Introduction
One of the most common operation in statistical signal processing systems is the calculation, either implicitly or explicitly, of the signal's Fourier spectrum. Thus, a good understanding of the Fourier transform is a necessary prerequisite for spectral estimation. Distinguishing between (discrete-time) energy and power signals, we will see how the spectrum can be obtained directly from the Fourier transform of the signal, or indirectly from the autocorrelation function. In practice, real-world signals have a finite duration and thus are not ideal, infinetely-long signals; this implicit signal windowing has consequences on the maximum attainable spectral resolution, which is the the degree to which a spectral estimate can express spectral details.
Module overview
This module covers the following topics:
- Recap: Fourier transforms - A good understanding of the Fourier transform and its variants is a necessary pre-requisite for spectral estimation. Here, a recap from previous courses is offered.
- Spectral distributions - The spectrum of discrete-time energy and power signals can be calculated directly from the signal samples in time, or inderictely from the autocorrelation function.
- Windowing - All real-world signals are windowed in time, either implicitly or explicitly. Different types of windows have different trade-offs in terms of spectral resolution and spectral leakage.