# Adaptive Noise Cancellation Based on Time Delay Estimation for Low Frequency Communication
## 1 Introduction  
Low frequency (LF) electromagnetic signals are less prone to multipath effect and provide good obstacle penetration in near field [1; 2]. Low frequency communication, taking advantage of these features, is an alternative solution for through-the-earth (TTE) wireless applications, such as subsurface wireless data acquisition [3], underground localization [4; 5] and magnetic resonance sounding [6]. To achieve two-way communication in long range, there still are many problems to solve. Low frequency signals have great path loss when propagating through the rock, water, and soil [7]. Therefore, huge antennas are used to obtain sufficient radiation gain, probably unsuitable for some applications [8]. In addition, the atmospheric noise mixed to the weak signals limit the signal-to-noise ratio (SNR) at the surface receiver [9]. In this paper, we focus on noise cancellation in low frequency communication. In the low-frequency radio range, the electromagnetic noise is caused by the natural electrical activity in the atmosphere and man-made electrical disturbances. The atmospheric noise is produced primarily by local and distant thunderstorms. The man-made noise arises from mining machines, transportation vehicles, power generators, transmission lines, etc. [10]. Because of all these factors, the low frequency noise tends to be highly impulsive and cannot be characterized by Gaussian distribution. The low frequency noise which is picked up by narrowband receivers, can be approximated by a symmetric \(\alpha\)-stable (\(SaS\)) model [11]. Usually, the receiver consists of multiple antennas, which areused for the direction of arrival (DOA) estimation [12; 13]. In some cases, the additive noise can be cancelled from the signal with some reference antennas [6; 14; 15]. The reference antennas are used to record only noise which is correlated with the noise in the signal [16]. Then the noise can be subtracted from the signal through adaptive noise cancellation, without prior knowledge of signal and communication systems. A popular adaptation technique is the least mean squares (LMS) adaptive algorithm which exploits the second order moments of the error between the desired signal and the filter output [17]. However, in the presence of impulsive of noise, the poor performance of LMS based methods cannot satisfy the requirements of TTE communication systems. To improve the performance, the least mean fourth (LMF) criterion [18] and least mean p-power (LMP) criterion [19] were proposed, using the fourth-order moment or p-order absolute moment of the error as the adaptation cost function. These algorithms are robust in impulsive noise, but the convergence speed and steady-state performance are not good enough [20]. In order to obtain faster convergence and smaller steady-state error, the information theoretic quantities can be used as cost functions to train adaptive systems [21]. There are two families of algorithms: minimum error entropy (MEE) based algorithms and maximum correntropy criterion (MCC) based algorithms. Using entropy of the error signal as a cost function, MEE based algorithms have good robustness and steady-state performance in non-Gaussian and impulsive noise. However, they are quite unsuitable for practical implementation because of their high computational complexity. As a compromise, MCC based algorithms have the robustness comparable to the MEE based algorithms, with similar computational complexity to LMS [22]. As a localized similarity measure, the correntropy places exponentially decreasing weights on impulsive samples. As shown in Figure 1, to obtain excellent performance with finite-length filters in practical circumstances, a delay \(\Delta\) should be inserted into the primary input [23]. Typically, the value of delay \(\Delta\) is chosen equal to about half time delay of the adaptive filter. In the practical TTE communication systems, the primary antenna and reference antennas are usually placed in different locations.