Frequency-domain LMS Algorithm Research Articles

An improved LMS algorithm for mode demultiplexing in frequency domain

In order to solve the problem that the traditional frequency domain least mean square (FD-LMS) algorithm will lose efficacy with the increase of differential mode group delay (DMGD) when the algorithm is used for demultiplexing of the 6×6 mode division multiplexing (MDM) system, an improved FD-LMS demultiplexing algorithm is proposed. By improving the error signal calculation method, the convergence performance of the output signal of the equalization filter is improved, and the steady-state error of the algorithm is reduced. Besides, the equalization performance of the traditional FD-LMS algorithm is compared with the improved FD-LMS algorithm. Simulation results show that the improved FD-LMS algorithm has great advantage over the traditional FD-LMS algorithm in demultiplexing performance on the premise that the computation complexity does not significantly increase. The optical signal to noise ratio (OSNR) penalty of the improved FD-LMS algorithm is 2.6 dB lower than that of traditional FD-LMS algorithm at a transmission distance of 80 km with DMGD is 50 ps/km.

Adaptive Frequency Response Calibration Method for Microphone Arrays

Frequency response mismatch among microphones inevitably leads to performance deterioration of signal processing algorithms of microphone arrays, especially for distributed microphone networks. An adaptive frequency response calibration method for microphone arrays based on frequency domain least mean square (FLMS) algorithm is proposed in this paper. Adaptive calibration filters are first introduced to calibrate the frequency response mismatch among microphones, which are concatenated with microphone channels. Then, the frequency response mismatch problem is formulated with a combined spectrum mean square error criterion of pairwise microphone outputs, and an unit-norm constraint is imposed to avoid undesired solutions. Finally, FLMS algorithm is used to design the calibration filter in each microphone channel. This method achieves good frequency response calibration performance even under severe ambient noise and reverberation conditions. Experimental results reveal the validity of the proposed method.

A Variable Step-Size Unconstrained Adaptive FD-LMS Algorithm for MDM Transmission

Mode-division multiplexing (MDM) over few-mode fibers has been proposed to break through the Shannon limit of the single-mode fiber. Mode coupling and differential mode group delay are two major drawbacks, which limit the performance of the system. In this paper, a variable step-size unconstrained adaptive frequency-domain least mean square (FD-LMS) algorithm is proposed for demultiplexing in a 6 × 6 MDM transmission. In an 80 km link, at the 7% FEC threshold, the least required optical signal noise ratio (OSNR) for the proposed algorithm is about 13.5 dB, which is the same as the constrained FD-LMS algorithm and the unconstrained FD-LMS algorithm. Besides, the proposed algorithm can improve the convergence speed by 45.1% and 56.9% in comparison with the constrained FD-LMS algorithm and the unconstrained FD-LMS algorithm. For the distance of 2000 km, the computational complexity of the proposed algorithm is 52.8% lower than that of the constrained FD-LMS algorithm. The effect of mode-dependent loss (MDL) on the proposed algorithm is also explored. The result shows that the MDL tolerance of the proposed algorithm is similar to the constrained FD-LMS algorithm, and the OSNR penalty of the proposed algorithm is 0.5 dB higher than that of the constrained FD-LMS algorithm at a transmission distance of 800 km.

Fast convergent frequency-domain MIMO equalizer for few-mode fiber communication systems

