## 1 Introduction  
Noise cancellation is desirable in many places. An active noise cancellation (ANC) system uses a microphone to sense noise, then uses a speaker to generate complementary waveforms (anti-noise), which are combined with the noise to cancel it by destructive interference [1; 2; 3; 4; 5]. Initially, analog ANC was used to generate the anti-noise, with a microphone and a speaker; this method cannot track environmental changes, such as change of the position of the microphone or speaker in the ear canal [6; 7; 8]. Digital ANC devices use digital signal processor (DSP) chips, to adaptively minimize noise, despite such environmental changes. The devices adaptively adjust coefficients of the finite impulse response (FIR) filter, to minimize noise. Digital ANCs are classified into feedback [8; 9; 10; 11; 12; 13; 14] and feedforward [15; 16; 17; 18; 19; 20; 14] types. A feedback digital ANC uses a microphone (error) and a speaker. A digital feedforward ANC uses two microphones (reference, error) and a speaker. The frequency range of the ANC operation is limited to \(\sim\)600 Hz [12] in the feedback ANC, to maintain loop stability, but is 1500 Hz [15] in the feedforward ANC, for in-ear headphones [15; 16; 17; 18; 19]. The requirement for convergence of the adaptive algorithm is tighter in the feedback ANC than in the feedforward ANC [8]. Due to the advantages in frequency range and convergence requirement, feedforward digital ANCs have become widely used. The feedforward ANC has an additional requirement, to satisfy the causality constraint [21; 22], i.e., the delay \(D_{SE}\) of the electrical processing circuit should be smaller than the primary path delay \(D_{P}\) minus the acoustic secondary path delay \(D_{SA}\). \(D_{P}\) refers to the acoustic propagation delay from the reference microphone input to the error microphone input. The secondary path refers to the combined electrical delay \(D_{SE}\) and the acoustic delay \(D_{SA}\) (Figure 1a). \(D_{SE}\) includes the delays of the reference microphone circuit, the electrical processing circuit, and the speaker driver circuit. \(D_{SA}\) includes the speaker delay of electrical-to-acousticconversion and the acoustic propagation delay from the speaker to the error microphone input. The electrical processing circuit includes an analog-to-digital converter (ADC), FIR filters, and a digital-to-analog converter (DAC); this circuit adds an additional delay, to match the secondary path delay to the primary path delay (Figure 1b). Thus, the causality constraint requires,  
\[D_{SE}<D_{P}-D_{SA}. \tag{1}\]  
The causality constraint problem does not occur in a large-scale feedforward ANC system, where the distance between speaker and microphone is on a scale of meters and \(D_{P}-D_{SA}\geq\sim\)3 ms, but in a small-scale feedforward ANC system, such as in-ear headphones [15], the distance between speaker and microphone is tens of millimeters or smaller, and \(D_{P}-D_{SA}\) is tens of microseconds; since this interval is so short, the algorithm thatis used in the electrical processing circuit of the ANC system must react quickly, and, therefore, cannot be too complex, to run within that interval. In in-ear headphones, where two microphones and a speaker are mounted in a small enclosure, the relative arrival times of the sound depend on the direction of the noise source, so the causality constraint, also, depends on this direction (Figure 2). \(D_{P}-D_{SA}\) is smaller when the noise is incident from the front (90\({}^{\circ}\)) than from the side (0\({}^{\circ}\)). The electrical processing circuit must finish operation within \(D_{P}-D_{SA}\), so this causality constraint is harder to meet, when the noise is an incident from the front rather than from the side. Feedforward ANC that uses the least mean squares (LMS) algorithm suffers from instability in convergence [5]. To solve this problem, a filtered-x LMS (FxLMS) algorithm is used  
Figure 1: Feedforward ANC (**a**) time delay and (**b**) block diagram. Blue: Primary path, Red: Secondary path.