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)
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 $\displaystyle S(f) = \frac{1}{\pi f_d \sqrt{1 - \left(\frac{f}{f_d}\right)^2}}$
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.
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.