## Doppler shift in wireless communications

The wave takes some time to reach the car.

- The distance between car and base station is ${x}_{0}$
- $c$ is the speed of light.
- ${T}_{0}$ is the delay of the wave. ${T}_{0}=\frac{{x}_{c}}{c}$ .

Therefore, the car receives a phase shifted signal $r\left(t\right)=\widehat{r}\mathrm{cos}\left(2\pi {f}_{c}t-{\phi}_{0}\right)$ .

Where ${\phi}_{0}$ is the phase shift. ${\phi}_{0}=2\pi {f}_{c}{T}_{0}$ .

$T=\frac{vt}{c}$

The received signal yields to:

$r\left(t\right)=\widehat{r}\mathrm{cos}\left(2\pi {f}_{c}\left(t-\frac{vt}{c}\right)-{\phi}_{0}\right)=\widehat{r}\mathrm{cos}\left(2\pi \left({f}_{c}-\frac{v}{c}{f}_{c}\right)t-{\phi}_{0}\right)$

Where the frequency shift - known as Doppler shift is given by:

${f}_{D}=\frac{v\text{\hspace{0.17em}}{f}_{c}}{c}$

An angle between wave propagation and velocity needs to be considered:

${f}_{D}=\frac{v\text{\hspace{0.17em}}{f}_{c}\mathrm{cos}\left(\alpha \right)}{c}$

## Video Doppler shift: Calculate the speed of the car!

All details you need for doing that are given in the video:

- ${f}_{c}=1\text{\hspace{0.17em}}GHz$
- ${f}_{D}=40\text{\hspace{0.17em}}Hz$
- $c=3\cdot {10}^{8}m/s$

The car's speed yields to:

$v=\frac{{f}_{D}c}{{f}_{c}}=\frac{40\text{\hspace{0.17em}}Hz\text{\hspace{0.17em}}3\cdot {10}^{8}m/s}{1\text{\hspace{0.17em}}GHz}=12m/s$

In channels where transmitter and receiver move relative to each other the signal frequency is shifted depending on the velocity. This so-called Doppler effect can be observed on passing cars, moving stars (redshift of light) and wireless communications.

## Start

In addition to the video ** Doppler shift** the received signal can be analyzed in the simulation.Start the simulation.
Note that the setup is exactly the scenario in the video:

## Experiment

In the video you were asked to calculate the speed of the car. Now check your result: The car moves with a speed of 12 m/s towards the transmitter.

That can be verified experimentally by changing the speed of the car. Click on *Doppler multipath channel* in the block diagram. Watch the impact on the Doppler shift.

Furthermore change the angle e.g. 180°, 90°, 30°.

## Next Steps

Due to reflections, fading and shadowing there are more transmission paths than the direct one. Think about how the spectrum looks now.

Start the simulation Doppler shift in wireless communications - 2 paths