Efficient computation of multiple diffracted short-pulses using ray fields

Author(s): Erricolo, D; Capolino, F; Albani, M | Abstract: 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.


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 uniform 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) 4tot. 6 = ,i,GO, 6 .+ 4 d . 6 + ,i,ddJ .+ G d d d J + ,,, .+ @d,6 + ,,.,
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.

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 (4) in which ddzv(t) is , 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, 1. Evaluate the field 4f-l(P,,t) incident on P,, due to the source R-1.

Approximate the incident field &-l(P,,t) as
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 at P,+l the field diffracted at P,

4W,+1,t) = 4:-l(Pi+l,n,T) 4Y(P;+lrt)
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 T D version of the double diffraction mechanism that is discussed in the next section.

IV. TIME DOMAIN DOUBLE DIFFRACTION (TD-DD)
The T D 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 )
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(ri,l,rz) @! , ( t -t d d ) , where A' is the incident spreading factor   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; = 4 2 = 320". In the second case observation is taken at its transition aspect, that is at grazing $2 = 290°, while the incidence is still out of transition 4; = 320". The last case considers both incidence and observation at 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°, f 13 = 80", @3 = 260°, f 14 = 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 environment have been presented in terms of multiple diffraction. Our T D 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.