Jakes' Model

Jakes' model is a deterministic model to simulate Rayleigh fading.
The classical paper is Dent' 1993 Modified Jakes's Model published in Electronic Letter.

Jakes model simplified the all angle inputs (oscillators) to a quadrant input (oscillators). The in-phase and quadrature components of each oscillator are cos(be_n) and sin(be_n). So the x-corr of I-Q is zero by the orthogonality.

The difference of Den't paper with Jakes' 1974 paper is using Walsh function in the azimuth domain to achieve orthogonality between waves (delay paths). So delay paths satisfies statistical independent, that actually is achieved by azimuth orthogonality.


In summary, there are three types of orthogonalities that are used in Dent Model.

1. Oscillator orthogonality: cos(w_1 t), cos(w_2 t), ... are orthogonal if w_1 is not equal to w_2. This is true when the running time is larger than the largest period (smallest frequency).
2. Azimuth orthogonal: waves inner product has oscillator components in all angle. By azimuth orthogonal functions, the inner product is zero.
3. I-Q orthogonality: similar to Azimuth orthogonal, IQ are selected to be cos and sin.

Given the three orthogonalities, it is obvious that Li-Guan's 2000 IEEE paper is incorrect.
The azimuth orthogonality is not achieved because the author selected half cycle is as azimuth function. The right theta should be 2*pi*n*j/N0. The paper missed the 2 factor.

The good thing of the paper is that the angle amplitude is not +-1 as Dent model. The azimuth amplitude is distributed over a range, that matches the better assumption that the amplitude distribution around the angle.

Issues in simulation:

1. The running time must be long enough to achieve oscillator orthogonality. The smallest angular frequency is wm * cos (al_n). al_n is close to pi/2 when n --> N0. The smaller angular frequency has longer period of time. The upshot is if we set N0 too big, the last few oscillator needs very long period to be orthogonal.
2. Random phase: the random phase can be included in the oscillator, as Dent model. Li-Guan's model used incorrect j*pi/4. That loses the flexibility of random phase in the oscillator.