Zubair Khalid

Distance Distributions in Regular Polygons

This paper derives the exact cumulative density function (cdf) of the distance between a randomly located node and any arbitrary reference point inside a regular L-sided polygon. Using this result, we obtain the closed-form probability density function of the Euclidean distance between any arbitrary reference point and its nth neighbor node when N nodes are uniformly and independently distributed inside a regular L-sided polygon. First, we exploit the rotational symmetry of the regular polygons and quantify the effect of polygon sides and vertices on the distance distributions. Then, we propose an algorithm to determine the distance distributions, given any arbitrary location of the reference point inside the polygon. For the special case when the arbitrary reference point is located at the center of the polygon, our framework reproduces the existing result in the literature.

An Optimal-Dimensionality Sampling Scheme on the Sphere With Fast Spherical Harmonic Transforms

We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at L using only L2 samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The number of samples required in our scheme is a factor of two or four fewer than existing techniques, which require either 2L2 or 4L2 samples. We note, however, that we do not recover a sampling theorem on the sphere, where spherical harmonic transforms are theoretically exact. Nevertheless, we achieve high accuracy even for very large band-limits. For our optimal-dimensionality sampling scheme, we develop a fast and accurate algorithm to compute the spherical harmonic transform (and inverse), with computational complexity comparable with existing schemes in practice. We conduct numerical experiments to study in detail the stability, accuracy and computational complexity of the proposed transforms. We also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications.

Spatio-Spectral Analysis on the Sphere Using Spatially Localized Spherical Harmonics Transform

This correspondence studies a spatially localized spectral transform for signals on the unit sphere, which we call spatially localized spherical harmonics transform (SLSHT). For a systematic treatment, we explicitly express the transform in terms of rotated versions of an azimuthally symmetric window function and introduce the spatio-spectral SLSHT distribution with a succinct matrix representation. We present guidelines for the choice of the window function in the SLSHT, based on the inherent tradeoff between the spatial and spectral resolution of different window functions from the perspective of the uncertainty principle. We demonstrate the use of an eigenfunction window, obtained from the Slepian concentration problem on the sphere, as a good choice for window function. As an illustration, we apply the transform to the topographic map of Mars, which can reveal spatially localized spectral contributions that were not obtainable from traditional spherical harmonics analysis.

Novel sampling scheme on the sphere for head-related transfer function measurements

This paper presents a novel sampling scheme on the sphere for obtaining head-related transfer function (HRTF) measurements and accurately computing the spherical harmonic transform (SHT). The scheme requires an optimal number of samples, given by the degrees of freedom in the spectral domain, for the accurate representation of the HRTF that is band-limited in the spherical harmonic domain. The proposed scheme allows for the samples to be easily taken over the sphere due to its iso-latitude structure and non-dense sampling near the poles. In addition, the scheme can be used when samples are not taken from the south polar cap region of the sphere as the HRTF measurements are not reliable in south polar cap region due to reflections from the ground. Furthermore, the scheme has a hierarchical structure, which enables the HRTF to be analyzed at different audible frequencies using the same sampling configuration. In comparison to the proposed scheme, none of the other sampling schemes on the sphere simultaneously possess all these properties. We conduct several numerical experiments to determine the accuracy of the SHT associated with the proposed sampling scheme. We show that the SHT attains accuracy on the order of numerical precision (10-14) when samples are taken over the whole sphere, both in the optimal sample placement and hierarchical configurations, and achieves an acceptable level of accuracy (10-5) when samples are not taken over the south polar cap region of the sphere for the band-limits of interest. Simulations are used to show the accurate reconstruction of the HRTF over the whole sphere, including unmeasured locations.

Fast Directional Spatially Localized Spherical Harmonic Transform

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.

Commutative Anisotropic Convolution on the 2-Sphere

We develop a new type of convolution between two signals on the 2-sphere. This is the first type of convolution on the 2-sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that 1) the new convolution admits anisotropic filters and signals and 2) the domain of the output remains on the sphere. Therefore, the new convolution well emulates the conventional Euclidean convolution. In addition to providing the new definition of convolution and discussing its properties, we provide the spectral analysis of the convolution output. This convolutional framework can be useful in filtering applications for signals defined on the 2-sphere.

3D Spatial Fading Correlation for Uniform Angle of Arrival Distribution

We derive a closed-form expression for the spatial fading correlation (SFC) between two arbitrary points in 3D-space for the uniform limited azimuth-elevation angle of arrival probability density function (pdf). This expression simplifies the computation of the SFC, can be used in any 3D antenna array geometry, and avoids the need to generate separate expressions for specific antenna array geometries. We corroborate the accuracy of the closed-form expression through application to 2D and 3D antenna array geometries. We expect the results presented in this letter to be of significant importance for performance evaluation and sensitivity analysis in multi-input multi-output (MIMO) systems.

Voronoi cell geometry based dynamic Fractional Frequency Reuse for OFDMA cellular networks

Interference Management (IM) is one of the major challenges of next generation wireless communication. Fractional Frequency Reuse (FFR) has been acknowledged as an efficient IM technique, which offers significant capacity enhancement and improve cell edge coverage with low complexity. In literature, FFR has been analyzed mostly with cellular networks described by Hexagon Grid Model, which is neither tractable nor scalable to the dense deployment of next generation wireless networks. Moreover, the perfect geometry based grid model tends to overestimate the system performance and not able to reflect the reality. In this paper, we use the stochastic geometry approach, FFR is analyzed with cellular network modeled by homogeneous Poisson Point Process (PPP). A dynamic frequency allocation scheme is proposed which take into account the randomness of the cell coverage area describe by Voronoi tessellation. It is shown that the proposed scheme outperforms the traditional fixed frequency allocation schemes in terms of per user capacity and capacity density.

Resource allocation for uplink M2M communication: A game theory approach

Machine-to-Machine (M2M) communication in cellular network is the driver for the future Internet of Things (IoT). The main challenge of M2M communication is the possibility of huge traffic in the uplink network that can cause problem in the network. This paper considers the problem of resource allocation among machines connecting in uplink to different femto base stations (FBSs). Resource allocation problem is analyzed through both non-cooperative and cooperative game to maximize their data rate and minimize utilization of power. Numerical result shows that by adapting non-cooperative game, all machines are getting data rate as per Nash Equilibrium (NE) or either they can set their strategy to maximize their data rate selfishly. On the other hand for coalitional game theory approach machines who participate in game are getting fair resource allocations.

A tractable framework for exact probability of node isolation and minimum node degree distribution in finite multihop networks

This paper presents a tractable analytical framework for the exact calculation of the probability of node isolation and the minimum node degree distribution when N sensor nodes are independently and uniformly distributed inside a finite square region. The proposed framework can accurately account for the boundary effects by partitioning the square into subregions, based on the transmission range and the node location. We show that for each subregion, the probability that a random node falls inside a disk centered at an arbitrary node located in that subregion can be analytically expressed in closed form. Using the results for the different subregions, we obtain the exact probability of node isolation and minimum node degree distribution that serves as an upper bound for the probability of k-connectivity. Our theoretical framework is validated by comparison with the simulation results and shows that the minimum node degree distribution serves as a tight upper bound for the probability of k-connectivity. The proposed framework provides a very useful tool to accurately account for the boundary effects in the design of finite wireless networks.