

UC IrvineUC Irvine Previously Published WorksTitleEfficient Computation of Multiple Diffracted Short-Pulses Using Ray FieldsPermalinkhttps://escholarship.org/uc/item/67p1n0c3AuthorsErricolo, DaniloCapolino, FilippoAlbani, MatteoPublication Date2002DOI10.1109/aps.2002.1018224Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaEfficient Computation of Multiple Diffracted Short-Pulses Using Ray Fields 'Danilo Erricolo', Filippo CapolinoZ and Matteo Albani3 'Dept. of ECE, University of Illinois at Chicago, 851 S. Morgan St, Chicago, IL, 60607-7053, USA. derricol@ece.uic.edu a Dept. Information Eng., University of Siena, Siena, Italy. Currently, Dept. of ECE, University of Houston, 4800 Calhoun Rd, Houston, TX 77004-4005, USA. capolinoOuh.edu, 3Dip. Fisica della Materia e Tecnologie Fisiche Avanzate, Univ. of Messina, salita Sperone 31, 98166 Messina, Italy. malbaniOingegneria. unime.it I. INTRODUCTION A numerically efficient representation for short-pulse (wideband) field propagation in an environment as in Fig.la is presented in terms of time-domain (TD) rays. Closed form uni- form vavefmnt approximations of the organized multiple diffracted fields are provided, using simple T D  transition functions. Focus is given on an arbitrary nth diffracted mechanism, as shown in Fig.la, with specialization on a second order diffraction mechanism (doubly diffracted (DD) fields) as in Fig.lb. The uniform asymptotic solutions for singly diffracted (SD) and DD fields are valid for early observation times (wavefront approximations). Their timerange of validity may be extended to late observation times when the exciting signal does not contain low frequency components. Higher order diffracted fields are approximated using an efficient numerical convolution between TD-SD fields. For simplicity we will not consider here hybrid mechanisms, such as reflected-diffracted rays, etc. The response of the total field to an impulsive excitation (Dirac delta function) is represented in terms of ray fields as in which @o,6 includes all the T D  geometrical optics (GO) fields, &d,6 includes all the TD-SD fields, G d d l 6  includes all the T D  double diffracted (DD) fields, G d d d J  includes all the T D  triply diffracted (DD) fields, and so on. When the source radiates a waveform G(t), the impulsive response (1) is used to construct the total radiated field &tot,G(t) = $W6( t )@G( t ) ,  in which @ denotes time convolution. Instead of representing the total diffracted field as a continuous superposition (convolution) of impulsive responses, it might be convenient to approximate it as a discrete superpositions of rectangular-pulse responses. Accordingly, the excitation waveform G(t) is expanded as a superposition of rectangular pulses of duration T (1) 4 to t .6  = ,i,GO,6 .+ 4 d . 6  + , i ,ddJ .+ G d d d J  + ,,, .+ @d,6 + ,,., N &t)=z g,rect(t-nT,T), "=l in which the weights gn = G(nT) sample the excitation waveform h(t) in the middle of the rectangular pulses. The total field is thus approximated as Fig. 1. (a) Multiple diffracted ray. ( b )  Double diffracted (DD) ray and notation used in [5] 0-7803-7330-8/02 $17.OO Q 2002 IEEE 342 (3) @ G ( t )  % C gn p t , r ( t  - n ~ ) ,  , p , r  = ,jGO,r + , j d ?  + , j d d , r  + ,,,, n=l in which the responses to the rectangular pulse, dd*. and Gdd,., are evaluated in closed form from the impulsive responses GGoz6, 4"v6 and Gdd~," in Sections I1 and IV. The present formulation has several advantages. First, the representation (3) of the total radiated field $"t,r is asymptotically uniform, as it was the total impulsive radiated response (1). Second, the various terms are easier to evaluate then (l), since they do not contain singularities due to the Dirac delta excitation. Indeed, the GO field for rectangular excitation GCoz'(t) =: rect(t,T)/(4nR), with R the distance between source and observer, is finite and band limited, imposing the same properties when it is multiply diffracted. 