Rayleigh Fading Simulation – Young’s model

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In the previous article, the characteristics and types of fading was discussed. Rayleigh Fading channel with Doppler shift is considered in this article.

Consider a channel affected by both Rayleigh Fading phenomena and Doppler Shift. Rayleigh Fading is caused due to multipath reflections of the received signal before it reaches the receiver and the Doppler Shift is caused due to the difference in the relative velocity/motion between the transmitter and the receiver. This scenario is encountered in day to day mobile communications.

A number of simulation algorithms are proposed for generation of correlated Rayleigh random variables. David J.Young and Norman C Beaulieu proposed a method in their paper titled “The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform”[1] based on the inverse discrete Fourier transform (IDFT). It is a modification of the Smith’s algorithm which is normally used for Rayleigh fading simulation. This method requires exactly one-half the number of IDFT operations and roughly two-thirds the computer memory of the original method – as the authors of the paper claims.

Rayleigh Fading can be simulated by adding two Gaussian Random variables as mentioned in my previous post. The effect of Doppler shift is incorporated by modeling the Doppler effect as a frequency domain filter.

The model proposed by Young et.al is shown below.

The Fading effect + Doppler Shift is simulated by multiplying the Gaussian Random variables and the Doppler Shift’s Frequency domain representation. Then IDFT is performed to bring them into time domain representation. The Doppler Filter used to represent the Doppler Shift effect is derived in Young’s paper.

The equation for the Doppler Filter is :

$\dpi{130} F_M[k] = \begin{cases} \0, &k = 0\\ \sqrt{\frac{1}{2\sqrt{1-\left (\frac{k}{Nf_m} \right )^{2}}}}, & k =1,2,...,k_m-1\\ \sqrt{\frac{k_m}{2}\left[\frac{\pi}{2}-\arctan \left(\frac{k_m-1}{\sqrt{2k_m-1}} \right ) \right ]}, & k=k_m\\ 0, & k=k_m+1,...,N-k_m-1\\ \sqrt{\frac{k_m}{2}\left[\frac{\pi}{2}-\arctan \left(\frac{k_m-1}{\sqrt{2k_m-1}} \right ) \right ]}, & k=N-k_m\\ \sqrt{\frac{1}{2\sqrt{1-\left (\frac{N-k}{Nf_m} \right )^{2}}}},& k =N-k_m+1,...,N-2,N-1\end{cases}$

Matlab Code

Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan

Reference:

[1] D.J. Young and N.C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete fourier transform,” IEEE transactions on Communications, vol. 48, pp. 1114-1127, July 2000.

External Resources

• Jasper Hatilima

Hey Mathuranathan, I am implementing differential space time modulation and it’s purpose is to do detection without CSI. I have managed to implement it with a time invariant channel but now I want to see how it performs in a slow time varying channel at various doppler spreads….and the theory says I should use the Jakes model to have a relationship between one channel matrix to the next…by multiplying one channel matrix by some value obtained through zeroth order bessel function of first kind….But it just doesn’t give sensible results 🙁 Kindly assist on this implementation of a time varying channel that depends on doppler spread using Jakes model.

• Upul

Hi,

Could you please explain how we can use FFT and IFFT in order to solve specific problem in discrete domain? I’m glad if you can publish another post for that.

Thanks.

• Mathuranathan

Can you elaborate ?

• Upul

Basically I just need to know how FFT and IFFT works to compute Fourier transform and Inv. Fourier transform in discrete domain.

As a example ,lets say we want to model OFDM Transmission using FFT/IFFT (lets say in MatLab). Can you explain the approach I should follow in order to use FFT and IFFT at receiver and transmitter with out using direct formulas?

Thank You.

• Mathuranathan

Hi Upul,