Finite Bit Research Articles

Uncloneable Quantum Advice

The famous no-cloning principle has been shown recently to enable a number of uncloneable cryptographic primitives, including the copy-protection of certain functionalities. Here we address for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation, but that is natively unkeyed: quantum advice. Remarkably, this is an application of the no-cloning principle in a context where the quantum states of interest are not chosen by a random process. We establish unconditional constructions for promise problems admitting uncloneable quantum advice and, assuming the feasibility of quantum copy-protecting certain functions, for languages with uncloneable advice. Along the way, we note that state complexity classes, introduced by Rosenthal and Yuen (ITCS 2022) — which concern the computational difficulty of synthesizing sequences of quantum states — can be naturally generalized to obtain state cloning complexity classes. We make initial observations on these classes, notably obtaining a result analogous to the existence of undecidable problems. Our proof technique defines and constructs ingenerable sequences of finite bit strings, essentially meaning that they cannot be generated by any uniform circuit family with non-negligible probability. We then prove a generic result showing that the difficulty of accomplishing a computational task on uniformly random inputs implies its difficulty on any fixed, ingenerable sequence. We use this result to derandomize quantum cryptographic games that relate to cloning, and then incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable advice. Applying this two-step process to a monogamy-of-entanglement game yields a promise problem with uncloneable advice, and applying it to the quantum copy-protection of pseudorandom functions with super-logarithmic output lengths yields a language with uncloneable advice.

A bit rate condition for feedback stabilization of uncertain switched systems with external disturbances based on event triggering

AbstractThis paper studies the input‐to‐state stability of an uncertain switched linear system with external disturbances, whose feedback loop is closed over a digital communication network. In the network, only a finite bit rate can be provided to transmit the feedback signal and unknown network delay always exists. By using reachable set approximation and propagation methods, it can well handle with the effects of model uncertainty, external disturbances and network delay on the stability of the system. To save network resources, event‐triggered sampling is taken here. Under the event‐triggered sampling strategies, a new encoder is proposed, which consumes less feedback bits than the conventional time‐triggered encoders. It is proven that even in the existence of model uncertainty, external disturbances and network delay, the proposed event‐triggered sampling strategies require a lower feedback bit rate to stabilize the switched system than conventional periodic sampling methods.

RIS-Assisted Self-Interference Mitigation for In-Band Full-Duplex Transceivers

The wireless in-band full-duplex (IBFD) technology can in theory double the system capacity over the conventional frequency division duplex (FDD) or time-division duplex (TDD) alternatives. But the strong self-interference of the IBFD can cause excessive quantization noise at the output of the analog-to-digital converters (ADCs), which hampers its real implementation. Fortunately, the reconfigurable intelligent surface (RIS) can reconfigure the wireless channels by dynamically adjusting the reflection coefficients, which makes it a promising solution to self-interference mitigation (SIM) for the IBFD system. This paper considers a RIS-assisted IBFD (RAIBFD) system where a full-duplex base station (BS), assisted by a co-located RIS, receives and transmits signals on the same frequency band simultaneously. We propose to jointly design the DL precoding matrix and the RIS coefficients to mitigate the strong self-interference before the ADCs of finite bit resolutions. To gauge the performance of the proposed RAIBFD system, we also consider an idealistic RIS-assisted IBFD (I-RAIBFD) system with ADCs of infinite bit resolutions and RIS-assisted FDD (RAFDD) system, and take them as performance benchmarks. The effectiveness of the proposed RAIBFD system is verified by simulation studies, even if the phases of the RIS have only finite resolutions.

