## Shift Covariant Time-Frequency Distributions of Discrete Signals (1997)

Venue: | IEEE Trans. on Signal Processing |

Citations: | 15 - 5 self |

### BibTeX

@ARTICLE{O'neill97shiftcovariant,

author = {Jeffrey C. O'neill and William J. Williams},

title = {Shift Covariant Time-Frequency Distributions of Discrete Signals},

journal = {IEEE Trans. on Signal Processing},

year = {1997},

volume = {47},

pages = {133--150}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many commonly used time-frequency distributions are members of the Cohen class. This class is defined for continuous signals and since time-frequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the class of all quadratic time-frequency distributions that are covariant to time shifts and frequency shifts. In this paper we extend this method to three types of discrete signals to derive what we will call the discrete Cohen classes. The properties of the discrete Cohen classes differ from those of the original Cohen class. To illustrate these properties we also provide explicit relationships between the classical Wigner distribution and the discrete Cohen classes. I. Introduction I N signal analysis there are four types of signals commonly used. These four types are based on whether the signal is continuous or discrete, and whether the signal is aperiodic or periodic. The four signal types ...

### Citations

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Citation Context ...ner distribution is more commonly presented in the following form: W x (t; !) = Z x(t + �� 2 ) x (t \Gamma �� 2 ) e \Gammaj !�� d�� (2.2) and was derived in the context of quantum mech=-=anics by Wigner [73]-=- and later extended to the signal processing context by Ville [72]. The Wigner distribution satisfies many of the desirable properties in Table 2.1 (all except positivity, strong time support, and str... |

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Citation Context ...ach of the four types of signals, there is an appropriate Fourier transform pair, so it seems plausible that there should exist four types of time-frequency distributions (TFDs). The Cohen class [1], =-=[2]-=- (with the restriction that the kernel is not a function of time and frequency and is also not a function of the signal) can derived axiomatically as the class of all quadratic TFDs for type I signals... |

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Citation Context ...orm, which has close ties to the wavelet bases, and the Wigner (or Wigner-Ville) distribution. The areas of time-frequency and time-scale analysis have seen many exciting developments in recent years =-=[35, 21, 22, 29, 16, 17, 18]-=-. The theory has been progressing rapidly and has been successfully applied to wide variety of signals including the following: speech, biological, biomedical, geophysical, machine monitoring, radar, ... |

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Citation Context ... is a member of the Cohen class, and that, under certain constraints, elements of the Cohen class can be decomposed into weighted sums of spectrograms [31]. The cross terms in the Wigner distribution =-=[32]-=- satisfy the following properties: ffl cross terms are centered exactly between two auto terms, ffl if two auto terms are separated in frequency by \Delta ! , then the rate of oscillation of the cross... |

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Citation Context ...forms are where the kernel of time-frequency distribution is allowed to be signal adaptive [5, 6, 44, 76]. Another approach that is close to quadratic is the reassignment method of Auger and Flandrin =-=[1]-=-. The close to quadratic time-frequency distributions often provide improved performance over the quadratic timefrequency distributions. Other time-frequency distributions are explicitly higher order ... |

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Citation Context ...actor of four the required computations. Second, even with oversampling, distributions in the type I Cohen class are not always straightforward to compute. For example, the Choi-Williams distribution =-=[45]-=- is often cited in the literature, but the kernel is not bandlimited and also does not have compact support. As a result, it is not clear how to sample the kernel, and thus provide an accurate impleme... |

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Citation Context ...; !) = x (t) X(!) e \Gammaj!t RD II x (n; !) = x (n) X(!) e \Gammaj!n A prominent distribution that is missing from the list in Table II is a type II Wigner distribution. Discretization methods [15], =-=[17]-=-, [16], [28] have failed to produce a satisfactory type II Wigner distribution since they require the signal to be oversampled by a factor of two. In [36], [37] we present an alternative definition of... |

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Citation Context ...nd affine classes, but not quadratic will be called "close" to the quadratic. Examples of close to quadratic forms are where the kernel of time-frequency distribution is allowed to be signal=-= adaptive [5, 6, 44, 76]-=-. Another approach that is close to quadratic is the reassignment method of Auger and Flandrin [1]. The close to quadratic time-frequency distributions often provide improved performance over the quad... |

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Citation Context ...d�� fi fi fi fi 2 : The Wigner, Rihaczek, and Page distributions are all quadratic functions of the signal, but there is no obvious connection between the three of them. In his pioneering work, Co=-=hen [20]-=- devised a method for creating an infinite number of time-frequency distributions. The method he used to do this is based in quantum mechanics and is derived using operator theory. For more details on... |

