# Notes on a New Structure of Active Noise Control Systems
## 2 Problem Statement  
The ideas presented in the paper are discussed below using as an example a classical feedforward active noise control system structure called FxLMS (filtered-reference least mean square) algorithm [4; 5; 7; 8; 23]. A block diagram of this control system structure is shown in Figure 1. Blocks appearing in the figure are defined below on an example of an active noise control system creating local zones of quiet in an enclosure. The terminology used is also explained in Figure 2 that shows a laboratory configuration applied as a model for simulations included in the paper. Figure 1: The classical FxLMS (filtered-reference least mean square) control system structure embedded in a simulation environment. An unwanted wide-sense stationary random noise propagates in an acoustic space of the enclosure from a primary source to an error microphone around which a local zone of quiet is created. In the error microphone, the noise to be attenuated meets a signal generated by the control loudspeaker driven by a control signal \(u(i)\) (where \(i=0,1,\ldots\) denotes consecutive discrete-time instants). These two signals interfere. The result of interference is picked up by the error microphone and is next processed by an electronic path consisting of an amplifier, analogue antialiasing filter, and A/D converter (the hardware forming an error signal measurement path). At the output of this path, numbers that are a discrete-time error signal \(e(i)\) are obtained. The acoustic space between the primary noise source and error microphone as well as elements of the error signal measurement path together with dynamics of a noise generation system form the disturbance path. The noise generation system is a device transforming the mathematical definition of the noise to be attenuated into a realization of a random process that is further processed into the corresponding acoustic wave. The disturbance path is introduced for the precise definition of the models used in simulations. The control signal \(u(i)\) is an output of a time-varying FIR (finite impulse response) filter \(W(z^{-1})\) (\(z^{-1}\) denotes one-step delay operator) that is adaptively tuned at each discrete-time instant \(i\). It reaches the error microphone after processing by the D/A converter equipped with a zero-order hold filter, analogue forming filter, amplifier, control loudspeaker, and acoustic space. These elements together with the components of the error signal measurement path like the error microphone, amplifier, analogue antialiasing filter, and A/D converter form the secondary path. An input signal of the filter \(W(z^{-1})\) is a signal that is returned by a reference microphone placed in a vicinity of the primary source of noise. Dynamics of this microphone together with dynamics of the electronic elements converting the noise at the primary source (its mathematical definition) into numbers forming a discrete-time reference signal \(x(i)\) is called a reference signal measurement path. This path similarly like the disturbance path was introduced for a simulation purpose only. The discrete-time signal \(x(i)\) processed additionally by a model of the secondary path together with the  
It follows from this nonlinear and time-varying nature of the adaptive system that noise reduction obtained with the adaptive active noise control system tuned with the discrete-time signals \(e(i)\) or \(e_{0}(i)\) may give different results. It implies also that obtaining minimum of \(\mathcal{E}\left\{e^{2}(i)\right\}\) does not guarantee obtaining a minimum of \(\mathcal{E}\left\{e^{2}_{0}(i)\right\}\) for the consecutive discrete-time instants and vice versa. In Figure 3 the error signal measurement path is presented as excluded from the disturbance and secondary paths. As a result the secondary path is shown as a serial connection of the secondary path control signal part and error signal measurement path. Consequently, the disturbance path consists of the acoustical disturbance path and error signal measurement path.