## Estimating and Interpreting the Instantaneous Frequency of a Signal (1992)

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Venue: | Proceedings of the IEEE |

Citations: | 146 - 5 self |

### BibTeX

@INPROCEEDINGS{Boashash92estimatingand,

author = {Boualem Boashash and Senior Member},

title = {Estimating and Interpreting the Instantaneous Frequency of a Signal},

booktitle = {Proceedings of the IEEE},

year = {1992},

pages = {520--538}

}

### Years of Citing Articles

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### Abstract

The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, the notion of frequency loses its effectiveness, and one needs to use a parameter which accounts for the time-varying nature of the process. This need has given rise to the idea of instantaneous frequency. The instantaneous frequency (IF) of a signal is a parame-ter which is often of significant practical importance. In many situations such as seismic, radar, sonar, communications, and biomedical applications, the IF is a good descriptor of some physical phenomenon. This paper discusses the concept of instantaneous frequency, its definitions, and the correspondence between the various mathe-matical models formulated for representation of IF. The paper also considers the extent to which the IF corresponds to our intuitive expectation of reality. A historical review of the successive attempts to define the IF is presented. Then the relationships between the IF and the group-delay, analytic signal, and bandwidth-time (BT) product are explored, as well as the relationship with time-frequency distribu-tions. Finally, the notions of monocomponent and multicomponent signals, and instantaneous bandwidth are discussed. It is shown that all these notions are well described in the context of the theory presented. I.

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Citation Context ...on of the signal, he arrived at a similar definition of IF to that of Carson and Fry [ll] who used a complex representation of the signal. The next important step in the study of IF was made by Gabor =-=[20]-=- who proposed a method for generating a unique complex signal from a real one. His method for doing so is first to find the FT of the real signal and then to “suppress the amplitudes belonging to nega... |

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Citation Context ... Mandel asks a question: Which of these two quantities is most closely related to measurements? He also provides the answer: It strongly depends “on the nature of the experiment.” Priestley indicated =-=[32]-=- that a nonstationary process in general cannot be represented in a meaningful way by the simple Fourier expansion as described by (5b). For example, consider the nonstationary signal with time-varyin... |

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Citation Context ...17), all odd moments would be zero since IS( f ) I 2 is even, and this would not fit well with physical reality. Gabor’s complex signal is referred to as the “analytic signal” (see Appendix A). Ville =-=[42]-=- unified the work done by Carson and Fry [ll] and Gabor [20] and defined the IF of a signal expressed by s(t) = a(t) cosq$(t) as 1 d fz(t) = --[argz(t)l (18) 27r dt where z(t) is the analytic signal g... |

69 |
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Citation Context ...one should adapt definitions in the way which is most appropriate for the particular question being considered (see [7] for more details). Several definitions are available for the measure of B and T =-=[37a]-=-, [22a]. A definition commonly used was proposed by Gabor [20]: Let s(t) be a zero-mean finite-energy signal; the effective duration, T,, and the effective bandwidth, B,, are respectively given by J -... |

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Citation Context ...age of the frequencies that exist in the time-frequency plane at a given time [17]. This can be expressed as (34) Several other authors have also contributed to the study of IF [l], [21], [22], [24], =-=[34]-=-. Rihaczek [34] investigated the points in the time-frequency plane of his complex energy distribution where the signal energy is concentrated. He showed that the energy was concentrated in frequency ... |

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Citation Context ...uld adapt definitions in the way which is most appropriate for the particular question being considered (see [7] for more details). Several definitions are available for the measure of B and T [37a], =-=[22a]-=-. A definition commonly used was proposed by Gabor [20]: Let s(t) be a zero-mean finite-energy signal; the effective duration, T,, and the effective bandwidth, B,, are respectively given by J --oo Bs2... |

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Citation Context ... us see under what conditions the following equation can be verified: a(t) cos 4(t) + j ~[a(t) cos 4(t)l = a(t)ej+(t). (39) The problem was investigated for complex finite-energy signals in [2], 191, =-=[30]-=-, and [33]. The solution is found by PROCEEDINGS OF THE IEEE, VOL. 80, NO. 4, APRIL IW2susing Bedrosian's product theorem (BPT) summarized in Appendix C. It leads to the following result: Equation (39... |

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Citation Context ...tained by using the HT, as Gabor and Ville have noted [20], [42], but whether or not it corresponds to any physical reality is another question. This point is discussed further in Section 111. Mandel =-=[28]-=- also challenged any physical interpretation of the IF. He discussed its relationship with spectral frequency in terms of Fourier decomposition. He argued that there is no one-to-one relationship betw... |

11 |
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Citation Context ..., then fig # 7;’; i.e., these two functions are not the inverse of each other. In this case, no physical meaning can be associated with these t - f laws, although they are mathematically well defined =-=[26]-=- (Fig. 6(a)). This question is also studied in [5]. Note that there are two requirements for the IF and GD laws to be the inverse of one another. The first is that the variations in time of the IF be ... |

11 |
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Citation Context ...s X fi(t) = fo + -m(t). 27T They argued that the notion of IF is a generalization of the definition of constant frequency, i.e., it is the rate of change of phase angle at time t. In 1946 Van der Pol =-=[41]-=- approached the problem of formulating a definition for the instantaneous frequency by analyzing an expression for simple harmonic motion: (8) s(t) = acos(27Tft + e) (9) where a is amplitude, f is fre... |