11. TIME DOMAIN SINGLE DIFFRACTION (TD-SD) The impulsive TD-SD field has been presented in 111, [Z], [3], while the response to the unit step function U ( t )  has been presented in [4]. After expressing the rectangular pulse as rect(t,T) = U ( t  + T / 2 )  - U ( t  - T/2) ,  the TD-SD field response to rectangular pulse is obtained as the difference 7pyt - T )  = 7p"( t  + T/Z) - 7 p U ( t  - T/2), @ U ( t )  = d L D , " : , " ( t  TT'(T + T') - t d )  (4) in which ddzv(t) is the response to the unit step function V ( t )  from [4], or obtained by integrating the results in (1],(2],(3]. td = (T + T')/c is the turn-on time (wavefront arrival time) of the diffracted field, and is the TD-SD diffraction coefficient. The TD transition function is evaluated from theconvolution fu(zm,t) = f(z,, t )@U(t )  where f(zm,t) = z m / [ m ( t +  zm/c)], in which f(zm, t )  is the transition function for impulsive excitation (we have used the notation in [3]). The parameters in (5) are dZh = cot(n- (-l)"'(@-@'))/(Zn), for m = 1,2, dkh = Fcot(7r - ( - I ) ~ ( @  + @'))/(Zn), for m = 3,4, z1 = 2Lcos2(2n?rN+ - (4 - $'))/z, z2 = 2Lcosz(2n?rN--(+-+'))/2, z3 = 2LcosZ(2nnN+-(@+@'))/2, z4 = 2Lcos2(2nnN-- (@+ @'))/Z, where L = TT ' / (T  +T') (for spherical wave incidence), N* are the integers that most nearly satisfy the equation 27rnN' - (4 f 4') = fn, 6' is the incidence angle, T' is the incidence distance, 4 is the observation angle, r is the observation distance and n is the wedge aperture angle factor. 111. T D  MULTIPLE DIFFRACTIONS Multiple diffractions may be computed by cascading TD-SD fields derived in the previous section. Let PO be the source, pN+1 the observation, and pl,...,pN the locations of N diffraction points (see Fig.la), a pulsed field emitted at  Po produces a transient field a t  P N + ~  computed repeating N times the following steps: 1. Evaluate the field 4f-l(P,,t) incident on P,,  due to the source R-1. 2. Approximate the incident field &-l(P,,t) as 343 and note that the field 4t-l(Pz,t) is sampled at  the center of the rectangles ($;-l(P;,njT)). 3. Using the previous incident fiel% evaluate a t  P,+l the field diffracted at  P, 4W,+1,t) = 4:-l(Pi+l,n,T) 4Y(P;+lrt) (8) n,=1 in which the diffracted field @'(Pi+~,t)  is the response to the rectangular pulse excitation In evaluating the field multiply diffracted by wedges, two peculiar situations may occur: a) two consecutive edges have a common face and the electric field is tangent to the common face (soft polarization); b) two consecutive edges are experiencing a double transition, i.e., two consecutive diffraction points Pi and Pi+1 are almost aligned with source Pi-1 and observer Pi+,. In such cases, more accurate results are accomplished by introducing a TD version of the double diffraction mechanism that is discussed in the next section. IV. TIME DOMAIN DOUBLE DIFFRACTION (TD-DD) The TD response of a double wedge to a rectangular pulse is derived from the uniform wavefront approximation for the impulsive TD-DD field presented in [ 5 ] ,  [SI. Similarly to the SD case (4), the TD-DD field response is obtained as the difference (4). p+( t  - T) = 4 d d J J ( t  + T / 2 )  - I j d d , U ( t  - T / 2 )  (9) where ddd*u is the response to a unit step function. Referring to Fig. lb,  the TD-DD field is given by - Ai(r;)  A(r i , l , r z )  @!,(t - t d d ) ,  where A' is the incident spreading factor [for a spherical source, A'(ri) = 1/(4m;)] at  4, A(r;,  e, TZ) = &/dt'r2(r; + I + r z )  is the DD field spreading factor. @,(t - t dd )  is the 'ID-DD coeficient, evaluated at the retarded time t - tdd ,  where tdd = ( r ;  + I  + T Z ) / C  is the turn-on time of the ,DD field. Similarly to the frequency domain case (5],[7],  the double diffraction coefficient is D$ = a:bu with and where upper/lower sign applies to the soft/hard case, respectively. The TD-DD transition functions (12) are obtained by the convolution ?{;z}*u(t)  = ?{;zI( t )  8 V ( t ) ,  ? { : I } ( t )  being the transi- tion functions for TD-DD field with impulsive excitation [5],[6],  that is calculated in closed form as of G(s, y, t) = {arctan (z/y) 1 +t/x2 - arctan(s/y)} V ( t ) .  The various pa- rameters are defined as in [5],[6], iipq = fld-sinP;sin[(QP - 2 ~ n l N : ~ ) / 2 ] ,  6,, = J s d m s i n P z s i n [ ( @ y  - 27rnlNy)/2],  in which N p  and NP are the integers that most nearly satisfy @;" = 2 ~ n ~ N ; ~  and @r = 27rnzN7, respectively; b e  ing @yq = 7r + ( -1)Pb; + (-l)q$m and @? = 7r ,+ (-l)rq52 t (-1!aq512. Furthermore w = &rZ/[(ri + ! ) ( T z  + e ) ]  and E I Z  = sign(&, . P ~ z )  = sign(& . &2). The second or- der contribution 6:;'" becomes particularly important in two possible cases: when two edges have a common face, and when source and observation are almost aligned with two consecutive edges. [ 7 1  344 Fig. 2. TD-DD field diffracted at the two edges shown in Fig. lb.  