KR101101095B1 - An equalizer using alternate adaptation algorithm - Google Patents

Abstract

PURPOSE: An equalizer which uses an alternation adaptive algorithm is provided to improve error performance in a ready state by alternatively performing adaption processes by a DD(Decision Directed) and vsCMA(variable step-size Constant Modulus Algorithm). CONSTITUTION: An initial convergence algorithm and steady state algorithm are operated in order to eliminate ISI(inter-symbol interference) with respect to a received signal. The initial convergence algorithm and steady state algorithm are alternately operated for each algorithm execution cycle. An estimation error is observed in an initial convergence algorithm operation procedure for every execution cycle. A convergence coefficient determining the renewal width of a tap coefficient is renewed according to a reduction of the estimation error. The initial convergence algorithm is corresponded to a CMA(Constant Modulus Algorithm). The steady state algorithm is corresponded to a DD(Decision Directed) algorithm.

Description

Equalizer Using Alternate Adaptation Algorithm

The present invention relates to an equalizer applied to a communication system, and more particularly, to an equalizer to which a performance improvement algorithm of a blind channel equalizer is applied.

The constant modulus algorithm (CMA) is a widely used blind channel equalization algorithm, particularly for blind equalization of received signals in systems using high-bandwidth, high-order quadrature amplitude modulation (QAM) modulation. The CMA considers multilevel signal constellations as signal points with a single constant modulus (CM), increasing the reliability of the error estimate and converging in the closed condition of the eye diagram. . However, since the error signal is formed using the CM determined by the statistic of the signal point instead of the original signal point as a reference signal, the error level is large even after the equalizer reaches a completely steady state. Large error levels cause errors in symbol determination by the judging device after equalization. In order to improve the steady-state characteristics of the CMA, a method of converting to a decision-directed (DD) algorithm after initial convergence is reached, that is, a CMA-DD conversion scheme is generally used. The DD algorithm converges to a minimum mean square error in open channel conditions. In other words, it is known that the decision algorithm provides the best performance in terms of convergence speed and bit error probability compared to other soft decision blind equalization algorithms under the condition that the equalization is performed and the eye model is opened.

On the other hand, accurate error estimation by hard decision such as DD under the condition that the eye model of the received signal is closed may result in performance degradation due to inaccuracy of error estimation. Therefore, in order to ensure convergence of the DD algorithm, it is necessary to appropriately select a time point for switching to DD, but capturing such a time point is not easy. Hereinafter, the characteristics of the CMA will be described in detail.

을 비용 함수(cost function)로 하고, 이를 최적화하는 블라인드 등화 알고리즘이다.CMA measures the following CM

Is a cost function, and is a blind equalization algorithm that optimizes this.

(1)

(One)

는 일정 모듈러스이다. CM criterion (1)은 등화기 출력의 크기의 제곱이

로부터 떨어진 정도에 비용을 적용하는 것이다. 그러면 탭 계수 벡터에 대해 비용 함수를 최적화하는 최소 평균 자승(least mean square: LMS) 알고리즘은 다음의 형태를 갖는다.here Silver equalizer output,

Is constant modulus. CM criterion (1) is the square of the magnitude of the equalizer output

The cost is applied to the extent away from it. The least mean square (LMS) algorithm that optimizes the cost function for the tap coefficient vector then takes the form:

(2)

(2)

는 탭 길이가 N인 등화기의 탭 계수 벡터,

는 등화기의 입력 데이터 벡터

의 복소 공액(complex conjugate)을 나타낸다.

는 벡터의 전치를 의미한다.

은 양의 수렴상수(step-size parameter)이며,

은 등화기 출력으로,From the stomach

Is the tap coefficient vector of the equalizer with tap length N,

Input data vector of equalizer

Represents a complex conjugate of.

Is the transpose of the vector.

Is a positive step-size parameter,

Is the equalizer output,

(3)

(3)

는 신호점의 통계적 기대값에 의해 결정되는 상수로서to be. From (2)

Is a constant determined by the statistical expected value of a signal point

(4)

(4)

는 통계적 기대값,

은 송신 심벌을 나타낸다.Is defined as

Is the statistical expectation,

Denotes a transmission symbol.

을Error signal of CMA from (2)

of

(5)

(5)

와 등화기 출력 크기의 제곱

과의 차이를 이용하여 오차 신호를 발생시킨다. Can be written as As can be seen in equation (4), the CMA is constant modulus

And squared equalizer output size

An error signal is generated using the difference between and.

