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## REVISED MANUSCRIPT 1 Superposition Frames for Adaptive Time-Frequency Analysis and Fast Reconstruction

### Citations

664 | Entropy-based algorithms for best basis selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ... that avoids direct (pseudo-)inversion of the corresponding analysis operator. While approaches such as modulated lapped transforms for audio coding [18], wavelet packet decompositions via best basis =-=[19]-=-, [20] and adaptive segmentation via dynamic programming [21]–[25] can lead to flexible tilings of the time-frequency plane, the general goal of efficient reconstruction from signal-adaptive, overcomp... |

315 |
An Introduction to Frames and Riesz Bases
- Christensen
- 2003
(Show Context)
Citation Context ...nd illustrate their application to superposition systems. We conclude with a brief discussion in Section VIII. II. PRELIMINARIES We first review some well-known properties of Gabor frames [28], [32], =-=[33]-=- and discuss their relationship to shorttime Fourier analysis. We take as our setting the space CL, and interpret its members as discrete-time L-periodic signals x ∈ `2(ZL), with ZL denoting the integ... |

276 | Continuous and discrete wavelet transforms
- Heil, Walnut
- 1989
(Show Context)
Citation Context ...ter products of each frame vector with itself; from (4) via the orthogonality relation M−1∑ m=0 e2piimb(t−t ′)/L = { M when M divides t− t′, 0 otherwise, we obtain the so-called Walnut representation =-=[34]-=- of a Gabor frame operator S, which will be used repeatedly throughout. Definition 5 (Discrete Walnut Representation): Denote by M\(t − t′) the condition that M divides t − t′, and by IM\(t−t′)[t − t′... |

275 |
Painless non-orthogonal expansions
- Daubechies, Grossmann, et al.
- 1986
(Show Context)
Citation Context ...cal Dual Superposition Frames We next develop the reconstruction properties of our superposition families in a frame-theoretic context, noting that they qualify as “painless nonorthogonal expansions” =-=[36]-=-, and that the development below also holds for more general nonstationary Gabor frames [31]. In analogy to Definition 4, we associate a superposition frame operator SI : CL → CL through its action on... |

127 | Oversampled filter banks
- Cvetković, Vetterli
- 1998
(Show Context)
Citation Context ...udoy, pbasu, patrick}@seas.harvard.edu). time-frequency representations remains an open problem. This issue is particularly important, given the recent interest in oversampled, modulated filter banks =-=[26]-=-–[28]. In this article we introduce a broad family of adaptive, linear time-frequency representations that admit a fast overlap-add reconstruction property akin to standard short-time Fourier techniqu... |

108 |
Lapped transforms for efficient transform/subband coding
- Malvar
- 1990
(Show Context)
Citation Context ...sentation, by which we mean any noniterative method that avoids direct (pseudo-)inversion of the corresponding analysis operator. While approaches such as modulated lapped transforms for audio coding =-=[18]-=-, wavelet packet decompositions via best basis [19], [20] and adaptive segmentation via dynamic programming [21]–[25] can lead to flexible tilings of the time-frequency plane, the general goal of effi... |

97 | Frame-theoretic analysis of oversampled filter banks - Bolcskei, Hlawatsch, et al. - 1998 |

83 |
Tilings of the timefrequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms
- Herley, Kovacevic, et al.
- 1993
(Show Context)
Citation Context ...while the neighbor overlap condition of (28) is also required in the case of lapped orthogonal transforms (see, e.g., [7]). While our construction retains the flavor of time-varying lapped transforms =-=[37]-=-, [38], we emphasize that the resultant frames can avoid the lack of translation invariance inherent in the orthogonal setting, while still ensuring fast reconstruction. We show below that if F (Ig) i... |

70 | Life beyond bases: The advent of frames (part
- Kovačević, Chebira
(Show Context)
Citation Context ... pbasu, patrick}@seas.harvard.edu). time-frequency representations remains an open problem. This issue is particularly important, given the recent interest in oversampled, modulated filter banks [26]–=-=[28]-=-. In this article we introduce a broad family of adaptive, linear time-frequency representations that admit a fast overlap-add reconstruction property akin to standard short-time Fourier techniques. W... |

