The **soft**-in, **soft**-out (SISO) decoding is used as **soft** decode algorithm. The SISO **decoder** receives a **soft** real value of the signal as **input**. The **decoder** then deduces an estimation for each data bit expressing the probability of the transmitted data bit. This probability is the **soft** **output**. The sign of **soft** **output** is a hard decision (H), and the magnitude is used as reliability value of a hard decision (L_value). Higher L_value means more reliable hard decision information. Lower L_value means less reliable decision information. When the L_value is equal to 0, the probability of the correctness of the decision is 0.5. The SISO decoding minimizes the average decoded symbol error rate. The SISO implements operations related to the maximum a posteriori (MAP) algorithm. The **decoder** is executed using a version of the classic MAP algorithm implemented in the log-domain [7], [8].

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spectral efficiency as well as their robust performance against fading. In MIMO inter-symbol interference (ISI) channels, a severe interference problem occurs due to the ISI, spatial and co-**channel** interference. Thus, the error propagation problem becomes more serious, and its mitigation has to be considered when designing the receiver. In this regard, a turbo equalizer which exchanges **soft** information between the equalizer and the **decoder** has been shown to be an effective method to combat the ISI caused by frequency-selective channels. By iteratively exchanging **soft** extrinsic information between a **soft**-**input** **soft**-**output** equalizer and a **decoder**, turbo equalizer can achieve large performance gains over a separated equalizer and **decoder** structure. In its original form, Douillard et al. employed maximum a posterior probability (MAP) equalization and decoding methods in an iterative fashion [2]. However, the computational complexity required to derive the a posteriori log likelihood ratio (LLR) for the MAP **decoder** is prohibitive. This is because the number of states in the trellis diagram for the frequency-selective MIMO channels increases exponentially with the product of the number of users and their **channel** memory length. Therefore, the design of low-complexity turbo equalizers based on minimum mean square error (MMSE) criterion has attracted considerable attention in the past few years. The existing approaches to MMSE-based turbo equalizers can be roughly classified into three categories.

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The concept of turbo equalizer was first proposed in [1], where **soft**-**output** Viterbi algorithm (SOVA) with maximum likelihood sequence estimation (MLSE) [2] - [4] is used for the calculation of extrinsic information in both SISO equalizer and SISO **channel** **decoder**. Replacing SOVA with maximum a posteriori probability (MAP) algorithm [5] - [6] leads to similar performance. SOVA algorithm or MAP algorithm can generate the best possible performance under the structure of turbo equalization. On the other hand, the computational complexity of MAP and SOVA algorithms grow exponen- tially with modulation constellation size M and equivalent discrete-time **channel** length L. Large values of M and L, which are common in modern communication systems, make equalization with MAP or SOVA algorithms intractable. As a consequence, a sub-optimum, reduced complexity, SISO equalizer with minimum performance degradation is highly desirable for efficient turbo equalization in practical systems. The design of low complexity SISO equalizer has attracted considerable attentions recently [7] - [18]. In [7], a linear SISO minimum mean square error (MMSE) based equalizer is employed for the cancellation of both ISI and multiple access interference in a code division multiple access (CDMA) system. The linear MMSE weight vectors are derived by using the first order statistics (mean) calculated from the **soft** **input**. The method is extended in [8] - [10], where both first order statistics and second order statistics collected from **soft** **input** are used for the formulation of the linear MMSE equalizer. A non-linear, SISO block decision feedback equalizer (BDFE) is proposed in [11], where hard decisions from decision feedback is used for ISI cancellation. These algorithms can achieve a reasonable performance at a computational complexity much lower compared to the optimum MAP or SOVA algorithms.

