A Sharʽiah Analysis of ʼijāra as a Financing Mode in the wake of Parallel Banking System of Bank of Khyber

The only religion Islam provides the thorough layout for human beings not only covers the spiritual needs but also shaped the materialistic desires in a better way. The most important aspect among all is the availability of proper, smooth flowing and practical financial system as it involve almost in all fields of life. Looking at the current situation and fast growing industry, Islamic Banking System emerged as solid and firm scheme for optimum utilization of finance in full conformity of Shariʽah. ʼijāra is one of the modes frequently used in Interest free banks. The Bank of Khyber claims as Parallel Banking System i.e. It practicing conventional Banking as well as Islamic Banking System. This study emphasis on analyzing the practice of ʼijāra as a financing mode in the wake of Parallel Banking system of Bank of Khyber in the light of Sharʽiah teachings and to portray the real picture of it to enhance the level of satisfaction of customer.   

Improved Matrix Pencil Methods for Parameters Estimation of Plane Wave Signals

The problem of parameters estimation of plane wave signals using an array of sen- sors has received a considerable attention from researchers and engineers during the last few decades. In general, the parameters of interest are the frequencies and direction-of-arrivals of incoming signals. Although, a number of methods have been proposed in literature, the subspace-based methods including MUSIC and ESPRIT are widely used to estimate the required parameters because of their relatively less computational cost and high resolution. In the presence of coher- ent signals, these covariance-based methods require an additional step of spatial smoothing. Another subspace based method is Matrix Pencil method that is a direct data domain method and analyzes the data on snapshot by snapshot bases; consequently, a non-stationary environment can be handled, easily. Moreover, Ma- trix Pencil method is directly applicable in scenarios where the input signals are fully correlated or coherent since it implicitly performs spatial smoothing while constructing the data matrix. The main focus in this thesis is on several improve- ments to the existing Matrix Pencil methods (especially related to reduction in computational complexity with comparable estimation accuracy). First of all, in Matrix Pencil method it is commonly assumed that noise is spatially uncorrelated. In the presence of correlated noise, matrix pencil method often fails to yield unbiased estimates of required parameters since the signal sub- space estimated through singular value decomposition of the (noisy) data matrix is biased. To combat with spatially correlated noise, we proposed a Generalized Matrix Pencil method which utilizes a generalized singular value decomposition to obtain unbiased estimates of the required parameters. Since, parameters estimation of plane wave signals is a real-time problem, it is vital to maintain the computational burden of parameters estimation algorithms as low as possible. Many efforts have already been made to minimize the com- putational burden of exiting methods. In the context of Matrix Pencil method algorithms, the (existing) Unitary Matrix Pencil method reduces the computa- tional complexity to about one-fourth by converting the complex data matrix in Matrix Pencil method into a real matrix using a unitary matrix transformation. If some a priori information about direction-of-arrival or frequency of incoming signals is available, as in radar and sonar applications, then a reduced dimensional processing of covariance/data matrix is possible. For such scenarios, a number of researchers have proposed the beamspace approach, which first projects the original data into a subspace of lower dimensions (using DFT) and then processes the beamspace data by using well known algorithms such as MUSIC and ESPRIT. In order to reduce the computational complexity of Matrix Pencil based algorithms, we propose Beamspace Matrix Pencil methods that transform the complex data matrix into a real and reduced dimensional matrix using selected rows of a DFT matrix. Depending upon the number of selected rows, the computational burden is reduced several times with comparable estimation accuracy to that of existing methods. Moreover, if there is no a priori information available then Beamspace technique can be applied via parallel processing with overlapped sectors as can be done in the case of Beamspace MUSIC and ESPRIT. In addition to this, we also propose a Multiple Invariance Beamspace Matrix Pencil method, which exploits the multiple invariance structure inherent in the DFT transformed manifold matrix to improve the estimation accuracy without any significant increase in computational burden. In order to enhance the estimation accuracy, multiple snapshots are often used in Matrix Pencil methods. However, a straight forward implementation of Beamspace technique on multiple snapshot cases may decrease its computational advantage. Therefore, we propose a Multiple Snapshot Beamspace Matrix Pencil method, which not only utilizes a priori information about direction-of-arrival but also uses the frequency information of incoming signals so as to further reduce the computational burden of Beamspace Matrix Pencil method. In the thesis, we also address the grouping problem of estimated parameters in already existing multi-dimensional Matrix Pencil and Unitary Matrix Pencil methods. In particular, we extend the (existing) two-dimensional Modified Matrix Pencil method for three-dimensional scenarios in which the required parameters are always estimated in an automatically grouped form. Moreover, it is shown that the matrices whose Eigenvalues yield the required parameters (in Unitary Matrix Pencil methods) bear the same Eigenstructure. By exploiting this same Eigenstructure property, we propose a Modified Unitary Matrix Pencil method in which the required parameters are obtained in a grouped form thereby eliminating the need of an extra grouping algorithm. This, not only reduces the computational burden but also alleviates the problem of wrong grouping. Finally, in this project we developed a direction-of-arrival estimation system consisting of a uniform linear array of six sensors. The hardware details of the developed system are reported in this thesis. This system is subsequently used to compare the performance of various Matrix Pencil methods for real-world data. Key words: Sensor array, array signal processing, direction-of-arrival estimation, parameters estimation, plane wave signals, coherent signals, spatially correlated noise, subspace based method, computational complexity, unitary matrix pencil, beamspace matrix pencil, DFT transformation, multiple invariance, multiple snap- shot matrix pencil, automatic grouping, comparison analysis, real-world data.