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( 1 ) NARROW BAND MODELS (Molisch, Section 7.2, pages 119-120)

(1.1) STATISTICAL NARROW BAND MODELS (Molisch, page 119)

• Modeling of small-scale fading: Within a sufficiently limited physical area, multipath amplitude variations admit to a random variable modeling of the associated multipath phase ensemble which is characterized by a Doppler spatial correlation and a Rayleigh distribution of the resulting absolute amplitude.
• Modeling of large -scale fading : small-scale Rayleigh-distributed amplitudes, if averaged over larger areas, are manifested as a log-normal distribution of the resulting field strength and the concomitant normal distribution of the associated log of field strength (path loss).
• Waveguide Model: Appropriate for low antennas in the mobile channel(which requires field distribution analysis ),the Waveguide Model imposes a Poisson distribution on an ensemble of fields in an urban environment modeled as a 3D wave guide to calculate path loss.
• Boltzman Model: Adapted from fluid flow models, modeling of urban channel as dynamics of particles on a 3D lattice, the Boltzman model is well adapted to complex boundary conditions and provides good agreement with path-loss measurements.

(1.2) DETERMINISTIC CHANNEL MODELING (SITE-SPECIFIC MODELS FOR PATH LOSS)

(1.2.1) EMPIRICAL DETERMINISTIC MODELS FOR THE URBAN CHANNEL

MODEL NAME	FREQUENCY RANGE/INTENDED IMPLEMENTATION	DEFICIENCIES/REMEDIATION
Okumura Hata	(150 MHz-2GHz) Valid only for base station (transmitter) above roof tops, path loss is modeled with three parametrically adjustable coefficients in the equation $L=A+B(\log(d))+C$.	Insensitivity to even dramatic variations in terrain profile. Not reliable for both antennas below roof top elevation.
Hata	(150 MHz -1.5 GHz) Empirical realization of Okumura-Hata Model with parametric adaptability to suburban and rural areas as well as urban areas.	Not reliable for smaller cell communications less than one kilometer in radius.
COST-231 Walfisch-Ikegami	Reliable for multiple (uniform) building diffraction for base station both above and below building elevation	Restricted to the vertical plane , over-estimates loss for shorter paths among tall buildings with both antennas below building elevation
Dual Slope Model	based on a two ray model in which the slope of the loss as a function of the log of the range coordinate is varied from an initial to a final value for values of the range (break-point) exceeding a critical optical configuration associated with sufficient penetration of the first Fresnel zone.	Underlying phenomenology associated with final value of slope (impedance boundary conditions effecting arguments of a complex complementary error function and phase of the reflection coefficient)too multi-variate for a single parametric fit.

OUTDOOR MODELS (Molisch, page 119)

• Okumura Hata- valid only for base station (transmitter) above roof tops, path loss is modeled with three parametrically adjustable coefficients in the equation $L=A+B\log(d)+C$.
• Hata Model- Empirical realization of Okumura-Hata Model with parametric adaptability to suburban and rural areas as well as urban areas, but not reliable for smaller cell communications less than one kilometer in radius.

• COST-231 Walfisch-Ikegami- Path loss is modeled as the sum of a multi-screen diffraction due to multiple buildings assumed to be of uniform spacing and height, a roof-to street loss due to diffraction from the mobile station horizon (final building of the multi- screen diffracter) and a reflection loss involving a surface provided by the building immediately adjacent to the mobile antenna in the direction of propagation.
• Dual slope model- based on a two ray model in which the slope of the loss as a function of the log of the range coordinate is varied from an initial to a final vale at the onset of a range-correlated critical optical configuration associated with sufficient penetration of the first Fresnel zone.

INDOOR MODELS ( Sarkar, et. al. page 57)

• Conventional Indoor Models
  • The Motley-Keenan Model: site-specific indoor model which requires explicit floor plans and antenna position coordinates and includes explicit wall attenuation contribution, but in its neglect of indirect but often less-attenuated paths, is considered unreliable.
• Computationally Efficient models
  • Improved Ray-Tracing: Adaptation of results from the combination of several ray tracing methods with UTD in a 2D environment are used in a novel 3D propagation prediction model which is more accurate than conventional 2D models and more efficient than full 3D ray tracing models.
  • Selective Use of FDTD-localized implementation to model effect of walls on indoor propagation channel. The inadequacy of ray-tracing methods to properly model the effects of the inner structure of walls and of the assumption of specular reflection may be addressed through the use of Floquet constructs in an FDTD simulation to describe periodic structures inherent in wall fabrication materials as required to properly model non-specular reflections and diffusive scattering effects.