Super-sensitivity full-field measurement of structural vibration with an adaptive incoherent optical method

Full-field measurements of structural vibrations have been achieved by incoherent optical methods with video cameras such as digital image correlation and optical flow. In contrast to coherent optical methods such as scanning laser vibrometers, incoherent optical methods are low-cost and easy to set up. However, the sensitivity (minimum measurable displacement) of incoherent optical methods is generally lower than that of coherent optical methods. Typically, the sensitivity of incoherent optical methods is essentially limited by the finite bit depth of the digital camera due to the quantization with round-off errors. Quantitatively, this theoretical sensitivity limit is determined by the bit depth B as δp=1/(2B−1) [pixel] which corresponds to a displacement causing an intensity change of one gray level. Fortunately, natural dithering may be leveraged to overcome the quantization and exceed the sensitivity limit, achieving super-sensitivity.In this work, we first study the mechanism of the sensitivity limit induced by the quantization and the critical limitation of existing super-sensitivity methods in the full-field measurement of structural vibration. Addressing such a limitation, we present a general adaptive weighted averaging method with dithering, achieving super-sensitivity incoherent optical measurement of full-field deformational vibration of flexible structures. Specifically, both spatial and temporal samplings of the pixels are leveraged simultaneously to adaptively identify the motion (deformation) shape of the structure by principal component analysis (PCA) of spatiotemporal pixel intensities. Then, the identified deformation shape is used to perform a deformation-shape-weighted averaging over spatial pixels with dithering, revealing the sub-sensitivity displacement while retaining spatial resolution at full field. Numerical simulations and laboratory experiments are performed for principle explanation and method validation. Particularly, we derive a mathematical model of the minimum measurable vibration amplitude A∗ for the developed super-sensitivity method, which is found to be determined by the number of spatial pixels for averaging Ns and the noise level σn in the imaging system. Thus, this work provides, for the first time, a rigorous quantification of the sensitivity of incoherent optical measurement for structural vibration, and a new quantitative method to achieve super-sensitivity full-field measurement of structural vibration.

Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning

End-to-end learning has become a popular method to optimize a constellation shape of a communication system. When the channel model is differentiable, end-to-end learning can be applied with conventional backpropagation algorithm for optimization of the shape. A variety of optimization algorithms have also been developed for end-to-end learning over a non-differentiable channel model. In this paper, we compare gradient-free optimization method based on the cubature Kalman filter, model-free optimization and backpropagation for end-to-end learning on a fiber-optic channel modeled by the split-step Fourier method. The results indicate that the gradient-free optimization algorithms provide a decent replacement to backpropagation in terms of performance at the expense of computational complexity. Furthermore, the quantization problem of finite bit resolution of the digital-to-analog and analog-to-digital converters is addressed and its impact on geometrically shaped constellations is analysed. Here, the results show that when optimizing a constellation with respect to mutual information, a minimum number of quantization levels is required to achieve shaping gain. For generalized mutual information, the gain is maintained throughout all of the considered quantization levels. Also, the results implied that the autoencoder can adapt the constellation size to the given channel conditions.

Super-sensitivity incoherent optical methods for full-field displacement measurements.

The sensitivity of incoherent optical methods using video cameras (e.g., optical flow and digital image correlation) for full-field displacement measurements, defined by the minimum measurable displacements, is essentially limited by the finite bit depth of the digital camera due to the quantization with round-off error. Quantitatively, the theoretical sensitivity limit is determined by the bit depth B as δp = 1/(2B - 1) [pixel] which corresponds to a displacement causing an intensity change of one gray level. Fortunately, the random noise in the imaging system may be leveraged to perform a natural dithering to overcome the quantization, rendering the possibility of breaking the sensitivity limit. In this work we study such a theoretical sensitivity limit and present a spatiotemporal pixel-averaging method with dithering to achieve super-sensitivity. The numerical simulation results indicate that super-sensitivity can be achieved and is quantitatively determined by the total pixel number N for averaging and the noise level σn as δ p ∗∝(σ n /N)δ p.

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

AbstractWe introduce novel methods for implementing generic quantum information within a scale‐free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically‐regulated information flows through distributed systems of finite sets of binary‐valued Barwise‐Seligman classifiers. Specifically, we construct a functor from the category of cone‐cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis too develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2‐complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah‐Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin‐networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.