62 |
Minimum entropy deconvolution
- Wiggins
- 1978
(Show Context)
Citation Context ...th Mr, br defined via (15). This ratio of powers of norms of short-time Fourier coefficients is suggestive of an “empirical spectral kurtosis,” and has also been used in minimum entropy deconvolution =-=[40]-=-; other choices are also possible [14]. As shown in [3], maximizing (32) favors short-time segments that concentrate local signal energy within the smallest regions of the time-frequency plane. Indeed... |

57 | Adaptative Signal Models: Theory, Algorithms and Audio Applications
- Goodwin
- 1997
(Show Context)
Citation Context ... n=0 ∑N−1 r=0 Ĩ[n, r] (r + 1) = ∑N−1 n=0 Ĩ[n, 0] = N = 8. Example 2 ((N − 3)-Part Partition): The ordered partition function associated to the bottom panel of Fig. 1(b) is Ĩ[n, r] = { Ĩ[2, 0] = Ĩ=-=[6, 0]-=- = Ĩ[7, 0] = 1 not merged, Ĩ[0, 1] = Ĩ[3, 2] = 1 merged. Note again that in accordance with (11), we have that∑N−1 n=0 ∑N−1 r=0 Ĩ[n, r](r + 1) = 3 + 2 + 3 = 8. B. Superposition Systems We now empl... |

49 |
A high resolution data-adaptive time-frequency representation
- Jones, Parks
- 1990
(Show Context)
Citation Context ...-off to adapt to the local spectral content of x[t]. Over the past two decades, this shortcoming has motivated the development of various linear and nonlinear adaptive time-frequency analysis methods =-=[2]-=-–[14], in applications ranging from biomedical engineering [15] to radar signal analysis [16] and speech processing [17]. Despite the recognized importance of overcomplete signaladaptive time-frequenc... |

34 | Bayesian variable selection and regularization for time-frequency surface estimation - Wolfe, Godsill, et al. |

24 |
Speech Database Development at MIT
- Zue, Seneff, et al.
- 1990
(Show Context)
Citation Context ...is and reconstruction procedures presented earlier. These examples—a stylized synthetic waveform akin to the example employed in [3] and a phonetically balanced speech utterance from the TIMIT corpus =-=[43]-=-— both exhibit varying time-frequency structure, which in turn motivates signal-adaptive analysis and reconstruction. Our first example signal x (Fig. 5, top left) comprises a local and global sinusoi... |

23 | Time-frequency jigsaw puzzle: adaptive and multilayered Gabor expansions
- Jaillet, Torrésani
- 2007
(Show Context)
Citation Context ...ices and windows are employed. However, neither of these schemes treats the use of subset selection to achieve a signaladaptive system in the manner of the present article. More recent approaches [9]–=-=[12]-=- address subset selection from a Gabor frame or union of Gabor frames, but do not consider the structure of the corresponding canonical dual. A very recent approach in this direction is the study of g... |

23 | Flexible tree-structured signal expansions using timevarying wavelet packets
- Xiong, Ramchandran, et al.
- 1997
(Show Context)
Citation Context ...nalysis operator. While approaches such as modulated lapped transforms for audio coding [18], wavelet packet decompositions via best basis [19], [20] and adaptive segmentation via dynamic programming =-=[21]-=-–[25] can lead to flexible tilings of the time-frequency plane, the general goal of efficient reconstruction from signal-adaptive, overcomplete Based upon work supported in part by DARPA Grant HR0011-... |

21 |
Discrete Gabor structure and optimal representation
- Qiu, Feichtinger
- 1995
(Show Context)
Citation Context ...thms and illustrate their application to superposition systems. We conclude with a brief discussion in Section VIII. II. PRELIMINARIES We first review some well-known properties of Gabor frames [28], =-=[32]-=-, [33] and discuss their relationship to shorttime Fourier analysis. We take as our setting the space CL, and interpret its members as discrete-time L-periodic signals x ∈ `2(ZL), with ZL denoting the... |

