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

Venue: | IEEE Trans. on Signal Processing |

Citations: | 16 - 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 ...

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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 ... 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 ...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 ... 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... |

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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... |

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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 ...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... |

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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: ... |

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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: \... |

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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... |

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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... |

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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... |

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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... |

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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 ...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 ... 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... |

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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 ...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 ... 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 ...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 - Williams - 1994 |

7 |
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(Show Context)
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... |

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6 |
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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 |
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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... |

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

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

6 |
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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 |
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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... |

4 |
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(Show Context)
Citation Context ...I x (t; !) = 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 definit... |

4 |
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(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 (... |

3 |
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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... |