Space division multiplexing using few-mode fibers has been extensively explored to sustain the continuous traffic growth. In few-mode fiber optical systems, both spatial and polarization modes are exploited to transmit parallel channels, thus increasing the overall capacity. However, signals on spatial channels inevitably suffer from the intrinsic inter-modal coupling and large accumulated differential mode group delay (DMGD), which causes spatial modes de-multiplex even harder. Many research articles have demonstrated that frequency domain adaptive multi-input multi-output (MIMO) equalizer can effectively compensate the DMGD and demultiplex the spatial channels with digital signal processing (DSP). However, the large accumulated DMGD usually requires a large number of training blocks for the initial convergence of adaptive MIMO equalizers, which will decrease the overall system efficiency and even degrade the equalizer performance in fast-changing optical channels. Least mean square (LMS) algorithm is always used in MIMO equalization to dynamically demultiplex the spatial signals. We have proposed to use signal power spectral density (PSD) dependent method and noise PSD directed method to improve the convergence speed of adaptive frequency domain LMS algorithm. We also proposed frequency domain recursive least square (RLS) algorithm to further increase the convergence speed of MIMO equalizer at cost of greater hardware complexity. In this paper, we will compare the hardware complexity and convergence speed of signal PSD dependent and noise power directed algorithms against the conventional frequency domain LMS algorithm. In our numerical study of a three-mode 112 Gbit/s PDM-QPSK optical system with 3000 km transmission, the noise PSD directed and signal PSD dependent methods could improve the convergence speed by 48.3% and 36.1% respectively, at cost of 17.2% and 10.7% higher hardware complexity. We will also compare the frequency domain RLS algorithm against conventional frequency domain LMS algorithm. Our numerical study shows that, in a three-mode 224 Gbit/s PDM-16-QAM system with 3000 km transmission, the RLS algorithm could improve the convergence speed by 53.7% over conventional frequency domain LMS algorithm.

A Step-Size Controlled Method for Fast Convergent Adaptive FD-LMS Algorithm in Few-Mode Fiber Communication Systems

Space-division multiplexing using few-mode fibers (FMF) has emerged as a promising technology to overcome the next capacity crunch. The key challenges of FMF systems are inter-modal crosstalk due to random mode coupling and large differential mode group delay (DMGD). Adaptive frequency domain least mean square (FD-LMS) algorithm has been proposed as the most hardware efficient multi-input multi-output equalization method for compensating large DMGD. Except for hardware complexity, the convergence speed of the adaptive FD-LMS algorithm is another important consideration. In this paper, we propose a noise power spectral density (PSD) directed adaptive FD-LMS algorithm, which adopts variable step size to render the posterior error of each frequency bin convergent to the background noise in an additive white Gaussian noise (AWGN) channel. In a 3000 km six-mode transmission system with 35 ps/km DMGD, compared with signal PSD dependent FD-LMS method and conventional FD-LMS method, our new algorithm could improve the convergence speed by 36.1% and 48.3%, but their hardware complexity will only increase 10.7% and 17.2%, respectively.

An Optimized Combination of Frequency Domain Adaptive Equalizers

In order to overcome the defects of the high computational loads and selecting the threshold of mean square error (MSE) for time domain decision-directed constant modulus blind equalization algorithm (DD+CMA), a frequency domain parallel decision multi-modulus blind equalization algorithm based on frequency domain MMA(FMMA) and frequency domain LMS (FLMS) algorithm is proposed. The proposed algorithm is composed of the FMMA and FLMS, and the FMMA and FLMS run automatically in soft switching parallel manner. In running process, it is not necessary to selecting the threshold of the MSE. Moreover, the computational loads can be reduced by circular convolution in the frequency domain signals instead of linear one of the time domain signals. Simulation results show that performance of the proposed algorithm outperforms the FLMS and the FMMA algorithm.

A better frequency domain adaptive algorithm for active noise control in a short duct

For the feedforward active noise control in a short duct, the noncausality of the whole system is often inevitable, since the reference sensor needs to be placed very close to the control source, and the acoustic transmission delay between them is often surpassed by the inherent AD/DA and anti-aliasing filters latency in the controller. The commonly used normalized frequency domain LMS algorithm possesses the benefit of fast convergence speed, but suffers from deteriorated steady-state performance when used in non-causal circumstances. In this paper, an efficient modification of the normalized frequency domain LMS algorithm is proposed, which can significantly improve wide-band noise reduction level in non-causal circumstances. Both simulations and experiments demonstrate the superiority of the proposed algorithm.