이므로, CM 척도는 등화기 출력을

를 반경으로 하는 원주 상에 놓이도록 함을 알 수 있다. 그러나

는 원래 신호점과는 다른, 전혀 새로운 형태의 등가 신호점이므로 원래 신호점과의 불일치로 인하여 CMA 등화기가 완전히 수렴에 도달한 상태에 이르러도 오차가 영이 되지 않는 결과가 불가피하다.From the tap coefficient update equation (2) of the CMA, the tap coefficient is no longer updated when full equalization is reached. Therefore, assuming that the error signal is zero at steady state

CM scales the equalizer output

It can be seen that it lies on the circumference of which radius is. But

Since is a completely new type of equivalent signal point, which is different from the original signal point, the error does not become zero even when the CMA equalizer reaches full convergence due to inconsistency with the original signal point.

As such, the constant modulus algorithm (CMA) blind channel equalization algorithm, in which the error remains at a considerable level, is error-prone because there is an error in symbol determination.

Accordingly, an object of the present invention is to provide a blind channel equalizer using a new algorithm that can improve the convergence characteristic of an initial state and the error performance of a steady state.

The present invention proposes an effective method of combining with DD without determining the time of DD conversion, which is a very sensitive factor in performance variation in CMA-DD conversion. In the proposed scheme, a variable convergence constant vsCMA (variable step-size CMA) and the DD algorithm are combined by alternate adaptation. The proposed alternating adaptation algorithm improves the steady-state performance of the CMA and has the ability to recover the phase rotation for any phase rotation of the received signal by the channel.

The equalizer provided in the receiving end of a communication system,

As an algorithm for eliminating intersymbol interference (ISI) with respect to a received signal, an equalizer using an alternating adaptive algorithm, which performs an initial convergence algorithm and a steady state algorithm at the same time, alternately performs the same, may be provided.

The term "initial convergence algorithm" is a term encompassing an algorithm having excellent blind convergence characteristics, and the term "steady state algorithm" is a term encompassing an algorithm having excellent steady state error performance. The latter example may include, but is not limited to, DD, examples of which are listed in the following description.

In addition, in the equalizer, when performing the initial convergence algorithm, a convergence constant that determines the update width of the tap coefficient is observed at every execution period, and convergence is estimated when the estimation error is small. It is possible to provide an equalizer using an alternating adaptation algorithm, characterized in that to perform an update that reduces the constant.

In addition, the present invention, in the equalizer, the initial convergence algorithm is a constant modulus algorithm (CMA), the steady state algorithm can provide an equalizer using an alternating adaptation algorithm, characterized in that the DD (decision-directed) algorithm. .

In addition, the present invention, in the equalizer,

은 아래 식에 의해 갱신되는 것을 특징으로 하는 교번 적응 알고리즘을 사용한 등화기를 제공할 수 있다.The convergence constant

May provide an equalizer using an alternating adaptation algorithm, which is updated by the following equation.

는 수렴상수 갱신의 속도를 조절하기 위한 상수로

이고,here

Is a constant to control the speed of convergence constant update.

ego,

는 윈도우 길이가 L인 이동 평균으로, 아래와 같이 주어지고,

Is a moving average of window length L, given by

는 추정 오차에 대한 제곱으로

이고,

Is the square of the estimation error

ego,

은 등화기 출력이고,

는 일정 모듈러스로,

Is the equalizer output,

Is the schedule modulus,

이고,

ego,

는 통계적 기대값,

은 송신 심벌이다.

Is the statistical expectation,

Is the transmit symbol.

In addition, the present invention, in the equalizer,

The tap coefficient update expression of the initial convergence algorithm is calculated by the following equation,

The tap coefficient update equation of the steady state algorithm may provide an equalizer using an alternating adaptive algorithm, which is calculated by the following equation.

은, In the above, the error signal

silver,

이고,

ego,

는 등화기의 입력 데이터 벡터

Input data vector of equalizer

의 복소 공액(complex conjugate)을 나타내고,

Represents a complex conjugate of

는 Error signal of steady state algorithm

Is

이다.

to be.

는 임계 판정 장치에 의한 심벌 판정을 나타낸다.here

Denotes symbol determination by the threshold determination apparatus.

