The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey’s Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time O(n logn). From a lower bound perspective, relatively little is known. Ailon shows in 2013 an Ω(n logn) bound for computing the normalized Fourier Transform assuming only unitary...

-M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. In this paper, we study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Speci...

We adopt the concept of channel diagonalization to time-frequency signal expansions obtained by DFT filter banks. As a generalization of the frequency domain channel representation used by conventional orthogonal frequency-division multiplexing receivers, the time-frequency domain channel diagonalization can be applied to time-variant channels and aperiodic signals. An inherent error in the cas...

In this paper we derive perfect reconstruction (PR) conditions for oversampled cosine-modulated lter banks. The results can be regarded as a generalization of the known work for critical subsampling. We show that in the over-sampled case we gain some additional degree of freedom, which can be exploited in the lter design process. This leads to PR prototypes with stopband attenuations being much...

We study the design of signal–adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi–infinite programming (SIP) ...