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From (2)X n-k M(x) mod g(x) is computed, the remainder polynomial(R(x)) is obtained which should be appended with the message polynomial (M(x)) to produce the codeword(c(x)) i.e., c(x) = X n-k M(x) + R(x). This codeword is then stored in flash memory, later for correction purpose in **decoder** stage. These encoder procedure is illustrated by considering M(x) = 10001. Therefore,

Some set theories such as theory of fuzzy sets [ 13 ] , rough sets [ 10 ] , intuitionistic fuzzy sets [ 3 ] , vague sets [ 5 ] etc. can be deal with unclear notions. But, these theories are not sufficient to solve some difficulties and problems. There are some vague problems in economics, medical science, social science, finance etc. Then, what is the reason of vague problems and difficulties? It is possible the insufficiency of the parametrization tool of the theories. In 1999, Molodtsov [ 9 ] introduced the idea of **soft** set theory as a general mathematical tool for coping with these difficulties. In 2001, Maji, Biswas and Roy [ 7 ] defined the concept of a fuzzy **soft** set and [ 8 ] an intuitionistic fuzzy **soft** set. In 2003, Maji et al. [ 6 ] studied the theoretical concepts of the **soft** set theory. In 2009, Ali et al. [ 1 ] investigated several operations on **soft** sets and defined some new notions such as the restricted union etc. In 2010, Xu et al. [ 12 ] introduced vague **soft** sets and studied some properties of them. In 2010, Feng et al. [ 4 ] studied **soft** sets combined with fuzzy sets and rough sets as a tentative approach. In 2011, Shabir et al. [ 2 ] introduced algebraic structures of **soft** sets via new notions. In 2011, Naz et al. [ 11 ] defined some notions such as **soft** topological space, **soft** interior, **soft** closure etc.

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ABSTRACT: In this paper a new class of **soft** sets called **Soft** ĝ-Closed sets in **Soft** Topological Spaces is introduced and studied. This new class is defined over an initial universe and with a fixed set of parameters. Some basic properties of this new class of **soft** sets are investigated. This new class of **Soft** ĝ-Closed sets contributes to widening the scope of **Soft** Topological Spaces and its applications.

Smart antenna aided broadband beamforming plays an increas- ingly important role in wireless communications. The paper inves- tigates blind space-time equalization/equalizers (STE) designed for single-**input** multi-**output** (SIMO) systems. Specifically, the con- stant modulus algorithm (CMA) and a **soft** decision-directed (SDD) scheme, originally derived for low-complexity blind equalization of single-**input** single-**output** (SISO) channels, are combined for em- ployment in the SIMO scenario.

Zadeh [44] in 1965. This theory brought a paradigmatic change in mathematics. But there exists difficulty, how to set the membership function in each particular case. The theory of intuitionistic fuzzy sets is more generalized concept than the theory of fuzzy sets, but this theory has the same difficulties. All the above mentioned theories are successful to some extent in dealing with problems arising due to vagueness present in the real world. But there are also cases where these theories failed to give satisfactory results, possibly due to inadequacy of the parameterization tool in them. As a necessary supplement to the existing mathematical tools for handling uncertainty, in 1999, Molodtsov [31] initiated the concept of **soft** set via a set-valued mapping. The theory of **soft** sets is free from the difficulties mentioned above. Since its introduction, the concept of **soft** set has gained considerable attention and this concept has resulted in a series of works [3], [11], [16], [17], [18], [19], [20], [25], [27], [30], [41], [42], [43], [45] including some successful applications in information processing [14], [15], [32], [47], decision-making [8], [12], [26], [29], [35], demand analysis [10], forcasting [40], relations [9], algebraic structures of the set theory [1], [3], [4], [11], [17], [18], [19], [20], [21], [22], [23], [24], [34], [38] , topology [6], [11], [37], [39], [46], theory of BCK/BCI-algebra [17], operation research [5], [13], [16] etc. In recent times, researchers have contributed a lot towards fuzzification of theory **soft** sets. Maji et al. [28], introduced the concept of fuzzy **soft** set and some properties regarding fuzzy **soft** union, fuzzy **soft** intersection, complement of fuzzy **soft** set, De Morgan Law etc. In section 3, properties of fuzzy **soft** closure, fuzzy **soft** interior are studied and investigated. Also in this section, concepts of fuzzy **soft** exterior, fuzzy **soft** boundary are introduced and some properties related to these structures are established.

