## Multiple window time-varying spectrum estimation (1996)

Venue: | in Conf. Info. Sci. and Sys. (CISS |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Bayram96multiplewindow,

author = {Metin Bayram and Richard Baraniuk},

title = {Multiple window time-varying spectrum estimation},

booktitle = {in Conf. Info. Sci. and Sys. (CISS},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

We overview a new non-parametric method for estimating the time-varying spectrum of a non-stationary random process. Our method extends Thomson’s powerful multiple window spectrum estimation scheme to the time-frequency and time-scale planes. Unlike previous extensions of Thomson’s method, we identify and utilize optimally concentrated Hermite window and Morse wavelet functions and develop a statistical test for extracting chirping line components. Examples on synthetic and real-world data illustrate the superior performance of the technique. 2

### Citations

1731 |
Ten Lectures on Wavelets
- Daubechies
- 1992
(Show Context)
Citation Context ...16 Bayram & Baraniuk frequency time Figure 9: The tear-drop shaped concentration region (6.2) for the Morse wavelets for β = γ = 1. orthogonal and maximally concentrated in the time-frequency region =-=[30,32]-=- { (t,f) : t 2 + 9 } 3C + 1 ≤ 4f2 |f| (6.2) of area A = 3π(C − 2) [32]. Figure 9 depicts this region. Just as a circular disk contains all points equidistant from the center point in the Euclidean dis... |

408 | Harmonic analysis in the phase space - Folland - 1989 |

274 |
Prolate spheroidal wave functions, Fourier analysis, and uncertainty V: The discrete case
- Slepian
- 1978
(Show Context)
Citation Context ...minimize variance) and optimally concentrated in frequency (to minimize bias). The optimal windows satisfying these requirements for signals of finite extent are the prolate spheroidal wave functions =-=[16, 19]-=- (see Figure 2). These orthogonal functions are the eigenfunctions of a localization operator that band limits and then time limits functions. As windows, they are perfectly suited to stationary spect... |

256 |
Modern Spectral Estimation: Theory and Application
- Kay
- 1988
(Show Context)
Citation Context ...ne components. Examples on synthetic and real-world data illustrate the superior performance of the technique. 2 Introduction Many methods exist for estimating the power spectra of stationary signals =-=[1]-=-. However, these methods are insufficient for the non-stationary signals that occur in important applications such as radar, sonar, acoustics, biology, and geophysics. These applications demand time-f... |

246 | Wavelets: Algorithms & Applications - Meyer - 1993 |

216 |
Spectrum estimation and harmonic analysis
- Thomson
- 1982
(Show Context)
Citation Context ...µi(t)e j2πγi(t) a deterministic “chirp” signal with instantaneous amplitude µi(t) and instantaneous phase γi(t). For inspiration, we turn to the seminal stationary spectrum estimation work of Thomson =-=[16]-=-. Realizing that random and deterministic spectral components must be dealt with separately, Thomson introduced a powerful multiple window (MW) spectrum estimator for stationary signals in [16] to obt... |

87 | Improving the readability of time-frequency and time-scale representations by the reassignment method - Auger, Flandrin - 1995 |

78 |
Time-frequency localization operators: a geometric phase space approach
- Daubechies
- 1988
(Show Context)
Citation Context ...r shape of the concentration region. The Hermite functions are eigenfunctions of the Fourier transform and also of a time-frequency localization operator over the circular time-frequency region (4.2) =-=[20]-=-. The eigenvalues in this latter case are a function of the area A = πR 2 of the region [20] λk(R) := 1 − e −R2 2 k∑ i=0 1 i! 2−i R 2i . (4.3)Multiple Window Time-Varying Spectrum Estimation 9 0 0 fr... |

74 | On the Principles of Elementary Quantum Mechanics, Physica 12 - Groenewold - 1946 |

67 | Adaptive covariance estimation of locally stationary processes
- Mallat, Papanicolaou, et al.
- 1998
(Show Context)
Citation Context ...an-square error between the true WVS and the estimate. Other estimation procedures for locally stationary time-frequency spectra that fit within these general frameworks include those of Mallat et al =-=[13]-=- and von Sachs et al [62,63]. 9 Conclusions In this chapter, we have overviewed two multiple-window time-frequency and time-scale spectrum estimators that extend Thomson’s seminal work [16] on multipl... |

