Show that the PSD is real-valued and positive.

Key Concepts

Power Spectral Density (PSD)

The Power Spectral Density is a function that represents how the power of a signal or process is distributed across different frequency components. It provides essential insight into the frequency content of a signal, particularly for analyzing stationary random processes, making it a central concept in signal processing and statistical analysis of signals.

Autocorrelation Function

An autocorrelation function measures the similarity or correlation between a signal and a time-shifted version of itself. For wide-sense stationary processes, this function depends only on the time difference, not on the specific time instance, and is inherently symmetric. This symmetry ensures that its Fourier transform is real-valued, which directly relates to proving that the corresponding PSD is also real-valued.

Fourier Transform

The Fourier Transform is a mathematical tool used to convert functions between the time domain and the frequency domain. In the context of power spectral density, the Fourier Transform is applied to the autocorrelation function to yield the PSD. The properties of the Fourier Transform, especially when applied to symmetric autocorrelation functions, guarantee that the result is a real-valued function, thereby satisfying part of the condition that the PSD is real-valued.

Positive Semidefiniteness

A function or matrix is said to be positive semidefinite if it does not produce any negative values when used in quadratic forms, which is a property of autocorrelation functions derived from physical signals. This inherent property ensures that when the autocorrelation function is transformed into the frequency domain via the Fourier Transform, the resulting Power Spectral Density is not only real but also non-negative everywhere. This attribute is critical because it aligns with the physical meaning of power as being non-negative.

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