Proportional Nonuniform Multi-Gabor Expansions
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Proportional Nonuniform Multi-Gabor Expansions Shidong Li College of Science and Engineering, San Francisco State University, San Francisco, CA 94132, USA Email: [email protected] Received 18 December 2003; Revised 10 May 2004; Recommended for Publication by Helmut Boelcskei A nonuniform multi-Gabor expansion (MGE) scheme is studied under proportional time and frequency (TF) shifts among different window indices m. In particular, TF parameters for each m are different, but proportional and relevant to windows’ TF patterns. The generation of synthesis waveforms for nonuniform MGE is generally difficult. We show constructively that there is a set of basic synthesis MGE waveforms at each window index under proportional parameter settings. Nonuniform MGE adapts to signal frequency dynamics effectively, and eliminates unnecessary overlapping redundancies of a uniform MGE. Examples of the evaluation of synthesis waveforms are provided. Efficiency comparison of TF analysis using nonuniform and uniform MGEs is also discussed. Keywords and phrases: nonuniform time-frequency shifts, multi-Gabor representations, time-frequency analysis, frames, dual frames.
1.
INTRODUCTION
Typical Gabor expansions use one fixed window and its translates and complex modulates as basic elements in an attempt to analyze the time-frequency (TF) information of a signal. Studies on Gabor expansions are intense; for example, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. There is also a book on Gabor analysis and algorithm by Feichtinger and Strohmer [17]. However, frequency-varying features of a signal or the multifrequency components in a signal require windows of different size (variance) for a refined TF resolution. Multi-Gabor expansions (MGEs) were developed specifically to meet such requirements; for example, see [11, 18], (cf. [19, 20]). MGEs are TF expansions using a set of multiple windows and their translates and modulates in a frame (overcomplete “basis”) system. The set of windows are custom-tuned. Typically, they range from a narrower window to a wider window to meet the requirements of TF representations of signals of timevarying frequency components. However, standard uniform MGEs apply the same TF shifts among all analysis windows. Such uniform MGEs do not take into consideration the distinct TF patterns of different windows, which gives rise to unnecessary redundancy. Nonuniform MGE schemes adapted to each window’s TF characteristics are more natural. Let 0 ≤ m ≤ M − 1 be the number of windows in an MGE system. Let j and k be the modulation and translation parameters, and let H L be an L-dimensional signal space. Then a discrete nonuniform multi-Gabor representation (of nonuni-
form TF shifts) is defined by, for all s ∈ HL , s=
M −1 N m −1 K m −1 m=0
j =0
k =0
s, γ(m, j, k) g (m) j/Nm ,kTm ,
(1)
where, for the window index m, Nm is the number of frequency bins (1/Nm is the frequency shift), Tm is the timetranslation parameter, Km ≡ L/Tm , g (m) is the mth window, and {γ(m, j, k) : m, j, k} is a set of synthes
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