Multidimensional multirate filters and filter banks derived from one-dimensional filters

We present a method by which every multidimensional (MD) filter with a n arbitrary parallelepiped-shaped passband support can be designed and implemented efficiently. We show that all such filters can be designed starting from a n appropriate one-dimensional prototype filter and performing a simple transformation. With D denoting the number of dimensions, we hence reduce the complexity of design as well as implementation of the MD filter from O ( N ” ) to O ( N ) . Furthermore, by using the polyphase technique, we can obtain an implementation with complexity of only 2N in the two-dimensional special case. With our method, the Nyquist constraint and zero-phase requirement can be satisfied easily. In the IIR case, stability of the designed filters is also easily achieved. Even though the designed filters a re in general nonseparable, these filters have separable polyphase components. One special application of this method is in MD multirate signal processing, where filters with parallelepiped-shaped passbands a re used in decimation, interpolation, and filter banks. Some generalizations and other applications of this approach, including MD uniform DFT quadrature mirror filter banks which achieve perfect reconstruction, a re studied. Several design examples are also given. No M EN c L AT U R E D Number of dimensions. 32 [a, b)’ Set of all D x 1 integer vectors. Set of D x 1 real vectors x with components x, in the range a I x, < b. nals. Note that n E 32. transform of MD discrete signals. Fourier transform pair. A D x D nonsingular integer matrix. Set of all integer vectors of the form n = InO n I “Time” index of MD discrete sig. . . rill, I ‘ w = [w,) wI Frequency variable of the Fourier * . . W L ) I1 ’ x(n) t-f X(W) M X ( M ) Mx. x E [O, 1)”. J ( M ) = /det MI = The number of elements in X ( M ) . A = J ( M ) * M ’ = [adjugate of M I . Note that is also an integer matrix. M a n u w r i p t recei\cd Fehruai-) 9. 19Y I : revised J u l ) 21. 1992. The a s sociate editor coordinating the re\ ieu 0 1 this paper and approving i t lor publication v.a> Prof’. Faqc Bouilrcaux-Bnrtels. This work was supported in part by the National Science Foundation Grants MIP 8604456. MIP 8919196. and h! matching! t’unds Iron1 Tehtronix. Inc.. Huphcs Aircratt Company. and Rochwcll Intcrnational. The authors are with the Depailnlcnt o l Elcctrical Engineering. Catitornia Insti tute o i Technology. Pasadena. CA 91 125. IEEE l q Numher 9207535. SPD (V) parallelepiped of V ) Set of all real vectors (symmetric V x , x E [ l , 1)Y I749