Three different c-: a) out of transitions; b) single transition and c) double transition aspects Fig. 3. Diffraction of a modulated rectangular pulse by the four con- secutive edges shown in Fig. la. v. NUMERICAL EXAMPLES AND CONCLUSIONS Numerical results for the diffraction of a rectangular pulse by the double wedge of Fig.lb are shown in Fig. 2. The different ray segments are assumed to be r; = 42cm, e = 45cm and r~ = 33cm long; while the involved angles are p: = p1 = looo, = f l ~  = 50", 412 = q5i2 = 110". Three cases are considered. In the first case both incidence on the first edge and observation w.r.t. the second edge are out of transition 4; = 42 = 320". In the second case observation is taken a t  its transition aspect, that is a t  grazing $2 = 290°, while the incidence is still out of transition 4; = 320". The last case considers both incidence and observation a t  grazing $; = $2 = 290", thereby DD experiences a double transition. The response to a T = 50ps rectangular pulse was computed in two different ways: 1) by cascading the single diffraction mechanism as explained in Sections I1 and I11 (dashed black line); and 2) by using the double diffraction mechanism of Section IV (continuous gray line). These results show that there is strong agreement between the responses obtained with the two methods outside the transition zone; the agreement is still very good when only one aspect is in transition, but greater differences are observed when a double transition occurs. The diffraction of the pulse cos(2xft)rect(t), which contains mostly high frequency components, by the four consecutive edges of Fig. la is reported in Fig. 3. The duration of the pulse is T=2Ons, the carrier frequency f=lGHz, the sampling interval At = 0.25ns and the polarization is soft. The geometrical parameters for this example are r-i = lm,  r-; = 2m, r2 = 3m, rh = 4m, r4 = 51x1, 4; = 81", = 260", flz = looo, q5z = 281°, f13 = 80", @3 = 260°, f14 = 95", $4 = 75". The results shown in Fig. 3 were obtained in a fraction of a second for 300 sample points. In conclusion, numerically efficient results for short pulse propagation in a complex envi- ronment have been presented in terms of multiple diffraction. Our T D  diffraction propaga- tors are based on discretization of a generic short-pulse in terms of narrow rectangles, and on closed form representations of diffracted fields when the excitation is a narrow rectangular pulse. REFERENCES [l] L. B. Felsen and N. Marcuvitz. Radiation and Scattering of Waves, IEEE Press, Piscataway, N. J., 1994, [Z] R. lanconescu and E. Heyman, "Pulsed beam diffraction by a perfectly conducting wedge: Exact solution," IEEE l h n s .  Antennas Pmpagat., vol. 42, no. 10, pp. 1377-1385, Oct. 1994. (31 P. R. Rousseau and P. H. Pathak, "Time-domain uniform geometrical theory of diffraction for a curved wedge," IEEE %ns. Antennas Pmpagat., vol. 43, no. 12, pp. 1375-1382, Dec 1995. [4] T. W. Veruttipong, "Time domain version of the uniform GTD," IEEE l h n s .  Antennas Propagot., MI. 38, no. 11, pp. 1757-1764, Nov 1990. [5] F. Capolino and M. Albani, "Time domain double diffraction at a pair of coplanar skew edges," Antennas Propagat. Soc. Intl. Symp., vol. 2, pp. 2062-2065, 2000. [6) F. Capolino and M. Alhani, "Impulsive pyramid-vertex and double wedge diffraction coefficients," Antennas Propagot. Soc. Intl. Synp., vol. 2, pp. 200-203, 2001. [7] F. Capolino, M. Albani, S. Maci, and R. Tiberio, "Double diffraction a t  a pair of coplanar skew edges," IEEE %ns. Antennas Pmpagat., vol. 45, no. 8, pp. 121%1226, Aug 1997. D. Erricolo's contribution to  this work was supported by the National Science Foundation under Grant ECS-9979413 345 

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Efficient Computation of Multiple Diffracted Short-Pulses Using Ray Fields

Efficient computation of multiple diffracted short-pulses using ray fields

Efficient computation of multiple diffracted short-pulses using ray fields

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