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Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
Figure 7.28 shows representations of a continuous wave (Figure 7.28A), a wave packet envelope function (Figure 7.28B), and two wave packets (Figure 7.28A,D). A radian frequency phasor representation is sufficient to describe a continuous wave (Figure 7.28A), and a neper frequency phasor is sufficient to describe the envelope of a wave packet (Figure 7.28B). However, a complex plane representation of the complete wave packet (Figure 7.28C) and a complex analytic signal representation (Figure 7.28D) are both nonstationary, whereas the representation would be stationary for a continuous wave. In the case of a nonstationary signal, the instantaneous frequency and the instantaneous bandwidth are changing over time. Therefore, the results are different when the signal is sampled at different points in time. On the other hand, in the case of a conventional, stationary signal, e.g., a sinusoid, sampling the signal at different points in time gives the same results.
Nonlinear Optics of Fibre Waveguides
Published in Yu. N. Kulchin, Modern Optics and Photonics of Nano and Microsystems, 2018
The instantaneous frequency is the derivative of the phase with respect to time. The full phase of the pulse Φ taking into account the linear and nonlinear parts is Φ(z,t)=-ω0T+zvg+|U(0,T)|2zeffLNL. $$ \Phi (z,t) = - \omega _{0} \left( {T + \frac{z}{{v_{g} }}} \right) + |U(0,T)|^{2} \left( {\frac{{z_{{{\text{eff}}}} }}{{L_{{NL}} }}} \right) . $$
Radar Micro-Doppler Signatures for Characterization of Human Motion
Published in Moeness G. Amin, Through-the-Wall Radar Imaging, 2017
Victor C. Chen, Graeme E. Smith, Karl Woodbridge, Chris J. Baker
While a point scatterer P of the target has translational motion with a velocity V and rotation with an initial Euler angle (ϕ0, θ0, ψ0) and an angular velocity vector Ω = [ωx, ωy, ωz]T rotating about the target local-fixed axes x, y, and z, then the Doppler frequency shift of the returned signal from the point scatterer P can be obtained by taking time derivative of the phase function in the returned signal. By definition, the time derivative of a phase function calculates the instantaneous frequency. Here, the instantaneous frequency is just what we want suitable for analysis of mono-component signal generated from a point scatterer P.
Instantaneous Frequency Selective Filtering Using Ensemble Empirical Mode Decomposition
Published in IETE Journal of Research, 2022
Rinki Gupta, Arun Kumar, Rajendar Bahl
Instantaneous frequency (IF) describes the temporal variation in the spectral content of a signal. Consider a real valued signal of the form where and denote the instantaneous amplitude (IA) and the instantaneous phase (IP) of the signal, respectively. The expression in (1) involves defining two unknowns, a(t) and , from a single observation s(t). The analytic signal associated with the real signal s(t) is defined using the Hilbert transform as [1] where H denotes the Hilbert transform. Then, the IF is defined as [1], and the IA is defined as Unlike Fourier frequency, IF is a function of time as seen from (3). This property makes the IF parameter particularly useful in analysing signals where useful information can be obtained by observing how the frequency content of the signal varies with time, such as in radar, sonar, bioacoustic, seismic and vibration signals [1]. Joint time–frequency representations, such as the spectrogram, also provide such information. However, the assumption of piecewise stationarity may not always hold and the underlying signal representation, which is again in terms of the Fourier frequencies, may not always reveal the true IF of the constituent signal components.
Variable mass loading effect on the long-term ambient response of a freeway bridge
Published in Structure and Infrastructure Engineering, 2022
Thomas Furtmüller, Christoph Adam, Robert Veit-Egerer
In Figure 15, the black line shows a sample time-history of the bridge acceleration at the sensor location, for one week. As observed, the daily amplitude increases in the morning and decreases at night, and the response on Sunday (7th day) is significantly smaller compared to the other days. To identify the resonance frequency, the acceleration time-history resulting from response history analysis can be evaluated in the same way as the experimentally obtained data, i.e. the estimation of PSDs and subsequent peak-picking. However, since the signal comprises only one modal contribution, in the present study the so-called instantaneous frequency can be directly determined. Typically, the instantaneous frequency is calculated using the derivative of the phase angle of the Hilbert transform of the considered signal (Mathworks Inc, 2018). A more robust method for determining this quantity is to calculate the first conditional spectral moment of the time-frequency distribution of the signal referred to as spectrogram P(t, f), according to (Mathworks Inc, 2018)
Performance Analysis of Fractional Fourier Domain Beam-Forming Methods for Sensor Arrays
Published in Smart Science, 2019
G. Sreekumar, Leena Mary, A. Unnikrishnan
where is the chirp rate, is the start frequency and c is the initial phase of the chirp. The term defines the phase and its first derivative represents the instantaneous frequency. Therefore three parameters define a chirp completely viz. start frequency, chirp rate and the duration over which t is defined.