Random signals, correlation and power spectral density

Describes random signals by mean, autocorrelation and power spectral density, links them by the Wiener–Khinchin theorem, and computes noise power through LTI systems.

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Why it matters

Every measurement carries noise — thermal noise in a bridge resistor, quantisation noise in an ADC, turbulence in a flow signal. You cannot predict its value, but you can predict its average power and how that power is spread over frequency. Correlation and power spectral density (PSD) are the tools for estimating noise in an instrument's bandwidth, recovering weak signals, and measuring delays — for example leak location or cross-correlation flow meters.

Key ideas

Random process. A random signal is one realisation of a random process X(t): a family of possible waveforms, each with some probability. It is described statistically — mean μX(t) = E[X(t)], mean square E[X²(t)], and autocorrelation RX(t1, t2) = E[X(t1)X(t2)].

Stationarity

  • Strict-sense stationary: all statistics are unchanged by a shift in time.
  • Wide-sense stationary (WSS): the mean is constant and the autocorrelation depends only on the lag τ = t2 − t1. Most engineering analysis assumes WSS.
  • Ergodic: time averages over one long record equal ensemble averages. This is what lets you measure the mean, power and autocorrelation from a single recording.

Autocorrelation RX(τ) of a WSS process — properties:

  • RX(0) = E[X²] = total average power (mean-square value).
  • Even: RX(−τ) = RX(τ); maximum at zero: |RX(τ)| ≤ RX(0).
  • If X has a non-zero mean and no periodic part, RX(τ) → μX² as τ → ∞.
  • A periodic component in X gives a periodic component in RX(τ) with the same period.
  • Variance σX² = RX(0) − μX².

Cross-correlation RXY(τ) = E[X(t)Y(t + τ)] measures how one signal resembles a shifted version of another. If y(t) is a delayed, noisy copy of x(t), RXY(τ) peaks at the delay. Note RXY(τ) = RYX(−τ); the definition order matters, so check your textbook's convention.

Power spectral density. SX(f) tells how the power is distributed over frequency (unit: V²/Hz for a voltage). Wiener–Khinchin theorem: the PSD is the Fourier transform of the autocorrelation. It is real, even and non-negative, and its total area equals RX(0). A DC mean shows up as an impulse μX² δ(f) at zero frequency.

White noise. A flat PSD SX(f) = N0/2 (two-sided) for all f, so RX(τ) = (N0/2) δ(τ): samples at any two different instants are uncorrelated. Ideal white noise has infinite power; real noise is white only over a band. Thermal noise in a resistor has a one-sided PSD 4kTR V²/Hz. In discrete time, a white sequence of variance σ² has R[m] = σ² δ[m] and a flat PSD equal to σ².

LTI systems with random inputs. If a WSS process passes through an LTI system H(f), the output is WSS with mean μY = μX·H(0) and PSD SY(f) = |H(f)|² SX(f). This is the basis of noise-bandwidth calculations.

Formulas

  • RX(τ) = E[X(t) X(t + τ)] — autocorrelation (WSS); RX(τ) = lim(T→∞) (1/2T) ∫ from −T to T of x(t)x(t + τ) dt for an ergodic process.
  • RXY(τ) = E[X(t) Y(t + τ)] — cross-correlation.
  • SX(f) = ∫ RX(τ) e^(−j2πfτ) dτ, RX(τ) = ∫ SX(f) e^(j2πfτ) df — Wiener–Khinchin.
  • P = RX(0) = ∫ SX(f) df (−∞ to ∞) — total power.
  • σX² = RX(0) − μX² — variance.
  • e^(−a|τ|) ↔ 2a/(a² + (2πf)²) — common autocorrelation–PSD pair.
  • (A²/2) cos(2πf0τ) ↔ (A²/4)[δ(f − f0) + δ(f + f0)] — sinusoid with random phase.
  • SY(f) = |H(f)|² SX(f), μY = μX H(0) — LTI system output.
  • PY = (N0/2) ∫ |H(f)|² df = N0·B_N |H(0)|² — white noise through a filter; B_N is the noise-equivalent bandwidth.
  • B_N = 1/(4RC) — noise bandwidth of a first-order RC low-pass.
  • v_rms² = 4kTRB — thermal (Johnson) noise in bandwidth B.

Symbols: τ lag (s); f frequency (Hz); μX mean (V); σX² variance (V²); RX (V²); SX (V²/Hz); N0/2 two-sided white-noise PSD (V²/Hz); a decay rate (s⁻¹); H(f) frequency response; B, B_N bandwidths (Hz); k = 1.38 × 10⁻²³ J/K; T absolute temperature (K); R resistance (Ω); C capacitance (F).

Worked examples

Example 1 (standard). A WSS voltage has RX(τ) = 9 + 4e^(−2|τ|) V². Find the mean, total power, variance and PSD.

  1. As τ → ∞, RX → 9 = μX², so |μX| = 3 V (the sign cannot be found from RX).
  2. Total power RX(0) = 9 + 4 = 13 V².
  3. Variance σX² = 13 − 9 = 4 V² (so σ = 2 V).
  4. PSD: SX(f) = 9 δ(f) + 4 × 2·2/(2² + (2πf)²) = 9 δ(f) + 16/(4 + 4π²f²) V²/Hz.
  5. Check: area of the continuous part = 4 × ∫ 4/(4 + 4π²f²) df = 4 ✓. μ = ±3 V, P = 13 V², σ² = 4 V².

Example 2 (GATE level). White noise with two-sided PSD N0/2 = 10⁻⁶ V²/Hz drives an RC low-pass filter with R = 1 kΩ, C = 1 µF. Find the output noise power, its rms value and the noise-equivalent bandwidth.

