How to implement Rayleigh fading channel in an FPGA – Part 1: Generation of correlated Rayleigh random variables

[1], Young et al [2], and Chung et al [3]. This blog post discusses the generation of discrete-time correlated Rayleigh samples using a simple algorithm based on the inverse discrete Fourier transform (IDFT) in [2].

Generation correlated Rayleigh fading

In the simulation of mobile wireless systems, the generation of discrete-time Rayleigh random samples need to be correlated due to the variation nature of the Doppler effect between the transmitter and the receiver as well as the propagation of the signal in a wireless environment. Therefore, the Rayleigh fading process must be correlated. Authors in [2] proposed a simple and efficient method to generate correlated Rayleigh fading based on an IDFT approach. The idea is to use Doppler filter sequences F[k] to filter identically distributed (i.i.d) Gaussian random variables in the frequency-domain. The resulting filtered samples are then transformed in time-domain by a complex IDFT to generate the discrete-time correlated Rayleigh samples x[k], as shown in Figure 1.

The Doppler filter coefficients F[k] are designed to approximate the theoretical autocorrelation function of the scattered received signal as follows. A discrete representation of the theoretical autocorrelation function of the scattered signal is a zero-order Bessel function of the first kind defined in [2].

Figure 1: Discrete-time correlated Rayleigh sample generation algorithm

where ƒ'_{m =} ƒ_{m /} ƒ_s is the normalized maximum Doppler shift ƒ_m by the sampling frequency ƒ_s and J₀(.) denotes a zero-order Bessel function of the first kind.

where N is number of i.i.d Gaussian random variables and .

As a result, simulation of this algorithm with different IDFT sizes using a normalized Doppler shift of  ƒ'_m = 0.05 is shown in Figure 2.

Figure 2: Auto-correlation of the correlated Rayleigh generation

It clearly shows that an IDFT size higher than 1024 provides the auto-correlation property closest to the theoretical discrete normalized autocorrelation function (blue curve).

Figure 3 (a) and (b) show the fading envelope/phase of the first 500 samples and its normalized density function compared with the theoretical Rayleigh probability density function (PDF) over 2²⁰ generated samples. It turns out that an IDFT size of 1024 is an efficient choice, offering a good trade-off between performance and complexity of implementation with a 1024-point inverse fast Fourier transform (IFFT).

(a) Fading envelope and phase

(b) PDF of generated Rayleigh fading

Figure 3: Correlated Rayleigh fading envelop, phase and its PDF

Conclusion

There are existing avenues of generating correlated Rayleigh random variables. This blog post discussed a simple and efficient method for generating the correlated Rayleigh random variables. The next blog post in this series will introduce an approach for implementing the discrete-time correlated Rayleigh samples on the Nutaq Perseus platform.

References

[1]	L.C. Tran and T.A. Wysocki and J. Seberry and A. Mertins, "A generalized algorithm for the generation of correlated Rayleigh fading envelopes in radio channels ," IEEE International Parallel and Distributed Processing Symposium, 2005.
[2]	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, no. 7, pp. 1114-1127, 2000.
[3]	Wei-Ho Chung and R.E. Hudson and K. Yao, "A unified approach for generating cross-correlated and auto-correlated MIMO fading envelope processes ," IEEE Transactions on Communications, vol. 57, no. 11, pp. 3481 – 3488 , 2009.