Random Processes, Autocorrelation and PSD

Random processes model noise and information signals statistically — the mathematical basis of the Communications section (~10 marks). GATE EC tests stationarity, autocorrelation and power spectral density (1–2 marks).

Key results

• Wide-sense stationary (WSS): mean constant, autocorrelation R_X(τ) depends only on lag τ.
• Autocorrelation R_X(τ) = E[X(t)X(t+τ)]; R_X(0) = average power = E[X²]; R_X(τ) is even, |R_X(τ)| ≤ R_X(0).
• Wiener-Khinchin: PSD S_X(f) = Fourier transform of R_X(τ); total power = ∫S_X(f)df = R_X(0).
• Ergodic: time averages = ensemble averages.
• Output of LTI system: S_Y(f) = |H(f)|²·S_X(f); R_Y = R_X ∗ h ∗ h(−).
• White noise: flat PSD S(f) = N₀/2, R(τ) = (N₀/2)δ(τ) (uncorrelated).

Quantity	Relation
R_X(0)	average power = E[X²]
S_X(f)	FT of R_X(τ) (Wiener-Khinchin)
LTI output PSD
White noise PSD	N₀/2 (flat)

Exam Tricks & Tips

• 🎯 R_X(0) equals the total average power — a fast way to get power from autocorrelation.
• 🎯 Wiener-Khinchin: PSD is the Fourier transform of the autocorrelation (and vice versa).
• 🎯 Through an LTI filter H(f), the output PSD is |H(f)|²·S_X(f) — magnitude-squared scaling.
• 🎯 Autocorrelation is even (R(−τ)=R(τ)) and peaks at τ=0.
• 🎯 White noise has a flat PSD N₀/2 and impulse autocorrelation → any two distinct instants are uncorrelated.
• ❌ Common mistake: using H(f) instead of |H(f)|² for the output PSD of an LTI system.

Expected exam pattern

1–2 marks: find average power from R_X(0), PSD via Wiener-Khinchin, output PSD/power through a filter, or test WSS.

Quick recap

WSS: constant mean, R_X(τ) lag-only. R_X(0)=power; S_X(f)=FT{R_X} (Wiener-Khinchin). LTI output S_Y=|H|²S_X. White noise: flat N₀/2, δ-autocorrelation. R_X even, peaks at 0.

Cover the answer, recall, then check. 11 cards on random processes, autocorrelation and PSD for GATE EC.

Q1. Define a wide-sense stationary (WSS) process.
A1. Its mean is constant and its autocorrelation depends only on the time lag τ, not absolute time.

Q2. What does R_X(0) represent?
A2. The average power of the process, E[X²(t)].

Q3. State the Wiener-Khinchin theorem.
A3. The power spectral density S_X(f) is the Fourier transform of the autocorrelation R_X(τ).

Q4. Two key properties of the autocorrelation function?
A4. It is even (R(−τ) = R(τ)) and maximum at τ = 0 (|R(τ)| ≤ R(0)).

Q5. Output PSD of an LTI system with transfer function H(f)?
A5. S_Y(f) = |H(f)|²·S_X(f).

Q6. What is the PSD and autocorrelation of white noise?
A6. Flat PSD S(f) = N₀/2 and autocorrelation R(τ) = (N₀/2)δ(τ).

Q7. Define an ergodic process.
A7. One whose time averages equal its ensemble (statistical) averages.

Q8. How do you get total power from the PSD?
A8. Integrate: P = ∫_{−∞}^{∞} S_X(f) df = R_X(0).

Q9. Why are two distinct samples of white noise uncorrelated?
A9. Its autocorrelation is an impulse at τ = 0, so R(τ) = 0 for any τ ≠ 0.

Q10. Is every WSS process stationary in the strict sense?
A10. Not necessarily — strict-sense stationarity implies WSS, but WSS does not imply strict-sense (except for Gaussian processes).

Q11. Units of power spectral density?
A11. Watts per hertz (W/Hz) — power distributed over frequency.

Real communication signals and noise are random — we can't write them as fixed functions, only describe their statistics. Random-process theory gives the tools: autocorrelation captures how a signal relates to a delayed copy of itself, and its Fourier transform (the power spectral density) shows where the power sits in frequency. GATE tests the Wiener-Khinchin relation and power/PSD calculations.

Core concept: A random process is an ensemble of signals; a wide-sense stationary process is characterised by its mean and autocorrelation R(τ), whose Fourier transform is the power spectral density S(f).

