# Convergence Analysis of the Phase-Scheduled-Command FXLMS Algorithm with Phase Error
## 1 Introduction  
In specific acoustic environments, such as the interior of a car, the customer-perceived quality of sound has gradually come to be considered important. It is well known that noise control involves some sounds being tuned to enhance the vehicle sound quality, invoking some desired emotional response, while others should be suppressed to reduce annoyance [1]. Therefore, in such a case, the goal of noise control is, not only to eliminate noise, but to control the noise at a specific level, which is known as sound quality control. With the benefit of the action of a secondary noise source in an active noise control (ANC) system, the amplitude of sound can be tuned to a specific level. Therefore, active sound quality control systems based on ANC algorithms provide the possibility of dealing with the problem by equalizing or profiling the sound [2; 3; 4; 5; 6; 7]. There are two schemes for the active sound quality control system based on the FXLMS algorithm: active noise equalizer (ANE) [8; 9; 10; 11; 12] and active sound profiling (ASP) algorithms [13; 14; 15; 16]. The ANE, first presented by [PERSON] et al. [8; 9], can either enhance or eliminate the sound level by the insertion of gain factors and the balancing branch, ensuring that the actual noise converges to a desirable value, tuned by gain factor \(\beta\). Based on the ANE technique, [PERSON] et al. [17; 18; 19] proposed an optimized filtered-error least mean square (FELMS) scheme, which improved the convergence speed and system stability, and a frequency domain delayless active sound quality control (ASQC) structure which was more efficient, faster and had lower computational complexity compared to time-domain algorithms. In view of this, [PERSON] et al. [20] developed an NEX-LMS-scheme ASQC algorithm by changing the gain factor with the frequency components for the application of equalization of engine noise. The performance of the NEX-LMS scheme was experimentally validated, and it worked effectively when the frequency was slowly increased. Additionally, [PERSON] et al. [21; 22; 23] introduced a frequency domain multichannel ANE scheme for independent sound control of multiple locations, and a system to control the loudness,roughness, sharpness and tonality of noise. The performance of the control schemes was verified through simulations and experiments. The ANE-based algorithms have been employed as a popular and important sound-equalizing method. Their performance has been studied, and it has been concluded that they are extremely sensitive to misestimation of the secondary path. The phase misestimation influence is dominant when \(\beta\) is small, and the amplitude misestimation influence becomes dominant when \(\beta>1\)[13; 24]. To fundamentally prevent the influence of the secondary model misestimation, [PERSON] et al. [13] proposed an alternative active sound profiling (ASP) scheme, which was characterized by converging the residual signal to a generated command value signal. The command value signal was set with the desirable value amplitude and the same frequency as the reference signal. A pseudo-error signal was used to update the adaptive filter weights, and the real residual noise signal converged to the command signal as the pseudo-error converged to zero. Further, to limit the control effort, the command signal was set to be in phase with the primary disturbance signal, and the phase-scheduled-command FXLMS (PSC-FXLMS) sound profiling algorithm can be derived. The command signal is created at the frequency corresponding to the reference signal, which ensures that its frequency components are the same as the disturbance signal. The command signal phase is set by the measured phase by performing a fast Fourier transform (FFT) on the modeled disturbance. However, the modeled disturbance may be influenced by broadband signals, such as road noise, resulting in an error of the measured phase.