20 -380 6-400 -WEKE --W/O EKF Time(seconds) 420 0 (a)Magnitude of the raw IR estimations 440 460 00 -380 380 -340 -320 -300 60 I(normalized) Figure 9:Path coefficients for finger reflection path. Time (seconds) (b)Magnitude of differential IR estimations with a constant attenuation of Ai in a short period.There- Figure 8:IR estimations for finger movement. fore,the trace of path coefficients should be a circle in the complex plane.However,due to additive noises,the trace in adaptively select one of these IR estimation to represent the reflection path so that noises introduced by side lobes of Figure 9 is not smooth enough for later phase measurements We propose to use the Extended Kalman Filter(EKF),a other paths can be reduced. Our heuristic to determine the delay of the non-linear filter,to track the path coefficient and reduce reflection path is based on the observation that the additive noise.The goal is to make the resulting path the reflection path will have the largest change of coefficient closer to the theoretical model so that the phase magnitude compared to other paths.Consider the change incurred by the movement can be measured with changes of magnitude in ht[ni]:h[ni]-ht-At[ni]= higher accuracy.We use the sinusoid model to predict and update the signal of both I/O components [8.To save the Ai le)e Here we assume computational resources,we first detect whether finger is that Ai does not change during the short period of At.When moving or not as shown in Section 6.2.When we find that the the delay ni is exactly the same as of the reflection path, finger is moving,we initialize the parameters of the EKF and the magnitude of h [ni]-h-Ar[ni]is maximized.This is perform EKF.We also downsample the path coefficient to because the magnitude of Ail is maximized at the peak 3 kHz to make the EKF affordable for mobile devices.Figure corresponds to the auto-correlation of the reflection path 9 shows that results after EKF are much smoother than the and the magnitude ofe)e) original signal. is maximized due to the largest path length change at the 6.4 Phase Based Movement Measurement reflection path delay. We use a curvature-based estimation scheme to measure In our implementation,we select l path coefficients with an the phase change of the path coefficient.Our estimation interval of three samples between each other as the candidate scheme assumes that the path coefficient is a superposition of reflection paths.The distance between these candidate of a circularly changing dynamical component,which is reflection paths and the structure path is determined by size caused by the moving finger,and a quasi-static component, of the phone,e.g..5~15 samples for the bottom Mic.We keep which is caused by nearby static objects [28,29,34].The monitoring the candidate path coefficients and select the algorithm estimates the phase of the dynamic component by path with the maximum magnitude in the time differential measuring the curvature of the trace on the complex plane IR estimations as the reflection path.When the finger is The curvature-based scheme avoids the error-prone process static,our system still keeps track of the reflection path.In of estimating the quasi-static component in LEVD [28]and this way,we can use the changes in the selected reflection is robust to noise interferences. path to detect whether the finger moves or not. Suppose that we use a trace in the two-dimensional plane y(t)=(I,i)to represent the path coefficient of the 6.3 Additive Noise Mitigation reflection.As shown in Figure 9,the instantaneous signed Although the adaptive reflection path selection scheme curvature can be estimated as: gives high SNR measurements on path coefficients,the addi- det(y'(t).y"(t)) tive noises from other paths still interfere with the measured k(t)= (6) path coefficients.Figure 9 shows the result of the trace of the y(t)3 complex path coefficient with a finger movement.In the ideal where y'(t)=dy(t)/dt is the first derivative of y(t)with case,the path coefficients is h[ni]=Aie-i(+2d(t)/A) respect to the parameter t,and det is taking the determinant01234 Time (seconds) 20 40 60 80 Path length (cm) (a) Magnitude of the raw IR estimations 01234 Time (seconds) 20 40 60 80 Path length (cm) (b) Magnitude of differential IR estimations Figure 8: IR estimations for finger movement. adaptively select one of these IR estimation to represent the reflection path so that noises introduced by side lobes of other paths can be reduced. Our heuristic to determine the delay of the reflection path is based on the observation that the reflection path will have the largest change of magnitude compared to other paths. Consider the changes of magnitude in ˆht[ni]:    ˆht[ni] − ˆht−∆t[ni]    =     Ai  e −j(ϕi+2π di (t ) λc ) − e −j(ϕi+2π di (t−∆t ) λc )      . Here we assume that Ai does not change during the short period of ∆t. When the delay ni is exactly the same as of the reflection path, the magnitude of    ˆht[ni] − ˆht−∆t[ni]    is maximized. This is because the magnitude of |Ai | is maximized at the peak corresponds to the auto-correlation of the reflection path, and the magnitude of     e −j(ϕi+2π di (t ) λc ) − e −j(ϕi+2π di (t−∆t ) λc )     is maximized due to the largest path length change at the reflection path delay. In our implementation, we selectl path coefficients with an interval of three samples between each other as the candidate of reflection paths. The distance between these candidate reflection paths and the structure path is determined by size of the phone, e.g., 5 ∼ 15 samples for the bottom Mic. We keep monitoring the candidate path coefficients and select the path with the maximum magnitude in the time differential IR estimations as the reflection path. When the finger is static, our system still keeps track of the reflection path . In this way, we can use the changes in the selected reflection path to detect whether the finger moves or not. 6.3 Additive Noise Mitigation Although the adaptive reflection path selection scheme gives high SNR measurements on path coefficients, the addi￾tive noises from other paths still interfere with the measured path coefficients. Figure 9 shows the result of the trace of the complex path coefficient with a finger movement. In the ideal case, the path coefficients is ˆht[ni] = Aie −j(ϕi+2πdi (t)/λc ) -400 -380 -360 -340 -320 -300 I (normalized) -460 -440 -420 -400 -380 Q (normalized) P O W EKF W/O EKF Figure 9: Path coefficients for finger reflection path. with a constant attenuation of Ai in a short period. There￾fore, the trace of path coefficients should be a circle in the complex plane. However, due to additive noises, the trace in Figure 9 is not smooth enough for later phase measurements. We propose to use the Extended Kalman Filter (EKF), a non-linear filter, to track the path coefficient and reduce the additive noise. The goal is to make the resulting path coefficient closer to the theoretical model so that the phase change incurred by the movement can be measured with higher accuracy. We use the sinusoid model to predict and update the signal of both I/Q components [8]. To save the computational resources, we first detect whether finger is moving or not as shown in Section 6.2. When we find that the finger is moving, we initialize the parameters of the EKF and perform EKF. We also downsample the path coefficient to 3 kHz to make the EKF affordable for mobile devices. Figure 9 shows that results after EKF are much smoother than the original signal. 6.4 Phase Based Movement Measurement We use a curvature-based estimation scheme to measure the phase change of the path coefficient. Our estimation scheme assumes that the path coefficient is a superposition of a circularly changing dynamical component, which is caused by the moving finger, and a quasi-static component, which is caused by nearby static objects [28, 29, 34]. The algorithm estimates the phase of the dynamic component by measuring the curvature of the trace on the complex plane. The curvature-based scheme avoids the error-prone process of estimating the quasi-static component in LEVD [28] and is robust to noise interferences. Suppose that we use a trace in the two-dimensional plane y(t) = (I hˆ t ,Qhˆ t ) to represent the path coefficient of the reflection. As shown in Figure 9, the instantaneous signed curvature can be estimated as: k(t) = det(y ′ (t),y ′′(t)) y ′ (t) 3 , (6) where y ′ (t) = dy(t)/dt is the first derivative of y(t) with respect to the parameter t, and det is taking the determinant