High-Selectivity Filter Banks for Spectral Analysis of Music Signals
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Research Article High-Selectivity Filter Banks for Spectral Analysis of Music Signals Filipe C. C. B. Diniz, Iuri Kothe, Sergio L. Netto, and Luiz W. P. Biscainho LPS-PEE/COPPE and DEL/Poli, Universidade Federal do Rio de Janeiro (UFRJ), Caixa Postal 68504, 21941-972 Rio de Janeiro, RJ, Brazil Received 7 December 2005; Revised 10 August 2006; Accepted 10 September 2006 Recommended by Masataka Goto This paper approaches, under a unified framework, several algorithms for the spectral analysis of musical signals. Such algorithms include the fast Fourier transform (FFT), the fast filter bank (FFB), the constant-Q transform (CQT), and the bounded-Q transform (BQT), previously known from the associated literature. Two new methods are then introduced, namely, the constant-Q fast filter bank (CQFFB) and the bounded-Q fast filter bank (BQFFB), combining the positive characteristics of the previously mentioned algorithms. The provided analyses indicate that the proposed BQFFB achieves an excellent compromise between the reduced computational effort of the FFT, the high selectivity of each output channel of the FFB, and the efficient distribution of frequency channels associated to the CQT and BQT methods. Examples are included to illustrate the performances of these methods in the spectral analysis of music signals. Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
This paper aims at describing tools for the spectral analysis of music signals that are characterized by high-selectivity filters, a channel frequency spacing that is more efficient for this kind of signals, and acceptable computational complexity. The paper includes a brief overview of some related techniques used in music spectral analysis. New tools which achieve a good compromise between computational complexity and component discrimination are then introduced. The standard spectral tool is the fast Fourier transform (FFT), which is the fast algorithm for the discrete Fourier transform (DFT). The FFT is widely used in several applications due to its simplicity [1]. Taking the FFT as a filter bank, it can be interpreted that such simplicity comes partly from the use of a low-order kernel filter, which results in poorly selective channels. As an attempt to solve this problem, Lim and Farhang-Boroujeny [2] took advantage of the FFT tree structure but with more complex kernel filters, resulting in the so-called fast filter bank (FFB). The FFB complexity is slightly higher than the FFT’s, but with highly selective channels in the frequency domain. The FFT and FFB channels are uniformly distributed along the frequencies, which means that all the channels present the same bandwidth, regardless of their center fre-
quencies. Depending on the envisaged application, this approach, shown in Figure 1(a), may not be efficient for music signals, due to the equal tempered scale used in Western music [3]. Focusing on this issue, Brown [4] created, based on the DFT, the constant-Q transform (CQT), in which the channel bandwidth Δ f varies propo
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