Doppler Spectrum of Wireless Channels

2. Regarding the Doppler spectrum of wireless channels, please answer the following questions:

(1) In certain wireless channels, Doppler frequency shift may cause aliasing in the wireless channel output signal. Describe various causes of aliasing, such as: movement of signal source or receiver, changes in signal propagation medium, etc. From a signal processing perspective, aliasing occurs when the Doppler spread exceeds half the sampling rate (Nyquist criterion). Implementation considerations include ensuring adequate sampling rates and anti-aliasing filters in channel simulation code.

(2) Wireless channels exhibit a classical Doppler spectrum shape. Explain the mechanism generating this spectral shape, such as: relationship between signal frequency variation and time, signal reflection and scattering phenomena. The classical spectrum typically follows a U-shaped (or bathtub) distribution due to isotropic scattering environments. Key algorithm aspects involve modeling multiple scatterers with uniform angle distribution using Jakes' model or similar statistical approaches in channel simulators.

(3) Use Simulink tools (employing S-Function methodology) to write code generating a single-path Rayleigh channel with classical Doppler spectrum shape, where the mobile speed is 120 km/h. Implementation should include proper parameterization of maximum Doppler frequency (f_d = v*f_c/c) and spectrum shaping filters. Additionally answer: Possible methods to verify the correctness of computer-generated classical spectrum, such as: comparison with theoretical calculations (e.g., analytical formulas for autocorrelation function), validation through practical measurements, spectral analysis using FFT-based techniques, and statistical tests for Rayleigh distribution properties.