(1.2.2)DETERMINISTIC CHANNEL MODELING (NUMERICAL SOLUTIONS OF EXACT VARIATIONS OF MAXWELL EQUATIONS)

SOLUTIONS BASED ON INTEGRAL EQUATIONS

• Method of Moments
  • Conventional MOM: Formally exact solution for fields as an integral over unknown sources (weighted by Green functions) appropriate for selected dimensionality in which a discretized representation of know field values is expressed as the product of an impedance matrix and a vector of unknown currents. The unknown currents may then be solved through a formal inversion of the impedance matrix or other standard methods of coupled linear algebraic equations.
  • Hybrid variations: Ray Tracing and Moment Method
  • Periodic Moment Method: Initial use of Moment Method adapted to periodic structure to prepare initiation rays which are then propagated with ray methods
• Fast Far-Field Approximation Method: Introduction of Green function feature accelerates traditional Integral Equation (IE) methods and has been applied effectively to undulating and hilly terrain and terrain with urban features; technical details involving terrain profile truncation and small-scale-roughness have been effectively addressed as discussed in the linked article: Media:FFFAM.pdf.

MODEL NAME	FREQUENCY RANGE/INTENDED IMPLEMENTATION	DEFICIENCIES/REMEDIATION/ENHANCEMENTS
Conventional Method of Moments	Formally exact solution for fields as an integral over unknown sources (weighted by Green functions) expressed in matrix form as the product of an impedance matrix and a column vector of (unknown) currents which may be computed through a formal inversion of the impedance matrix or other standard methods of solving coupled linear algebraic equations.	Hybridization with ray methods: Initial use of Moment Method adapted to periodic structure to prepare initiation rays which are then propagated with ray methods
Integral Equation (IE) Models	Numerical integration of (current) sources as weighted by Green function appropriate for desired dimensionality	Fast Far-Field Enhancement: Introduction of Green function feature accelerates traditional Integral Equation (IE) methods and has been applied effectively to undulating and hilly terrain and terrain with urban features; technical details involving terrain profile truncation and small-scale-roughness have been effectively addressed as discussed in the linked article

SOLUTIONS BASED ON DIFFERENTIAL EQUATIONS

• Finite Element Methods: General numerical method in which partial differential equations for fields are reformulated as ordinary differential equations by first expressing solutions in a discretzied space in terms of localized basis functions to thus provide a bilinear set of coupled equations with implicit boundary conditions which may be solved using standard techniques of linear algebra including matrix inversion. For problems in the frequency domain,the bi-lnear matrix equations reduced to an eignevlue equation which may be solved by matrix diagonalization to provide resonant frequencies (eigenvalues)and an (orthogonal) set of basis functions for general field solutions (eigenvectors).
• FDTD : coupled time-dependent equations for electric and magnetic fields are solved simultaneously on a temporal-spatial grid in which time derivatives of a given field are computed as spatial derivatives of the conjugate field as required by the Maxwell equations.
  • Reduced dimensionality formulations
  • Hybrid formulation of FDTD with geometrical optics
• (Vector) Parabolic Equation: Vector variation of scalar versions permits modeling of 3D scattering; especially effective for single building or groups of buildings at microwave frequencies.
• Novel Unitary (Time-Dependent) Propagator expanded in Chebychev basis

MODEL NAME	FREQUENCY RANGE/INTENDED IMPLEMENTATION	DEFICIENCIES/REMEDIATION/ENHANCEMENTS
Finite Element Methods	Finite Element Methods: General numerical method in which partial differential equations for fields are reformulated as ordinary differential equations by first expressing solutions in a discretized space in terms of localized basis functions to thus provide a bilinear set of coupled equations with implicit boundary conditions which may be solved using standard techniques of linear algebra including matrix inversion.	For frequency domain formulations, the bilinear set of equations reduces to an eigenvalue matrix equastion which can be solved by matrix diagonalization to provide resonant frequencies (eigenvalues)and an (orthogonal) set of basis functions for general field solutions (eigenvectors).
FDTD (Finite Difference (Time Domain))	coupled time-dependent equations for electric and magnetic fields are solved simultaneously on a temporo-spatial grid in which time derivatives of a given field are computed from spatial derivatives of the conjugate field in an explicit implementation of Maxwell's time dependent curl equations.	Reduced dimensionality formulations: polarizations consistent with (TM) or (TE) confined to a vertical plane permit a formal exact dimensionality reduction of Maxwell equations involving one field component normal to the plane and two tangential components of the conjugate field lying within the plane for a geometric factor performance enhancement. Hybrid formulation of FDTD with geometrical optics in which detailed analysis of optically reduced features not exceeding several wavelenghts in linear dimensions or manifesting inhomogeneities or anisotropies are reserved for FDTD implementation.
Scalar Parabolic Equation Solutions	Approximating time independent Laplace Wave Equation in cylindrical coordinates permits construction of a solution obtained from a spatial marching algorithm at a specified frequency.	Vector variation of scalar versions permits modeling of 3D scattering; especially effective for single building or groups of buildings at microwave frequencies.
Unitary Propagator, Chebychev Expansion	Explicit time dependent unitary propagator provides numerically exact solution of Maxwell time dependent field equations in a marching algorithm iterativley applied at each time step to permit time evolution of a general class of initial field values including those constructed in a broad band frequency domain continuum basis.	Expansion of the (unitary) matrix operator in a Chebychev basis permits the evaluation of field spatial derivatives with Fast Discrete Fourier Transform techniques to provide propagational dynamics scaling as N log N compared to N^3 for conventional matrix methods.

