Question

What is the variance of the difference between two successive signal samples, d(n) = x(n) – x(n-1)?

(a) \(σ_d^2=2σ_x^2 [1+γ_{xx} (1)]\)

(b) \(σ_d^2=2σ_x^2 [1-γ_{xx} (1)]\)

(c) \(σ_d^2=4σ_x^2 [1-γ_{xx} (1)]\)

(d) \(σ_d^2=3σ_x^2 [1-γ_{xx} (1)]\)

This question was posed to me in an interview for internship.

This is a very interesting question from Oversampling A/D Converters in section Sampling and Reconstruction of Signals of Digital Signal Processing

Answer 1

The correct answer is (b) \(σ_d^2=2σ_x^2 [1-γ_{xx} (1)]\)

Explanation: \(σ_d^2=E[d^2 (n)] = E{[x(n)- x(n-1)]^2}\)

= \(E [x^2 (n)]-2E{x(n)x(n-1)}+E[x^2 (n-1)]\)

= \(2σ_x^2 [1+γ_{xx} (1)]\).