N dimensional signals are characterized as values in an N dimensional space. Each point in the space is assigned a value, possibly complex. Each dimension in the space can be discrete, continuous, or on a time scale. A black and white movie can be modelled as a three dimensional signal.Acolor picture can be modelled as three signals in two dimensions, one each, for example, for red, green and blue. This chapter explores Fourier characterization of different types of multidimensional signals and corresponding applications. Some signal characterizations are straightforward extensions of their one dimensional counterparts. Others, even in two dimensions, have properties not found in one dimensional signals. We are fortunate to be able to visualize structures in two, three, and sometimes four dimensions. It assists in the intuitive generalization of properties to higher dimensions. Fourier characterization of multidimensional signals allows straightforward modelling of reconstruction of images from their tomographic projections. Doing so is the foundation of certain medical and industrial imaging, including CAT (for computed axial tomography) scans. Multidimensional Fourier series are based on models found in nature in periodically replicated crystal Bravais lattices [987, 1188]. As is one dimension, the Fourier series components can be found from sampling the Fourier transform of a single period of the periodic signal. The multidimensional cosine transform, a relative of the Fourier transform, is used in image compression such as JPG images. Multidimensional signals can be filtered. The McClellan transform is a powerful method for the design of multidimensional filters, including generalization of the large catalog of zero phase one dimensional FIR filters into higher dimensions. As in one dimension, the multidimensional sampling theorem is the Fourier dual of the Fourier series. Unlike one dimension, sampling can be performed at the Nyquist density with a resulting dependency among sample values. This property can be used to reduce the sampling density of certain images below that of Nyquist, or to restore lost samples from those remaining. Multidimensional signal and image analysis is also the topic of Chapter 9 on time frequency representations, and Chapter 11 where POCS is applied signals in higher dimensions.
Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution
