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Probabilistic Distribution in Communication Technology

This tutorial visualizes the relationship between Gaussian and Rayleigh distributions in radio communications by simulating a complex baseband signal affected by Additive White Gaussian Noise (AWGN). The GaussianRayleighenvelope (magnitude R) when there is no line-of-sight (NLOS). Adding a direct path yields the Rician

Mathematical Foundation

Complex baseband. The received sample is z = I + j Q. With no LOS, I and Q are independent Gaussian random variables with mean 0 and variance σ2. The magnitude R = √(I2 + Q2) then follows a Rayleighphase θ = atan2(Q, I) is uniform on [0, 2π).

Gaussian (I or Q). p(x) = (1 / (σ√(2π))) exp(−x2/(2σ2)).

Rayleigh (magnitude R). p(r) = (r/σ2) exp(−r2/(2σ2)), r ≥ 0. This is the distribution of the distance from the origin when (I, Q) is circular-symmetric Gaussian.

Rician (LOS present). When a line-of-sight component of amplitude A is added (e.g. I = A + nI, Q = nQ), the magnitude R follows a Ricianp(r) = (r/σ2) exp(−(r2 + A2)/(2σ2)) I0(rA/σ2), where I0 is the modified Bessel function of the first kind. The K-factor (in dB) is the ratio of LOS power to scattered power: K = 10 log10(A2/(2σ2)). As K → 0, the Rician becomes Rayleigh; as K → ∞, the envelope approaches a Gaussian around A.

Box–Muller transform. To generate Gaussian samples, we use I = σ√(−2 ln U1) cos(2πU2), Q = σ√(−2 ln U1) sin(2πU2) with U1, U2 uniform on (0, 1].

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0.0 dB
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Mean power: 0 PAPR: 0 dB Current R: 0
Constellation: R density (top-left), I distribution (top), Q distribution (left), 2D I/Q cloud; red vector R, θ
Time domain: I (red), Q (blue), Envelope R (yellow)
PDF: Gaussian (I) and Rayleigh/Rician (R) with theoretical curves

 

Usage

σ (sigma): Standard deviation of the Gaussian noise on I and Q. Increasing σ spreads the constellation cloud and widens the Rayleigh/Rician PDF. K (dB): K-factor (0–20 dB). At 0 dB the channel is Rayleigh (NLOS); as K increases, a line-of-sight component is added and the constellation shifts right; the magnitude PDF changes from Rayleigh to Rician. Freeze: Pause sampling to inspect the current frame; click again to resume.

Constellation: Composite plot: top-left shows R density (envelope histogram, real-time); top strip shows I distribution (marginal, real-time); left strip shows Q distribution (marginal, real-time, S-curve); center is the 2D I/Q scatter and density. The red vector and dot mark the latest sample (magnitude R and phase θ). With K = 0 the cloud is centered at the origin (Rayleigh); with K > 0 it shifts toward (A, 0) (Rician). I/Q axes use a fixed scale so σ effect is visible. Time domain: Auto-scrolling trace of I (red), Q (blue), and envelope R (yellow); always shows the most recent 400 samples. Deep fades (R near zero) are common in Rayleigh and rarer in Rician. PDF: Left histogram is the I component with the theoretical Gaussian curve overlay; right histogram is the magnitude R with the theoretical Rayleigh (K = 0) or Rician (K > 0) curve. Histograms are normalized so area under bars is 1.

Key Concepts

  • Gaussian I, Q (from Box–Muller) → Rayleigh magnitude when no LOS; phase is uniform.
  • Rayleigh: typical for NLOS multipath; envelope has a single peak away from zero.
  • Rician: LOS + scattered paths; K-factor is LOS power / scattered power; as K increases, the envelope PDF becomes more Gaussian around the LOS amplitude.
  • PAPR (Peak-to-Average Power Ratio) in dB indicates how much the instantaneous power fluctuates; important for power amplifier design.