Interpretation of Clarke's Doppler power spectral density 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.

R. H. Clarke, A Statistical Theory of Mobile-Radio Reception, The Bell System Technical Journal, July/Aug 1968, p. 957ff


  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.
A simple, "non-technical" way of thinking of it is the fact that the Doppler frequency is proportional to $\cos\theta$. The amplitudes of cosine, however, are not uniformly distributed, but are heavily weighted towards $\pm 1$. 

Example plot to demonstrate, using Python/Pylab code:

theta = linspace(0, 2*pi, 1001)
x = cos(theta)
hist(x)




More rigor can be seen by noting that 
\begin{align}
f &amp;= f_d \cos\theta\\
\theta &amp;= \cos^{-1}\left(\frac{f}{f_d}\right)
\end{align}
and the power received at any angle is proportional to a small angle increment $d\theta$:

$$
P(\theta) \propto d\theta = \frac{-1}{f_d\sqrt{1-\left(\frac{f}{f_d}\right)^2}} df
$$

And the total power can be determined by integrating the above quantity, which is identically what defines a power spectral density.