65 |
Time-Frequency Analysis. Englewood Cliffs
- Cohen
- 1995
(Show Context)
Citation Context ...)], (2.1) the WVS is defined as its Fourier transform ∫ Wx(t,f) := rx(t,τ)e −j2πfτ dτ. (2.2) Alternatively, the WVS can be defined as the expected value of the empirical Wigner distributions (WDs) Wx =-=[14,15]-=- of the realizations of the process [∫ Wx(t,f) = E[Wx(t,f)] = E x ∗ (t − τ/2)x(t + τ/2)e −j2πfτ ] dτ . (2.3) 12 Bayram & Baraniuk In this framework, the problem of time-varying spectrum estimation ca... |

56 |
Wigner-Ville spectral analysis of nonstationary processes
- Martin, Flandrin
- 1985
(Show Context)
Citation Context ...rum for stationary random processes, there is no unique definition for the time-varying spectrum of a nonstationary random process x. Perhaps the best compromise is the Wigner-Ville spectrum (WVS) Wx =-=[9]-=-. Given the instantaneous auto-correlation function rx(t,τ) := E[x ∗ (t − τ/2)x(t + τ/2)], (2.1) the WVS is defined as its Fourier transform ∫ Wx(t,f) := rx(t,τ)e −j2πfτ dτ. (2.2) Alternatively, the W... |

45 | Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. The Annals of Statistics
- Neumann, Sachs
(Show Context)
Citation Context ...he true WVS and the estimate. Other estimation procedures for locally stationary time-frequency spectra that fit within these general frameworks include those of Mallat et al [13] and von Sachs et al =-=[62,63]-=-. 9 Conclusions In this chapter, we have overviewed two multiple-window time-frequency and time-scale spectrum estimators that extend Thomson’s seminal work [16] on multiple-window spectrum estimation... |

35 |
Time-scale energy distributions: A general class extending wavelet transforms
- Rioul, Flandrin
- 1992
(Show Context)
Citation Context ...low frequency components of long duration), standard time-frequency techniques are not appropriate. These types of processes are better matched by the time-scale representations from the affine class =-=[15,29]-=-. The smoothing kernels in the affine class change with frequency to accommodate component scaling. The smoothing regions in different parts in the time-frequency plane for Cohen’s class and the affin... |

31 | Wavelet smoothing of evolutionary spectra by non-linear thresholding
- Sachs, Schneider
- 1996
(Show Context)
Citation Context ...he true WVS and the estimate. Other estimation procedures for locally stationary time-frequency spectra that fit within these general frameworks include those of Mallat et al [13] and von Sachs et al =-=[62,63]-=-. 9 Conclusions In this chapter, we have overviewed two multiple-window time-frequency and time-scale spectrum estimators that extend Thomson’s seminal work [16] on multiple-window spectrum estimation... |

21 |
Myoelectric teleoperation of a complex robotic hand
- Farry, Walker, et al.
- 1996
(Show Context)
Citation Context ...mponents in the data before performing (4.4) and then reshape the estimate accordingly. A straightforward application of Thomson’s sinusoid extraction algorithm to a signal from the model (2.4) as in =-=[3, 4]-=- relies on an assumption that the chirp functions e j2πγi(t) can be closely approximated locally as sinusoids. Unfortunately, this is not the case for rapidly chirping components, as we saw in Figure ... |

21 |
and P.Bertrand, “A class of affine Wigner functions with extended covariance properties
- Bertrand
- 1992
(Show Context)
Citation Context ...bution ∫∫ Ωy(t,f) = ( Wy(τ,ν)Π f(τ − t), ν f ) dτ dν (6.5) with kernel Π centered at time zero and frequency f0 = 1. The affine class can also be defined in terms of the unitary Bertrand distribution =-=[38, 39]-=-. Interestingly, the Klauder wavelet ψ0 has a positive Bertrand distribution, just as the Gaussian h0 has a positive Wigner distribution [40]. As in the time-frequency, Cohen’s class case, the MW time... |

21 | de Villedary, “Analysis of Time-Varying Signals with Small BT Values - Kodera, Gendrin, et al. - 1986 |

20 |
The discrete polynomial-phase transform
- Peleg, Friedlander
- 1995
(Show Context)
Citation Context ...he instantaneous frequencies of polynomial-phase line components [52]. Pitton has furthermore extended the F test in [46]. Further afield, we could extract chirps using the polynomial phase transform =-=[53]-=-, the reassignment method [54–57], the ridge and snakes method [58], or the squeezing method [59].Multiple Window Time-Varying Spectrum Estimation 21 Alternative frameworks: Multiple window estimates... |