The theory

Beginner — describing randomness

A random process X(t) assigns a random signal to each outcome. Key statistics:

• Mean μ = E[X(t)].
• Autocorrelation R(τ) = E[X(t)·X(t+τ)] — correlation between the process now and τ later.
• Wide-Sense Stationary (WSS): mean is constant and R(τ) depends only on the lag τ (not absolute time).
• Ergodic: time averages equal ensemble averages — lets us estimate statistics from one long record.

Intermediate — autocorrelation and PSD

For a WSS process, the Wiener-Khinchin theorem:
S(f) = FourierTransform{R(τ)} and R(τ) = inverse FT of S(f).
Properties:

• R(0) = E[X²] = average power (also = ∫S(f)df).
• R(τ) is even, and |R(τ)| ≤ R(0) (max at zero lag).
• S(f) ≥ 0 and is even for a real process; its area is the total power.

Advanced — through LTI systems and white noise

• Output of an LTI system with frequency response H(f): the output PSD is S_out(f) = |H(f)|²·S_in(f). Output power = ∫|H(f)|²S_in(f)df.
• White noise: flat PSD S(f) = N0/2 (two-sided) at all frequencies, so R(τ) = (N0/2)δ(τ) — samples at different times are uncorrelated. It has infinite total power (an idealisation), band-limited in practice.
• Cross-correlation and cross-PSD describe two jointly WSS processes; input-output cross-PSD S_xy(f) = H(f)S_x(f) is used in system identification.

Worked example

White noise of PSD N0/2 = 10⁻⁹ W/Hz passes through an ideal low-pass filter of bandwidth B = 10 kHz and gain 1. Find the output noise power.
Output PSD = |H|²·S_in = 1·(N0/2) over ±B (two-sided) → power = (N0/2)·(2B) = N0·B.
Power = 2·(10⁻⁹)·(10⁴) = 2×10⁻⁵ W (20 µW).
Equivalently, using one-sided: N0 = 2×10⁻⁹ W/Hz, power = N0·B = 2×10⁻⁹×10⁴ = 2×10⁻⁵ W. ✔

GATE relevance

1–2 mark NAT/MCQ: autocorrelation ↔ PSD (Wiener-Khinchin), average power from R(0) or ∫S(f)df, output PSD/power through an LTI filter (|H|²S), and white-noise properties. Feeds directly into SNR and probability-of-error problems.

Exam tricks and shortcuts

• Power = R(0) = ∫S(f)df; S(f) = FT of R(τ).
• Through a filter: S_out = |H(f)|²·S_in; integrate for power.
• White noise: S = N0/2 (two-sided), R(τ) = (N0/2)δ(τ), uncorrelated samples.
• Mnemonic: "PSD is the Fourier transform of autocorrelation" (Wiener-Khinchin).

Key formulas

• Autocorrelation: R_X(τ) = E[X(t)X(t+τ)]; R_X(0) = average power.
• Power spectral density: S_X(f) = FT{R_X(τ)} (Wiener–Khinchin).
• WSS process: mean constant, autocorrelation depends only on τ.
• Ergodic: time averages = ensemble averages.
• Power: P = ∫S_X(f)df = R_X(0).

The PSD (Fourier transform of autocorrelation) describes how a random signal's power is distributed over frequency.

Worked Example

Problem: A wide-sense-stationary random process X(t) has autocorrelation function R_X(τ) = 5·e^(−2|τ|). Find (a) the average (total) power of the process, and (b) its power spectral density S_X(ω).

Solution:
(a) The average power of a WSS process equals its autocorrelation at zero lag:
P = R_X(0) = 5·e^(0) = 5 W.
(This is E[X²(t)], the mean-square value.)

(b) By the Wiener–Khinchin theorem the PSD is the Fourier transform of the autocorrelation:
S_X(ω) = ∫_(−∞)^(∞) R_X(τ) e^(−jωτ) dτ.

Use the standard transform pair: A·e^(−a|τ|) ↔ 2Aa/(ω² + a²).
Here A = 5 and a = 2, so:
S_X(ω) = (2)(5)(2)/(ω² + 2²) = 20/(ω² + 4).

Check by inverse relation: total power = (1/2π)∫S_X(ω)dω = (1/2π)·(20·π/2) = 5 W ✓ (using ∫dω/(ω²+4) = π/2).

Answer: (a) Average power = 5 W; (b) S_X(ω) = 20/(ω² + 4).