## 2 Theory
If the total angular spectrum in the direction \({\bf k}\) is defined as the cost function  
\[J_{2}\ =\ \big{|}P_{\rm p}({\bf k},0)+P_{\rm s}({\bf k},0)\big{|}^{2} \tag{5}\]  
then the optimized strength of the secondary source is  
\[q_{\rm s,2}\ =\ -q_{\rm p}{\rm e}^{-{\rm j}{\bf k}({\bf r}_{\rm s}-{\bf r}_{ \rm p})} \tag{6}\]  
If the receiver is placed sufficiently far away, it can be assumed that \(\,|{\bf r}_{0}-{\bf r}_{\rm p}|\approx|{\bf r}_{0}-{\bf r}_{\rm s}|\,\) and \({\bf k}\) (\({\bf r}_{\rm s}-{\bf r}_{\rm p}\)) \(\approx\)\(k\) (\(\,|{\bf r}_{0}-{\bf r}_{\rm p}|-|{\bf r}_{0}-{\bf r}_{\rm s}|\,\)), then  
\[q_{\rm s,2}\approx-q_{\rm p}{\rm e}^{-{\rm j}{\bf k}(|{\bf r}_{0}-{\bf r}_{ \rm p}|-|{\bf r}_{0}-{\bf r}_{\rm s}|)}\approx q_{\rm s,1}. \tag{7}\]  
The optimized strengths obtained with Equations (2) and (6) are the same, which indicates that the angular spectrum control is equivalent to the performing sound pressure control at the far-field point in the target direction. In practice, the sum of the weighted sound pressures at \(M\) microphones spaced \(d\) apart in the \(z=0\) error plane is used to approximate the integral result in Equation (3) [10]  
\[\overset{\wedge}{P}({\bf k},0)\ =\ \sum_{m\ =\ 1}^{M}p(x_{m},y_{m},0)W(x_{m},y_{m}) \tag{8}\]  
where \(\overset{\wedge}{P}({\bf k},0)\) and \(p(x_{m},y_{m},0)\) are the estimated angular spectrum and the sound pressure at the \(m\)-th microphone located at \((x_{m},y_{m},0)\). \(W(x_{m},y_{m})\) is the weighting factor, which is defined as  
\[W(x_{m},y_{m})\ =\ {\rm e}^{{\rm j}(k_{x}x_{m}+k_{y}y_{m})}. \tag{9}\]If the sum of the weighted sound pressures at the \(M\) microphones is defined as the cost function, the optimized strength of the secondary source can be obtained as [10]  
\[q_{\rm s,3}\ =\ -\Big{(}\mathbf{Z}_{\rm se}{}^{\rm H}\mathbf{W}^{\rm H}\mathbf{W} \mathbf{Z}_{\rm se}\Big{)}^{-1}\mathbf{Z}_{\rm se}{}^{\rm H}\mathbf{W}^{\rm H} \mathbf{W}\mathbf{Z}_{\rm pe}\mathbf{\prime}_{\rm p} \tag{10}\]  
where \(\mathbf{Z}_{\rm se}\) is the transfer function column vector from the secondary source to the microphones; \(\mathbf{W}\) is the row vector composed of weighting factors from each error microphone; and \(\mathbf{Z}_{\rm pe}\) is the transfer function column vector from the primary source to the microphones. Defining the estimated angular spectrum of the primary source and secondary source as \(\overset{\wedge}{P}_{\rm p}(\mathbf{k},0)\) and \(\overset{\wedge}{P}_{\rm s}(\mathbf{k},0)\), the discrepancy between the estimated and accurate value can be formulated as  
\[ \begin{array}{ll}\delta_{\rm p}&=\ \frac{\overset{\wedge}{P}_{\rm p}( \mathbf{k},0)-P_{\rm p}(\mathbf{k},0)}{P_{\rm p}(\mathbf{k},0)}\\ \delta_{\rm s}&=\ \frac{\overset{\wedge}{P}_{\rm s}(\mathbf{k},0)-P_{\rm s}( \mathbf{k},0)}{P_{\rm s}(\mathbf{k},0)}\end{array}  \tag{11}\]  
In the following investigations, \(\mid\delta_{\rm p}\mid\) and \(\mid\delta_{\rm s}\mid\) are defined as the angular spectrum estimation errors of the primary and secondary source, respectively. These errors are calculated using the logarithm base 10 and scaled by a factor of 20. Equation (10) can be written as  
\[q_{\rm s,3}\ =\ -q_{\rm p}{\rm e}^{-\rm j\mathbf{k}(\mathbf{r}_{\rm s}-\mathbf{ r}_{\rm p})}\cdot\frac{1+\delta_{\rm p}}{1+\delta_{\rm s}} \tag{12}\]  
The noise reduction at the far-field point \(\mathbf{r}_{0}\) is  
\[\rm NR\ =\ 20\rm lg\left|\frac{1+\delta_{\rm s}}{\delta_{\rm s}-\delta_{\rm p}}\right| \tag{13}\]  
It is obvious that the noise reduction in the far field is affected by the angular spectrum estimation errors of both the primary and secondary sources. When the estimation errors of the primary and secondary sources are both zero, or the two estimation errors are exactly the same, perfect noise reduction can be achieved.