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Cognitive Radio Spectrum-Sharing Technology
Published in Krzysztof Iniewski, Wireless Technologies, 2017
Danijela Cabric, Robert W. Brodersen
If pulses are processed as real signals, the perfect estimation of an impulse phase is not possible. Commonly, the phase is extracted from the analytic signal composed of the in-phase and quadrature components. In narrowband receiver, the analytic signal is obtained via mixing with sine and cosine at the intermediate frequency (IF) stage of the receiver. In case of wideband, it can be obtained by performing a Hilbert transform on the received real signal after A/D conversion. Note that this approach effectively does not require I and Q mixers; thus, only one A/D converter is sufficient. The Hilbert transformers can be implemented in digital domain as an finite impulse response (FIR) or fast Fourier transform (FFT). The real and imaginary parts of the analytic signal are orthogonal, and the phase information can be studied on the Euler plane. A coordinate rotation digital computer (CORDIC) block can be used to calculate the phase and magnitude of the complex signal, which are used for impulse detection. For antipodal signaling the constellation plot allows threshold-based detection, similar to binary phase shift keying (BPSK) detection [8].
Fourier Optics and Image Processing
Published in Daniel Malacara-Hernández, Brian J. Thompson, Advanced Optical Instruments and Techniques, 2017
The relationship between the real and imaginary parts of an analytic signal can be described by the Hilbert transform. The discrete Hilbert transform (DHT) matrix can be written as hmn={2sin2[π(m−n)/2]π(m−n)m−n≠0,0,m−n=0.
Estimation of ToA/TDoA/FDoA
Published in Prabhakar S. Naidu, Distributed Sensor Arrays Localization, 2017
Example 3.7In Figure 3.22a, the plot of a sinusoidal signal with a sudden jump of frequency from 0.125 Hz to 0.16 Hz (normalized frequency) is shown. The jump is at time units of 65 (64 sec). This is the transmitted signal. It reaches the first sensor after 8 sec and the second sensor after 12 sec. There is a 4 sec TDoA. We compute the analytic signal using the Hilbert transform of each received signal, which becomes the real part of the analytic signal. From the analytic signal, we compute the phase function. A numerical differentiation of the phase function after unwrapping is carried out. We obtain instantaneous frequency (must be positive) as a function of time. We have plotted both instantaneous frequency variations in the same graph (see Figure 3.22b) to emphasize relative shift, which in this case is four time units. The spectrum shown in red refers to the near sensor and that in black refers to the farther sensor. This is a noise-free example. With background noise added, the results deteriorate as shown in Figure 3.22c for snr = 100. For still higher noise levels, say, for snr = 10 the lateral shift is barely discernable.
Detection and classification of internal defects in limestone blocks based on a deconvolution technique with SI-PLCA applied to GPR signals
Published in Research in Nondestructive Evaluation, 2019
Maria Violeta Montiel-Zafra, F. Canadas-Quesada, P. Vera-Candeas, N. Ruiz-Reyes, J. Rey Arrans, J. Martínez López
The Hilbert Transform is a widely used tool for expresing a band-pass signal as a low-pass signal and a phasor [37]. Each trace (A-Scan) of the pre-processed radargram, , is a signal with a certain envelope, frequency, and phase and can be converted to an analytic signal , where is the HT of the input signal . Besides, the imaginary part of the analytic signal is the real part with a phase shifted by 90°. Thus, and are the complex signal and the phase of the signal , respectively. Then, is the envelope. Thus, the whole B-Scan can be processed using the SI-PLCA algorithm since the inputs are non-negative values.
Differences in inter-segment coordination between high- and low-calibre ice hockey players during forward skating
Published in Sports Biomechanics, 2020
Caitlin M. Mazurek, David J. Pearsall, Philippe J. Renaud, Shawn M. Robbins
Phase angles were computed for the foot, shank and thigh using the Hilbert transform approach. The Hilbert transform allows for a clear assessment of the phase difference through the transformation of a real signal into a complex, analytic signal (Ippersiel et al., 2018; Lamb & Stöckl, 2014). A double reflection method was employed to pad the signal to address issues with data distortion associated with Hilbert transform (Ippersiel et al., 2019). Next, CRP was calculated by determining the absolute difference in the phase angles between body segments (proximal minus distal) in specific planes (Burgess-Limerick et al., 1993) including shank versus thigh in the sagittal plane (shank-sagittal/thigh-sagittal), shank in the sagittal versus thigh in the frontal plane (shank-sagittal/thigh-frontal) and foot versus shank in the sagittal plane (foot-sagittal/shank-sagittal). These segments generally produce the largest amplitudes in those specific directions during skating. A value of 0 degrees indicates the segments are moving completely in-phase with one another (e.g., windshield wipers moving side to side together), while 180 degrees indicates the segments are moving completely anti-phase (e.g., windshield wipers rotating to the centre at the same time). Values closer to 0 or 180 may be relatively in-phase or anti-phase, respectively (Oullier et al., 2002). Cubic spline interpolation was used to normalise CRP waveforms to 100% of stride from first ice contact within the capture area to the successive ice contact of the same skate. CRP calculations were performed using Matlab (version R2018a, MathWorks Inc., Natick, USA).
Reconstruction of analytic signal in Sobolev space by framelet sampling approximation
Published in Applicable Analysis, 2018
where is the characteristic function of the set I, and the Fourier transform of any tempered distribution g is defined by . In addition to the discard of negative frequency, analytic signal is a commonly accepted tool of defining instantaneous features such as instantaneous amplitude (IA), instantaneous phase (IP) and instantaneous frequency (IF).[1,2] Specifically,