By applying the Fourier transform to the individual taps, i.e. transforming with respect to $t$ and not $\tau$, the Doppler spectrum $$ S_i(f) = \mathcal{F}\left\{a_{i}(t)\right\} $$ can be obtained [2]. By squaring the absolute value of $S_i(f)$, the power density is derived. The Doppler power spectrum is shaped like a bath tub. Ideally, there is no power outside of the range $[-f_\mathrm{D}, f_\mathrm{D}]$. As such, $S_i(f) = 0$ if $|f| > f_\mathrm{D}$. If there is an LoS component with angle $\theta_\text{LoS}$ and a power determined by the $K\text{-Factor}$, then a Dirac peak will occur at the frequency [2,7] $$ f_\mathrm{peak} = \cos(\theta_{\mathrm{LoS}})f_{\mathrm{D}}. $$ Ideally, the Doppler power spectrum is described using the following function [2,7,9] $$ |S_i(f)|^2 = \dfrac{1}{K+1}\left(\dfrac{1}{\pi f_{\mathrm{D}}\sqrt{1-\left(\dfrac{f}{f_{\mathrm{D}}}\right)^2}} + K \delta(f-f_\mathrm{peak})\right) \quad \mathrm{with} \quad f \lt f_\mathrm{D}. $$ The factor $\dfrac{1}{K+1}$ is based on the assumption that $a_i\left(t\right)$ is normalized to unit power. The next slide shows the power density spectrum of the first tap $S_0(f)$. The visualized power density spectra are normalized such that $\mathrm{E}\left[|S_0(f)|^2\right] = 1$ in the range $[-f_\mathrm{D}, f_\mathrm{D}]$. To calculate $S_0(f)$, a fast Fourier transform (FFT) with a Blackman window is used. This causes the Dirac peak to bleed power into neighboring frequencies. The user may compare the result of the simulation to the ideal curve.

[2] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall PTR, 2002.
[7] F. Vatalaro and A. Forcella, “Doppler spectrum in mobile-to-mobile communications in the presence of three-dimensional multipath scattering”, IEEE Transactions on Vehicular Technology, vol. 46, no. 1, pp. 213–219, 1997. [9] R. H. Clarke, “A statistical theory of mobile-radio reception,” in The Bell System Technical Journal, vol. 47, no. 6, pp. 957-1000, 1968.