(1.2.3)DETERMINISTIC CHANNEL MODELING: RAY METHODS AS NUMERICAL SOLUTIONS OF THE HIGH FREQUENCY APPROXIMATION OF THE MAXWELL EQUATIONS (EIKONAL EQUATIONS)

CONVENTIONAL RAY METHODS

The Maxwell Equations admit to an exact solution expressed as a summation in powers of the wavelength which in the high frequency limit reduces to the first term which defines the (ray) elements of geometrical optics. Geometrical optics (GO) provides a suitable approximation for propagation involving distances and environmental features with linear dimensions large with respect to the wavelength. If generalized with the elements of the Unified Theory of Diffraction (UTD), the resulting methodology provides high frequency approximations of LOS propagation, reflection , refraction and diffraction.

• Image Method: Secondary and higher order sources are located as images of the transmitter with respect to all potential reflecting surfaces and all such rays along rays paths to the receiver are superimposed to construct the received field. The methods are systematic, complete and practical in an optically simple environment, but require an establishment of an upper bound of the number of rays to be considered in each mode of propagation (reflection and

diffraction) and are inadequate in moderately complex environments.

• Ray Launching (single source to omni-directional targets) (Molisch, Section 7.5.1)
  • free space loss
  • reflection
  • diffraction
  • diffuse scattering
• Ray tracing (point-to-point raypath ) (Molisch , Section 7.5.2, pages 132-133)
  • Longley-Rice Model (ITM): valid in the range 40 MHz-100 GHz, the model includes (1) an area mode prediction option to predict path parameters which does not require a terrain profile and (2) a point-to-point ray tracing option which requires a terrain profile as input, modeling terrain-induced diffractions according to the Fresnel-Kirchoff knife edge model.

ENHANCEMENTS TO IMPROVE EFFICIENCY OF RAY METHODS

• Ray Splitting- Acknowledging a fundamental assumption of Geometrical optics that a given ray represents only a finite solid angle into which it is directed, ray splitting introduces a companionate ray at a critical distance associated with exceeding an upper bound of solid angle for valid single ray representation. (Molisch , page 131)
• Reduced dimensionality methodology
  • Restriction to the vertical plane if transmitter is high and loss is dominated by diffraction from the mobile station's horizon to the receiver and by reflection from the closest building in the direction of propagation.
  • Restriction to horizontal (slant) plane if both antennas are below roof-top level
  • (2.5D)for larger distance urban channels : sum of contributions from vertical plane for horizontal diffracting edges and slant(horizontal) plane to negotiate vertical diffracting edges.
• Geographical Databases: The accuracy of deterministic methods is ultimately limited by the accuracy of relevant geographical and morphological features of the environment. (Molisch, page 134)
  • Building plans- available in digital form are of particular relevance to the indoor propagation channel.
  • Geographical and morphological data bases (land use), variably accessible according to nationality , are of particular relevance to propagation models selected for rural areas.
  • Urban propagation data base optimization: vector data providing critical coordinates of building structural features and pixel data area coverage providing land use descriptions which are directly complementary to coexisting structural features.

MODEL NAME/CATEGORY	FREQUENCY RANGE/INTENDED IMPLEMENTATION	DEFICIENCIES/REMEDIATION/ENHANCEMENTS
Coventional Ray Methods	The Maxwell Equations admit to an exact solution expressed as a summation in powers of the wavelength which in the high frequency limit reduces to the first term which defines the (ray) elements of geometrical optics. Geometrical optics (GO) provides a suitable approximation for propagation involving distances and environmental features with linear dimensions large with respect to the wavelength.	If generalized with the elements of the Unified Theory of Diffraction (UTD), the resulting methodology provides high frequency approximations of LOS propagation, reflection , refraction and diffraction
Method of Images	Secondary and higher order sources are located as images of the tranmsitter with respect to all potential reflecting surfaces and all such rays along rays paths to the receiver are superimposed to construct the received field.	The methods are systematic, complete and practical in an optically simple environment, but require an establishment of an upper bound of the number of rays to be considered in each mode of propagation (reflection and diffraction) and are inadequate in moderately complex environments.
Ray Launching	single source to omni-directional targets provides comprehensive assessment of general electromagnetics environment with direct access to propagation elements free space loss, reflection, diffractions and diffuse scattering.	Restriction to horizontal (slant) plane if both antennas are below roof-top level (2.5D)for larger distance urban channels : sum of contributions from vertical plane for horizontal diffracting edges and slant(horizontal) plane to negotiate vertical diffracting edges.
Ray Tracing	Ray tracing (point-to-point raypath ) such as Longley-Rice Model (ITM): valid in the range 40 MHz-100 GHz, the model includes (1) an area mode prediction option to predict path parameters which does not require a terrain profile and (2) a point-to-point ray tracing option which requires a terrain profile as input, modeling terrain-induced diffractions according to the Fresnel-Kirchoff knife edge model.	Ray Splitting- Acknowledging a fundamental assumption of Geometrical optics that a given ray represents only a finite solid angle into which it is directed, ray splitting introduces a companion ray at a critical distance associated with exceeding an upper bound of solid angle for valid single ray representation. Restriction to the vertical plane if transmitter is high and loss is dominated by diffraction from the mobile station's horizon to the receiver and by reflection from the closest building in the direction of propagation.