19 | Spectrum estimation by wavelet thresholding of multitaper estimators
- Walden, Percival, et al.
- 1995
(Show Context)
Citation Context ...sing a least squares procedure [48,49], while Pitton exploits knowledge of the windows’ leakage characteristics [45]. Other stationary adaptation schemes such as that of Hansson [50] and Walden et al =-=[51]-=- could also prove useful in the time-frequency setting. Extended chirp extraction algorithms: Many alternatives exist to our simple linear chirp extraction algorithm. Çakrak and Loughlin employ a mult... |

18 | Characterization of signals by the ridges of their wavelet transforms
- Carmona, Hwang, et al.
- 1997
(Show Context)
Citation Context ...2]. Pitton has furthermore extended the F test in [46]. Further afield, we could extract chirps using the polynomial phase transform [53], the reassignment method [54–57], the ridge and snakes method =-=[58]-=-, or the squeezing method [59].Multiple Window Time-Varying Spectrum Estimation 21 Alternative frameworks: Multiple window estimates succeed when the timefrequency spectrum can be approximated as “lo... |

17 | Optimal kernels for nonstationary spectral estimation
- Sayeed, Jones
- 1995
(Show Context)
Citation Context ...within timefrequency regions larger than the concentration region of the orthogonal windows. As such, they can be viewed as special cases of the more general estimation frameworks of Sayeed and Jones =-=[10]-=- and Kozek et al [11,12,60,61]. Rather than assuming a parametric Cohen’s class kernel that is the sum of several Wigner distributions of Hermite functions (recall (4.5)–(4.7)), Sayeed and Jones [10] ... |

17 |
Time-frequency and time-scale analysis
- Flandrin
- 1998
(Show Context)
Citation Context ...low frequency components of long duration), standard time-frequency techniques are not appropriate. These types of processes are better matched by the time-scale representations from the affine class =-=[15,29]-=-. The smoothing kernels in the affine class change with frequency to accommodate component scaling. The smoothing regions in different parts in the time-frequency plane for Cohen’s class and the affin... |

16 | Beyond time-frequency analysis: Energy densities in one and many dimensions
- Baraniuk
- 1998
(Show Context)
Citation Context ...bution ∫∫ Ωy(t,f) = ( Wy(τ,ν)Π f(τ − t), ν f ) dτ dν (6.5) with kernel Π centered at time zero and frequency f0 = 1. The affine class can also be defined in terms of the unitary Bertrand distribution =-=[38, 39]-=-. Interestingly, the Klauder wavelet ψ0 has a positive Bertrand distribution, just as the Gaussian h0 has a positive Wigner distribution [40]. As in the time-frequency, Cohen’s class case, the MW time... |

15 |
Wiener measures for path integrals with affine kinematic variables
- Daubechies, Klauder, et al.
- 1987
(Show Context)
Citation Context ...rse wavelet is commonly known as the Klauder wavelet [34], although it goes by 2 While Morse defined only a special case of these wavelets for γ = 1 [33], we will refer to the entire class derived in =-=[31,32]-=- as the Morse wavelets.Multiple Window Time-Varying Spectrum Estimation 15 0 0 frequency 0 time 0 frequency 0 0 time (a) Klauder wavelet ψ0 0 0 frequency 0 time 0 frequency 0 0 time (b) Morse wavelet... |

15 |
Multiwavelet spectral and polarization analysis of seismic records
- Lilly, Park
- 1995
(Show Context)
Citation Context ...nctions that balance the advantages of prolate and Hermite windows [44–46]. Lilly and Park have also considered multi-wavelet time-scale spectrum estimation using a specially designed set of wavelets =-=[47]-=-.20 Bayram & Baraniuk frequency (a) time (b) time frequency frequency (c) time (d) time Figure 12: (a) Test signal composed of two singularities in additive white Gaussian noise. (b) Scalogram of noi... |

14 |
Quadratic time-varying spectral estimation for underspread processes
- Kozek, Riedel
- 1994
(Show Context)
Citation Context ...regions larger than the concentration region of the orthogonal windows. As such, they can be viewed as special cases of the more general estimation frameworks of Sayeed and Jones [10] and Kozek et al =-=[11,12,60,61]-=-. Rather than assuming a parametric Cohen’s class kernel that is the sum of several Wigner distributions of Hermite functions (recall (4.5)–(4.7)), Sayeed and Jones [10] design an optimal kernel φ tha... |

