Spectral Estimation and Array Processing
Spectral estimation finds applications in a wide range of fields, and has received a vast amount of interest in the literature over the last century. As early as 1898, Sir Arthur Schuster derived the periodogram for the purpose of determining hidden periodicities in the monthly sunspot numbers. This classical problem is still attracting significant interest as the sunspots affect the Earth in many ways, having connections to areas as diverse as the climate on Earth and the channel fading in mobile phones. Similar problems occur in a wide range of fields, and accurate spectral estimation is often a key problem in many applications; often, one is then interested in finding high resolution spectral estimates for measured sequences or images, or recursive implementations of such estimators.
During recent decades, there has also been significant interest in examining the spectral content of temporally and spatially sampled data sequences. Using multiple sensors, one is, for instance, able to determine the direction to an emitting source, or a reflecting target, or to focus the sensors so that signals resulting from a particular direction are given more attention. This is done by forming a so-called beampattern (an example is shown above), indicating how signals from some directions are given a higher gain than signals from other directions.
The beampattern is closely related to temporal windowing, and, indeed, the same sequences are often used for both purposes. Just as in temporal windowing, there are strict limitations on how one is able to form appropriate beampatterns, and the area has received significant interest in the literature. One difficult aspect is how to handle coherent signals, typically resulting from various reflections along the propagation path. Other important aspects to consider are how to deal with array calibration errors, or with time-delays and Doppler shifts in the transmitted signal.
Another interesting problem is the estimation of the magnitude squared coherence (MSC) spectrum of two data sets. The figure above shows a data-dependent Capon-based MSC estimate of two images, commonly offering preferable estimates as compared to traditional Welsh-based estimators.