Objetivo General
4. Resultados de detección molecular en puntos de muestreo de superficies
Before generalizing the above results in a heuristic manner to model the UWB channel (seeSection 2.3.5), we must introduce the method in the literature for modeling large radar objects. The same method will be still valid for UWB pulse propagation in communications environment. Physically and mathematically the two problems are equivalent. Radar interrogation (per-path pulse response mod- eling) is essentially a transient process [34]. In order to simplify the calculation of radar reflectivity (per-path pulse response), one usually assumes that values ob- tained for a monochromatic source (time-harmonic solutions) are representative of the target (per-path pulse response) and can be used with little or no modi- fication, to predict the return for a pulsed signal with the same carrier frequency. This assumption cannot be made for ultrashort pulses or any other UWB signaling waveform. To obtain a target reflectivity model (per-path pulse response model) applicable to all types of signal, time-domain concepts suggest use of an impulse- response waveform to characterize each radar target (per-path pulse response).
The first such attempt to estimate an impulse-response waveform in the radar scattering context was perhaps that of Kennaugh and Cosgriff(1958). Using ex- tremely simple approximations for high- and low-frequency backscatter from con- ducting sphere and spheroids, estimates of the impulse-response waveform were obtained. The mathematical physical tools to attack the problems involving finite three-dimensional targets are described inSection 2.3.2. As suggested in (2.56), a linear system can be used to describe the object. We will follow [34] in the follow- ing derivations of this subsection.
To apply linear system analysis to scattering analysis problems [54,44,60], we first define input and output quantities as electrical field intensities at a pair of points in space and in fixed directions. As shown inFigure 2.2, the inputx(t) is the x-directed electric field intensities at the origin produced by a plane wave Ei = xx (t−z/c). If, now, this plane wave is incident upon an arbitrary scatter-
ing object (per-path obstacles), a scattered electromagnetic fieldEsis produced.
We may select an arbitrary component of this vector field at any pointP(r) to be the output ofy(t). Thus, restricting the direction and polarization of the incident
plane wave, the relationship between thex(t) andy(t) is that of a one-dimensional linear system. If we assume that the scatterer does not move, so that the system is time-invariant. The impulse-response waveformh0(t) is that obtained when the incident plane wavex(t) is impulsive, that is,x(t)=δ(t), the Dirac delta-function. This approach is essentially the same as inSection 2.2.
Several features of the impulse-response waveform can be inferred immedi- ately if the scattering body is of finite size:
(i) h0(t) is identically zero fort <0 if the scatterer lies to the right of thexy plane, as inFigure 2.2. The linear system must be causal,
(ii) h0(t) decays exponentially for large values ofτ =t−r/c, whereris the distance to the observation point from the origin, as inFigure 2.1, (iii) whenr L, whereLis the maximum dimension of the scattering ob-
ject,h0(t) varies withr in a given scattering direction asr−1f(t−r/c) for transverse components of the scattered field.
Properties of impulse-response waveformh(t). The normalized impulse-response h(t) and the transfer functionH(jω) are a Laplace transform pair. HereH(jω) is related to a transverse componentEs(t−r/c) of the scattering field at largerby
Re)H(jω)ejω(t−r/c)*= 2r c Es t−r c , (2.75)
the incident plane wave being given byEi = xcosω(t−z/c). We introduce the
notation H(s)=
∞
0 h
(t)e−stdt, H(s)=L)h(t)*, h(t)=L−1)H(s)*. (2.76) In practice, we usually first obtainH(jω) using the GTD/UTD asymptotic solu- tions, then we can replaces=jωinH(jω) to obtainH(s). Secondly, we apply the Laplace transform pair to obtain theh(t). Not all the functions have their inverse Laplace transform in closed forms. Due to the essential property of causality, a Laplace transform rather than a Fourier transform must be used, in order to guar- antee that the so obtained impulse-response is physically sound. Such obtained impulse-response must be a real function. This can serve as sanity check for the analysis.