14 |
On the wavelet transform of fractal objects
- Holschneider
- 1987
(Show Context)
Citation Context ...requency domain via Wigner distribution. (a) ψ0 (Klauder wavelet), (b) ψ1 (since the Fourier transform of ψ1 is purely imaginary, we plot the imaginary part), (c) ψ2. other names as well [35, p. 25], =-=[36,37]-=-. Figure 8 shows the first three Morse wavelets in time, their Fourier transforms, and their Wigner distributions. The Morse wavelets are the eigenfunctions of a localization operator over a tear-drop... |

13 |
Correlative time-frequency analysis and classification of nonstationary random processes
- Kozek, Hlawatsch, et al.
- 1994
(Show Context)
Citation Context ...regions larger than the concentration region of the orthogonal windows. As such, they can be viewed as special cases of the more general estimation frameworks of Sayeed and Jones [10] and Kozek et al =-=[11,12,60,61]-=-. Rather than assuming a parametric Cohen’s class kernel that is the sum of several Wigner distributions of Hermite functions (recall (4.5)–(4.7)), Sayeed and Jones [10] design an optimal kernel φ tha... |

10 | A new method for the numerical analysis of non-stationary signals - Kodera, Velledary, et al. - 1976 |

9 |
A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models
- Daubechies, Maes
- 1996
(Show Context)
Citation Context ...ended the F test in [46]. Further afield, we could extract chirps using the polynomial phase transform [53], the reassignment method [54–57], the ridge and snakes method [58], or the squeezing method =-=[59]-=-.Multiple Window Time-Varying Spectrum Estimation 21 Alternative frameworks: Multiple window estimates succeed when the timefrequency spectrum can be approximated as “locally stationary” within timef... |

9 |
Boudreaux-Bartels, “On the optimality of the Wigner distribution for detection
- Kay, F
- 1985
(Show Context)
Citation Context ...w function. While the periodogram suffers from large variance, this variance can be reduced by cutting the data into blocks, computing a periodogram of each block, and then averaging the periodograms =-=[18]-=-. However, this procedure also smears and biases the resulting spectrum estimate. The bias/variance tradeoff is clear: reducing the variance necessitates averaging over a larger number of shorter bloc... |

8 | Multiple window timefrequency analysis - Bayram, Baraniuk - 1996 |

8 |
Diatomic molecules according to the wave mechanics II vibrational levels,” Phys
- Morse
- 1929
(Show Context)
Citation Context ...f flatness at f = 0 and γ > 0. The zero-th order Morse wavelet is commonly known as the Klauder wavelet [34], although it goes by 2 While Morse defined only a special case of these wavelets for γ = 1 =-=[33]-=-, we will refer to the entire class derived in [31,32] as the Morse wavelets.Multiple Window Time-Varying Spectrum Estimation 15 0 0 frequency 0 time 0 frequency 0 0 time (a) Klauder wavelet ψ0 0 0 f... |

7 | Time frequency/time-scale reassignment - Chassande-Mottin, Auger, et al. - 2003 |

6 | Multiple window spectrogram and time-frequency distributions - Frazer, Boashash - 1994 |

6 |
Approximating time-frequency density functions via optimal combinations of spectrograms
- Loughlin, Pitton, et al.
- 1994
(Show Context)
Citation Context ...trograms using a geometric rather than arithmetic mean. This is closely related to Loughlin, Pitton, and Hannaford’s generation of positive time-frequency distributions using products of spectrograms =-=[28]-=-. 5 Extracting Line Components As in Thomson’s method for stationary signals, the averaging inherent in (4.4) will degrade the resolution of chirping line components. Following Thomson’s programme, we... |

5 | positivity, and minimum uncertainty in time-frequency energy distributions
- Flandrin
- 1998
(Show Context)
Citation Context ... defined in terms of the unitary Bertrand distribution [38, 39]. Interestingly, the Klauder wavelet ψ0 has a positive Bertrand distribution, just as the Gaussian h0 has a positive Wigner distribution =-=[40]-=-. As in the time-frequency, Cohen’s class case, the MW time-scale spectrum estimator can be interpreted as a member of the affine class with a kernel that is a weighted sum of Wigner (or Bertrand) dis... |