20 | Improved Instantaneous Frequency Estimation Using an Adaptive Short-Time Fourier Transform - Kwok, Jones - 2000 |

19 | A Simple Scheme For Adapting Time-Frequency Representations
- Jones, Baraniuk
- 1994
(Show Context)
Citation Context ... 0] = N = 8. Example 2 ((N − 3)-Part Partition): The ordered partition function associated to the bottom panel of Fig. 1(b) is Ĩ[n, r] = { Ĩ[2, 0] = Ĩ[6, 0] = Ĩ[7, 0] = 1 not merged, Ĩ[0, 1] = Ĩ=-=[3, 2]-=- = 1 merged. Note again that in accordance with (11), we have that∑N−1 n=0 ∑N−1 r=0 Ĩ[n, r](r + 1) = 3 + 2 + 3 = 8. B. Superposition Systems We now employ the above construction to arrive at a variab... |

19 | Adapted local trigonometric transforms and speech processing
- Wesfreid, Wickerhauser
- 1993
(Show Context)
Citation Context ...avoids direct (pseudo-)inversion of the corresponding analysis operator. While approaches such as modulated lapped transforms for audio coding [18], wavelet packet decompositions via best basis [19], =-=[20]-=- and adaptive segmentation via dynamic programming [21]–[25] can lead to flexible tilings of the time-frequency plane, the general goal of efficient reconstruction from signal-adaptive, overcomplete B... |

18 | Multi-Gabor dictionaries for audio time-frequency analysis
- Wolfe, Dörfler, et al.
- 2001
(Show Context)
Citation Context ...lattices and windows are employed. However, neither of these schemes treats the use of subset selection to achieve a signaladaptive system in the manner of the present article. More recent approaches =-=[9]-=-–[12] address subset selection from a Gabor frame or union of Gabor frames, but do not consider the structure of the corresponding canonical dual. A very recent approach in this direction is the study... |

18 | Gabor analysis for a class of signals called musisc - Dörfler - 2003 |

15 |
The order of computation for finite discrete Gabor transforms
- Orr
- 1993
(Show Context)
Citation Context ...window is the identity, and hence these latter MN operations are avoided. An in-depth discussion of the general (non-diagonal) case, including methods based on the Zak transform, is given in [32] and =-=[39]-=-. In the case that windows are merged, note that a global superposition frame F (Ig) derived from G (w, a, b) comprises Mg = L/bg ≥M modulations of Ng ≤ N windows: Ng = N−1∑ n=0 N−1∑ r=0 Ig[n, r]. (31... |

11 | R/D Optimal Linear Prediction
- Prandoni, Vetterli
- 2000
(Show Context)
Citation Context ...it minimizes the sum of individual segment costs J(xn,r[t]): Ĩ∗[n, r] , argmineI[n,r] ∑ n,r:eI[n,r]=1̃ I[n, r]J(xn,r[t]). Many choices for J(·) are possible, including rate-distortion cost functions =-=[22]-=-, sparsity-inducing measures [14], and the well-known entropy cost of [19], which we employ below. To formalize our approach, define J∗n as the minimum cost among ordered partition functions on {0, 1,... |

10 |
Discrete multiwindow Gabor-type transforms
- Zibulski, Zeevi
- 1997
(Show Context)
Citation Context ...nal x[t] via superpositions of neighboring translates of a single window function, to yield the superposition frames of the article title. Related procedures include the multi-window constructions of =-=[29]-=-, in which multiple systems are defined on the same time-frequency lattice; and the multi-Gabor expansions of [30], in which multiple time lattices and windows are employed. However, neither of these ... |

8 |
A unified time-frequency parametrization of EEGs
- Durka, Blinowska
- 2001
(Show Context)
Citation Context ...past two decades, this shortcoming has motivated the development of various linear and nonlinear adaptive time-frequency analysis methods [2]–[14], in applications ranging from biomedical engineering =-=[15]-=- to radar signal analysis [16] and speech processing [17]. Despite the recognized importance of overcomplete signaladaptive time-frequency analysis, the above methods generally fail to admit fast reco... |