Digital Sine Oscillators Based on Higher Order Sigma–Delta Modulation

In this paper, a new class of digital sine oscillators based on higher-order multibit sigma-delta modulation is presented. The samples of a given pure sine signal are quantized, resulting in a pulse-code modulated (PCM) sequence. A numeric condition for oscillation (NCO) based on the PCM signal is disclosed. If the NCO is fulfilled, a system constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> with a finite bit width can be derived and a hardware-efficient practical realization of a digital oscillator based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> -order sigma-delta modulation is possible. The oscillator output sequence is an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> -times differentiated version of the PCM signal and thus the PCM signal can be reconstructed by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> -subsequent accumulations. All signals are absolutely periodic and are represented by binary numbers with well-defined finite bit widths. Initial conditions for the registers of the oscillator are determined. An example for the generation of a high-purity PCM signal with 2048 samples per period is demonstrated. The noise spectrum of this signal is white and the ratio between the mean noise amplitude and the signal amplitude is about one over one billion. The resulting oscillator requires a third order sigma-delta modulator. The chip size is estimated for the oscillator being integrated in a commonly used CMOS technology.

Graph Signal Compression by Joint Quantization and Sampling

Graph signals arise in various applications, ranging from sensor networks to social media data. The high-dimensional nature of these signals implies that they often need to be compressed in order to be stored and transmitted. The common framework for graph signal compression is based on sampling, resulting in a set of continuous-amplitude samples, which in turn have to be quantized into a finite bit representation. In this work, we study the joint design of graph signal sampling along with quantization, for graph signal compression. We focus on bandlimited graph signals, and show that the compression problem can be represented as a task-based quantization setup, in which the task is to recover the spectrum of the signal. Based on this equivalence, we propose a joint design of the sampling and recovery mechanisms for a fixed quantization mapping, and present an iterative algorithm for dividing the available bit budget among the discretized samples. Furthermore, we show how the proposed approach can be realized using graph filters combining elements corresponding the neighbouring nodes of the graph, thus facilitating distributed implementation at reduced complexity. Our numerical evaluations on both synthetic and real world data shows that the joint sampling and quantization method yields a compact finite bit representation of high-dimensional graph signals, which allows reconstruction of the original signal with accuracy within a small gap of that achievable with infinite resolution quantizers.

Modular Computation of Restoration Entropy for Networks of Systems: A Dissipativity Approach

The problem of state estimation based on information received over a finite bit rate channel gives rise to the study of minimal bit rate above which state can be estimated with any desired accuracy. In the past few years, researchers have studied the minimal average bit rate which is sufficient enough for state estimation such that the estimation error stays within a given factor of its initial value. The notion of restoration entropy characterizes this type of bit rate. Recent results proposed numerical schemes to estimate restoration entropy by the computation of singular values of the linearized systems. Such schemes are either complex to implement or suffer severely from computational complexity and the size of the state dimension. In this paper, we describe a modular approach to compute an upper bound of the restoration entropy of a large network by decomposing the network to an interconnection of smaller subsystems. Then, we formulate a distributed optimization problem which is solved for each subsystem separately and then their optimization results are composed to get an upper bound of the restoration entropy for the overall network. We illustrate the effectiveness of our results by two examples.

Qualifications of Instruction Sequence Failures, Faults and Defects: Dormant, Effective, Detected, Temporary, and Permanent

Starting out from the survey of instruction sequence faults from [6] program faults are classified according to the conventional criteria of being dormant, effective, detected, temporary, and permanent. Being retrospectively approved is introduced as an additional qualification. For this theoretical investigation the context is simplified by contemplating instruction sequences as a theoretical model for programs, and by assuming that instruction sequences are supposed to compute total transformations on finite bit sequences of a fixed length only. The main conclusion which can be drawn from this work concerns the notion of dormancy. First of all it is noticed that the unconventional notion of a dormant failure is both plausible and amenable to a straightforward and convincing definition. The conventional notion of a dormant fault, however, is much harder to grasp and the definition of a dormant fault which is provided in the paper may be disputed. The notion of a dormant fault seems to admit no convincing intuition. All faults are defects but not the other way around. The idea of a fault exclusively depends on an instruction sequence and a specification of which it is considered to be a candidate implementation. In the presence of a design, however, in addition to faults, the notion of a deviation from design (DFD) defect arises, which constitutes a class of defects many of which are not faults. For DFD defects the notion of dormancy admits a straightforward and convincing definition.