(1.3 ) HYBRID DETERMINISTIC AND STOCHASTIC MODELS

While retaining a deterministic treatment of path loss (of a specified fade margin), inaccessible causal parameters are replaced by a statistical modeling of multi-component phase contributors to thus statistically model impulse responses

(1.4 ) ARTIFICIAL NEURAL NETWORKS MODELS

Addressing accuracy deficiencies of statistical models and efficiency limitations of site specific models, Artificial Neural Network models provide accurate field strength estimates of noisy data while presenting the additional benefit of intrinsic parallelism. Developmental calculations have explored exhaustive data bases of physical surroundings and while requiring a lengthy learning phase, the model was fast in the final field level prediction phase and underscored the significance of the quality of the data base in determining the accuracy of the final result. Similar investigative studies introducing topological and morphological data have attempted to facilitate the learning process with supplementary heuristic and deterministic equations. The slow convergence and unpredictability of field-strength predictions has been addressed through the introduction of radial basis functions which, though linear in the fundamental input parameters, provide a best non-linear-approxmation capability for the model.

( 2 )WIDE-BAND MODELS (Molisch, Section 7.3)

• Tapped delay
• Models for power delay profile
• Arrival times of rays and clusters
• Standardized channel model

(3) DIRECTIONAL MODELS (Molisch, Section 7.4)

• General model structure and factorization
• Angular dispersion at base station
• Angular dispersion at base station
• Polarization
• Model Implementation
• Standardized directional models
• MIMO models

REFERENCES

1 Molisch, Andreas F., Part II Wirelss Communicaton Channels, in Wireless Communications, John Wiley and Sons, (2005).

2 Sarkar, T.k., Ji, Z., Kim, K., Medouri, A., Salazar-Palma, M., A Survey of Various Propagation Models fro Mobile Communciation, IEEE Antenna and Propagation Magazine 45(3), 51-82 (2003).

TABULATED OBJECTIVES

PROJECT DESCRIPTION	INTERMEDIATE OBJECTIVES	ESTIMATED COMPLETION DATE/STATUS
VTRPE ENHANCEMENT	FIRST DERIVATIVE OPERATOR : discrete Fourier transformation of wave function operand produces representaion in reciprocal (spectral ) space in which differentiation is multiplicative such that multiplication of Fourier transfrom coefficient vector by corresponding spectral components followed by Fourier inversion to cooordinate space produces grid representation of (arbitrary) derivative of original wave function. Higher derivatives are obtained in k-space by effective repeated multiplication of transform vector by the diagonal spectral component matrix. The elements of the spectral components are given by K_k = k 2 i/ (N dx) where N is the number of grid points in the discretization of, for VTRPE, the vertical coordinate. SECOND DERIVATIVE OPERATOR: The expeonentiated kernel of the formal propagator may be expressed as sqrt(1 + (d^2/ d z^2)/k0^2) which is implemented in k space as the multiplicative factor sqrt( 1+ (p/k0)^2 to produce a result which is inverted to coordinate space. The proper constuction of such an operator can presumably be assured by first demonstrating that the implicit operation F^(-1) (d^2/d z^2)F| psi> is correct. APPLICATION OF HOMOGENEOUS PROPAGATOR e^( i sqrt( 1 + d^2/d z^)): For small displacements in range , the expansion of the formal propagator as U ~ 1 + i dx ( A + B) is expected to be valid and provides a testing procdure by demonstrating approximate agreement of the Fourier implemenation of the operator as its exponenetiated components and the more conventional approximate expansion. SIMPLE NON -REFRACTIVE VTRPE APPLICATION REPLICATING KNIFE EDGE DIFFRACTION	completed 02JUN2008(MO): Using N=128, numerical values of first and second derivatives obtained by the Fourier grid method were found to be in agreement with analytic values of test function to machine precision. completed 03JUN2008(TU) completed 05JUN2008(WE): The Magnus approximation U(x0,x,z)= e^(i H) = 1 + i H = 1 + i (A + B), where A= d^2/ (d z^2)/k0^2 and B=0, will be tested by comparing the action of the propagator e^(i H) on a test wave function with that of the approximate expansion expression U ~ 1 + i A. The action of the propagator upon the wave function for small dx is to add the quantity .5 (i) dx D^2/2 |c> to the original wave function . Thus U|C> = |C> + i dx D^2/2 |C> where |C> is the original wave function and D^2 is the second derivative operator in coordinates space. expected completion: 10JUN2008(MO); An optical potential absorbing element must be added to the refractivity correction function B(x,z) such that B(x,z) = B0(x,z) + i zk0 (1/2) V_abs whereby exponentiation of the imaginary optical potenital produces a damping effect on the wave function in vertical (z space) since the potential function is applied as the direct product of its coordinate-space components and the corresponding element of the wave function as it approaches either or both finite endpoints of the vertical coordinate. Imprecision in the description of the parameters required for the standard optical potential modeled as exp^{i k0 V0 sech^2( (z-z0)/L))} has motivated an alternative implementation of a more recent adaptation of the PML of Berenger as an effective optical potential for numerical solutions of the standard parabolic equation utilizing the split-step SPE Fourier propagator, (Levy, Chapter 8).
TIREM INSTALLATION/ENHANCEMENT	COLLECTION AND PROVISION OF VALIDATION DATA FORMULATION OF COORDINATES OF GREAT CIRCLE PATH BETWEEN ARBITARY ENDPOINTS EXTRACTION OF TERRAIN ELEVATION PROFILE FROM DTED DATA AT INTERMEDIATE POINTS ALONG PROFILE	completed 15MAY2008: representaive data format shown below completed 22MAY2008: representaive data in which endpoints are permuted for a reversal of great circle direction between fixed endpoints encounters identical intemediate loci requires acquisition of classified terrain data
INSTALLATION/VALIDATION OF NORTON-BASED SMOOTH (SPHERICAL) EARTH PROPAGATOR	Replication of ITU Recommendations for surface waves 30 kHz < fmhz< 30 MHz Replication of figures in Norton's book and Meek's book Replication of ITU Atlas of Ground wave Curves	completed 24MAY2008 completed 17MAY2008 expected completion data: 09JUN2008