8 | Adaptive time segmentation for improved speech enhancement
- Hendriks, Heusdens, et al.
- 2006
(Show Context)
Citation Context ...lopment of various linear and nonlinear adaptive time-frequency analysis methods [2]–[14], in applications ranging from biomedical engineering [15] to radar signal analysis [16] and speech processing =-=[17]-=-. Despite the recognized importance of overcomplete signaladaptive time-frequency analysis, the above methods generally fail to admit fast reconstruction of the signal x[t] from its time-frequency rep... |

7 | Adaptive short-time analysis-synthesis for speech enhancement
- Rudoy, Basu, et al.
(Show Context)
Citation Context ... Research Fellowship. Preliminary versions of material in parts of Sections VI-A and VII first appeared in the 2008 IEEE International Conferences on Acoustics, Speech, and Signal Processing (ICASSP) =-=[1]-=-. The authors are with the Statistics and Information Sciences Laboratory, Harvard University, Cambridge, MA 02138 USA (e-mail: {rudoy, pbasu, patrick}@seas.harvard.edu). time-frequency representation... |

7 | Adaptive short-time Fourier analysis - Czerwinski, Jones - 1997 |

7 |
Nonstationary Gabor frames
- Jaillet, Balazs, et al.
- 2009
(Show Context)
Citation Context ...abor frame or union of Gabor frames, but do not consider the structure of the corresponding canonical dual. A very recent approach in this direction is the study of general nonstationary Gabor frames =-=[31]-=-, and indeed our contribution can be viewed as one possible instantiation of this framework. However, as we show below, the additional structure induced by our superposition construction yields severa... |

7 | A nonparametric test for stationarity based on local Fourier analysis
- Basu, Rudoy, et al.
- 2009
(Show Context)
Citation Context ...pplication-specific considerations are likely to play a role in superposition frame selection, and to this end we note that a variety of other algorithms and approaches are possible (see, e.g., [17], =-=[41]-=-, [42]). B. Illustrative Examples To conclude our investigation of superposition frames, we now consider two illustrative examples that combine Algorithms 1 and 2 with the analysis and reconstruction ... |

6 |
Discrete multi-Gabor expansions
- Li
- 1954
(Show Context)
Citation Context ...es of the article title. Related procedures include the multi-window constructions of [29], in which multiple systems are defined on the same time-frequency lattice; and the multi-Gabor expansions of =-=[30]-=-, in which multiple time lattices and windows are employed. However, neither of these schemes treats the use of subset selection to achieve a signaladaptive system in the manner of the present article... |

6 | Lapped tight frame transforms
- Chebira, Kovac̆ević
(Show Context)
Citation Context ...the structure of canonical superposition duals in such cases. Second, establishing additional connections to lapped transform constructions, in particular the tight lapped frames recently proposed in =-=[44]-=-, seem a promising avenue for further investigation. APPENDIX Proof of Lemma 2: To establish (9) for all w ∈ CL and M ≥ len(w), expand the left-hand side of (9) as M−1X m=0 |〈x,Mmbw〉|2 = M−1X m=0 〈x,M... |

5 |
An Adaptive-Window-Width Short-Time Fourier Transform for Visualization of Radar Target Substructure Resonance
- Rothwell, Chen, et al.
- 1998
(Show Context)
Citation Context ...ming has motivated the development of various linear and nonlinear adaptive time-frequency analysis methods [2]–[14], in applications ranging from biomedical engineering [15] to radar signal analysis =-=[16]-=- and speech processing [17]. Despite the recognized importance of overcomplete signaladaptive time-frequency analysis, the above methods generally fail to admit fast reconstruction of the signal x[t] ... |

5 |
The most general time-varying filter bank and time-varying lapped transforms
- Wang
(Show Context)
Citation Context ...the neighbor overlap condition of (28) is also required in the case of lapped orthogonal transforms (see, e.g., [7]). While our construction retains the flavor of time-varying lapped transforms [37], =-=[38]-=-, we emphasize that the resultant frames can avoid the lack of translation invariance inherent in the orthogonal setting, while still ensuring fast reconstruction. We show below that if F (Ig) is a la... |

