Questions: Autocorrelation Function Properties and Estimation

Question 1

You compute the ACF of a signal and observe that it decays slowly, remaining significantly nonzero for lags up to several seconds. What does this pattern indicate?

• A. The signal is white noise, since white noise has a broad ACF spread across many lags
• B. The signal has long-range temporal correlation — samples separated by seconds are still statistically related
• C. The ACF computation has an error; a correct ACF always decays to zero within a few samples
• D. The signal is periodic with a period equal to the lag at which the ACF first crosses zero

Answer: (B)

A slowly decaying ACF indicates long-range dependence — the signal has memory, and distant samples are still correlated. This is the signature of an AR process with poles near the unit circle. White noise has an impulsive ACF (maximum at lag zero, zero everywhere else) because its samples are uncorrelated by definition. A periodic signal would show an oscillating ACF, not a monotone decay.

Question 2

For a finite data record, why is the biased ACF estimator (dividing by N regardless of lag) generally preferred over the unbiased estimator (dividing by N − |τ|)?

• A. The biased estimator is mathematically unbiased at all lags, making it more accurate
• B. The biased estimator always produces a valid positive semi-definite result, preventing nonsensical negative power spectral estimates
• C. The unbiased estimator underestimates the ACF at all lags, making it systematically wrong
• D. The biased estimator requires less computation because it avoids counting available pairs

Answer: (B)

The biased estimator (dividing by N) guarantees positive semi-definiteness — a mathematical requirement for any valid autocorrelation function. When Fourier-transformed to obtain a PSD, it always yields non-negative spectral values. The unbiased estimator (dividing by N−|τ|) corrects bias but inflates variance at large lags where few sample pairs exist, sometimes violating positive semi-definiteness and producing negative PSD values — a physically impossible result. Trading some bias for a valid estimate is the practical engineering choice.

Question 3

The autocorrelation function R_x(τ) of any real, stationary signal satisfies R_x(τ) = R_x(−τ) — it is an even (symmetric) function of lag.

• T. True
• F. False

Answer: True

Even symmetry follows directly from the definition: R_x(τ) = E[x(t)x(t+τ)]. Replacing τ with −τ gives R_x(−τ) = E[x(t)x(t−τ)], which by stationarity equals E[x(t+τ)x(t)] = R_x(τ). The expectation of a product doesn't depend on which sample leads. Geometrically, the ACF is symmetric about lag zero, so you only need to compute and interpret it for non-negative lags.

Question 4

White noise has a flat (constant) autocorrelation function, reflecting that most lags contribute equally to its power.

• T. True
• F. False

Answer: False

White noise has an impulsive ACF — R_x(0) equals average power, and R_x(τ) = 0 for all τ ≠ 0. White noise samples are uncorrelated by definition: knowing the value at one time gives no information about any other. A flat ACF would imply constant nonzero correlation at all lags. The confusion may arise from white noise's flat power spectral density — but the ACF (the Fourier transform of the PSD) of a flat spectrum is an impulse, not a flat function.

Question 5

What does it mean that the ACF of a periodic signal is itself periodic, and why is this property practically useful?

Model answer: If a signal x(t) has period T, then x(t) and x(t+T) are identical, so their product averages to R_x(0) — the ACF repeats at lags that are multiples of T. The ACF is therefore periodic with the same period T. This is useful because even when a periodic signal is buried in broadband noise, the noise's ACF decays to zero while the signal's periodic oscillation persists in the combined ACF, revealing the hidden period.

This turns the ACF into a periodicity detector robust to additive noise. In radar, sonar, or vibration analysis, you may need to detect a periodic component (engine rotation, target echo) in a noisy environment. Direct spectral analysis may show a peak buried in the noise floor, but the ACF will show a persistent oscillation at the signal's period while the noise contribution decays away. This is one of the key practical motivations for computing the ACF rather than working directly with the raw signal.