High-frequency impulse-response based on scattering centers. Let us present such a model based on the concept of scattering centers. Extensive development of high- frequency scattering theory (developed in Sections2.3.2and2.3.3) traces back at least to 1940s–1960s. Experimental conditions and analytic tools were mature un- til 1960s, mainly driven by the study of short pulses caused by nuclear explosions. Two categories of work were performed. One was to model the impulse-response of simple shapes and composite shapes (finite three-dimensional objects). Another category was to study the radio propagation of pulse in a dispersive medium like plasma at a distance of kilometers. Here we are only interested in methods devel- oped for the first category, especially based on experimental work [68,69].
It is postulatedthat for a sufficiently short pulse, the solution can be rep- resented as a sequence of pulses scattered from the resolved scattering centers. (Section 2.3.2has discussed the conditions of validity of the time-domain GTD to model the resolved scattering centers.) This postulated solution has an approx- imate equivalent in the frequency domain. The back scattering amplitude at the frequencyωwill be of the form
H(jω) n AnHn(jω)e−j2kLn= n AnHn(jω)e−jωτn, τn=Ln c, (2.77)
whereAn is the scattering amplitude,Hn(jω) is the frequency response for the nth center, and Ln is the distance along the ray path from the first to thenth
scattering center. The so-called per-path transfer function (heuristically defined inSection 2.3.5),Hn(jω), plays the central role in characterizing the pulse wave-
form. The per-path delay,τn=2Ln/c, describes the intrapath delay that connects
the localized responses. When an inverse Laplace transform is applied in (2.77) (see (2.70) and (2.71)), the early-time approximation to the impulse-response is obtained as h(τ)= n Anhn(τ)∗δ τ−τn . (2.78)
BothHn(jω) andhn(τ) have a unit energy for convenience. In general, (2.77) and
(2.78) are connected by the Laplace transform. We assume thatHn(jω) is centered
about f0with a normal bandwidthB. IfHn(jω) is slowly varying for−B/2 +f0≤ f ≤B/2 + f, the terme−j2kLnvaries rapidly in the same band. For a special case, a simple relation can be obtained as
h(τ) n AnHn ω0p τ−2Ln c , (2.79)
wherep(t) is the incident pulse. Thus thenth scattering center returns a pulse of amplitudeAnHn(ω0), delayed byτn=2Ln/cfrom the first term. From a physical
point of view, (2.78) and (2.79) imply that the dimensions of the scattering centers are of the order one wavelength or less. Thus, if the incident pulse contains more than several wavelengths (at the center frequency), the distortion of the scattered pulses is negligible. Refinements of (2.79) could be included by accounting for the frequency dependency ofHn(jω) near the center frequency, as in (2.78). It should
be noted that this representation in (2.78) and (2.79) may also include terms due to multiple scattering.
For high frequencies, one may approximate H(s) by an expansion called Luneberg-Kline series in negative powers ofs(as discussed inSection 2.3.2and used inSection 2.3.5). Individual terms in such an expansion are related to im- pulse, step, ramp, and low-order discontinuities inh(t). Thus, if the asymptotic
δ(τ) τ−τ1 τ−τ2 τ−τN A1 A2 AN h1(τ) h2(τ) hN(τ) + · · · h(τ)=+N n=1Anhn(τ)⊗δ(τ−τn) ⊗ ⊗ ⊗
Figure2.3. The structure of the generalized multipath model.
expansion ofH(s) contains a termAns−ne−sτ, the impulse-response waveform un-
dergoes a discontinuous jump of magnitudeAnin its (n−1)th-order derivative at t =τ. Terms of the formA0e−sτ in the asymptotic expansions are contributions
predicted by geometric optics, giving impulses inh(t) att=τ, of magnitudeA0. It is clear that low-frequency approximation serves to determine waveform shape and size, whereas high-frequency information relates to fine structure and detail in the waveform. Both are thus interpreted in a single concept that serves to predict the scattering behavior in the intermediate or resonance range of frequencies. The intuitive hypothesis that the ramp response is most closely associated with the vol- ume and size of a scatterer suggests that, as our knowledge of response waveforms for special objects increases, we will ultimately be able to sketch, by inspection, a reasonably accurate estimate of the ramp response of any target (per-path re- sponse).