Rayleigh and Ricean Fading

When there is no line-of-sight (LOS) path, the channel coefficient is:

$$h \sim \mathcal{CN}(0, 1) \implies |h| \sim \text{Rayleigh}$$

The envelope PDF is $f(r) = 2r\,e^{-r^2}$ for $r \ge 0$ (unit power), and the power $|h|^2 \sim \text{Exp}(1)$.

rng = np.random.default_rng(42)
N = 100000
h = (rng.standard_normal(N) + 1j*rng.standard_normal(N)) / np.sqrt(2)
# |h| is Rayleigh, |h|^2 is Exponential(1)
print(f"Mean |h|^2 = {np.mean(np.abs(h)**2):.4f}")  # ~1.0

When a LOS component exists, the channel is Ricean:

$$h = \sqrt{\frac{K}{K+1}} + \sqrt{\frac{1}{K+1}}\,w, \quad w \sim \mathcal{CN}(0,1)$$

where $K$ is the Ricean $K$-factor (ratio of LOS to scattered power). $K = 0$ gives Rayleigh; $K \to \infty$ gives AWGN.

def ricean_channel(K, N, rng=None):
    if rng is None:
        rng = np.random.default_rng()
    w = (rng.standard_normal(N) + 1j*rng.standard_normal(N)) / np.sqrt(2)
    h = np.sqrt(K/(K+1)) + np.sqrt(1/(K+1)) * w
    return h

The Nakagami-m distribution generalizes Rayleigh ($m=1$) and approximates Ricean ($m = (K+1)^2/(2K+1)$). The power PDF is:

$$f(x) = \frac{m^m x^{m-1}}{\Gamma(m)\Omega^m} e^{-mx/\Omega}, \quad x \ge 0$$

where $\Omega = E[|h|^2]$. Useful when fitting measured data.

from scipy.stats import nakagami
m, Omega = 2.0, 1.0
samples = nakagami.rvs(m, scale=np.sqrt(Omega), size=10000)

For a mobile receiver moving at speed $v$, the maximum Doppler frequency is $f_d = v f_c / c$. The Clarke-Jakes autocorrelation is:

$$R_h(\Delta t) = J_0(2\pi f_d \Delta t)$$

and the Doppler power spectrum is:

$$S(f) = \frac{1}{\pi f_d \sqrt{1 - (f/f_d)^2}}, \quad |f| < f_d$$

from scipy.special import j0
fd = 100  # Hz (60 km/h at 1.8 GHz)
dt = np.linspace(0, 0.01, 200)
R_h = j0(2 * np.pi * fd * dt)

The coherence time $T_c$ is the time duration over which the channel remains approximately constant:

$$T_c \approx \frac{0.423}{f_d} = \frac{0.423 c}{v f_c}$$

The coherence bandwidth $B_c$ is the frequency range over which the channel has approximately flat gain:

$$B_c \approx \frac{1}{5\tau_{\text{rms}}}$$

where $\tau_{\text{rms}}$ is the RMS delay spread.