7 |
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(Show Context)
Citation Context ... the signal; this property explains the importance of the IF in signal recognition, tracking, estimation, and modeling. However, the interpretation of the IF is often a subject of controversy. Shekel =-=[35]-=-, for example, argued that the IF defined by 1 fi(t) = -- 27~ dt is not a unique function of time, since any amplitude modulated (AM) wave, written in complex form, may be expressed as either m(t)ejZa... |

7 |
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Citation Context ...xtent and size being dependent on the interaction of the auto-term ambiguity functions and the window-the greater the separation of the ambiguity functions the more diminished will be the cross-terms =-=[43]-=-. Thus one could define a multicomponent signal as being one which has cross-terms present. Such a definition, however, is hard to test in practice. Perhaps a more practical way of testing for whether... |

6 | Rapid Measurement of Digital Instantaneous Frequency - Griffiths - 1975 |

5 |
Advanced Calculus for Engineers
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Citation Context ...t) is Gabor’s complex signal, s(t) is the real signal and H is the Hilbert transform (HT) (see Appendix A for its properties), defined as where p.v. denotes the Cauchy principle value of the integral =-=[22b]-=-. Signals, s(t) and H[s(t)], are often said to be in quadrature, because in theory they are out of phase by ~/2. However, in reality this is true only under certain conditions (discussed in Section 11... |

4 |
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(Show Context)
Citation Context ...der what conditions the following equation can be verified: a(t) cos 4(t) + j ~[a(t) cos 4(t)l = a(t)ej+(t). (39) The problem was investigated for complex finite-energy signals in [2], 191, [30], and =-=[33]-=-. The solution is found by PROCEEDINGS OF THE IEEE, VOL. 80, NO. 4, APRIL IW2susing Bedrosian's product theorem (BPT) summarized in Appendix C. It leads to the following result: Equation (39) is valid... |

3 | Copson, Asymptotic Expansions - T - 1965 |

2 |
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(Show Context)
Citation Context ... be the average of the frequencies that exist in the time-frequency plane at a given time [17]. This can be expressed as (34) Several other authors have also contributed to the study of IF [l], [21], =-=[22]-=-, [24], [34]. Rihaczek [34] investigated the points in the time-frequency plane of his complex energy distribution where the signal energy is concentrated. He showed that the energy was concentrated i... |

2 |
Timefrequency signal analysis and instantaneous frequency: their inter-relationship and applications”, presented at the
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(Show Context)
Citation Context ...ese two theorems relate the apparent difference in approach between the two methods of defining and measuring the IF: the IF describing the FM law in communication theory, and the signal analyst's IF =-=[23]-=-. Although the same quantity is being estimated, the different approaches adopted may lead to different results: i) Consider the typical FM transmission signal in communications systems: s(t) = a(t). ... |

2 |
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Citation Context ...bution yields the IF, regardless of how fast the IF varies. The problems associated with defining the IF are analogous to the problems associated with defining the instantaneous amplitude or envelope =-=[34a]-=-. A review of frequency, amplitude and phase in the context of oscillation theory is provided in [40a]. Other authors have contributed to the development of the discrete IF, which will be dealt with i... |

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Citation Context ... product. We will consider signals with a BT product sufficiently high (BT > 10) so that the approximation error involved in assuming band and time limited functions is very small. (Landau and Pollak =-=[38]-=- have found that more than 99% of signal energy is preserved within the limits of B and T if BT > 5). Note that FM signals used in communications and seismic belong to this class. All signals consider... |

1 | Instantaneous and time varying spectra-An introduction - Ackroyd - 1963 |

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Citation Context ...e average of the frequencies that exist in the time-frequency plane at a given time [17]. This can be expressed as (34) Several other authors have also contributed to the study of IF [l], [21], [22], =-=[24]-=-, [34]. Rihaczek [34] investigated the points in the time-frequency plane of his complex energy distribution where the signal energy is concentrated. He showed that the energy was concentrated in freq... |

1 |
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Citation Context ...ed for a Gaussian signal [20], [36]. Several authors have tried to develop a theoretical basis for the imposition of finite duration and finite bandwidth. A review is given in [7]. Lacoume and Kofman =-=[27]-=- formally defined a signal class CBT as the class of finite energy signals whose energy is approximately localized in the band [-B/2, B/2] and in time [-T/2, T/2]. Almost all signals encountered in pr... |

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Citation Context ...f equivalent bandwidth and equivalent duration given in (53) and (54) respectively, we know that this product is a constant, and that the minimum value of 1/4~ is attained for a Gaussian signal [20], =-=[36]-=-. Several authors have tried to develop a theoretical basis for the imposition of finite duration and finite bandwidth. A review is given in [7]. Lacoume and Kofman [27] formally defined a signal clas... |

1 | An investigation of whistling atmospherics - R - 1953 |

1 |
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(Show Context)
Citation Context ... was used by Carson and Fry [Ill to estimate the limits of IF variations for FM signals, by Rihaczek [34] to show that the energy is concentrated in frequency about the IF. It was also used by Vakman =-=[39]-=- to explain a seemingly paradoxical situation of instantaneous frequency: although the IF is a local concept (indicated by term “instantaneous”), to calculate it is necessary to determine the HT of th... |

1 |
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Citation Context ...IF are analogous to the problems associated with defining the instantaneous amplitude or envelope [34a]. A review of frequency, amplitude and phase in the context of oscillation theory is provided in =-=[40a]-=-. Other authors have contributed to the development of the discrete IF, which will be dealt with in the sequel [8]. C. Model, Notations, Definitions The review in the previous section shows that there... |