REPRESENTATIVE UNIFIED-PARAMETER VALIDATION DATA FOR COLORADO PLAINS

The following data is typical of that for the Colorado Plains in which the receiver site at a fixed location is varied from one to 13 meters with the expected decrease in excess loss which in the example below recovers from 15.3 dB to a gain of 6.8 dB. Unlike the data at NTIA, all relevant parameters including geographic coordinates of both antennas, propagation parameters and structural heights required for a given measurement are provided in a single record.

                  TRANSMITTER           RECEIVER 
TX SITE     CODE  LAT    LON         LAT      LON       FMHZ       H_TX        H_RX   XS LOSS(DB) RANGE (KM)

R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    1.0000   15.3000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    2.0000    9.8000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    3.0000    6.1000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    4.0000    4.1000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    5.0000    2.2000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    6.0000    0.8000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    7.0000   -0.9000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    8.0000   -2.5000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    9.0000   -3.7000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   10.0000   -4.9000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   11.0000   -5.6000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   12.0000   -6.3000    0.5700
R1-0_5-T1   O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   13.0000   -6.8000    0.5700

$cf={{\mu \omega } \over {\pi }}$

Consider construction of a text reference in the text 1. The following data is typical of that for the Colorado Plains in which the receiver site at a fixed location is varied from one to 13 meters with the expected decrease in excess loss which in the example below recovers from 15.3 dB to a gain of 6.8 dB. Unlike the data at NTIA, all relevant parameters including geographic coordinates of both antennas, propagation parameters and structural heights required for a given measurement are provided in a single record.

ALTERNATE MECHANISMS FOR ENFORCING IMPEDANCE BOUNDARY CONDITIONS USING THE SPLIT-STEP FOURIER APPROACH TO SOLVING PARABOLIC EQUATIONS

In his original modern implementation of the Split-Step Fourier Operator in solving the parabolic approximation of the Helmholtz equation, Ryan [2] was able to satisfy the impedance boundary conditions by introducing a modified index of refraction in the coordinate space portion of the SSFPE unitary propagator, a technique which was also employed by Barrios [4] who noted the essential difference of her work and that of Dockery who has since [3] introduced the mixed Fourier (MFT) transform formalism for satisfying the impedance boundary conditions, a technique which has also since been employed by Frank Ryan [5] . Dockery however reports [6] that the MFT was unstable for high frequencies and that it introduced a factor of two in the computational time, difficulties which have since been removed by Dockery and Kuttler. However in a recent review article, [8] Media:Dockery_2007.pdf, Dockery reported that the method remained problematic for small scale roughness at certain frequencies but that these problems have been effectively solved by Kuttler and Janaswami [7]. Both schools (NOSC) and (JHU-APL) are in accord that for homogeneous smooth earth, the boundary conditions can be solved for vertical and horizontal polarization by expressing the solution as linear combination of even(V) or odd functions(H) which can be respectively propagated with (fast ) Fourier cosine or sin transforms and that for realistic electrical ground constants, a linear combination of these fundamental solutions may be analytically employed to solve the more general problem. As an exercise in the enforcement of impedance surface boundary conditions, work at the PSU ARL will first develop computational proficiency in the elementary Fourier projection and propagation techniques by reproducing the work in Dockery's paper[3] for propagation over the smooth earth.

