The Per Tone Equalization algorithm is a novel discrete multiTone (DMT) Equalization method with practical applications in subscriber digital loop systems. Unlike time domain DMT equalizers (TEQ), which equalize all DMT Tones in a combined fashion, the Per Tone equalizer equalizes each Tone separately. In this paper, the performance and complexity of this technique is investigated and compared with that of other TEQs. Furthermore, the behavior of this technique in different simulation conditions, such as over different standard CSA loops and in the presence of additive white Gaussian noise with variable power spectral density level and near end crosstalk (NEXT), is studied and compared with that of other TEQs. Simulation results show that Per Tone has a reduced sensitivity to synchronization delay and a much better performance compared to other conventional DMT equalizer design algorithms.

In this paper, a combination of channel, receiver frequency-dependent IQ imbalance, and carrier frequency offSet estimation under short cyclic prefix length is considered in orthogonal frequency division multiplexing system. An adaptive algorithm based on the Set-membership filtering algorithm is used to compensate for these impairments. In short CP length, per-Tone Equalization Structure is used to avoid inter-symbol interference. Due to CFO impairment and IQ imbalance in the receiver, we will expand the PTEQ Structure to a two-branch Structure. This Structure has high computational complexity, so using the Set-membership filtering idea with variable step size while reducing the average computation of the system can also increase the convergence speed of the estimates. On the other hand, applying wavelet transform on each branch of this Structure before applying adaptive filters will increase the estimation speed. The proposed algorithm will be named SMF-WP-NLMS-PTEQ. The results of the simulations show better performance than the usual adaptive algorithms. Besides, estimation and compensation of channel effects, receiver IQ imbalance and carrier frequency offSet under short CP can be easily accomplished by this algorithm.

An effective and flexible method for encoding ambiguous data is using cubic Sets. The concept of incline algebraic sub-Structure is considered and is interlinked with the notation of the cubic Set to define cubic subincline. The sense of cubic sub incline of algebra is established with relevant results. Additionally, the results such as homomorphic image, preimage, cartesian product and level Sets of cubic sub incline are worked out in this study, and several of its associated findings were looked into.

Multiple Sets is a recently developed mathematical framework designed to manage uncertainty and multiplicity simultaneously. They are characterized by membership matrices, which allow them to represent multiple uncertain features of objects and their corresponding multiplicities. This paper presents an in-depth study of the topological Structure of multiple Sets, extending existing theories of basis, interior and closure in a multiple topological spaces (MTS). We introduce the notions of subbasis, local basis, $C_1$ space and $C_{11}$ spaces, neighbourhoods, limit points, derived Sets, compactness, multiple closure spaces (MCS), sequences of multiple Sets and $M$-continuous functions within multiple topological space (MTS). Several results related to these concepts have been proven. Additionally, we provide illustrative examples of multiple topological space (MTS) and analyze their key characteristics.

In this paper, we present Structure of the fixed point Set results for condensing Set-valued maps. Also, we prove a generalization of the Krasnosel’ skii-Perov connectedness principle to the case of condensing Set-valued maps.