Typo

3-perception/vhar-system/2-method.tex

@@ -88,8 +88,8 @@ A sample $r(t_j, t_k)$ of the audio signal at sampling time $t_k$, with $t_k >=
 \begin{subequations}
   \label{eq:signal}
   \begin{align}
-    r(t_j, t_k) & = A\, \text{sgn} ( \sin (2 \pi \frac{\dot{x}\,(t_j)}{\lambda} t_k + \phi_j ) ) & \label{eq:signal_speed} \\
-    \phi_j        & = \phi_{j-1} + 2 \pi \frac{\dot{x}\,(t_j) - \dot{x}\,(t_j - 1)}{\lambda} t_k                 & \label{eq:signal_phase}
+    r(t_j, t_k) & = A\, \text{sgn} ( \sin (2 \pi \frac{\dot{x}\,(t_j)}{\lambda} t_k + \phi(t_j) ) ) & \label{eq:signal_speed} \\
+    \phi(t_j)        & = \phi(t_{j-1}) + 2 \pi \frac{\dot{x}\,(t_j) - \dot{x}\,(t_{j-1})}{\lambda} t_k                 & \label{eq:signal_phase}
   \end{align}
 \end{subequations}
 
@@ -97,7 +97,7 @@ This rendering preserves the sensation of a constant spatial frequency of the vi
 %
 %Note that the finger position and velocity are transformed from the camera frame $\poseFrame{c}$ to the texture frame $\poseFrame{t}$, with the $x$ axis aligned with the texture direction.
 %
-%However, when a new finger position is estimated at time $t_j$, the phase $\phi_j$ needs to be adjusted as well with the frequency to ensure a continuity in the signal as described in \eqref{signal_phase}.
+%However, when a new finger position is estimated at time $t_j$, the phase $\phi(t_j)$ needs to be adjusted as well with the frequency to ensure a continuity in the signal as described in \eqref{signal_phase}.
 %
 The phase matching avoids sudden changes in the actuator movement thus affecting the texture perception in an uncontrolled way (\figref{phase_adjustment}) and, contrary to previous work \cite{asano2015vibrotactile,ujitoko2019modulating}, it enables a free exploration of the texture by the user with no constraints on the finger speed.
 A square wave is chosen to get a rendering closer to a real grating texture with the sensation of crossing edges \cite{ujitoko2019modulating}, and because the roughness perception of sine wave textures has been shown not to reproduce the roughness perception of real grating textures \cite{unger2011roughness}.