Questions: Gaussian and Colored Noise Characterization

Question 1

A sensor produces noise that a colleague describes as 'Gaussian.' They conclude it must also have a flat power spectral density. Is this reasoning correct?

• A. Yes — Gaussian random processes always produce white (flat-spectrum) noise by definition
• B. Yes — the central limit theorem guarantees both Gaussian amplitude distribution and flat PSD when many sources are summed
• C. No — spectral shape and amplitude distribution are independent properties; Gaussian noise can have any spectral color
• D. No — Gaussian noise is always pink (1/f), not white, because biological and physical systems produce 1/f distributions

Answer: (C)

Spectral character (white, pink, brown, etc.) and amplitude distribution (Gaussian, uniform, Laplacian, etc.) are completely independent properties of a noise process. Saying a noise is Gaussian tells you its amplitude values follow a normal distribution; it says nothing about how power is distributed across frequencies. You can have Gaussian white noise, Gaussian pink noise, or Gaussian noise with any spectral shape. Conversely, white noise can have Gaussian or non-Gaussian amplitudes. The CLT guarantees Gaussian amplitude when many independent sources are summed, but has no direct implication for spectral shape.

Question 2

Which of the following correctly describes what 'white noise' means in signal processing?

• A. White noise has very small amplitude — it is 'white' because it is faint and barely detectable
• B. White noise has a power spectral density that is flat (equal power per unit bandwidth at every frequency)
• C. White noise has a Gaussian amplitude distribution — the term 'white' refers to its statistical purity
• D. White noise contains only high-frequency components, analogous to high-frequency light

Answer: (B)

The term 'white' is an analogy to white light, which contains all visible frequencies in equal measure. White noise has equal power per unit bandwidth at every frequency — a flat PSD. It says nothing about the amplitude being small or large, and nothing about the amplitude distribution being Gaussian. White noise's autocorrelation function is a scaled impulse: R(τ) = σ²δ(τ), meaning samples at different times are uncorrelated. The color of noise (white, pink, brown/red, blue) refers entirely to the spectral shape, not to amplitude statistics.

Question 3

Thermal noise from a resistor is Gaussian in its amplitude distribution because it arises from the sum of many independent random electron motions.

• T. True
• F. False

Answer: True

This is a direct application of the central limit theorem. Thermal (Johnson-Nyquist) noise arises from the superposition of a vast number of independent random electron motions in the resistor. Even if each individual electron's motion has some non-Gaussian distribution, their sum converges to a Gaussian distribution. This makes thermal noise an excellent example of 'Gaussian' amplitude statistics arising from a physical mechanism — and it is also approximately white in spectrum over the relevant frequency range.

Question 4

Colored noise should have a non-Gaussian amplitude distribution — if a noise process is Gaussian, it is necessarily white.

• T. True
• F. False

Answer: False

This is a key misconception. Colored noise is simply white noise that has been filtered (shaped in frequency), and filtering a Gaussian process produces another Gaussian process. Therefore Gaussian colored noise is entirely possible and common — for example, passing Gaussian white noise through a lowpass filter produces Gaussian pink-ish or brown noise. The amplitude distribution is preserved under linear filtering (Gaussianity is a distributional property), but the spectral content is reshaped. Spectral color and amplitude distribution are orthogonal axes of description.

Question 5

What are the two independent axes on which a noise process must be characterized? Give an example that shows they are truly independent of each other.

Model answer: A noise process must be characterized by (1) its spectral character — how power is distributed across frequencies (white, pink, brown, etc.) — and (2) its amplitude distribution — the probability distribution of its instantaneous values (Gaussian, uniform, Laplacian, etc.). These are independent: for example, Gaussian white noise has flat spectrum and Gaussian amplitudes, while Gaussian pink noise has 1/f spectrum but still Gaussian amplitudes (produced by filtering Gaussian white noise). Alternatively, non-Gaussian white noise (e.g., uniformly distributed white noise) has flat spectrum but non-Gaussian amplitudes.

The independence matters practically: a matched filter for signal detection assumes a specific noise PSD (spectral axis) to set the filter shape, while the optimal detector structure depends on the amplitude distribution (amplitude axis). Assuming noise is both white and Gaussian when it is actually colored and Laplacian will cause you to design both the wrong filter and the wrong decision rule. Correct noise characterization requires checking both properties separately — typically via Welch PSD estimation (spectral axis) and Q-Q plots or KS tests (amplitude axis).