  1. RC = 10³ × 10⁻⁶ = 1 ms; |H(f)|² = 1/(1 + (2πfRC)²).
  2. ∫ |H(f)|² df over all f = 1/(2RC) = 500 Hz.
  3. PY = (N0/2) × 1/(2RC) = 10⁻⁶ × 500 = 5 × 10⁻⁴ V².
  4. v_rms = √(5 × 10⁻⁴) = 0.0224 V = 22.4 mV.
  5. Noise bandwidth B_N = 1/(4RC) = 250 Hz (check: N0 B_N = 2 × 10⁻⁶ × 250 = 5 × 10⁻⁴ ✓), larger than the 3 dB bandwidth 1/(2πRC) = 159 Hz.
  6. PY = 5 × 10⁻⁴ V², 22.4 mV rms, B_N = 250 Hz.

Common mistakes

  • Forgetting that RX(0) is the total power, including DC.
  • Taking the PSD of e^(−a|τ|) as 1/(a + j2πf) — that is the one-sided exponential; the two-sided pair is 2a/(a² + (2πf)²).
  • Mixing one-sided (N0) and two-sided (N0/2) PSDs — a factor-of-2 error in noise power.
  • Using the 3 dB bandwidth instead of the noise-equivalent bandwidth for noise power.
  • Assuming "uncorrelated" means "independent" — it does only for Gaussian processes.
  • Writing SY = H(f) SX(f); the PSD transfers through |H(f)|².

For GATE IN

Expect: PSD from a given autocorrelation (and vice versa), mean, variance and power from RX(τ), output PSD and noise power when white noise passes through an RC or ideal filter, properties of valid autocorrelation and PSD functions, and delay estimation from a cross-correlation peak. Practise the e^(−a|τ|) pair and the RC noise-bandwidth result.

Quick check

  1. What is the PSD at f = 0 of RX(τ) = e^(−2|τ|)?
  2. Can RX(τ) = e^(−τ) (for all τ) be an autocorrelation function?
  3. White noise of N0/2 = 1 µV²/Hz through an ideal low-pass of 1 kHz — output power?
  4. What does a periodic component in RX(τ) indicate?
  5. RX(0) = 10 V², μX = 2 V — variance? Answers: 1. 2·2/(4 + 0) = 1 V²/Hz. 2. No — it is not even and grows for negative τ. 3. 10⁻⁶ × 2000 = 2 × 10⁻³ V². 4. A periodic component in the signal. 5. 10 − 4 = 6 V².

Interview questions

All Signals and Systems interview questions

Try answering each one aloud before you open it.

  1. 1.What is a random signal in the context of signals and systems?Concept

    A random signal is a type of signal whose values cannot be precisely predicted and are characterized by some probability distribution. Unlike deterministic signals, random signals have inherent uncertainty and are often described using statistical measures such as mean, variance, and autocorrelation.

  2. 2.Explain the concept of correlation in signals and systems.Concept

    Correlation in signals and systems refers to the measure of similarity between two signals as a function of the time-lag applied to one of them. It is used to identify the degree to which two signals are related. The correlation can be autocorrelation (a signal with itself) or cross-correlation (between two different signals).

  3. 3.What is power spectral density (PSD) and why is it important?Concept

    Power spectral density (PSD) is a measure of a signal's power content versus frequency. It provides insight into how the power of a signal is distributed across different frequency components. PSD is important for analyzing the frequency characteristics of signals, especially in the presence of noise, and is widely used in telecommunications and signal processing.

  4. 4.How is the autocorrelation function related to the power spectral density?Concept

    The autocorrelation function and the power spectral density are related through the Wiener-Khinchin theorem. This theorem states that the power spectral density of a stationary random process is the Fourier transform of its autocorrelation function. This relationship allows us to analyze signals in both time and frequency domains.

  5. 5.Why is the concept of correlation used in communication systems?Application

    Correlation is used in communication systems to detect signals in the presence of noise, synchronize signals, and perform channel estimation. By measuring how similar a received signal is to a known reference signal, systems can effectively extract the desired information even when the signal is corrupted by noise.

  6. 6.How can you determine if a signal is wide-sense stationary?Application

    A signal is considered wide-sense stationary if its mean and variance are constant over time, and its autocorrelation function depends only on the time difference between two points, not on the actual time at which the function is evaluated. This implies that the statistical properties of the signal do not change over time.

  7. 7.Calculate the autocorrelation of a signal x(t) = A cos(ωt + φ).Numerical

    The autocorrelation of x(t) = A cos(ωt + φ) is given by R(τ) = (A²/2) cos(ωτ). This result is derived by considering the periodic nature of the cosine function and using trigonometric identities to simplify the expression.

  8. 8.A discrete-time white-noise sequence has zero mean and variance 4. What are its autocorrelation and power spectral density?Numerical

    For white noise, samples at different times are uncorrelated, so the autocorrelation is R[m] = σ²δ[m] = 4δ[m]. Its DTFT is flat: S(e^(jω)) = 4 for all ω, and (1/2π)∫S dω over one period gives back the total power 4. In continuous time, ideal white noise is instead specified by its PSD N0/2 and has infinite variance, so a finite variance only makes sense for discrete-time or band-limited noise.

  9. 9.Explain how the cross-correlation function can be used to measure the time delay between two signals.Application

    The cross-correlation function measures the similarity between two signals as a function of the time-lag applied to one of them. By finding the time-lag at which the cross-correlation function reaches its maximum value, we can determine the time delay between the two signals. This is useful in applications like radar and sonar for determining the distance to an object.

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