REFERENCES

[1] Dockery, G.D., "Modeling Electromagnetic Wave Propagation in the Troposphere Using the Parabolic Equation", IEEE TAP 36(10), 1464-1470 (1988).

[2] Ryan, F., "Analysis of Electromagnetic Propagation Over Variable Terrain Using The Parabolic Wave Equation", NOSC Technical Report 1453, (1991)

[3] Kuttler, J.R. and Dockery G.D., "Theoretical description of the parabolic approximation /Fourier split-step methods of representing electromagnetic propagation in the troposphere", Radio Science 26(2), 381-392 (1991).

[4] Barrios, A. E., " A Terrain Parabolic Equation Model for Propagation in the Troposphere" , IEEE TAP 42(1), 90-98 (1994)

[5] Ryan, F.J., "VTRPE: A Variable Terrain Electromagnetic Parabolic Equation Model", Proc. 11th Annual Review of Progress in Applied Computational Electromagnetics II, 816-823 (1995)

[6] Dockery, G.D. and Kuttler J.R. , " An Improved Impedance-Boundary Algorithm for Fourier Split-Step Solutions of the Parabolic Wave Equation" IEEE TAP 44 (12), 1592-1599 (1996)

[7] Kuttler, J.R., and Janaswami, R., " Improved Fourier Transform Methods for Solving the Parabolic Wave Equation", Radio Science 37(2), (5-1)-(5-11), (2002)

[8] Dockery, G.D, "An Overview of Recent Advances for the TEMPER Radar Propagation Model", Proceedings of the Radar Conference, 2007 IEEE, Publication Date: 17-20 April 2007 ,896-905 (2007)

PREPARATION OF INITIAL FIELD DISTRIBUTION FUNCTIONS

Since the most general impedance boundary conditions are enforced by expressing the field as linear combinations of even and odd functions (with respect to vertical coordinate z) , it becomes desirable to derive the symmetry components from an arbitrary function such as an initial field distribution which may be constructed from an antenna amplitude field pattern confined to the permissible values of the k-space coordinate. Given a suitable antenna radiation pattern in p space such as f(p)= sin( ap)/ap, the associated even and odd functions are constructed as

ce = [f(p) + f (-p)]/2

co= [f(p) - f(-p) ]/2

The original field pattern , f_{+}(p) and its mirror image (about z=0) f_{-} (p) may be recovered from the even and odd projections as

f_{+} (p) = ce + co

f_{-} (p) = ce - co

With suitably constructed wave forms in momentum space which exhibit the desired even and odd symmetry characteristics, the corresponding z-space representations of the even and odd functions are recovered as the cosine and sine transforms of the corresponding momentum space wave functions. Adjustments for the desired maximal momentum coordinate amplitude (such as the selection of beam directivity according to the relation p_0 = k_0 sin (theta_0) ) assures that the momentum values p_0 so constructed will correspond to that value in p-space with the largest amplitude in the p-space representation. Adjustments for antenna height z0 are also made in momentum space by introducing the phase factors e^((+/-) i p z0) to the corresponding terms in Eqs(1) .

ce = [f(p)e^(-i p z0) + f (-p)e^(i p z0)]/2

co= [f(p)e^(-i p z0) - f(-p)e^( i p z0)]/2

The z-space representations of the even and odd components are obtained as cosine and sine inversions of the corresponding even and odd p-space functional forms .

psi_e( z) = F_{C}^{-1} c_e (p)

psi_o( z) = F_{S}^{-1} c_o (p)

TABULATED OBJECTIVES, 23 JUNE, 2008

PROJECT DESCRIPTION	INTERMEDIATE OBJECTIVES	ESTIMATED COMPLETION DATE/STATUS
VTRPE ENHANCEMENT	Construction of initial field distribution function in z-space , psi(x=x0,z) from the generic antenna amplitude function f(p)= sin (p-p0)/( a p-p0))	construction of even and odd components of the p-space amplitude function demonstration of recovery of f(p) as sum of even and odd components transformation of even and odd p-space symmetry functions to corresponding z -space function derivation of exact analytic functional forms and second derivatives (see below) in p and z-space representations appropriate for a Gaussian antenna amplitude factor as required to validate the action of the propagator for a small incremented range parameter propagation of resulting odd z-space function as initial wave function for replication of Dockery's Fig.(2)[3, page 388] in which the the unitary propagator acts upon the (odd) p-space representations of the field pattern generated by sine transforms thus: $|\psi _{0}(x+dx,z)>=F_{S}^{-1}e^{(-i(\delta x)p^{2}/(2k_{0}))}F_{S}|\psi _{0}(x,z)>$

ANALYTIC INITIAL FIELDS FOR SPLIT-STEP FOURIER PROPAGATOR

Using the methods described above for an initial Gaussian antenna radiation pattern, the general form for the initial wave function in p-space is given by