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An implication of the SPA that reduces the complexity of the parity check update at the cost of some loss in performance. This implication has been derived by operating in the log-likelihood domain. Recently, a new reduced complexity decoding algorithm that also operates entirely in the log-likelihood domain. It bridges the gap in performance between the optimal SPA and finally, low complexity software and hardware implementations of an iterative **decoder** for LDPC codes suitable for multiple access application.

A lot of work has been carried out in this regard and number of papers is submitted so as to solve the problems of the conventional converter circuits [2], [3], [5], [8]. But the main disadvantage is the increase in voltage and current stress across the circuit components and reduced efficiency. In order to overcome these above mentioned problems, a new ZVT-ZCT-PWM dc-dc converter that combines ZVT-ZCT methods are also suggested [7], [9], and [10]. Various advancements in semiconductor technologies to produce high power devices have facilitated numerous applications where high power density is very important for practical and sophisticated solutions. In these converters, the turn on or off process takes place under ZVS and/or ZCS during a very short period of Zero Voltage Transition (ZVT) or ZCT time provided by a resonance. Hence, the resonant converters are used to operate at higher switching frequencies. Most of the new **soft**-switching converters reduce switching loss only at the expense of much increased voltage/current stresses of the switches, which leads to a substantial increase in conduction loss.

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A novel encryption system to increase security in a three tier manner without any additional complexity is proposed in this paper. The encryption block here is a Shrinking generator which is a Linear Feedback Shift Register (LFSR) based stream cipher system in which controlled randomness provides security. The **channel** coding technique used is Turbo code that performs very well and provides results near Shannon’s Limit. The design of interleaver used in turbo code provides security while **channel** coding. Puncturing pattern designed for **channel** coding further increases the security of the system and improves the code rate also. Security of the system is achieved by hiding the keys used in code generation and puncturing from unintended users. For an intended user, performance of the **channel** coding system is further improved by using **Soft** **Input** Decryption (SID) technique. The hardware complexity of the proposed Shrinking Generator Based Cipher (SGBC) is compared with joint coding cryptographic schemes available in literature. Improved Linear Consistency Attack is mounted to analyze the security of the proposed system and the results show that a significant increase in security could be achieved without any additional increase in complexity.

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While long polar codes can achieve the capacity of arbitrary binary-**input** discrete memory less channels when decoded by a low complexity successive cancelation (SC) algorithm, the error performance of the SC algorithm is inferior for polar codes with finite block lengths. The cyclic redundancy check (CRC) aided successive cancelation list (SCL) decoding algorithm has better error performance than the SC algorithm. However, current CRC aided SCL (CA-SCL) decoders still suffer from long decoding latency and limited throughput. In this paper, a reduced latency list decoding (RLLD) algorithm for polar codes is proposed. Our RLLD algorithm performs the list decoding on a binary tree, whose leaves correspond to the bits of a polar code.

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detection layer and performed for several times once new branches are accessed. Furthermore, the authors in [17] investigate the practical performance of a novel sphere **decoder** (Geosphere) for multiuser detection. A novel two-dimensional zig-zag ordering strategy has been studied in the sense that the number of path metric cal- culations is reduced. Additionally, the lower bound of the path metric is employed to eliminate the branches if the path metric is smaller than the lower bound. Another efficient ordering and pruning scheme is stud- ied in [18], which performs the horizontal pruning and vertical pruning with a novel tight lower limit for the path metric. These two schemes discussed above could also be used in the complex-valued SDs with simple modifications.

