In this dissertation, we present novel batch algorithms to tackle the multi-path fading effect of the wireless channels using convex optimization tools. We consider two major problems: channel equalization and channel shortening.;Blind channel equalization has been widely investigated in the past decade. Blind algorithms are preferred because of their ability to equalize the channel without spending extra bandwidth. Existing works have proposed various blind channel equalization costs and characterized their convergence. Most of the blind signal recovery algorithms are implemented as stochastic gradient descent based adaptive schemes making them attractive to applications where the channel is slow varying. However, existing solutions for blind channel equalization often suffer from slow convergence and require long data samples. On the other hand, packet based data transmission in many practical digital communication systems makes it attractive to develop steepest descent implementation in order to speed up convergence. We focus on developing steepest decent implementation of several well-known blind signal recovery algorithms for multi-channel equalization and source separation. Our steepest descent formulation is more amenable to additional parametric and signal subspace constraints for faster convergence and superior performance.;Most of the well-known blind channel equalization algorithms are based on higher-order statistics making the corresponding cost non-linear non-convex functions of the equalizer parameters. Therefore, the steepest descent implementations often converge to local optima. We develop batch algorithms that use modern optimization tools so that the global optima can be found in polynomial time. We convert our blind costs of interest into fourth-order functions and apply a semi-definite formulation to convert them into convex optimization problems so that they can be solved globally. Our algorithms work well not only for removing multipath fading effect in channel equalization problem but also for mitigating inter-channel interference in source separation problem.;Nevertheless, in practical communication systems, pilot symbols are inserted to the packet for various purposes including channel estimation and equalization. Hence, the use of the pilot in conjunction with blind algorithms is more preferred. We investigate simple and practical means for performance enhancement for equalizing wireless packet transmission bursts that rely on short sequence as equalization pilots. Utilizing both the pilot symbols and additional statistical and constellation information about user data symbols, we develop efficient means for improving the performance of linear channel equalizers. We present two convex optimization algorithms that are both effective in performance enhancement and can be solved efficiently. We also propose a fourth-order training based cost so that it can be combined with other fourth-order blind costs and be solved efficiently using semi-definite programming. The simulation results show that with the help of very few pilots, the equalization can be done under very short packet length.;Many modern communication systems adopt multicarrier modulation for optimum utilization of multi-path fading channel. Under this scenario, a cyclic prefix which is not shorter than the channel length is added to enable equalization. We study the problem of channel shortening in multicarrier modulation systems when this assumption is not met. We reformulate two existing second-order statistic based methods into semidefinite programming to overcome their shortcoming of local convergence. Our batch processor is superior to the conventional stochastic gradient algorithms in terms of achievable bit rate and signal to interference and noise ratio (SINR). Addressing the shortcoming of second-order statistic based costs, we propose a new criterion for blind channel shortening based on high order statistical information. The optimization criterion can be achieved through either a gradient descent algorithm or a batch algorithm using the aforementioned convex optimization for global convergence.
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