### Introduction

Doppler effect has been widely used in radar to measure the relative velocity between source and the target. The radar and lidar that use Doppler effect to measure relative velocities are called Doppler radar. Many autonomous or semi-autonomous machines, such as air-plane, autonomous vehicle, are often equipped with Doppler radar.

In this blog post, I would like to discuss the physics and mathematics of Doppler effect for Doppler radar.

### Ordinary Radar

In the ordinary radar configurations, the transmitter of the radar sends out a wave, the wave hits a target object and gets reflected, a small portion of the reflected wave will be received by the receiver of the radar. By measuring the time gap between signal transmission and receipt $\Delta t$, we could determine the distance between the radar and the target object $r$ easily.

\[\begin{align} r &= \frac{c \Delta t}{2} \\ \end{align}\]where $c$ is the wave velocity. Both radar and the target object could be moving. But as long as the wave velocity $c$ is much greater than the relative velocity between the radar and the target object $\Delta v$, the above equation holds.

The ordinary radar is also called time-of-flight radar.

### Doppler Radar

Different from ordinary radar, Doppler radar could also be used for measuring the the relative velocity between the radar and the target object $\Delta v$.

#### Doppler Effect

Let’s consider the Doppler effect in the simplest 1D scenario. The wave frequency that the receiver measured $f^{\prime}$ is

\[\begin{align} \lambda^{\prime} &= \lambda - \frac{\Delta v}{f} \\ &= \frac{c}{f} - \frac{\Delta v}{f} \\ &= \frac{c - \Delta v}{f} \\ &= \frac{c}{f^{\prime}} \\ \end{align}\]where $c$ is the wave velocity, $f$ is the source wave frequency, $\Delta v$ is the relative velocity between the source and the target, $\Delta v > 0$ when the source and the target are moving closer, $\Delta v < 0$ when the source and the target are moving farther. Therefore,

\[\begin{align} f^{\prime} &= \frac{c}{c - \Delta v} f \\ \end{align}\]Apparently, when the source and target are moving closer, i.e., $\Delta v > 0$, $f^{\prime} > f$; when the source and target are moving farther, i.e., $\Delta v < 0$, $f^{\prime} < f$.

The Doppler frequency $\Delta f$, which is the difference between the receiver frequency and the transmitter frequency is

\[\begin{align} \Delta f &= f^{\prime} - f \\ &= \frac{c}{c - \Delta v} f -f \\ &= \frac{f \Delta v}{c - \Delta v} \\ \end{align}\]Assuming $c \gg \Delta v$, we have

\[\begin{align} \Delta f &= \frac{f \Delta v}{c} \\ \end{align}\]The Doppler effect will also happen to a Doppler radar when the radar and the object are relatively moving. Let’s still consider the simplest 1D scenario. The target object will reflect the wave transmitted from the source. In this case, the target object becomes the source of wave, and the radar becomes the target object since the radar has a receiver. The frequency of reflected wave will be the same as the frequency of the wave the target objects receives, which is $f^{\prime}$. Then the frequency of the wave measured from the receiver on the radar $f^{\prime\prime}$ will be

\[\begin{align} f^{\prime\prime} &= \frac{c}{c - \Delta v} f^{\prime} \\ \end{align}\]Assuming $c \gg \Delta v$, the Doppler frequency $\Delta f$, which is the difference between the receiver frequency and the transmitter frequency is

\[\begin{align} \Delta f &= f^{\prime\prime} - f \\ &= (f^{\prime\prime} - f^{\prime}) - (f^{\prime} - f) \\ &= \frac{f \Delta v}{c} + \frac{f \Delta v}{c} \\ &= \frac{2 f \Delta v}{c} \end{align}\]With this mathematical relationship, by measuring the the Doppler frequency $\Delta f$ on the radar, we could determine the relative velocity between the radar and the target object easily.

#### Phase Shift

In practice, instead of measuring the Doppler frequency $\Delta f$, many Doppler radar measure phase shift $\Delta \varphi$.

Assuming $c \gg \Delta v$, the phase difference between the transmitted signal and the received signal, i.e., phase shift $\Delta \varphi$, is

\[\begin{align} \Delta \varphi &= \frac{2r}{\lambda} 2\pi \\ &= \frac{4 \pi r}{\lambda} \\ &= \frac{4 \pi r f}{c} \\ \end{align}\]where $r$ is the distance between the radar and the target object, and $\lambda$ is the wavelength.

\[\begin{align} \frac{d \Delta \varphi}{dt} &= \frac{4 \pi f}{c} \frac{d r}{dt} \\ &= \frac{4 \pi f}{c} \Delta v \\ \end{align}\]Note that $\Delta v = \frac{d r}{dt}$.

This means that if we could measure $\frac{d \Delta \varphi}{dt}$, we could calculate the relative velocity between the radar and the target object. In practice, $\frac{d \Delta \varphi}{dt}$ could be measured easily by transmitting lots of signals in unit time, and measuring the $\frac{\Delta \Delta \varphi}{\Delta t}$.

### Conclusions

Doppler radar is not that sophisticated to understand. We determine the relative velocity by measuring the change rate of Doppler effect phase shift.