Random Processes and Noise

Communication signals are received with noise. Describing both the message and the noise as random processes lets us predict the performance of a system.

Random processes

A random process X(t) is a family of random signals.

• Mean μ_X(t) = E[X(t)]; autocorrelation R_X(t₁, t₂) = E[X(t₁)X(t₂)].
• Wide-sense stationary (WSS): constant mean, and R_X depends only on τ = t₁ − t₂.
• Ergodic: time averages equal ensemble averages.
• R_X(0) = E[X²] = average power; R_X(τ) is even and maximum at τ = 0.

Power spectral density

Wiener–Khinchin theorem: the PSD S_X(f) is the Fourier transform of R_X(τ).

• Power = ∫ S_X(f) df = R_X(0).
• Through an LTI system H(f): S_Y(f) = |H(f)|² S_X(f); output mean = H(0) × input mean.

White noise: S(f) = N₀/2 for all f; R(τ) = (N₀/2)δ(τ). Through an ideal low-pass filter of bandwidth B, the noise power is N₀B.

Gaussian processes stay Gaussian through LTI systems.

Physical noise

• Thermal (Johnson) noise: available power P = kTB; open-circuit RMS voltage v_n = √(4kTRB). k = 1.38 × 10⁻²³ J/K. At 290 K, kT = −174 dBm/Hz.
• Shot noise: i_n² = 2qIB (diodes, transistors).

Noise figure and temperature

• Noise figure F = (SNR_in)/(SNR_out) ≥ 1 (NF in dB = 10 log F).
• Equivalent noise temperature T_e = (F − 1)T₀, with T₀ = 290 K.
• Friis formula for cascaded stages: F = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁G₂) + …. The first stage dominates, so receivers use a low-noise amplifier first.

Worked example

Find the thermal noise power in a 1 MHz bandwidth at 290 K.

• P = 1.38 × 10⁻²³ × 290 × 10⁶ = 4.0 × 10⁻¹⁵ W = −114 dBm.

Practice