## 3 Real-Time Interference Cancellation  
To reduce the \"blind time,\" a weakness of MPM-based active noise cancellation, the time to analyze the signal must be reduced. MPM uses some of the sampled information to make various calculations. Various calculations are required, such as performing matrix operations or determining singular values, which require some time, that is, a certain number of samplings. To minimize this sampling, the technique presented in this paper was used. This method assumes that the amplitude, phase, and sampling time step of the initial IEMI are known, and the externally incident IEMI is assumed to be a complex exponential wave. At this time, the complex sinusoidal signal can be expressed as the following Equation. \[y(t)=A_{I}\exp(j(2\pi f_{I}t+\phi_{I})) \tag{7}\]  
Next, the signals produced by the internal active power supply are summed, which has a frequency difference of \(\Delta f\) compared to the original signal. This is expressed as the formula below. \[y_{s}(t)=A_{I}\exp(2\pi f_{I}t+\phi_{I})+A_{ac}\exp(2\pi(f_{I}+\Delta f)t+\phi ^{\prime}) \tag{8}\]  
Figure 5 shows the combined signal when \(A_{ac}\) is 2.5% of \(A_{I}\) and \(\phi_{I}\) = 30\({}^{\circ}\). Figure 4: Active cancellation result using the MPM. The time axis was sampled at 0.001 s, and the active signal was applied at 0.5 s. From this time, it was confirmed that there was no cancellation due to the frequency difference, and it was fluctuating slightly. Since the fundamental frequency was very high compared to the sampling frequency, it could be seen that the section before 0.5 s, where only IEMI, exists looked like DC. Therefore, the sampled signal for the section after 0.5 s is shown as a function of \(\Delta f\). This is expressed as the following formula. \[y_{sp}(t)=A_{ac}\exp(j(2\pi\Delta ft+\phi)) \tag{9}\]  
For this signal, the measured value at \(t_{1}\) s and the measured value at \(t_{2}\) s are compared. \[y_{sp}(t_{1})=A_{ac}\exp(j(2\pi\Delta ft_{1}+\phi)) \tag{10}\]  
\[y_{sp}(t_{2})=A_{ac}\exp(j(2\pi\Delta ft_{2}+\phi)) \tag{11}\]  
At this time, \(t_{2}\) is equal to \(t_{1}+\Delta t\),  
\[y_{sp}(t_{2})=A_{ac}\exp(j(2\pi\Delta f(t_{1}+\Delta t)+\phi)) \tag{12}\]  
\[y_{sp}(t_{2})=A_{ac}\exp(j(2\pi\Delta ft_{1}+2\pi\Delta f\Delta t+\phi)). \tag{13}\]  
Here, use [PERSON]'s formula to expand the measurement at \(t_{2}\) s,  
\[y_{sp}(t_{2})=A_{ac}\Big{[}\cos\Big{(}\Big{(}2\pi\Delta ft_{1}+\phi\Big{)}+2 \pi\Delta f\Delta t\Big{)}+j\sin\Big{(}\Big{(}2\pi\Delta ft_{1}+\phi\Big{)}+2 \pi\Delta f\Delta t\Big{)}\Big{]}. \tag{14}\]  
To simplify the expression, replace \(2\pi\Delta ft_{1}+\phi=A\), \(2\pi\Delta f\Delta t=B\), and then apply the trigonometric formula. \[y_{sp}(t_{2})=A_{ac}[\cos A\cos B-\sin A\sin B+j(\sin A\cos B+\cos A\sin B)] \tag{15}\]  
It is assumed here that the sampling time, \(\Delta t\), is much shorter than \(t_{1}\), and frequency difference, \(\Delta f\), is also less than 100. This also assumes that the generated frequency from the active source is close to the externally measured frequency. From this assumption, \(\Delta f\Delta t\) is close to zero. That means \(B\) has a very small value. The series expansion of the sine and cosine functions is shown below. \[ \begin{array}{l}\mbox{\it sps},\Delta t\ll 1<t_{1},\,\Delta f<100\rightarrow \Delta f\Delta t\approx 0\\ \Rightarrow\sin x=x-\frac{x^{3}}{6}+\cdots,\,\cos x=1-\frac{x^{2}}{2}+\frac{x^ {4}}{24}-\cdots\end{array}  \tag{16}\]  
Figure 5: The synthesized signal at 0.5 s.  
\(\Delta f\Delta t\) is assumed to be close to zero. As mentioned earlier, the substituted \(B\) value above also comes close to zero. This assumption allows approximating the sine function up to the first term and the cosine function up to the second term. The signal sampled at \(t_{2}\) can be arranged using this assumption as follows. \[