$\psi _{+}(x=0;p)=e^{-\alpha {(p-p_{0})}^{2}}e^{-ip(z_{0})}$

where the quantity p0 is related as stated above to the angle of inclination of the main beam above the horizontal and where the phase factor in $z_{0}$ is the p-space manifestation of antenna heights above the surface of the (smooth) earth. Given this analytic form for the positive argument p-function,the corresponding negative argument function follows from the transformation $p\mapsto -p$, to yield

$\psi _{-}(x=0;p)=e^{-\alpha {(-p-p_{0})}^{2}}e^{ip(z_{0})}$

The even (odd) function ( with respect to p=0) is obtained from $\psi _{+}$ and $\psi _{-}$ as their sum (difference) to yield

$\psi _{e}(x=0;p)=\psi _{+}+\psi _{-}=e^{-\alpha {(p^{2}+p_{0}^{2})}}\cosh(2\alpha pp_{0}+ipz_{0})$

and

$\psi _{o}(x=0;p)=\psi _{+}-\psi _{-}=e^{-\alpha {(p^{2}+p_{0}^{2})}}\sinh(2\alpha pp_{0}+ipz_{0})$

File:Psi0 plot.gif

,

The associated even(odd) functions in z-space are obtained as the Fourier cosine(sine)inversions of the corresponding even(odd) functions in p-space and the total solution in z -space may either be constructed from these transformed components or obtained directly as the general Fourier inversion of a composite complex p-space wave function constructed from the even and odd functions and extended to the negative half space as appropriate for the respective understood symmetry. A typical initial function n p-space modeled as a displaced Gaussian (centered at p0, black) and its associated even(red) and odd(blue) components and the cooresponding representations in coordinates space ( as a shifted Gaussian centered at z0, same color scheme) is shown at right.

$\psi _{e}(x=0;z)={\sum _{j=1}^{N}}{\cos(\pi jk/N)\psi _{e}(0,p)}={(1/2\alpha )}^{1/2}{\cosh(zz_{0}/2a-ip_{0}z)e^{(-(z^{2}+z_{0}^{2})/(4a))}e^{ip_{0}z_{0}}}$

$\psi _{o}(x=0;z)={\sum _{j=1}^{N}}{\sin(\pi jk/N)\psi _{o}(0,p)}={(1/2\alpha )}^{1/2}{\sinh(zz_{0}/2a-ip_{0}z)e^{(-(z^{2}+z_{0}^{2})/(4a))}e^{ip_{0}z_{0}}}$

$\psi _{+}(x=0;z)={\sum _{j=1}^{2N}}{e^{(i2\pi jk/(2N))}\psi _{+}(0,p,-p)}={(1/2\alpha )}^{1/2}e^{-{(z-z_{0})^{2}/(4\alpha )}}e^{-ip_{0}(z-z_{0})}$

The even function $\psi _{e}(x=0;z)$ can may be rewritten as

$\psi _{e}(x=0;z)={(1/2\alpha )}^{1/2}{{\big (}\cosh(zz_{0}/2a)\cos(p_{0}z)-i\sinh(zz_{0}/2a)\sin(p_{0}z){\big )}e^{(-(z^{2}+z_{0}^{2})/(4a))}e^{ip_{0}z_{0}}}$

which can be more transparently written as

$\psi _{e}(x=0;z)={(1/2\alpha )}^{1/2}{\big (}e^{-{(z+z_{0})}^{2}/(4a)}e^{ip_{0}(z+z_{0})}+e^{-{(z-z_{0})}^{2}/(4a)}e^{-ip_{0}(z-z_{0})}{\big )}$

which is the sum of two Gaussians, centered at $\pm z_{0}$ and partitioned between real an imaginary sinusoidal factors also centered at $\pm z_{0}$, whereas the odd function is of a similar form but wth a center of inversion through the origin

$\psi _{o}(x=0;z)={(1/2\alpha )}^{1/2}{\big (}-e^{-{(z+z_{0})}^{2}/(4a)}e^{ip_{0}(z+z_{0})}+e^{-{(z-z_{0})}^{2}/(4a)}e^{-ip_{0}(z-z_{0})}{\big )}$

A quadratic coefficient of $\alpha =128$ was observed numerically to provide conjugate functions well localized within their respective discrete representations and therefore permits a calculation of the associated beam width from the relation

$\alpha (p-p_{0})^{2}={\big (}(\log(2)/2)/{{p_{1/2}}^{2}{\big )}(p-p_{0})^{2}}=>p_{1/2}={(\log(2)/(2\alpha ))}^{1/2}=.0520347.$

Analytic expressions for the second derivatives (in z) of these initial wave functions are of interest for validation of a single action of the unitary propagator as described in the next section and are given below:

INFINITESIMAL VALIDATION OF SPLIT-STEP FOURIER PROPAGATOR

The Split-Step Fourier Propagator approach to solving the parabolic equation resulting from an application of the paraxial approximation upon the (scalar) Helmholtz equation relevant to a transverse electromagnetic field normal to the propagation plane defining the dimensionally reduced 2-D problem, involves iterative application of a unitary propagator in which the coordinates-space representation of the field is first transformed to k-space where the action of the square root operator is excuted through a multiplicative process to produce an intermediate operand which is then inverted to coordinate space.