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Citation Context ... II Rihaczek distribution that has been defined above. The validity of the Moyal formula [3], [39] is useful in several applications including signal synthesis [40] and detection /estimation problems =-=[41]-=-. Given two type II signals,sx(n) and y(n), the Moyal formula can be formulated for type II signals as: 7 X n Z C II x (n; !) \Theta C II y (n; !) d! = fi fi fi fi fi X n x(n) y (n) fi fi fi fi fi 2 T... |

30 |
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Citation Context ...orm, which has close ties to the wavelet bases, and the Wigner (or Wigner-Ville) distribution. The areas of time-frequency and time-scale analysis have seen many exciting developments in recent years =-=[35, 21, 22, 29, 16, 17, 18]-=-. The theory has been progressing rapidly and has been successfully applied to wide variety of signals including the following: speech, biological, biomedical, geophysical, machine monitoring, radar, ... |

27 | Time-frequency signal analysis - Boashash - 1990 |

25 |
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Citation Context ...sses will be denoted the type II, III, and IV Cohen classes. There are three common methods for deriving TFDs for type I signals. The first uses operator theory [1], [2], the second uses group theory =-=[6]-=-, and the third uses covariance properties [3], [4], [5]. In this paper we choose to use the covariance based approach to investigate TFDs for signals of types II, III, and IV, because of the simplici... |

25 | The interference structure of the Wigner distribution and related time-frequency signal representations”, in: “The Wigner Distribution - Hlawatsch, Flandrin - 1997 |

23 | Kernel design for reduced interference distributions - Jeong, Williams - 1992 |

21 |
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Citation Context ...UENCY DISTRIBUTIONS OF DISCRETE SIGNALS 141 Third, the discrete Cohen classes provide the framework for relating discrete TFDs to other discrete-time processing such as linear, time-varying filtering =-=[46]-=- and signal detection [41]. Boashash [13] has created a class of discrete TFDs for type II signals called the Generalized Discrete TimeFrequency Distributions (GDTFDs). While the implementation of thi... |

21 |
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Citation Context ...nd affine classes, but not quadratic will be called "close" to the quadratic. Examples of close to quadratic forms are where the kernel of time-frequency distribution is allowed to be signal=-= adaptive [5, 6, 44, 76]-=-. Another approach that is close to quadratic is the reassignment method of Auger and Flandrin [1]. The close to quadratic time-frequency distributions often provide improved performance over the quad... |

18 | Discrete-time, discrete-frequency timefrequency analysis
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Citation Context ...lin, Atlas, and Droppo [7], [8] have investigated the formulation of a type IV TFDs using operator theory. Richman, Parks, and Shenoy have investigated type IV Wigner distributions using group theory =-=[9]-=-. There has also been much other work investigating methods for computing TFDs from sampled signals [10]--[30]. The results presented here are more comJ.C. O'Neill is with Boston University. Address: ... |

17 |
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Citation Context ...ratic timefrequency distributions. Other time-frequency distributions are explicitly higher order functions of the signal. Some examples are the polynomial Wigner distributions by Boashash and O'Shea =-=[12]-=-, the L-Wigner distributions by Stankovi'c [70, 69, 71], and the Wigner higher order moment spectra by Fonollosa and Nikias [31]. These higher order methods provide remarkable results for very specifi... |

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Citation Context ...41 Third, the discrete Cohen classes provide the framework for relating discrete TFDs to other discrete-time processing such as linear, time-varying filtering [46] and signal detection [41]. Boashash =-=[13]-=- has created a class of discrete TFDs for type II signals called the Generalized Discrete TimeFrequency Distributions (GDTFDs). While the implementation of this method is slightly simpler conceptually... |

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Citation Context ...t are currently not well understood. It is often desired that a time-frequency distribution be positive valued, and Cohen has created a general class of time-frequency that are always positive valued =-=[23]-=-. Since this class is very general, it intersects with every class (quadratic and non-quadratic) of timefrequency distributions listed above. The main difficulty with this class is that it is not clea... |

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Citation Context ... as a means for finding "good" signal-adaptive kernels. It has been observed that, in the ambiguity function, the auto terms lie near the origin while the cross terms tend to lie away from t=-=he origin [27]-=- and the methods of Baraniuk and Jones exploit this observation. The signal dependent kernel introduced in [5] is called a radially Gaussian kernel (RGK) and is formulated in the ambiguity plane as: \... |

14 |
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Citation Context ...tly, quadratic time-frequency distributions have been defined that are covariant to operators other than the three mentioned above. One example is the hyperbolic class of time-frequency distributions =-=[62]-=-. Time-frequency distributions in the hyperbolic class are covariant to scales and hyperbolic time shifts. The hyperbolic class intersects with the affine class but not with Cohen's class. Baraniuk [2... |