In the present invention, a new blind equalization concept called alternating adaptive equalization is proposed, and in particular, a vsCMA-DD alternating adaptive equalization algorithm which alternates between adaptation by vsCMA and adaptation by DD is proposed. The proposed alternating adaptive equalization improves the steady-state error level of the CMA by vsCMA with variable step size, and by combining it gently with DD by alternating adaptation, it not only converges to the steady state quickly but also shows the error performance in the steady state. It can be improved. Therefore, the proposed algorithm is used instead of the CMA-DD conversion scheme when applied to blind equalization for higher-order QAM signals, which can effectively reduce the symbol error rate of the system.

1 is a graph comparing conventional CMA for 16-QAM signal with SER performance according to embodiments of the present invention.
2 and 3 are graphs comparing conventional CMA for 16-QAM signal with MSE performance according to embodiments of the present invention.
4 and 5 are constellations of a conventional CMA and a 16-QAM signal according to embodiments of the present invention.

Hereinafter, preferred embodiments of the present invention will be described in detail.

1. Convergence Constant Variable Algorithm

In general, as equalization proceeds, as the eye model is opened, the error level decreases, and when the steady state is reached, the error signal is theoretically zero. Thus, the tap coefficient increment term becomes zero, leaving the equalizer no longer updated. However, in the CMA, which is an initial convergence algorithm, the actual signal point and the reference signal point are different from each other, especially for higher-order QAM signal points, so that the error signal does not become zero in the steady state. This causes the CMA to converge to a certain level of error in steady state.

We therefore investigated other possibilities for the tap coefficient increment term of the CMA to be zero at steady state. One way would be to reduce the convergence constant to near zero at steady state. By bringing the convergence constant close to zero, the tap coefficient increment term approaches zero, which causes the tap coefficient to be updated very small or almost stops tap coefficient update. As a result, the steady state error level can be improved.

In the proposed method, the estimation error was used as a criterion for determining whether to change the convergence constant at each execution cycle in order to gradually reduce the convergence constant in the steady state. In other words, when the estimation error is observed and the trend decreases, the convergence constant is changed according to the ratio of the estimation error. At this time, in order to increase the reliability of observation, the moving average of the estimation error was taken as the observation target. In summary, the convergence constant in the nth iteration is changed by the following equation.

(6)

(6)

는 수렴상수 갱신의 속도를 조절하기 위한 상수로

이다.here

Is a constant to control the speed of convergence constant update.

to be.

는 윈도우 길이가 L인 이동 평균이며, 다음으로 주어진다.

Is a moving average of window length L, which is given by

(7)

(7)

는 추정 오차에 대한 제곱으로

이다.

Is the square of the estimation error

to be.

The logarithm of the moving average of the estimation error in Eq. (6) is intended to mitigate the large fluctuation range of the estimation error and the speed of the convergence constant variable. The tap coefficient update expression in vsCMA with the proposed variable convergence constant is

(8)

(8)

Therefore, the proposed convergence constant variable algorithm can sufficiently reduce the error level of CMA by bringing the convergence constant to zero in steady state.

2. Alternate Adaptation Algorithm (AAA)

The error level of CMA in steady state is greatly reduced by vsCMA, but in general, it is possible to use the well-known fact that hard decision guarantees the minimum mean square error under open eye model, and also facilitates the combination of vsCMA and DD. To this end, we propose an alternate adaptive vsCMA-DD algorithm in which vsCMA and DD operate in alternating fashion.

Unlike the CMA-DD conversion method, the alternating adaptation algorithm does not completely convert to DD. Instead, the vsCMA and DD algorithms complement each other because vsCMA and DD alternately adapt the equalizer. Therefore, in the proposed alternating adaptation method, there is no problem according to the switching time point experienced in the CMA-DD conversion method.

The alternating adaptive equalizer alternates vsCMA and DD. In vsCMA adaptation mode, the error signal applies (5) and in DD adaptation mode, the error signal applies (9).

(9)

(9)

는 임계 판정 장치에 의한 심벌 판정을 나타낸다. 교번 적응 등화에서 vsCMA에 의한 탭 계수 갱신은 (8)을, DD에 의한 탭 계수 갱신은 다음 식을 따른다. here

Denotes symbol determination by the threshold determination apparatus. In the alternating adaptive equalization, tap coefficient update by vsCMA is (8), and tap coefficient update by DD is given by the following equation.

