## 2 Description of the LMS-Based Algorithms  
The LMS algorithm was introduced by [PERSON] et al. in 1960 [PERSON] et al. (1960) and nowadays represents one of the most popular adaptive algorithms for its simplicity of implementation and robustness. It consists of the minimization of a cost function, which is usually defined as the energy measured at error microphones for ANC applications. This operation is mathematically translated in an explicit algorithm based on a single time step method ([PERSON]'s forward formula), used to iteratively update the adaptive filter coefficients at each time instant, starting from the previous instant. This method is conditionally stable due to its explicit nature; thus, it is necessary to properly set the step-size parameter in order to achieve numerical stability. Moreover, another type of stability must be considered in relation to the fact that the basic LMS algorithm does not take into consideration the delay introduced by the wave propagation from the control loudspeaker to the error microphone. To take into account this delay, the reference signal is filtered for the impulse response calculated between the control source and the microphone, named the secondary path. This algorithm takes the name of filtered-X least mean squares (FXLMS) algorithm ([PERSON] and [PERSON], 1960), and it will be illustrated in Section 2.1. The adaptive filter is introduced as a finite impulse response (FIR) filter, so in the z-transform domain, it is represented by a polynomial. However, in some cases, for example, when acoustic feedback from the control source to the reference microphone is present, it is necessary to model the poles of the physical system to avoid instability. On the other hand, the implementation of an infinite impulse response (IIR) filter, represented by a rational function in the z-transform domain, is able to model the poles of the system and overcome this problem, although the algorithm could converge to a solution that is not the absolute optimum. This kind of algorithm is called filtered-U recursive LMS (FURLMS) and is described in detail in Section 2.3.  
In many applications, the disturbance source is not stationary and an increment of power can lead to the instability of the algorithm, if the step-size parameter is fixed to a large value. In this case, an FXLMS algorithm with step-size normalized with the power of the filtered reference signal (FXLMS with NSS) turns out to be useful, as will be described in Section 2.2.  
When it is not possible to sense directly the signal of the source, the reference signal must be internally computed. The FXLMS algorithm applied to a feedback system of this kind is described in Section 2.4.  
These algorithms will be applied to the multichannel ANC system schematized in Figure 1, composed of one reference microphone (for the feed-forward configurations only), two control loudspeakers, and two error microphones. This system works as described in the following:  
* Two error microphones measure the error signals \(e_{1}(n)\) and \(e_{2}(n)\) and define the zone of quiet where the noise generated by the disturbance source has to be eliminated;
* For the case of feed-forward systems, the input microphone near the noise source measures a reference signal \(x(n)\) before noise reaches the control loudspeakers; for feedback systems, this signal is internally modeled starting from \(e_{1}(n)\) and \(e_{2}(n)\);
* The control unit processes the measured data in real-time with the adaptive algorithm and calculates the output signals \(y_{1}(n)\) and \(y_{2}(n)\), which drive the two control loudspeakers;* The control sources produce the computed signals that will be superimposed to the existing noise and will cancel it for interference.  
In the following paragraphs this procedure, with the previously described algorithms, is illustrated in detail.