assume we are tracking a vehicle with a radar/lidar. The radar/lidar is placed some where to observe the vehicle. the state of the vehicle is

$$ \mathbf{x}=\begin{bmatrix} x \\ v \end{bmatrix},P=\begin{bmatrix} cov(x,x) && cov(x,v) \\ cov(v,x) && cov(v,v) \end{bmatrix} $$

from the radar/lidar, we can read the time of echo signal, not the distance, but the time. In order to connect the state vector and the measurement, we need the measurement matrix $H$. In this case,

$$ H=\begin{bmatrix} 1/c && 0 \end{bmatrix} $$

where $c$ is the light speed.

thus, the measurement is about time, when we project the estimated state to measurement unit, which is

$$ \mathbf{z}_{k}=H_k \mathbf{x}_{k-1}=\begin{bmatrix} 1/c && 0 \end{bmatrix}\begin{bmatrix} x_{k-1} \\ v_{k-1} \end{bmatrix}=x_{k-1}/c $$

assume we are tracking a vehicle with a radar/lidar. The radar/lidar is placed some where to observe the vehicle. the state of the vehicle is

$$ \mathbf{x}=\begin{bmatrix} x \\ v \end{bmatrix},P=\begin{bmatrix} cov(x,x) && cov(x,v) \\ cov(v,x) && cov(v,v) \end{bmatrix} $$

from the radar/lidar, we can read the time of echo signal, not the distance, but the time. In order to connect the state vector and the measurement, we need the measurement matrix $H$. In this case,

$$ H=\begin{bmatrix} 1/c && 0 \end{bmatrix} $$

where $c$ is the light speed.

thus, the measurement is about time, then we project the estimated state to measurement unit.

here is an example, $\mathbf{x}_{k\vert k-1}$ is the predicted state at time k based on the state of time k-1. We project the predicted state into measurement unit system with observation model $H_k$, in order to merge the distribution in the next step.

$$\begin{eqnarray} \mathbf{z}_{k}&=&H_k \mathbf{x}_{k\vert k-1}=\begin{bmatrix} 1/c && 0 \end{bmatrix}\begin{bmatrix} x_{k\vert k-1} \\ v_{k\vert k-1} \end{bmatrix}=x_{k\vert k-1}/c \\ \text{var}(H_k\mathbf{x}_{k\vert k-1})&=&H_kP_{k\vert k-1}H_k^T=\text{var}(x_{k\vert k-1})/c^2 \end{eqnarray}$$

assume we are tracking a vehicle with a radar/lidar. The radar/lidar is placed some where to observe the vehicle. the state of the vehicle is

$$ \mathbf{x}=\begin{bmatrix} x \\ v \end{bmatrix},P=\begin{bmatrix} cov(x,x) && cov(x,v) \\ cov(v,x) && cov(v,v) \end{bmatrix} $$

from the radar/lidar, we can read the time of echo signal, not the distance, but the time. In order to connect the state vector and the measurement, we need the measure matrix $H$. In this case,

$$ H=\begin{bmatrix} 1/c && 0 \end{bmatrix} $$

where $c$ is the light speed.

thus, the measurement is about time, which is

$$\begin{eqnarray} \mu_{k+1}&=&H_k \mathbf{x}_k=\begin{bmatrix} 1/c && 0 \end{bmatrix}\begin{bmatrix} x_k \\ v_k \end{bmatrix}=x_k/c \\ \Sigma_{k+1}&=&H_k P_k H_k^T=\begin{bmatrix} 1/c && 0 \end{bmatrix}\begin{bmatrix} \text{cov}(x_k,x_k) && \text{cov}(x_k,v_k) \\ \text{cov}(v_k,x_k) && \text{cov}(v_k,v_k) \end{bmatrix}\begin{bmatrix} 1/c \\ 0 \end{bmatrix}=\text{cov}(x_k,x_k)/c^2\\ \Sigma_{k+1}&=&=\text{var}(x_k/c) \end{eqnarray}$$

assume we are tracking a vehicle with a radar/lidar. The radar/lidar is placed some where to observe the vehicle. the state of the vehicle is

$$ \mathbf{x}=\begin{bmatrix} x \\ v \end{bmatrix},P=\begin{bmatrix} cov(x,x) && cov(x,v) \\ cov(v,x) && cov(v,v) \end{bmatrix} $$

from the radar/lidar, we can read the time of echo signal, not the distance, but the time. In order to connect the state vector and the measurement, we need the measurement matrix $H$. In this case,

$$ H=\begin{bmatrix} 1/c && 0 \end{bmatrix} $$

where $c$ is the light speed.

thus, the measurement is about time, when we project the estimated state to measurement unit, which is

$$ \mathbf{z}_{k}=H_k \mathbf{x}_{k-1}=\begin{bmatrix} 1/c && 0 \end{bmatrix}\begin{bmatrix} x_{k-1} \\ v_{k-1} \end{bmatrix}=x_{k-1}/c $$