Development of a method for synthesis the FIR filters with a cascade structure based on genetic algorithm

The object of research is the process of digital signal processing. The subject of research is methods of synthesis of digital filters with a finite impulse response based on a genetic algorithm. Digital filtering is one of the tasks of digital signal processing. FIR filters are always stable and provide a constant group delay. There are various methods for synthesizing digital filters, but they are all aimed at synthesizing filters with a direct structure. One of the most problematic areas of a digital filter with a direct structure in digital processing is the high sensitivity of the filter characteristics to inaccuracies in setting the filter coefficients. Genetic algorithm-based filter synthesis methods use an ideal filter as the approximated filter. This approach has a number of disadvantages: it complicates the search for an optimal solution; computation time increases. The study used random search method, which is the basis of genetic algorithm (used for solving optimization problems); theory of digital filtering in filter analysis; numerical methods for modeling in a Python program. Prepared synthesis method FIR filter with the cascade structure, which is less sensitive to the effect of finite bit width. Computation time was reduced. This is due to the fact that the proposed method searches for the most suitable filter coefficients based on a genetic algorithm and has a number of features, in particular, it is proposed to use a piecewise-linear function as an approximated amplitude-frequency response. This makes it possible to reduce the number of populations of the genetic algorithm when searching for a solution. The synthesis of an FIR filter with a cascade structure based on a genetic algorithm showed that for a 24-order filter it took about 30–40 generations to get the filter parameters close to the optimal values. In comparison with classical methods of filter synthesis, the following advantages are provided: calculations of the coefficients of a filter with a cascade structure directly, the possibility of optimizing coefficients with limited bit depth.

An event-triggered consensus protocol for quantized second-order multi-agent systems with network delay and process noise

This paper focuses on the quantized consensus problem of discrete-time second-order multi-agent systems which suffer from bounded process noise. In such systems, neighboring agents exchange information through a digital network that has only a finite bit rate and is subject to bounded network delay. Network delay may exacerbate the effects of quantization error and degrade the control performance by reducing the information to be extracted from receiving time instants of exchange packets. To handle network delay and save communication bit rates, an event-triggered distributed control protocol based on a dynamic encoding–decoding policy is proposed. Under the proposed event-triggered protocol, the asymptotic consensus of the concerned noise-free multi-agent systems can be achieved at a lower bit rate than that of periodic sampling control. This bit rate superiority results from the fact that our protocol makes use of the extra information being extracted from the event-triggering time instants. Moreover, it is shown that no higher bit rate is required to guarantee the desired consensus in the existence of bounded process noise. Simulations are done to demonstrate the effectiveness of our consensus protocol. As shown by our numerical examples, the occupied bit rate to ensure the desired consensus can be saved by up to 50% compared with periodic sampling strategies.

Event‐triggered feedback stabilisation of switched linear systems under finite bit rates

This study considers to asymptotically stabilise a switched linear system which transmits its feedback signals through a digital communication network. Such network can only provide a finite bit rate to represent feedback signals. In the concerned switched linear system, the mode mismatch may occur between the sensor and the controller due to the unknown switching time instants. It assumes that the mode switching is slow enough in the sense of combined dwell time and average dwell time, each individual mode is stabilisable and the available feedback bit rate is high enough, but still finite. In order to save network bandwidth resources, novel event-triggering strategies are implemented, i.e. only when pre-defined events are triggered, the system samples and transmits feedback signals. Then under the proposed event-triggering strategies, a stabilising bit rate condition for the concerned switched linear system is derived. Moreover, this study proves that the obtained stabilising bit rate under the proposed event-triggering strategies can be strictly lower than the stabilising bit rate required by the earlier work based on periodic sampling. Simulations are done to illustrate the bit rate superiority of the proposed event-triggering strategies.

