## 4 Enhanced LMS Methods with IMC
### Novel Adaptive Filtering Algorithm Based on the IMC 1: Non-Recursive Method
#### 4.1.2 Numerical Validation  
The proposed technique was implemented in numerical simulations and was validated by comparison with the conventional LMS algorithm. An adaptive digital filter with a FIR filter structure has been defined with two coefficients. The target disturbance plant is assumed to be a second order system with a natural frequency of \(\omega_{n}=50\) Hz and a damping ratio of \(\zeta=0.1\). A sinusoidal input signal \(x(t)=A_{1}\sin(\omega_{1}t+\theta_{1})\) with one frequency component at \(\omega_{1}/2\pi=100\) Hz is used as both the input reference and the disturbance plant input (\(A_{1}=1\), \(\theta_{1}=0\)). The \(\mu\) value was chosen as 0.4 for both cases. In the first case, suppose that the input filter is not used in an adaptive filtering system when the secondary path dynamics \(P_{s}(z)\) cannot be neglected. The secondary path is presumed to be a first-order system, with a time constant 0.01, as in Equation (14b), and the exact model is assumed to be available. The internal model controller \(Q(z)\) can be designed as Equation (14c), with the model of the target plant \(\hat{P}_{s}(z)\) following the procedure in Section 3. The value of \(\lambda\) is chosen as 0.002. \[P_{s}(z)=\frac{1}{0.01s+1} \tag{14a}\] \[\hat{P}_{s}(z)=\frac{1}{0.01s+1}\] (14b) \[Q(z)=\frac{0.01s+1}{0.002s+1} \tag{14c}\]  
Figure 4a compares the time domain tracking of the desired signal with a sinusoidal input under the control of the LMS and the internal model LMS method 1 (IM-LMS 1). Figure 4b shows the estimation error. The conventional system with LMS becomes unstable and diverges without the filtered reference signal, but IM-LMS 1 can manage this condition just as with the filtered-X LMS system, and the whole system can follow the reference signal with stability. \[P_{s}(z)=\frac{1}{0.01s+1} \tag{15a}\] \[\hat{P}_{s}(z)=\frac{1}{0.05s+1}\] (15b) \[Q(z)=\frac{0.05s+1}{0.002s+1} \tag{15c}\]  
Figure 4: \(Cont.\)Assuming the same conditions as the previous case, Figure 5a compares the time-domain tracking of the desired signal under the control of IM-LMS 1, with an exact model and with uncertainty. Figure 5b shows the estimation error. In the transient state, up to around 0.1 s, uncertainty in the model induces more error than the exact model. However, after that point, the error converges to zero in the steady state. Although a significantly mistuned plant model has been used in the control system, the IM-LMS 1 method made the system converge and showed an acceptable tracking performance. This suggests that the new algorithm has more self-adaptability. Figure 4: Simulation results in time domain with a single sinusoid at 100 Hz. (**a**) Time domain tracking with LMS and internal model LMS 1 methods; (**b**) estimation error; (**c**) predicted spectra. Figure 5: _Cont._  
Next, to check the tracking performance of the algorithm, the IM-LMS 1 algorithm was evaluated for the amplitude-modulated (AM) signal \(x_{a}(t)\), which is defined as follows:  
\[x_{a}(t)=A_{0a}(1+\gamma_{rand})\cos(\omega_{ca}t+\theta_{c})\cdot[1+B_{0a}\cos (\omega_{oa}t+\theta_{o})] \tag{16a}\] \[\hat{P}_{s}(z)=\frac{1}{0.01s+1}\]0 (16b) \[\hat{P}_{s}(z)=\frac{1}{0.01s+1}\]1 \[\hat{P}_{s}(z)=\frac{1}{0.01s+1}\]2  
\(x_{a}(t)\) contains a sinusoid at \(\omega_{oa}/2\pi=12\) Hz, which is modulated with a carrier signal at \(\omega_{ca}/2\pi=150\) Hz and has a modulation depth of \(B_{0a}=0.5\). For convenience, \(A_{0a}\) is normalized as 1.0, and the phase angles \(\theta_{c}\) and \(\theta_{o}\) are ignored. A sinusoid at \(\omega_{ca}\) is used as a reference input. Equation (16b), \(F[\;]\) illustrates the Fourier transform, and \(\delta\) indicates the Dirac delta function that is located at frequency \(\omega\). The LMS parameter \(\mu\) is chosen as 0.5. For comparison, the same signal is also tracked by the conventional FX-LMS. Figure 6a compares the time domain tracking of the desired signal with the filter output, for both FX-LMS and IM-LMS 1.