(10)

10

Since the proposed adaptive equalization adapts the equalizer using two algorithms alternately, the performance of the alternating adaptive equalization will eventually be a compromise of the performance of the two algorithms, so the vsCMA and DD will reach the average of the performance that can be reached. However, in the adaptive adaptive vsCMA-DD equalization, the steady state error level of the CMA is sufficiently lowered by applying the vsCMA first, and this characteristic also contributes to the smooth coupling with the DD by the alternate adaptation. Therefore, the proposed alternating adaptation method improves the steady-state characteristics without degrading the initial blind convergence characteristics of the CMA. In addition, as a result of combining CMA with DD by alternating adaptation, an arbitrary phase rotation of a received signal point due to channel characteristics is recovered even without a carrier phase recovery circuit.

3. Simulation and Analysis

에서

의 조건에 대해 심벌 오율(symbol error rate: SER)과 평균 제곱 오차(mean square error: MSE) 성능 및 수신 신호의 성상도(scatter diagram)를 분석하였다. Signal-to-noise ratio under multipath propagation channels and additive noise for 16-QAM signal points

in

The symbol error rate (SER), mean square error (MSE) performance, and scatter diagram of the received signal were analyzed for the conditions of.

를 사용하였다. 심벌당 신호 대 잡음비는 비트당 신호 대 잡음비

를 사용하여 나타내면,

가 된다. 여기서 k는 심벌당 비트수이다. The signal-to-noise ratio (SNR) is the normalized signal-to-noise ratio, or SNR per symbol.

Was used. Signal-to-noise ratio per symbol is the signal-to-noise ratio per bit

If you use

Becomes Where k is the number of bits per symbol.

는 다음으로 계산하였다.In performance evaluation

Was calculated as:

(11)

(11)

는 송신 심벌에 대한 추정

으로 대체될 수 있다. 여기서

는 등화기 출력

에 대응하는 송신 심벌로, 채널과 등화기에 의한 지연

가 고려되었다. 본 모의실험 조건에서는 채널 임펄스 응답에서 3 심벌 지연을 갖는 프리고스트(preghost)와 11 심벌 지연의 등화기 중심 탭으로 인하여 총 14 심벌 지연을 고려하였다.This is an MSE using transmit symbols.

Is estimated for the transmitted symbol

Can be replaced by here

Equalizer output

Is the transmission symbol corresponding to, the delay by the channel and equalizer

Was considered. In this simulation condition, a total of 14 symbol delays are considered due to the preghost with 3 symbol delay and 11 equalizer center taps in the channel impulse response.

, 교번 적응 알고리즘(AAA)에서는 vsCMA에서 초기값을

으로, DD에서

으로 설정하였다.

는

으로 하였다. MA 윈도우 길이는 이동평균의 응답 속도와 성능을 고려하여 L=2로 하였다.The number of taps of the equalizer was N = 23 for the CMA, vsCMA, and DD algorithms. The convergence constant is the CMA

In the alternate adaptation algorithm (AAA), the initial value is determined by vsCMA.

, From DD

Set to.

Is

It was made. The MA window length was L = 2 considering the response speed and performance of the moving average.

The channel used in the simulation produces phase rotation, so it is necessary to correct it for proper performance evaluation. In particular, since the transmission symbol corresponding to the equalizer output is used to obtain the SER and MSE performance (11), the phase rotation by the channel must be corrected in advance in the CMA. To this end, CMA uses a phase tracking loop after equalization. On the other hand, the phase tracking circuit is unnecessary because the proposed method restores phase rotation by alternating adaptation.

실행주기(iterations) 이후의 심벌을 조사하였다. 아울러 등화 알고리즘이 도달할 수 있는 최선의 성능, 즉 다중경로 채널 영향이 없는 조건에서의 SER 성능을 가늠하기 위해 AWGN(additive white Gaussian noise) 채널에 대한 SER성능을 함께 제시하였다. 도 1에서 AAA-1은 CMA와 DD의 교번 적응 알고리즘이며, AAA-2는 vsCMA와 DD의 교번 적응 알고리즘을 나타낸다. First, the SER performance was evaluated to find the probability of reception error at the equalizer output. To find the SER with enough equalizer

The symbols after the iterations were examined. In addition, the SER performance for the additive white Gaussian noise (AWGN) channel is presented to estimate the best performance that the equalization algorithm can achieve. In FIG. 1, AAA-1 is an alternating adaptive algorithm of CMA and DD, and AAA-2 is an alternating adaptation algorithm of vsCMA and DD.

