## 4 Enhanced LMS Methods with IMC
### Novel Adaptive Filtering Algorithm Based on the IMC 2: Recursive Method
#### 4.2.2 Numerical Validation  
The internal model LMS method 2 (IM-LMS 2) was validated for two different kinds of multi-spectral signals, namely, AM and frequency-modulated (FM) signals. The same AM signal is used for evaluating the method. Figure 8 compares the tracking performance and estimation error of the conventional LMS and IM-LMS 2. In the transient region, up to around 0.05 s, the IM-LMS 2 method shows a much better tracking performance, as shown in Figure 8a. However, in the steady-state region after 0.05 s, the IM-LMS 2 has a slower convergence than FX-LMS, for around 0.4 s (0.05 \(\leq t\leq\) 0.45), and finally, the estimation errors of both of the methods have similar peak-to-peak values, as shown in Figure 8b. Since FX-LMS is targeted only to the 150 Hz frequency component \(\omega_{\alpha\alpha}\), the time-domain tracking in steady state would have been superior to that of IM-LMS 2. This is clear in the frequency spectrum plot of Figure 9, which shows a difference of in magnitude of almost 30 dB at 150 Hz. This figure compares the predicted spectra of the uncontrolled AM signal and controlled AM signal with both of the methods. Although a broadband reduction cannot be achieved with either FX-LMS or IM-LMS 2, they have shown significant reductions at primary peaks (\(\omega_{\alpha\alpha},\omega_{\alpha\alpha}-\omega_{\alpha\alpha},\omega_{ \alpha\alpha}+\omega_{\alpha\alpha}\)). The amplitude reduction levels of FX-LMS at the primary peaks are higher than those of IM-LMS 2, while FX-LMS shows a severe spillover at higher frequency, from around 180 Hz to 270 Hz. This indicates that the conventional LMS algorithm with a feedforward control structure is insufficient to control the multi-spectral signals, without full knowledge of the original signal. Figure 8: Control of an amplitude-modulated (AM) signal. (**a**) Time-domain tracking with LMS and internal model LMS 2 methods; (**b**) estimation error. Figure 9: Predicted spectra for an amplitude-modulated (AM) signal with LMS and internal model LMS 2 methods. In addition, the IM-LMS 2 method shows considerable attenuation at around 50 Hz, which is the natural frequency of the target plant, whereas the FX-LMS case demonstrates similar amplitude levels around this frequency. This condition comes from the benefit of the recursive structure of the adaptive control system, where untargeted peaks could be directly handled by feeding the estimated plant output \(\hat{d_{k}}\) back to the reference input of the LMS algorithm. Therefore, the whole control system identifies the frequency components other than \(\omega_{\alpha}\). In contrast, the conventional FX-LMS still recognizes only the \(\omega_{\alpha}\) component of \(x(t)\). Overall, IM-LMS 2 uses recursive control techniques and exploits both the conventional LMS algorithm and IMC simultaneously. However, there is some trade-off in the tracking performance with the conventional FX-LMS. The FM signal was investigated similarly, as shown in Equation (25a,b). \[x_{f}(t)=A_{0f}\cos(\omega_{cf}t+B_{0f}\sin(\omega_{of}t+\gamma_{rand})+\theta _{f}) \tag{25a}\] \[F[x_{f}(t)] = A_{0f}I_{0}(B_{0f})\pi[\delta(\omega-\omega_{cf})+\delta(\omega+ \omega_{cf})]\] \[+\tfrac{1}{2}A_{0f}\sum\limits_{n=1}^{\infty}{(-1)^{n}I_{n}(B_{0 f})\pi[\delta(\omega-\omega_{cf}+n\omega_{of})+\delta(\omega+\omega_{cf}-n \omega_{of})}\] \[+(-1)^{n}\delta(\omega-\omega_{cf}-n\omega_{of})+(-1)^{n}\delta( \omega+\omega_{cf}+n\omega_{of})]\]  
\(J_{n}\) indicates a Bessel function of order \(n\), and the modulation depths \(A_{0f}\), \(B_{0f}\), and \(\theta_{f}\) are selected as 1.0, 0.5, and 0, respectively. The reference input and other parameters that were used are the same as those for the AM case. Figure 10 compares the tracking performance and estimation error of the conventional LMS and IM-LMS 2 methods.