# Directional Suppression of Monotone Noises with A Parametric Array Loudspeaker
## 2 Theory and Methods  
### FxLMS Algorithm for Adaptive Noise Filter Design  
The least mean squares (LMS) algorithms are based on the fundamental principles of Wiener filtering following the steepest gradient descent to produce optimized finite impulse response (FIR) filters. The FxLMS algorithm is an iterative adaptation of the LMS method and is now considered the baseline algorithm in ANC, known for its simplicity, low computational cost, and stable convergence [26]. Figure 1a shows a complete diagram of the FxLMS algorithm for the single-channel noise reduction considered in this paper. One key element in the algorithm is determining the secondary path filter \(\mathbf{s}(n)\), which depicts the acoustic path for the projection of counter-noise sound from the secondary source to the target where the error microphone locates. Neglecting the response from the secondary path can result in misjudgment of the optimum noise-canceling signal and affect the algorithm's convergence [8]. Consider an adaptive FIR filter for the ANC control \(\mathbf{w}(n)\); the residual noise measured at the error microphone \(e(n)\) can be expressed as  
\[e(n)=d(n)-[\mathbf{w}(n)\times x(n)]\times\mathbf{s}(n), \tag{1}\]  
where \(x(n)\) is the ambient noise signal and \(d(n)\) is the measured ambient noise signal at the reference microphone through the primary path \(\mathbf{p}(n)\). The LMS constraint requires the cost function \(\xi(n)=e^{2}(n)\) to be minimum by letting \(\
abla\xi(n)\) approach zero. Using the gradient descent method, the iteration for updating \(\mathbf{w}(n)\) is expressed as  
\[\mathbf{w}(n+1)=\mathbf{w}(n)+\frac{\mu}{2}\
abla\xi(n), \tag{2}\]  
where \(\mu\) is the filter learning rate to control the iteration step size. By derivation of \(\
abla\xi(n)\), the above iteration can be expanded as  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu e(n)\mathbf{s}(n)\times x(n). \tag{3}\]  
Since \(\mathbf{s}(n)\) is unknown, \(\mathbf{s}(n)\) is applied as a legitimate estimation of the secondary path. \(x^{\prime}(n)\) represents the output from \(\mathbf{s}(n)\) as  
\[x^{\prime}(n)=\mathbf{s}(n)\times x(n), \tag{4}\]  
which updates Equation (3) into  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu e(n)x^{\prime}(n). \tag{5}\]  
The aim of this paper is focused on the elimination of monotone noises, which can cause more harm to people's mental health in modern urban or industrial environments. Hence, we adopt an offline approach to identify the secondary path for its straightforwardness in acquiring \(\mathbf{s}(n)\). The monotone noise features a concentrated power spectrum to represent an acoustic environment with its spectrum characteristic that does not change over time; so, we also use fixed step sizes in the algorithm for the iteration of the FIR filters. Since offline identification is performed as a standalone process, it does not introduce extra computation complexity into the main ANC algorithm to compromise its robustness. However, if environmental ambient noises such as Gaussian white noise are considered in ANC, the adaptive filter iteration step size should be considered due to the variation in  
Figure 1: Block diagram of the implemented FxLMS algorithm with offline identification of the secondary path \(\mathbf{s}(n)\). (**a**) The complete FxLMS algorithm, where \(\mathbf{w}(n)\) is the adaptive ANC filter constrained by the estimated secondary path \(\mathbf{s}(n)\) and the residual error \(e(n)\). (**b**) The offline identification of the secondary path for estimation of the LMS-constrained filter \(\mathbf{s}(n)\). the noise power spectrum. To conclude, the complete FxLMS algorithm implemented in this paper can be summarized into two steps. The first step is the offline identification of the secondary path following the diagram shown in Figure 1b. By monitoring the ambient noise \(x_{0}(n)\) and the residual error \(c_{0}(n)\) at the target position, an LMS-constrained \(\$(n)\) can be estimated. The second step is to update the adaptive filter \(w(n)\) to achieve ANC following Equation (5) using \(\$(n)\).