What I understand of Doppler spread is that the relative motion between Transmitter (TX) and Receiver (RX) change the exposing time of signal. In rapport to a constant-distance TX-RX, a moving toward each other TX-RX "compresses" signal in time (signal takes less time to propagate), then signal is "expanded" in frequency domain. Similarly, a moving away RX-TX "expands" signal in time and "compresses" its spectrum. In short, that is scaling Fourier Transform. These two extreme cases set the left and right bounds of spreading an original frequency between $-f\_d$ and $+f\_d$ where $f\_d$ is max Doppler spread.

In looking at the Clarke model, it is just multiple propagation model with rich scattering environment and equal angle of arrival. (link for more details
[Clarke model](https://en.wikipedia.org/wiki/Rayleigh_fading#Doppler_power_spectral_density))

If I understand well, there are two assumptions which are reasonale in urban environment:

* Rayleigh fading
* equal angle of arrival, or equal receiver sensitivity

I have followed the math from the original article, it seems ok. The final Doppler power spectrum is then
$\displaystyle S(f) = \frac{1}{\pi f\_d \sqrt{1 - \left(\frac{f}{f\_d}\right)^2}}$

[![Clarke Doppler power spectrum](https://i.stack.imgur.com/vwDcT.png)](https://i.stack.imgur.com/vwDcT.png)

What I don't understand is that **why energy is concentrated to the two extreme spread frequency $-f\_d$ and $f\_d$ while angles of arrival are uniform. Is there any physical interpretation ? What am I missing from the famous Clarke model ?** Personally, this model seems well-model the typical urban environment.

[R. H. Clarke, *A Statistical Theory of Mobile-Radio Reception*, The Bell System Technical Journal, July/Aug 1968, p. 957ff](http://faculty.poly.edu/~tsr/EL5013_fall12/docs/bstj47-6-957.pdf)

> 
> **Answers** Although the answer of Carlos captures the most fundamental mathematical part, the real answer is in his comment about "mapping
>  between angle and frequency". Moreover, the answer of Maximilian is
>  interesting too.
> 
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