10 | Regularity and unitarity of bilinear time-frequency signal representations
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Citation Context ...! 0 ) = 0. This property can be satisfied, and an example of a type II TFD that satisfies this is the type II Rihaczek distribution that has been defined above. The validity of the Moyal formula [3], =-=[39]-=- is useful in several applications including signal synthesis [40] and detection /estimation problems [41]. Given two type II signals,sx(n) and y(n), the Moyal formula can be formulated for type II si... |

10 |
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Citation Context ...e II TFD that satisfies this is the type II Rihaczek distribution that has been defined above. The validity of the Moyal formula [3], [39] is useful in several applications including signal synthesis =-=[40]-=- and detection /estimation problems [41]. Given two type II signals,sx(n) and y(n), the Moyal formula can be formulated for type II signals as: 7 X n Z C II x (n; !) \Theta C II y (n; !) d! = fi fi fi... |

10 |
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Citation Context ...nd affine classes, but not quadratic will be called "close" to the quadratic. Examples of close to quadratic forms are where the kernel of time-frequency distribution is allowed to be signal=-= adaptive [5, 6, 44, 76]-=-. Another approach that is close to quadratic is the reassignment method of Auger and Flandrin [1]. The close to quadratic time-frequency distributions often provide improved performance over the quad... |

9 | Covariant time-frequency representations through unitary equivalence - Baraniuk - 1996 |

9 | Applications of operator theory to timefrequency analysis and classification,” submitted for publication
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Citation Context ... use the covariance based approach to investigate TFDs for signals of types II, III, and IV, because of the simplicity and directness of the mathematics. Narayanan, McLaughlin, Atlas, and Droppo [7], =-=[8]-=- have investigated the formulation of a type IV TFDs using operator theory. Richman, Parks, and Shenoy have investigated type IV Wigner distributions using group theory [9]. There has also been much o... |

9 |
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Citation Context ...orm in the time-frequency domain. IV. The Type II Cohen Class The above proof for the type I Cohen class extends directly to form the type II Cohen class, which is identical to the class of AF-GDTFDs =-=[21]-=-. The AF-GDTFDs were known to be covariant to time and frequency shifts, but it was not known until this point that the AF-GDTFDs include all type II TFDs that are covariant to time and frequency shif... |

9 |
A centrosymmetric kernel decomposition for time-frequency distribution computation
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Citation Context ...l). This form makes it clear that the spectrogram is a member of the Cohen class, and that, under certain constraints, elements of the Cohen class can be decomposed into weighted sums of spectrograms =-=[31]-=-. The cross terms in the Wigner distribution [32] satisfy the following properties: ffl cross terms are centered exactly between two auto terms, ffl if two auto terms are separated in frequency by \De... |

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Citation Context ...ed positive time-frequency distributions by iterating quadratic time-frequency distributions. There has also been recent work in constructing distributions of quantities other than time and frequency =-=[3, 4, 39, 22]-=-. The most common distributions outside of time and frequency are those of time and scale. The concept of scale is closely related to the concept of frequency and some time-frequency distributions, li... |

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Citation Context ... sampled signal of the form x(n) = A(n) e j'(n) A(n) ? 0 (4.9) and uses the following definition for the periodic frequency moment: h! 2 i n = arg Z 2�� 0 e j! C II x (n; !; /)d! (4.10) Lovell et.=-= al [48]-=- has used this periodic moment in defining frequency estimators and shown that, for sampled signals, it provides improved performance over non-periodic moments. This definition for the periodic freque... |

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Citation Context ... X(!) e \Gammaj!t RD II x (n; !) = x (n) X(!) e \Gammaj!n A prominent distribution that is missing from the list in Table II is a type II Wigner distribution. Discretization methods [15], [17], [16], =-=[28]-=- have failed to produce a satisfactory type II Wigner distribution since they require the signal to be oversampled by a factor of two. In [36], [37] we present an alternative definition of the type I ... |

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Citation Context ...se to use the covariance based approach to investigate TFDs for signals of types II, III, and IV, because of the simplicity and directness of the mathematics. Narayanan, McLaughlin, Atlas, and Droppo =-=[7]-=-, [8] have investigated the formulation of a type IV TFDs using operator theory. Richman, Parks, and Shenoy have investigated type IV Wigner distributions using group theory [9]. There has also been m... |

7 | Fast implementations of generalized discrete time-frequency distributions - Cunningham, Williams - 1994 |