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China’s rapid economic growth has proceeded at considerable cost of resources and environment. To find out how to reduce the cost while maintaining economic growth, we took panel data of 28 provinces from 2000 to 2009 as our samples and conducted comparative analysis on the said cost among 28 provinces. From the perspective of **soft**-**input**, we subsequently examined **soft**-input’s influence on the cost of re- sources and environment of in economic growth (CREIEG). Results show that: government spending on scientific research and environmental protections has a significant role in reducing energy consumption and wastewater discharge, but it has limited impact on gas emissions and solid waste emissions reduction. Moreover, financial development also has a positive role in reducing energy consumption. Finally, we propose a number of initiatives to reduce the cost of resources and environment of economic growth based on the analysis results.

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Core and memory **input** switch shown in Fig.9 have the ability to to provide the data vectors arranged in the correct order to and from the processing cores in a single cycle. This data is logically divided into a number of complex―matrix variables‖ of size Nrx by Nrx. When an instruction is executed for a subchannel, the chunk of data associated with the subchannel is retrieved and then delivered to the core-**input** switch. As shown in Fig.7. The core-**input** switch is a two level multiplexing circuit that selects and properly arranges the complex vectors needed by the processing core—whether they are row vectors, column vectors, matrix diagonals, or a combination thereof.

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The **soft** set theory is a rapidly processing field of mathematics. Molodtsov’s [10] **soft** set theory was originally proposed as general mathematical tool for dealing with uncertainty problems. He proposed **soft** set theory, which contains sufficient parameters such that it is free from the corresponding difficulties, and a series of interesting applications of the theory instability and regularization, Game Theory, Operations Research, Probability and Statistics. Topological structure of **soft** sets was initiated by Shabir and Naz [12] and studied the concepts of **soft** open set, **soft** interior point, **soft** neighborhood of a point, **soft** separation axioms and subspace of a **soft** topological space. Many researchers extended the results of generalization of various **soft** closed sets in many directions. Athar Kharal and B. Ahmad [3] defined the notion of a mapping on **soft** classes and studied several properties of images and inverse images of **soft** sets.

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metric spaces. Metric space is widely used in mathematics and **soft** metric space is a generalization of metric spaces. That is to say, **soft** metric spaces can be developed in a metric view with **soft** sets. While these important results caused most mathematicians to think of **soft** metric spaces as just a rather convenient tool to define and to deal with **soft** topological spaces, a few began to study **soft** metric spaces for their own sake as [11]. The development of the theory of **soft** metric spaces has proceeded in the following main directions: General theory of **soft** metric spaces and **soft** topological theory of metric spaces. For each of the cases the simplest and most fruitful method which the **soft** metric proposed was the introduction of the notion of **soft** distance and **soft** spheres. Rather than discuss the **soft** metric spaces in full generality, let us look at a particular situation of the view of **soft** spheres.

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b) (𝐹, 𝐴) is said to be a **soft** subset of (𝐺, 𝐴 ) if ∀ λ ∈ 𝐴, 𝐹(λ) ⊆ 𝐺(λ) and it is denoted by (𝐹, 𝐴) ⊆� (𝐺, 𝐴) . (𝐹, 𝐴) is said to be a **soft** upperset of (𝐺, 𝐴) if (𝐺, 𝐴) is a **soft** subset of (𝐹, 𝐴) . We denote it by (𝐹, 𝐴) ⊇� (𝐺, 𝐴) . (𝐹, 𝐴) and (𝐺 , 𝐴) is said to be equal if (𝐹, 𝐴) is a **soft** subset of (𝐺, 𝐴) and (𝐺, 𝐴) is a **soft** subset of (𝐹, 𝐴) . c) The union of (𝐹, 𝐴) and (𝐺, 𝐴) over 𝑉 is (𝐻, 𝐴) defined as 𝐻(λ) = 𝐹(λ) ∪ 𝐺(λ), ∀ λ ∈ A . We write

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Multiple-**Input** Multiple-**Output** i.e. MIMO technique is based upon the use of multiple antennas concept or exploitation of spatial diversity technique. It was programmed such that it provides high speed communication links even in harsh environment. But it spread amongst the WAN, base station etc. Now, since the WSN depends on the energy of the node, we have to deal with the energy constraints in the WSN.

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