Doppler shift was discovered by a physicist named Christian Doppler long time ago. Doppler shift is used to characterize the frequency change due to movement. A typical example is that when a fire truck drives toward you, the tone of its siren changes over the time because its speed towards you varies.
Doppler shift for wireless communication is
\[f_d=v/c*f_c\]
whereas v is the speed of the object, c is the speed of light, and \(f_c\) is carrier frequency. For instance, if you sit on a high-speed training moving at 350km/h browsing Internet and your phone is using a 3GHz band, then
\[f_d = (350*1000/3600)/(3*10e8)*3*10e9 = 972 Hz\]
Based on this equation, moving faster as well as using a band with higher carrier frequency will make Doppler shift larger. Usually the highest Doppler shift is observed in high-speed train. Larger Doppler shift makes channel estimation harder in wireless communication.
Rayleigh fading is a commonly used model for a wireless channel, and it is used when the transmitter can't see the receiver such as in Manhattan island. Rayleigh fading model can be derived from joint Gaussian distribution. Assuming \(x\) and \(y\) are two zero-mean Gaussian R.V. orthogonal to each other, their joint distribution becomes:
\[f_{X,Y}(x,y)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{x^2+y^2}{2\sigma^2}}\]
Rayleigh fading tracks the envelope, \(r\). With \(r^2=x^2+y^2\), we have \(x=r*cos(\theta)\) and \(y=r*sin(\theta)\). To transfer a \(f_{X,Y}(x,y)\) to distribution of \(r\), the determinant of the Jacobian matrix needs to be found:
\[J(r,\theta)=det\begin{bmatrix}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r}&\frac{\partial y}{\partial \theta}\end{bmatrix}=det\begin{bmatrix}cos(\theta)&-r*sin(\theta)\\sin(\theta)&r*cos(\theta)\end{bmatrix}=r\]
Therefore,
\[f_{R,\Theta}(r,\theta)=\frac{J(r,\theta)}{2\pi\sigma^{2}}e^{-\frac{r^2}{2\sigma^2}}=\frac{r}{2\pi\sigma^{2}}e^{-\frac{r^2}{2\sigma^2}}\]
After integrating over \(2\pi\) to remove \(\theta\), Rayleigh fading model is:
\[f_{R}(r)=\frac{r}{\sigma^{2}}e^{-\frac{r^2}{2\sigma^2}}\]
for \(r\geq0\).
In reality, fading channel is always correlated in time. That means the wireless channels your phone sees in this moment and the next moment are always more or less similar. Fourier transmission of wireless channel's time-domain auto-correlation becomes its Power Spectral Density (PSD). For Rayleigh fading with vertical antenna and signal coming from all angles in uniform distribution, it has a famous PSD called Jakes model. Jakes model also depends on Doppler shift, and it is:
\[S(f)=\frac{1}{\pi f_d}\frac{1}{\sqrt{1-(f/f_d)^2}}\]
when \(|f|\leq f_d\) and \(S(f)=0\) otherwise. Jakes model has a famous U shape and the width of the "U" depends on Doppler shift, \(f_d\). When \(f_d=0\), Jakes model is an impulse and it means Rayleigh channel never changes.
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