[9] The UWB beam summation formulation is constructed in the framework of the windowed Fourier transform (WFT) frames, where the aperture field _{0}(**x**) is expanded using a set of WFT functions. The general theory of frames is given by *Daubechies* [1992], while a theoretical overview for the present context is given by *Shlivinski et al.* [2004]. The formulation is structured upon a discrete phase space lattice

where (_{j}, _{j}), *j* = 1, 2 are the unit cell dimensions in the *x*_{j} coordinates, and the index **μ** = (**m**, **n**) = (*m*_{1}, *m*_{2}, *n*_{1}, *n*_{2}) is used to tag the lattice points. The unit cell dimensions in *x*_{1} and *x*_{2} need not be identical as long as they satisfy the overcompleteness criterion (6) in each coordinate, yet for the sake of simplicity we chose here _{1} = _{2} = and _{1} = _{2} = . Since each lattice point will be associated with a beam propagator emerging from **x**_{m} in the aperture in a direction that is determined by **ξ**_{n} via (3), we use the same lattice for all the frequencies. In addition, **X**_{μ} should be overcomplete for all ω ∈ Ω (see (6)). To this end we choose a reference frequency > ω_{max} (typical parameters are discussed after (14)) and define **X**_{μ} to be complete at , that is, the unit cell area in the *x*-*k*_{x} phase space is 2π:

Equation (5) implies that **X**_{μ} is overcomplete for all ω < since

Here ν is the overcompleteness parameter: The lattice is overcomplete for ν < 1 and critically complete in the Gabor limit ν 1. Thus scaling ν with ω as in (6) yields a frequency-independent beam lattice [*Shlivinski et al.*, 2004]. Having constructed the overcomplete lattice, we proceed by choosing a proper window (**x**) and construct the WFT frame in *L*_{2}(^{2}) for all ω < , upon the phase space lattice **X**_{μ},

Here (**x**) are obtained as a Cartesian multiplication of two one-dimensional windows _{j}(*x*_{j}), *j* = 1, 2, each one yielding a proper WFT frame in *L*_{2}() associated with the lattice (_{j}, _{j}). The frame _{μ}(**x**) is associated with a “dual frame” _{μ}(**x**) that has the same form in (7) but with the “dual window” (**x**). The latter needs to be calculated for a given and lattice (, ) for each ω ∈ Ω (see *Daubechies* [1992] and *Shlivinski et al.* [2004] for calculation algorithms for ). For small ν, however, one may use the analytic approximation

The frame representations of _{0}(**x**) for ω < are given by

Here _{μ} are recognized as the WFT of _{0} around (**x**_{m}, **ξ**_{n}) with respect to the window . In view of their role in (9) the frame functions (**x**) and (**x**) are thus identified as the “analysis” and “synthesis” windows, respectively. The radiated field (**r**) for *z* > 0 is obtained by propagating the windows _{μ} in (9a), giving

where the “beam propagator” _{μ}(**r**) are the radiated fields due to the distributions _{μ}(**x**) in the aperture. They may be described, for example, by the plane wave representation (2),

where _{μ} = (**ξ** − **n**)*e*^{−ikξ·m} is the spectrum of _{μ}, with being the spectrum (1) of the “mother” window . If is wide on a wavelength scale then _{μ}(**r**) behave like collimated beams emerging from **x**_{m} in the *z* = 0 plane in the directions (θ_{n}, ϕ_{n}) defined by ξ_{n} via (3). Propagating beams occur only for ∣**ξ**_{n}∣ ≲ 1 − Δ_{ξ}, where Δ_{ξ} is the spectral width of (Δ_{ξ} ≪ 1 for collimated beams). For ∣**ξ**_{n}∣ ≳ 1, _{μ} decay exponentially with *z* and are therefore neglected in (10). Thus, in practice (10) sums only terms with ∣**ξ**_{n}∣ < 1, thus expressing the field as a discrete superposition of beams emerging from all lattice points **x**_{m} and real directions **ξ**_{n}. The number of terms in the beam summation (10) can be further reduced by applying source and observation localizations in the phase space. Source localization is affected by the WFT in (9b) that extracts the local feature of _{0}, and therefore enhances only the excitation amplitudes _{μ} of the beams that match the local radiation direction of _{0}. Viewed from a phase space perspective, these _{μ} reside near the Lagrange submanifold, which is the phase space skeleton of geometrical optics, and near the diffraction submanifold, which correspond to diffracted rays that emerge from discontinuity points in _{0}. These properties have been established theoretically by *Steinberg et al.* [1991] and *Arnold* [1995] via an asymptotic evaluation of the WFT (9b), as well as numerically [*Shlivinski et al.*, 2004].