5 |
Pseudo affine Wigner distributions: Definition and kernel formulation
- Gonçalvès, Baraniuk
- 1998
(Show Context)
Citation Context ...ich the line extraction algorithm locks on to each component. In Figure 12, we illustrate the performance of the time-scale MW method using a 256-point test signal containing two Hölder singularities =-=[41, 42]-=- in additive white Gaussian noise n(t) x(t) = |t − 64| −0.1 + |t − 180| −0.1 + n(t). (7.1) Unlike the scalogram in Figure 12(c), the MW estimate of Figure 12(d) clearly captures the cone-like time-fre... |

5 |
Multiple window time-varying spectral analysis
- Cakrak, Loughlin
- 2001
(Show Context)
Citation Context ...ias/variance tradeoff of the estimator. The improvement can be dramatic [16]. In the time-frequency setting, Çakrak and Loughlin adaptively weight Hermite eigenspectra using a least squares procedure =-=[48,49]-=-, while Pitton exploits knowledge of the windows’ leakage characteristics [45]. Other stationary adaptation schemes such as that of Hansson [50] and Walden et al [51] could also prove useful in the ti... |

4 |
Optimized weighted averaging of peak matched multiple window spectrum estimates
- Hansson
- 1999
(Show Context)
Citation Context ...Hermite eigenspectra using a least squares procedure [48,49], while Pitton exploits knowledge of the windows’ leakage characteristics [45]. Other stationary adaptation schemes such as that of Hansson =-=[50]-=- and Walden et al [51] could also prove useful in the time-frequency setting. Extended chirp extraction algorithms: Many alternatives exist to our simple linear chirp extraction algorithm. Çakrak and ... |

3 |
Optimally Karhunen-Loève-like STFT expansion of nonstationary processes
- Kozek
- 1993
(Show Context)
Citation Context ...regions larger than the concentration region of the orthogonal windows. As such, they can be viewed as special cases of the more general estimation frameworks of Sayeed and Jones [10] and Kozek et al =-=[11,12,60,61]-=-. Rather than assuming a parametric Cohen’s class kernel that is the sum of several Wigner distributions of Hermite functions (recall (4.5)–(4.7)), Sayeed and Jones [10] design an optimal kernel φ tha... |

3 | Boudreaux-Bartels, "On the optimality of the Wigner distribution for detection - Kay, F - 1985 |

3 | Time-frequency concentrated basis functions - Parks, Shenoy - 1990 |

3 |
Path integrals for affine variables
- Klauder
- 1980
(Show Context)
Citation Context ... −fγ /2 dβ df β [ fγ dβ+k e df β+k ( f β+k e −fγ)] , k = 0,1,2,... (6.1) with β > 0 the degree of flatness at f = 0 and γ > 0. The zero-th order Morse wavelet is commonly known as the Klauder wavelet =-=[34]-=-, although it goes by 2 While Morse defined only a special case of these wavelets for γ = 1 [33], we will refer to the entire class derived in [31,32] as the Morse wavelets.Multiple Window Time-Varyi... |

3 |
Instantaneous frequency estimation of polynomial phase signals
- akrak, Loughlin
- 1998
(Show Context)
Citation Context ...alternatives exist to our simple linear chirp extraction algorithm. Çakrak and Loughlin employ a multiple window estimate to estimate the instantaneous frequencies of polynomial-phase line components =-=[52]-=-. Pitton has furthermore extended the F test in [46]. Further afield, we could extract chirps using the polynomial phase transform [53], the reassignment method [54–57], the ridge and snakes method [5... |

3 | de Bruijn, Uncertainty principle in Fourier analysis - G - 1967 |

2 |
Issues in Myoelectric Teleoperation of Complex Artificial Hands
- Farry
- 1994
(Show Context)
Citation Context ...mponents in the data before performing (4.4) and then reshape the estimate accordingly. A straightforward application of Thomson’s sinusoid extraction algorithm to a signal from the model (2.4) as in =-=[3, 4]-=- relies on an assumption that the chirp functions e j2πγi(t) can be closely approximated locally as sinusoids. Unfortunately, this is not the case for rapidly chirping components, as we saw in Figure ... |

2 | de Bruijn, "Uncertainty principles in Fourier analysis - G - 1967 |

2 |
Jacknifing multiple-window spectra
- Thomson
- 1994
(Show Context)
Citation Context ...MW WVS estimate is manifestly positive for all signals. And the connection with positive distributions does not stop here. Computation of the jackknife estimate of the variance of the MW WVS estimate =-=[27]-=- leads naturally to the concept of combining eigenspectrograms using a geometric rather than arithmetic mean. This is closely related to Loughlin, Pitton, and Hannaford’s generation of positive time-f... |