7 |
Reduced Interference Time-Frequency Distributions," to appear in Time-Frequency Signal Analysis: Methods and Applications
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Citation Context ...to the spectrogram, BornJordan, Rihaczek, Page, and Levin distributions are all direct discretizations of the corresponding kernels for the type I Cohen class TFDs [2], [3], [32]. The binomial kernel =-=[35]-=- satisfies many desirable properties of TFDs and the recursive structure also allows the implementation of fast algorithms. The type II Cohen class also provide the framework for the discrete formulat... |

6 |
Duality and classification of bilinear time-frequency signal representations
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Citation Context ...nd frequency and is also not a function of the signal) can derived axiomatically as the class of all quadratic TFDs for type I signals that are covariant to time shifts and frequency shifts [3], [4], =-=[5]-=-. In this paper, we will investigate the quadratic, time and frequency shift covariant classes of TFDs for the other three types of signals. The original class will be renamed the type I Cohen class a... |

6 | Interpreting and estimating the instantaneous frequency of a signal-part 1: fundamentals - Boashash - 1992 |

6 |
An analysis of instantaneous frequency representation using time-frequency distribution-generalized wigner distribution
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Citation Context ...requency distributions are explicitly higher order functions of the signal. Some examples are the polynomial Wigner distributions by Boashash and O'Shea [12], the L-Wigner distributions by Stankovi'c =-=[70, 69, 71]-=-, and the Wigner higher order moment spectra by Fonollosa and Nikias [31]. These higher order methods provide remarkable results for very specific classes of signals, but produce extremely unsatisfact... |

5 |
The aliasing problem in discrete-time Wigner distributions
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Citation Context ... x (t) X(!) e \Gammaj!t RD II x (n; !) = x (n) X(!) e \Gammaj!n A prominent distribution that is missing from the list in Table II is a type II Wigner distribution. Discretization methods [15], [17], =-=[16]-=-, [28] have failed to produce a satisfactory type II Wigner distribution since they require the signal to be oversampled by a factor of two. In [36], [37] we present an alternative definition of the t... |

5 | New Properties for Discrete, Bilinear Time-Frequency Distributions - O'Neill, Williams - 1996 |

5 | Quadralinear time-frequency representations - O'Neill, Williams - 1995 |

5 |
A multitime definition of the Wigner higher order distribution: L-Wigner distribution
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(Show Context)
Citation Context ...requency distributions are explicitly higher order functions of the signal. Some examples are the polynomial Wigner distributions by Boashash and O'Shea [12], the L-Wigner distributions by Stankovi'c =-=[70, 69, 71]-=-, and the Wigner higher order moment spectra by Fonollosa and Nikias [31]. These higher order methods provide remarkable results for very specific classes of signals, but produce extremely unsatisfact... |

3 |
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(Show Context)
Citation Context ...uous functions. As a means for representing these functions, one can compute samples of these two-dimensional functions such that the continuous function could be recovered through sinc interpolation =-=[23]-=-, [24]. The method presented in [23], [24] is unnecessarily complicated and simpler method that uses an oversampled signal is presented in [33]. Note that these methods only provide accurate results w... |

3 | Kernel design techniques for alias-free time-frequency distributions - O'Hair, Suter - 1994 |

3 |
Boudreaux-Bartels, "A comparative study of alias-free time-frequency representations
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(Show Context)
Citation Context ...alf the Nyquist rate, so aliasing occurs and the frequency axis has a period of �� rather than 2��. To avoid aliasing the signal must be sampled at twice the Nyquist rate. Costa and Boudreaux-=-=Bartels [24]-=- and also Morris and Wu [50] have claimed a third type of aliasing. They point out that in the AF-GDTFDs, there are extra components that are not present in TFDs in Cohen's class. These extra componen... |

3 |
Signal representations geometry and catastrophes in the time-frequency plane
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(Show Context)
Citation Context ...cond, the cross terms take negative values which a true density function should not. The structure of the cross terms in the Wigner distribution has been investigated in detail and is well understood =-=[30, 36]-=-. Suppose two auto terms are separated in time by \Deltat and in frequency by \Delta! as shown in Figure 2.5. There will be a cross term centered between the two auto terms in the time frequency plane... |

3 |
A method for improved distribution concentration in the time-frequency analysis of multicomponent signals using the L-Wigner distribution
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- 1995
(Show Context)
Citation Context ...y represent. Although the Wigner higher order moment spectra have a similar appearance to the Q function defined in this chapter, there is no clear relation between the two. Stankovi'c and Stankovi'c =-=[69, 70, 71] defined the cla-=-ss of L-Wigner distributions as: L x (t; !; `) = Z [x(t + �� 2` )] ` [x (t + �� 2` )] ` e j!�� d�� Boashash and O'Shea [12] defined a class of polynomial Wigner distributions as: P x (... |