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Clarke-Jakes model Explained

 

Understanding Clarke-Jakes Model: The Foundation of Rayleigh Fading

A comprehensive guide to the Clarke-Jakes model, the Doppler Bathtub Spectrum, and precision Coherence Time calculations in wireless communications.

What is Clarke’s Model?

Clarke’s Model (often called the Clarke-Jakes model) is the mathematical framework used to describe small-scale fading in mobile wireless channels. It explains how a signal behaves when a receiver moves through a dense multipath environment where there is no direct Line-of-Sight (NLOS).

Key Assumptions:

  • A fixed transmitter and a moving receiver.
  • An infinite number of scatterers (rich multipath).
  • Signals arrive from all horizontal directions (360°) with equal probability.
  • The received signal envelope follows a Rayleigh Distribution.

The Mathematical Core: Bessel Functions

While simple models use 1/fd, Clarke’s model proves that the time auto-correlation of the channel follows a zeroth-order Bessel function of the first kind:

Rh(Δt) = J0(2π fd Δt)

Where J0 is the Bessel function, fd is the maximum Doppler shift, and Δt is the time lag.

The Doppler "Bathtub" Spectrum (Jakes Spectrum)

Clarke’s model predicts that the Power Spectral Density (PSD) of a faded signal is not flat. Instead, it forms a U-shape, with power peaks at the edges of the Doppler spread (fc ± fd).

Why the "Bathtub" shape?

As the receiver moves, components arriving from directly ahead or directly behind experience the maximum Doppler shift, causing power to "pile up" at the frequency boundaries.



Jakes PSD Illustration
Access Simulation →

Redefining Coherence Time (Tc)

In standard engineering, we use approximations for Coherence Time based on Clarke’s Bessel correlation function. The "accurate" constant depends on the correlation threshold:

Correlation Threshold Tc Formula Best Use Case
0.5 (50%) 0.423 / fd Standard Modern Wireless Engineering
0.9 (90%) 1 / (8 fd) High-Precision / Sensitive Systems
Rule of Thumb 1 / fd Quick back-of-the-envelope estimations

Example Calculation (LTE/5G)

If a car is moving at 120 km/h (33.3 m/s) using a 3.5 GHz 5G signal:
1. Doppler Shift (fd) = (33.3 * 3.5e9) / 3e8 ≈ 388 Hz.
2. Coherence Time (50% Correlation) = 0.423 / 388 ≈ 1.09 milliseconds.

This means the 5G channel changes completely every ~1ms, requiring very fast channel estimation!

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