Spectral Density and Autocorrelation Function of White Noise

White noise is a random signal that has a constant power spectral density across all frequencies. It is characterized by its autocorrelation function and spectral density. Let's understand the spectral density and autocorrelation function of white noise in detail.

Spectral Density of White Noise
The spectral density of a random signal describes how the power of that signal is distributed across different frequencies. For white noise, the spectral density is constant across all frequencies. This means that the power is evenly distributed across the entire frequency spectrum.

Autocorrelation Function of White Noise
The autocorrelation function of a random signal describes how the signal correlates with itself at different time lags. For white noise, the autocorrelation function is a delta function. A delta function is a mathematical function that is zero everywhere except at one point, where it is infinite. In the case of white noise, the autocorrelation function is a delta function at zero lag, indicating that the signal is uncorrelated with itself at any time lag.

Relationship Between Spectral Density and Autocorrelation Function
The spectral density and autocorrelation function of a signal are Fourier transform pairs. This means that they are mathematically related to each other through the Fourier transform. Specifically, the spectral density is the Fourier transform of the autocorrelation function, and vice versa.

For white noise, since the autocorrelation function is a delta function and the spectral density is constant, their Fourier transform relationship is as follows:
- The Fourier transform of a delta function is a constant, which represents the spectral density of white noise.
- The inverse Fourier transform of a constant is a delta function, which represents the autocorrelation function of white noise.

Therefore, the correct answer is option 'B': The spectral density of white noise is uniform (constant across all frequencies), and the autocorrelation function is a delta function (zero except at zero lag).