A Frequency Domain LMS Algorithm with Dynamic Selection of Frequency Bins

Frequency-domain (FD) adaptive filter algorithms are able to achieve a low computational complexity by using the overlap-and-save implementation means compared to time-domain (TD) ones. In this article, we propose a new FD least-mean-square (FD-LMS) algorithm which dynamically selects frequency bins in order to reduce the computational complexity and maintain the convergence performance of the conventional FD-LMS. The optimal selection of frequency bins is derived by the largest decrease between the successive FD mean square deviations (MSDs) at every data block. Simulation results show that the proposed algorithm provides a low steady-state normalized MSD (NMSD) and similar convergence rate compared to the conventional FD-LMS algorithm. In addition, it gains a low computational complexity.

Generalized Robust Multichannel Frequency-Domain LMS Algorithms for Blind Channel Identification

Recently, several noise-robust adaptive multichannel LMS algorithms have been proposed based on the spectral flatness of the estimated channel coefficients in the presence of additive noise. In this work, we propose a general form for the algorithms that integrates the existing algorithms into a common framework. Computer simulation results are presented and demonstrate that a new proposed algorithm gives better performance compared to existing algorithms in noisy environments.

Generalized Robust Multichannel Frequency-Domain LMS Algorithms for Blind Channel Identification

Recently, several noise-robust adaptive multichannel LMS algorithms have been proposed based on the spectral flatness of the estimated channel coefficients in the presence of additive noise. In this work, we propose a general form for the algorithms that integrates the existing algorithms into a common framework. Computer simulation results are presented and demonstrate that a new proposed algorithm gives better performance compared to existing algorithms in noisy environments.

Adaptive inverse control of random vibration based on the filtered-X LMS algorithm

Random vibration control is aimed at reproducing the power spectral density (PSD) at specified control points. The classical frequency-spectrum equalization algorithm needs to compute the average of the multiple frequency response functions (FRFs), which lengthens the control loop time in the equalization process. Likewise, the feedback control algorithm has a very slow convergence rate due to the small value of the feedback gain parameter to ensure stability of the system. To overcome these limitations, an adaptive inverse control of random vibrations based on the filtered-X least mean-square (LMS) algorithm is proposed. Furthermore, according to the description and iteration characteristics of random vibration tests in the frequency domain, the frequency domain LMS algorithm is adopted to refine the inverse characteristics of the FRF instead of the traditional time domain LMS algorithm. This inverse characteristic, which is called the impedance function of the system under control, is used to update the drive PSD directly. The test results indicated that in addition to successfully avoiding the instability problem that occurs during the iteration process, the adaptive control strategy minimizes the amount of time needed to obtain a short control loop and achieve equalization.

Variable step-size multichannel frequency-domain LMS algorithm for blind identification of finite impulse response systems

The choice of step size is a critical factor in blind identification of single-input multi-output systems using adaptive multichannel frequency-domain least-mean-squares (MCFLMS) algorithm. The proper step size is dependent on the signal power and it influences the speed of convergence and the final misalignment error. Applying normalised MCFLMS (NMCFLMS) algorithm, the dependency of the step-size parameter on signal power can be eliminated; however, it is shown that even for a moderate signal-to-noise ratio, the algorithm fails to converge to the eigenvector corresponding to the minimum eigenvalue of the data correlation matrix and hence misconverges to a fictitious solution. A self-adaptive variable step-size MCFLMS (VSS-MCFLMS) algorithm that optimises the performance of the algorithm in each iteration in order to achieve minimum misalignment with the true channel impulse response is proposed. The convergence analysis of the proposed algorithm is performed and it is shown that the VSS-MCFLMS algorithm converges, both in noise-free and noisy conditions, to the eigenvector corresponding to the minimum eigenvalue and is therefore more noise robust when compared with the NMCFLMS. The algorithm also shows the optimal speed of convergence in a noise-free case and the proposed variable step size guarantees the stability of the algorithm. Simulation results are presented to demonstrate that the proposed VSS-MCFLMS algorithm gives lower final misalignment when compared with the NMCFLMS algorithm without sacrificing the speed of convergence and computational efficiency.

A computationally efficient frequency-domain LMS algorithm with constraints on the adaptive filter

The frequency domain implementation of the LMS algorithm is attractive due to both the reduced computational complexity and the potential of faster convergence compared with the time domain implementation. Another advantage is the potential of using frequency-domain constraints on the adaptive filter, such as limiting its magnitude response or limiting the power of its output signal. This paper presents a computationally efficient algorithm that allows the incorporation of various frequency domain constraints into the LMS algorithm. A penalty function formulation is used with a steepest descent search to adapt the filter so that it converges to the new constrained minimum. The formulation of the algorithm is derived first, after which the use of some practical constraints with this algorithm and a simulation example for adaptive blind equalization are described.

Generalized sliding FFT and its application to implementation of block LMS adaptive filters

This paper has two contributions. First, the concept of the generalized sliding fast Fourier transform (GSFFT) as an efficient implementation of the hopping FFT is introduced. Application of the GSFFT is broad and not limited to what has been considered in this paper. The frequency domain block LMS (FBLMS) adaptive filters are then revised, and their implementations for block lengths less than the length of the adaptive filter are studied. The GSFFT and the available pruned FFTs are used to give an efficient implementation of these filters. In the particular case of the block length equal to one, where the FBLMS algorithm reduces to the frequency domain LMS (FLMS) algorithm, it is shown that the latter can be implemented with the order of M complexity, where M is the length of the adaptive filter. >

Frequency-domain implementation of Griffiths–Jim adaptive beamformer

It is well known that the drawback of the time-domain least-mean-square (LMS) algorithm proposed by Widrow [Proc. IEEE 63, 719–720 (1975)] in adaptive filter applications is that its convergence speed decreases as the eigenvalue spread of the input autocorrelation matrix increases. However, this problem can be overcome by employing the transform-domain LMS method [Narayan et al., IEEE Trans. ASSP 31, 609–615 (1983)] which first transforms input time-domain signals into another transform-domain signals through DFT or some orthogonal transforms and then uses the self-orthogonalizing algorithm [Gitlin and Magee, IEEE Trans. Commun. 25, 666–672 (1977)] to optimize the variable weights of an adaptive filter. This method has been shown to offer great improvement in convergence rate over the time-domain LMS method. The objective of this paper is to apply the frequency-domain LMS algorithm with the self-orthogonalizing technique, called FLMS, to the Griffiths–Jim adaptive beamformer to accelerate the convergence rate for real-time processing of adaptive array signals. It is shown that the two minimum mean-square errors of the beamformer implemented in the time and frequency domains are identical. Computer simulations show that the adaptive beamformer using the FLMS exhibits faster convergence behavior and better performance of nulling jammers than that using the LMS, especially for the larger eigenvalue spread.

A weighted normalized frequency domain LMS adaptive algorithm

A general filtering scheme is presented for obtaining an input power estimate for setting the convergence parameter mu separately in each frequency bin of a frequency-domain LMS adaptive filter (FDAF) algorithm. A linear filtering operation is performed on the magnitude square of the input data and incorporated directly into the algorithm as a data-dependent time-varying stochastic mu (n). The mean performance of the weighted normalized frequency domain LMS algorithm (WNFDAF) is analyzed using independent and identically distributed Gaussian data, and the results are validated by the Monte Carlo simulations of the algorithm. The simulations are also used to study the weight transient behavior. The simulations suggest that sort smoothing times are sufficient for rapid weight convergence without large fluctuations in the power estimates significantly affecting transient weight behavior.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A fast frequency-domain adaptive algorithm

A time-varying convergence factor mu is proposed for the frequency-domain LMS (least-mean-square) adaptive algorithm, which results in the frequency-domain optimal block algorithm (FOBA). The FOBA is the frequency-domain implementation of the recently proposed optimum block algorithm (OBA). The FOBA results in computational savings in comparison to the OBA and in performance enhancement relative to the frequency-domain LMS algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comparison of several frequency-domain LMS algorithms

In this letter, we investigate and compare several frequencydomain LMS adaptive algorithms. These are Dentino's LMS (DLMS) algorithm, the Fast LMS (FLMS) algorithm, and the proposed Optimal FLMS (OFLMS) algorithm. Computer simulations are carried out to evaluate these algorithms under different conditions. It is shown that the OFLMS produced excellent results, particularly in the case of colored-noise excitation.