Necessary and Sufficient Bit Rate Conditions to Stabilize a Scalar Continuous-Time LTI System Based on Event Triggering

This paper considers a scalar continuous-time linear time-invariant system, whose feedback signal is transmitted through a communication network. Such a network has only finite bit rate and suffers from transmission delay which is characterized by both lower and upper delay bounds. The concerned system implements event-triggering strategies, i.e., only when certain events are triggered, the system samples and transmits feedback signals. This paper first derives some lower bounds on the feedback bit rate required to achieve the input-to-state stability under arbitrary event-triggering strategies. Then this paper proposes some constructive methods to design the event-triggering strategy and the controller, and can achieve the input-to-state stability at a bit rate being arbitrarily close to these obtained lower bit rate bounds. Moreover, this paper proves that the stabilizing bit rates under the proposed event-triggering strategies can be strictly lower than the stabilizing bit rate required by any time-triggering strategy. This bit rate superiority comes from the fact that under event triggering, the state information can be freely extracted from the receive time instants of feedback packets without consuming any bit rate. Simulations are done to demonstrate the bit rate superiority of the proposed event-triggering strategies.

Bit rate conditions to stabilize an uncertain scalar continuous‐time linear system with clock offset

SummaryThis paper aims to stabilize a scalar continuous‐time linear system that transmits its feedback information through a digital communication network. The finite bit rate of the feedback network plays a critical role in the input‐to‐state stability of that system. Stabilizing bit rate conditions depend on not only the system matrix but also the uncertainty of the system matrix and the unknown clock offset between the transmitter (sensor) and the receiver (controller). Compared with the current literature, this paper can handle the system matrix uncertainty and clock offset through implementing an event‐triggered sampling strategy and derive stabilizing conditions that require lower bit rates than those of periodic sampling strategies. Such bit rate saving mainly comes from the fact that our event‐triggered strategy can extract state information from the receive time instants of feedback packets without any extra communication cost. Simulations are done to confirm the effectiveness of the obtained stabilizing bit rate conditions.

Bit-Rate Conditions for the Consensus of Quantized Multiagent Systems With Network-Induced Delays Based on Event Triggering.

This paper is mainly concerned with bit-rate conditions to guarantee the consensus of unstable scalar multiagent systems by event-triggering strategies. In such systems, state information is quantized and transmitted among agents through digital communication networks which suffer from network-induced delays. In order to save communication resources, we implement a periodic event-triggering scheme, under which agents quantize and transmit their own states when a predefined event is triggered. By introducing a well-designed internal saturation function, the proposed strategy can place a priori bounds on the control inputs of agents and guarantee the asymptotic consensus of agents at a finite bit rate, even under network-induced delays. By extracting extra information from the receive time instants of data packets, our strategy can require less information carried by data packets and a lower bit rate to maintain the desired asymptotic consensus than some state-of-the-art time-triggered consensus strategies. Under our strategy, the obtained bit-rate conditions depend on the network-induced delays, the unstable eigenvalue of agent dynamics, and the network topology. The simulation results are provided to illustrate the effectiveness of these bit-rate conditions.

Robust Quantized Consensus of Discrete Multi-Agent Systems in the Input-to-State Sense

In this paper, we consider the quantized consensus problem of multiple discrete-time integrator agents which suffer from additive noise. Due to the limited communication resources, each agent can only exchange quantized information of finite length with its neighbors through unreliable channels, where the transmitted signals may be delayed or even lost. Quantization errors, network delay, and packet loss result in quantization mismatch. To handle the additive noise and quantization mismatch, a quantized consensus protocol is proposed by implementing dynamic encoding and decoding policies at a finite bit rate. Particularly, the proposed consensus protocol introduces an internal saturation function into all the controllers of agents so that the control inputs of neighbor agents can be predicted locally by each agent even under distributed control law and quantization mismatch. Based on such shared control input information, the proposed consensus protocol can guarantee the achievement of the approximate consensus of all agents in the input-to-state sense, i.e., the consensus errors of agents are bounded from above by the sum of a class $\mathcal {K}_\infty $ function of the upper bound of the additive noise and a class $\mathcal {KL}$ function of the upper bound on the initial states of agents. It is shown that such approximate consensus in the input-to-state sense can be guaranteed at an as low bit rate as 1 bit per time step for each agent without requiring any global information, except an upper bound of the number of agents. Two specific situations, namely packet loss, and network delay, are further analyzed with the explicit expressions of quantization mismatch. The simulations are done to confirm the effectiveness of the proposed quantized consensus protocol.

