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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 Gaussian distribution describes the noise on the Real (I) and Imaginary (Q) components; the Rayleigh distribution describes the envelope (magnitude R) when there is no line-of-sight (NLOS). Adding a direct path yields the Rician distribution.

Sections

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 Rayleigh distribution, and the phase θ = atan2(Q, I) is uniform on [0, 2π).

The component (Gaussian) and envelope (Rayleigh) densities are:

Gaussian (I or Q): p(x) = (1 / (σ√(2π))) exp(−x2/(2σ2))
Rayleigh (R): p(r) = (r/σ2) exp(−r2/(2σ2)),  r ≥ 0

The Rayleigh density 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 (I = A + nI, Q = nQ), the magnitude follows a Rician distribution, parameterized by the K-factor (ratio of LOS to scattered power):

p(r) = (r/σ2) exp(−(r2 + A2)/(2σ2)) I0(rA/σ2),   K = 10 log10(A2/(2σ2)) dB

where I0 is the modified Bessel function of the first kind. As K → 0 the Rician becomes Rayleigh; as K → ∞ the envelope approaches a Gaussian around A.

Box–Muller transform. Gaussian samples are generated from two uniforms U1, U2 on (0, 1]:

I = σ√(−2 ln U1) cos(2πU2),  Q = σ√(−2 ln U1) sin(2πU2)

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

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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.

Limitations

  • Statistics only, no channel dynamics. Each sample is drawn independently, so the demo shows the amplitude/phase distributions of a fading channel but not its time correlation, Doppler spectrum, or delay spread (see [[jakesi]] / [[fadingi]] for time-varying fading).
  • Flat, single-tap fading. One complex coefficient per sample — frequency-selective fading and multipath delay profiles are not represented.
  • Idealized Gaussian noise. Box–Muller produces perfect i.i.d. Gaussian I/Q with no impulsive noise, phase noise, or quantization; real receivers see additional impairments.
  • Rician via a static LOS amplitude. The LOS component is a fixed real amplitude A on the I axis; a rotating/random-phase specular component and multiple specular paths are not modeled.
  • Empirical histograms vs. theory. Finite sample counts make the histograms fluctuate around the theoretical curves; convergence to the exact PDF requires many samples.
  • Normalized units. σ, A, and power are dimensionless display values, so PAPR and "mean power" are illustrative rather than tied to a physical link budget.