Multidimensional Signal Processing

Abstract

A one-dimension a 1 w 1 ndow 1s chosen from the 1 arge cat a 1 og of those available primarily due to its leakage-resolution tradeoff (LRT>. Is it possible to generalize a 1-D window to higher dimensions such that the window's 1-D properties are homogeneously preserved? If we require that the window be continuous and bounded the answer is usually no. Bounded (projection window) general 1zations do exist for the Parzen and TukeyHann1ng windows. The resulting windows, however, are very close to that window obtained by simply rotating the 1-D window into two dimensions. IKTROOUCTION When choosing from the large catalog of standard one-dimensional windows [1-2], one is largely motivated by the window's leakage-resolution tradeoff (LRT). Is 1t possible to generalize these windows to two and higher dimensions such that the 1-D window properties are preserved in each 1-0 slice? If we require these multidimensional windows to be bounded and continuous, the answer is usually negativ~ In the two cases considered 1n this correspondence where bounded two dimensional generalizations do exist, the resulting windows are close to those obtained by the rotation generalization of 1-D windows [3]. A short review of the outer product and rotation of 1-D window generalization methods is given in the next section. In both cases, the LRT is altered in the transformation. In order to homogeneously maintain the 1-D window properties, the higher dimension window must be chosen so that its projection onto one dimension results in the 1-D window. Unfortunately, this requires unbounded generalizations in many cases of interest. The Parzen and Tukey-Hamm1 ng windows are the exceptions. For the discrete case, bounded projection windows can be formed such that desired LRT is preserved inhomogeneously at a number of angular orientations. ·I I PRELIHINABIES There are a wealth of one-dimensional windows with various 1eat<.ageresolut1on tradeoffs. A one-dimensional window, w1 has finite extent: (where IT Ct> = 1 for It I~ 1/2. and is zero elsewhere>, fs normalized with and is even function, i.e., The spectrum of a window is defined by wl ( w) = t!l (t)exp(-j w t)dt The area of a window 1$ = W1(0) The magnitude of a typical window spectrum is shown in Figure 1. For good resolution, the main lobe width, 6, should be small, and for minimal spectra 1 1 eakage, the norma 1 i zed side 1 obe magn 1tude, o , shou 1 d a 1 so be small. Invariably, however, decreasing one of these parameters increases the other. A two dimensional window w2ct1, t 2>, with spectrum fX> f:'2ctl' t 2 > exp [-JJdt 1dt 2 oo -:oo fs commonly generated from a 1-0 counterpart by either the outer product or window rotation techniques [3). The outer product window is and the rotated window, initially suggested by Huang [4J, fs In either case, if w1 1s a "good" window, then so is w2• For certain applications, (e.g. "good" filter design) such dimensional generalizations are acceptable. In other cases, such as spectral estimation, a smal 1 perturbation in window shape can significantly alter results [SJ. Both the outer product and the rotated window s1gn1ffcantly alter the LRT of the corresponding 1-0 window. To illustrate the effects of outer product and rotational dimensional generalization, we choose a boxcar window