# Frequency-Domain Filtered-x LMS Algorithms for Active Noise Control: A Review and New Insights
## 1 Introduction  
Acoustic noise control is essential for reducing the noise level in modern society since noise seriously affects human health [1; 2]. Passive noise control, which is based on using reactive devices, e.g., Helmholtz resonators and quarter wavelength resonators, and using resistive materials, e.g., acoustic linings and porous membranes, is very effective for reducing high-frequency noise, but not so effective for low-frequency noise reduction. Active noise control (ANC) based on the principle of superposition is an appealing method for low-frequency noise reduction [3; 4; 5; 6]. In practice, passive noise control and ANC methods are usually combined to provide wideband noise reduction. In ANC systems, the filtered-x least mean-square (FxLMS) algorithm is widely used to update the weights of the control filter, but the complexity of the FxLMS algorithm increases linearly with the filter length. In certain applications, the control filter is on the order of several thousand. Therefore, the FxLMS and other time-domain algorithms, e.g., the filtered-x affine projection (FxAP) [7; 8; 9], are too complex, which is prohibitive for real-time systems. The frequency-domain adaptivefilter (FDAF) [10; 11; 12; 13; 14; 15; 16; 17] has been successfully used in echo cancellation [18; 19; 20; 21], acoustic feedback cancellation [22], and beamforming [23] due to its good convergence behavior and low complexity. In [24; 25; 26; 27], the block LMS (BLMS) was extended to ANC applications and the corresponding frequency-domain implementation was provided. The experiment in [27] used both the periodic and band-limited signals as the reference signals, but the delay problem was not discussed. In [28], a more efficient implementation of BLMS was presented, where the filtering of the reference signal by the secondary path was also implemented using FFT. Meanwhile, the fast Hartley transform (FHT) was adopted to reduce the complexity. The frequency-domain implementation was extended to the multichannel case in [26; 29]. The aforementioned algorithms are obtained by applying the FDAF originally used in echo cancellation to the ANC problem. However, the algorithm requirements for echo cancellation and ANC are quite different. Specifically, the direct use of the FDAF for ANC introduces two kinds of delays. First, there is at least one-block delay between the input of the reference signal and the output of the cancelling signal because the FDAF algorithm is implemented on a block-by-block basis. This delay can violate the causality constraint, which is a major concern for broadband ANC [4; 30; 31]. Second, the delay between the weight adaptation and the observation of the error signal is introduced because of the effect of the secondary path. The behavior of the filtered-x algorithm is similar to that of the delayed-LMS algorithm, which reduces the upper bound of the step size and slows the convergence and reconvergence rate [32]. In addition, the peak complexity of the FDAF algorithms presented in [28] is quite high because the FFT operation should be completed in one sampling interval; therefore, these algorithms are impractical for real-time implementation. Many approaches have been proposed to reduce the two kinds of delays. The modified filtered-x structure [33; 34] was used in the frequency-domain ANC in [35], which makes the adaptive algorithm behave like the standard FDAF and thus removes the delay in the weight adaptation. The partitioned-block FDAF (PBFDAF) algorithm was introduced for ANC to reduce the delay in the signal path [36; 37; 38; 39]. This is achieved by partitioning the whole impulse response into several small sections, but this method does not completely eliminate the input-output delay. Several delayless algorithms were then proposed to totally remove the delay in the signal path. The calculation of the adaptive filter output can be implemented directly using the time-domain convolution and hence the delay in the signal path is removed [40; 41; 42; 43].