SER에서 CMA보다 요구

를 8dB정도, 그리고 CMA를 적용한 AAA-1은 CMA에 비해 6dB정도 개선하였다. 한편 CMA는

근처에서

가 증가하여도 더 이상 SER의 개선이 없는 오류 마루(error floor)에 도달하는 반면, 교번 적응 알고리즘은

이 커짐에 따라 지속적으로 성능 개선을 보이고 있다. 특히 vsCMA를 적용한 AAA-2의 경우

가 증가함에 따라 SER이 급격히 낮아져서 CMA나 AAA-1에 비해 상당한 개선을 나타내었다. 아울러 AAA-2는 AWGN 채널의 SER에 대해 3 내지 4dB 이내의 차이를 유지하고 있다.Alternate Adaptive Algorithm AAA-2 with vsCMA

SER demands more than CMA

The AAA-1, which is about 8dB and the CMA, is improved by 6dB compared to the CMA. CMA

Near

Increases to reach an error floor where there is no further improvement in SER, while the alternating adaptive algorithm

As it grows, performance continues to improve. Especially for AAA-2 with vsCMA

As increased, the SER drastically lowered, indicating a significant improvement over CMA or AAA-1. The AAA-2 also maintains a difference of less than 3 to 4 dB for the SER of the AWGN channel.

이하의 SER을 얻기 위한 요구

는 AAA-2의 경우 22dB, CMA의 경우 30dB이상이므로

와

의 두 조건에서 평가하였다.Next, the MSE after equalization was examined to evaluate the performance by reflecting the noise extended by the equalization. As seen in the previous SER performance,

Request to get the following SER

Is 22dB for AAA-2 and 30dB for CMA.

Wow

Was evaluated under two conditions.

조건에서 16-QAM 신호점에 대해 100회의 독립적인 수행을 통해 얻어진 MSE 결과를 나타내었다. 1회의 수행에

심벌을 처리하였으며, MSE의 급격한 변동을 완화하기 위해 32 탭 길이의 이동 평균 필터(moving average filter)를 거쳤다. 2

MSE results obtained from 100 independent runs of 16-QAM signal points under conditions were shown. In one performance

The symbol was processed, and a 32 average tap moving average filter was used to mitigate MSE fluctuations.

In the simulation, the convergence constants of the three algorithms, CMA, AAA-1, and AAA-2, were set identical to each other. In steady state performance, two alternating adaptive algorithms AAA-1 and AAA-2 improved the performance of CMA, and AAA-2 with vsCMA showed better performance than AAA-1 with CMA. AAA-2 showed a 3dB MSE improvement over CMA.

조건에서의 실험 결과를 제시하였다. 도 2의 결과에서와 마찬가지로 vsCMA를 적용한 교번 적응 알고리즘 AAA-2는 CMA에 비해 MSE 성능을 5dB 이상 개선하였고, AAA-1에 비해 2dB 이상 향상되었다.In Figure 3

The experimental results at the conditions are shown. As in the result of FIG. 2, the alternating adaptive algorithm AAA-2 using vsCMA improved MSE performance by 5 dB or more compared to CMA, and improved 2 dB or more compared to AAA-1.

와

에서 16-QAM 신호점에 대한 성상도(scatter diagram)를 나타내었다. 도 4 및 도 5에서는 완전한 수렴에 도달한

실행주기 (iterations) 이후의 데이터를 제시하였다. 도 4와 도 5에서 (a)는 등화 전, 즉 등화기의 수신 신호점, (b)는 CMA에 의해 등화된 결과, (c)와 (d)는 각각 AAA-1과 AAA-2 교번 적응 등화의 결과이다. (a)의 등화 전 수신 신호는 채널에 의해 위상 회전되고 넓은 범위에 퍼진 성상도(scatter diagram)을 보이고 있으며, (b)의 CMA의 경우는 눈모형은 어느 정도 열렸으나 위상 복원이 필요한 상태로 수렴하였다. (c)와 (d)의 AAA-1과 AAA-2 교번 적응 알고리즘의 경우에는 눈모형이 더욱 열린 심호점을 형성하였으며, 별도의 위상 복원 과정 없이 위상 복원된 신호점을 보이고 있다. 두 조건에서 vsCMA를 적용한 AAA-2가 가장 뚜렷하게 모아진 성상도(scatter diagram)를 형성하였다.Finally in FIGS. 4 and 5, respectively

