The covariance function of a band limited white noise is :

Question

Question

This question was previously asked in

NIELIT Scientific Assistant CS 5 Dec 2021 Official Paper

Answer 1

Concept:

The covariance function of a random process is the inverse Fourier transform of its power spectral density (PSD).

For band-limited white noise, the PSD is constant (flat) within a limited frequency band and zero outside that band.

This kind of spectrum corresponds to a rectangular function in the frequency domain.

The inverse Fourier transform of a rectangular function is a sinc function.

Calculation:

Given:

Band-limited white noise: flat PSD in range \( -B \leq f \leq B \)

The covariance function \( R(\tau) \) is given by:

\( R(\tau) = \int_{-B}^{B} \frac{N_0}{2} e^{j2\pi f\tau} df = N_0 B \cdot \text{sinc}(2B\tau) \)

Where, \(\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}\)

Final Answer:

The covariance function of a band-limited white noise is a sinc function.

Correct Option: (3) A sinc function

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