5 | Time-varying autoregressive tests for multiscale speech analysis
- Rudoy, Quatieri, et al.
(Show Context)
Citation Context ...tion-specific considerations are likely to play a role in superposition frame selection, and to this end we note that a variety of other algorithms and approaches are possible (see, e.g., [17], [41], =-=[42]-=-). B. Illustrative Examples To conclude our investigation of superposition frames, we now consider two illustrative examples that combine Algorithms 1 and 2 with the analysis and reconstruction proced... |

4 | Benchmarking flexible adaptive time-frequency transforms for underdetermined audio source separation
- Nesbit, Vincent, et al.
- 2009
(Show Context)
Citation Context ... to adapt to the local spectral content of x[t]. Over the past two decades, this shortcoming has motivated the development of various linear and nonlinear adaptive time-frequency analysis methods [2]–=-=[14]-=-, in applications ranging from biomedical engineering [15] to radar signal analysis [16] and speech processing [17]. Despite the recognized importance of overcomplete signaladaptive time-frequency ana... |

4 | Jointly optimal time segmentation, component selection and quantization for sinusoidal coding of audio and speech - Heusdens, Jensen - 2005 |

3 | Optimal time segmentation for overlapadd systems with variable amount of window overlap - Niamut, Heusdens - 2005 |

2 |
adaptive optimal-kernel time-frequency representation
- “An
- 1995
(Show Context)
Citation Context ...sion whether or not to merge adjacent windows can be made based on any suitable cost function. As one example, we employ the time-frequency concentration measure appearing in the popular work of [3], =-=[4]-=- on adaptive optimal-kernel timefrequency representations. Specifically, consider a short-time segment xn,r[t] , {Tnaw[t]x[t]Isupp(Tnaw)[t]}, and define its time-frequency concentration in the manner ... |

2 |
A Wavelet Tour of Signal Procesing, 2nd ed
- Mallat
- 1999
(Show Context)
Citation Context ...=0 Ĩ[n, r] (r + 1) = ∑N−1 n=0 Ĩ[n, 0] = N = 8. Example 2 ((N − 3)-Part Partition): The ordered partition function associated to the bottom panel of Fig. 1(b) is Ĩ[n, r] = { Ĩ[2, 0] = Ĩ[6, 0] = Ĩ=-=[7, 0]-=- = 1 not merged, Ĩ[0, 1] = Ĩ[3, 2] = 1 merged. Note again that in accordance with (11), we have that∑N−1 n=0 ∑N−1 r=0 Ĩ[n, r](r + 1) = 3 + 2 + 3 = 8. B. Superposition Systems We now employ the abov... |

2 | Adaptive windowed Fourier transform - Djurovic, Stankovic - 2003 |

2 |
Rate-distortion optimal timesegmentation and redundancy selection for VoIP
- Rødbro, Jensen, et al.
- 2000
(Show Context)
Citation Context ...is operator. While approaches such as modulated lapped transforms for audio coding [18], wavelet packet decompositions via best basis [19], [20] and adaptive segmentation via dynamic programming [21]–=-=[25]-=- can lead to flexible tilings of the time-frequency plane, the general goal of efficient reconstruction from signal-adaptive, overcomplete Based upon work supported in part by DARPA Grant HR0011-07-1-... |

2 |
Short-time Fourier transform,” in Advanced topics in signal processing
- Nawab, Quatieri
- 1988
(Show Context)
Citation Context ...exity of these procedures. A. Reconstruction via the Constant Overlap-Add Method The classical “overlap-add” approach to signal reconstruction from short-time Fourier coefficients proceeds as follows =-=[35]-=-. Recall the covering condition of (6) which is necessary for a Gabor system G (w, a, b) to form a frame for CL, and also sufficient if M = L/b ≥ len(w). Clearly, this covering condition holds if tran... |