Wow

The scatter diagram for the 16-QAM signal point is shown in. 4 and 5 the complete convergence is reached

Data after the iterations is presented. In Figures 4 and 5, (a) is the equalization signal, i.e. the received signal point of the equalizer, (b) is equalized by CMA, and (c) and (d) are AAA-1 and AAA-2 alternating adaptations, respectively. The result of equalization. The received signal before equalization in (a) is phase rotated by the channel and shows a scatter diagram spreading over a wide range.In the case of CMA of (b), the eye model is open to some extent but converges to a state requiring phase recovery. It was. In the case of the AAA-1 and AAA-2 alternating adaptive algorithms of (c) and (d), the eye model formed a deeper open point, and the signal point was recovered without phase reconstruction. Under both conditions, AAA-2 with vsCMA formed the most distinct scatter diagram.

The simulation results show that the proposed algorithm can sufficiently improve the steady-state error performance without degrading the convergence characteristics of the CMA by varying the convergence constant of the CMA and combining it with the DD by alternating adaptation. In addition, the proposed scheme avoids the problems of the CMA-DD switching scheme and has the ability to recover any phase rotation of the received signal generated by the channel.

In the embodiment of the present invention, the CMA is used as an example of an initial convergence algorithm, and the decision basis (DD) is used as an example of a steady state algorithm. However, the initial convergence algorithm is a Sato algorithm, a generalized Sato algorithm, a Reduced Constellation Algorithm (RCA), You can use any one of Stop-and-Go Algorithm, Stop-Caution-Go Algorithm, Godard Algorithm, Constant Modulus Algorithm (CMA), Modified CMA (Modified CMA), Multilevel Modulus Algorithm (MMA). Decision-Directed (DD) Algorithm, Hard Decision-Directed (HDD) Algorithm, Generalized Decision-Directed (GDD) Algorithm, Soft Decision-Directed (SDD) Algorithm, Gaussian Clustering Algorithm (GCA), MAP Equalizer (Maximum a Posteriori Probability Equalizer), Dual-Mode Algorithm, Concurrent Equalizer, Decision Feedback E qualizer: DFE) can be used.

In other words, applying the alternating adaptive algorithm may alternately perform the initial convergence algorithm and the steady state algorithm for each one of the algorithms and the steady state algorithms listed above. The equalizer can be implemented.

The rights of the present invention are not limited to the embodiments described above, but are defined by the claims, and those skilled in the art can make various modifications and adaptations within the scope of the claims. It is self-evident.

No reference sign

Claims (5)

In the equalizer installed at the receiving end of the communication system,
An algorithm for eliminating intersymbol interference (ISI) for a received signal, wherein the initial convergence algorithm and the steady state algorithm are simultaneously executed, and the initial convergence algorithm and the steady state algorithm are alternately performed at every iteration of the algorithm. Equalizer using alternating adaptive algorithm. The method according to claim 1, wherein when performing the initial convergence algorithm, the convergence constant that determines the update width of the tap coefficients is observed at every execution period, and the update that reduces the convergence constant when the estimation error becomes small is performed. An equalizer using an alternating adaptive algorithm, characterized in that it executes. The equalizer of claim 1, wherein the initial convergence algorithm is a constant modulus algorithm (CMA) and the steady state algorithm is a decision-directed (DD) algorithm. The method of claim 2,
The convergence constant

The equalizer using the alternating adaptive algorithm, characterized in that is updated by the following equation.

here

Is a constant to control the speed of convergence constant update.

ego,

Is a moving average of window length L, given by

Is the square of the estimation error

ego,

Is the equalizer output,

Is the schedule modulus,

ego,

Is the statistical expectation,

Is the transmit symbol. The method of claim 4, wherein
The tap coefficient update expression of the initial convergence algorithm is calculated by the following equation,

The tap coefficient update expression of the steady state algorithm is calculated using the following equation.

In the above, the error signal

silver,

ego,

Input data vector of equalizer

Represents a complex conjugate of
Error signal of steady state algorithm

Is

ego,
here

Denotes symbol determination by the threshold determination apparatus.

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