A robust quantized consensus protocol for discrete-time multi-agent systems with additive noise

This paper investigates the consensus problem of multiple discrete-time integrator agents with communication constraints and additive process noise. It proposes a protocol to achieve the approximate consensus of agents over inter-agent communication networks with finite bit rates. Under that protocol, dynamic encoding and decoding algorithms are implemented for each pair of neighboring agents to transmit quantized states at a finite bit rate. With received quantized states of neighboring agents, the control input of each agent is locally computed. Particularly input saturation is introduced into the local controllers of agents and places both lower and upper bounds on the local control inputs of agents. These control input bounds can be known in advance and greatly enhance the robustness of the consensus protocol. Under the proposed protocol, the approximate consensus can be guaranteed at any finite bit rate to encode the states of agents. It is shown that even a single bit per time step is enough for the desired approximate consensus. The additive process noise does not increase the bit rate required for that approximate consensus. Moreover, the proposed consensus protocol can be designed with only an upper bound on the number of agents and is more robust than some previous consensus protocols which may require the global information of the inter-agent network topology, such as the second largest eigenvalue of the Laplacian matrix. Even when some communication links are broken due to communication failure or some nodes leave, the same set of consensus parameters can still robustly guarantee the expected approximate consensus. Simulations are conducted to illustrate the effectiveness of the proposed quantized consensus protocol.

Nonconstant modulus 신호의 적응 등화를 위한 SE-MMA와 QE-MMA 알고리즘 성능 비교

본 논문은 16-QAM과 같은 스펙트럼 효율적인 nonconstant modulus 신호 전송에서 채널에 의한 부호간 간섭을 보상하기 위한 SE-MMA (Signed Error-Multiple Modulus Algorithm)와 QE-MMA (Quantized Error-Multiple Modulus Algorithm) 적응 등화 알고리즘의 성능을 비교하였다. 기존 MMA 적응 등화기의 탭 계수 갱신시 오차 신호가 필요하게 되는데, SE-MMA는 연산량을 줄이기 위해 오차 신호의 극성만을 이용하며, QE-MMA는 오차 신호의 극성에 유한 비트의 2의 승수 양자화 성분까지 고려하게 되므로 이로 인하여 서로 상이한 등화 성능을 갖게 된다. 이들의 성능을 비교하기 위하여 동일한 채널과 환경에서 등화기 출력 성상도, 잔류 isi, 최대 찌그러짐과 MSE, SER을 적용하여 컴퓨터 시뮬레이션을 수행하였으며, 결과 모든 성능 지수에서 QE-MMA가 SE-MMA보다 우월함을 확인하였다. This paper compares the SE-MMA (Signed Error-MMA) and QE-MMA (Quantized Error-MMA) adaptive equalization algorithm in order to compensates the intersymbol interference due to channel in the transmission of spectral efficient nonconstant modulus signal such as 16-QAM. In the currently MMA adaptive equalizer, the error signal is needed for the updating the tap coefficient. The SE-MMA uses the polarity of error signal for reduce the computational operation in that process, the QE-MMA consider the this polarity and finite bit power-of-2 quantized component in that process, so they has different equalization performance. In order to comparing these performance, the computer simulation was performed in the same channel and environment, the output signal constellation of equalizer, residual isi and maximum distortion, MSE, SER were applied. As a result of computer simulation, the QE-MMA have more superior performance than the SE-MMA in every performance index.