$|\psi (x+\delta x,z)>=\mathbf {F} ^{-1}(p\mapsto z){\big (}e^{-i(\delta x)\mathbf {H} }{\big )}\mathbf {F} (z\mapsto p)|\psi (x,z)>$

For strictly vertical or horizontal polarization, the resulting boundary conditions are enforced by first extracting even or odd symmetry representations of the wave fnction and replacing the general Fourier transforms $\mathbf {F}$ by the Fourier cosine $(\mathbf {C} )$ or sine transform $(\mathbf {S} )$ thus:

$|\psi _{e}(x+\delta x,z)>=\mathbf {C} ^{-1}(p\mapsto z){\big (}e^{-i(\delta x)\mathbf {H} }{\big )}\mathbf {C} (z\mapsto p)|\psi _{e}(x,z)>$

$|\psi _{o}(x+\delta x,z)>=\mathbf {S} ^{-1}(p\mapsto z){\big (}e^{-i(\delta x)\mathbf {H} }{\big )}\mathbf {S} (z\mapsto p)|\psi _{o}(x,z)>$

(General boundary conditions for finite ground constants involve linear combinations of the above symmetry forms where the general Fourier transformation is obtained as the sum of the two half-transforms acting upon their respective symmetry-specific operands.)

$\mathbf {F} =\mathbf {C} +i\mathbf {S}$

$\mathbf {F} ^{-1}=\mathbf {C} ^{-1}-i\mathbf {S} ^{-1}$

The unitary propagator may be expanded for small arguments to produce expressions which may readily evaluated numerically to replicate the effective action of the operator for a single iteration, which for propagation in a vacuum reduces to the expression

$e^{-i(\delta x)\mathbf {H} }\approx \mathbf {1} -i(\delta x){\hat {\mathbf {P} }}^{2}/(2k_{0})$

which in the coordinate representation assumes the form

$e^{-i(\delta x)\mathbf {H} }\approx \mathbf {1} -i(\delta x){\boldsymbol {{\hat {\nabla }}_{z}}}^{2}/(2k_{0})$

$e^{-i(\delta x)\mathbf {H} }\approx \mathbf {1} -i(\delta x){\big (}\partial ^{2}/{\partial {z^{2}}{\big )}/(2k_{0}})$

$e^{-i(\delta x)\mathbf {H} }-\mathbf {1} \approx -i(\delta x){\big (}\partial ^{2}/{\partial {z^{2}}{\big )}/(2k_{0}})$

which indicates that the single application of the unitary propagator for small $\delta x$ can be approximated by the sum of the unit matrix and a (diagonal) correction which is proportional to the second derivative with respect to the elevation coordinate, z. The exact expressions for the (second derivatives in z ) of the initial wave functions can presumably be very closely approximated by subtracting the unit matrix from the action of the propagator upon the wave function.

$\partial ^{2}/\partial {z^{2}}\psi _{e}(z)=e^{-(1/4\alpha )(z^{2}+z_{0}^{2})+ip_{0}z_{0}}{\big (}{\big (}2a_{v}+b_{u}^{2}+4{(a_{v}z)}^{2}{\big )}\cosh(b_{u}z)+{\big (}4a_{v}b_{u}z\sinh(b_{u}z{\big )}{\big )}$

$\partial ^{2}/\partial {z^{2}}\psi _{o}(z)=e^{-(1/4\alpha )(z^{2}+z_{0}^{2})+ip_{0}z_{0}}{\big (}{\big (}2a_{v}+b_{u}^{2}+4{(a_{v}z)}^{2}{\big )}\sinh(b_{u}z)+{\big (}4a_{v}b_{u}z\cosh(b_{u}z{\big )}{\big )}$

where the constants $a_{v}$ and $b_{u}$ are given by

$a_{v}=-(1/4\alpha )$

$b_{u}=(z_{0}/2\alpha )-ip_{0}$

$e^{-i(\delta x)\mathbf {H} }-\mathbf {1} \approx -i(\delta x){\big (}\partial ^{2}/{\partial {z^{2}}{\big )}/(2k_{0}})$

which exhibits a diagonal form with matrix elements given by

$\delta _{ij}{\big (}e^{-i(\delta x)\mathbf {H_{ij}} }-1{\big )}\approx -i(\delta x){\big (}\partial ^{2}/{\partial {z^{2}}{\big )}/(2k_{0}})$

An effort will be made to reproduce the work of Dockery for propagation over the smooth earth at a frequency of 3000 MHz with a transmitter height of 30 m and a receiver at fixed ranges of 40 km and 80 km and of a variable height between 0 and 500 m

File:Dockery F2.gif

, which has been replicated below using an application of Norton's adaptation of Sommerfeld's methods for propagation over the smooth earth.

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