Hyperbolic Chirp Research Articles

A flue gas velocimeter based on the hyperbolic chirp modulation method of acoustic source signal

Accurate measurement of flue gas velocity is of great significance for the safe production, energy saving, and emission reduction of coal-fired power plants. In this paper, a method of flue gas velocimetry based on a hyperbolic chirp-modulating acoustic source was proposed, and its superiority was verified by numerical simulations and field experiments. For an acoustic velocimeter, it is crucial to ensure the accuracy of the acoustic Time-of-Flight (TOF), which directly affects the flue gas velocity measurements. The improvement of the accuracy of estimation of the TOF directly determines the accuracy of the flue gas velocity measurement. Simulation on the different types of acoustic sources at different signal-to-noise ratios (SNRs) was conducted, and the results proved that the hyperbolic chirp signal is more accurate in the measurement of TOF. In the strong noise range of − 10 dB to − 20 dB, the average relative root mean square error (RRMSE) obtained using this hyperbolic chirp modulating acoustic source is 0.60 dB and 0.36 dB lower than obtained using the linear and logarithmic modulating modes, respectively. The average relative error (RE) is approximately 0.87 times that of the linear chirp. Power plant flue field experiments show that the measurement error using this hyperbolic modulating mode is about 0.52 times and 0.74 times that of using linear and logarithmic modulating modes, respectively. In harsh measurement environments with strong noise, the proposed acoustic source modulation scheme can significantly improve the measurement SNR and effectively enhance the accuracy and robustness of flue gas velocimetry.

A prefix‐based approach for joint Doppler and channel estimation in underwater communication

SummaryDeveloping a reliable and robust underwater acoustic communication system is a difficult task due to the complicated nature of the underwater channel, non‐stationary noise, and several other factors. Indeed, channel estimation or equalization presents numerous challenges in this non‐stationary, highly Doppler, multipath environment; as a result, traditional equalizers and PLL‐based methods have limited performance. Generally, communication over such time‐varying channels is accomplished via packets that contain a prefix/preamble signal for training, a payload containing the actual data, and a silent period for proper alignment. The prefix signal must be designed properly because it is used to estimate the channel and determine the start of the packet. In this paper, we propose a new prefix signal based on the hyperbolic chirp signal, which has Doppler invariance properties. These properties enable the extraction of the entire packet even under severe multipath and Doppler. Our new proposed prefix signal can accurately and efficiently characterize an underwater channel by estimating the multipath delay, amplitude, and Doppler scales. Our new method has been validated through extensive simulations using various channel models for its robustness and effectiveness under various conditions. It has also been tested on a real‐world channel.

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

Hyperbolic FRFT‐FBMC for 5G signals

This study considers a novel type of signal for 5G communications based on filter bank multi-carrier-quadrature amplitude modulation (FBMC-QAM). In this new method, the innovative type of fractional Fourier transform on the basis of hyperbolic chirps that denominated as hyperbolic fractional Fourier transform (FRFT) is implemented. The closed form of bit error probability of the introduced signal in Rayleigh and Rician selective fading channels is derived. According to simulations results, utilisation of the HFRFT-FBMC system can attain the better bit error probability in noisy channels, especially in the scenarios of high Doppler frequency rates. Furthermore, the result of calculations with simulations is evaluated.

Ambiguity function and accuracy of the hyperbolic chirp: comparison with the linear chirp

In this study, the authors derive the ambiguity function (AF) of a narrowband and a wideband hyperbolic chirp. They calculate the second derivatives of the squared amplitude of the narrowband complex AF and use them to calculate the Fisher information matrix (FIM) of the estimators of the target range and velocity. The FIM is then used to calculate the Cramer-Rao lower bounds (CRLBs) of the variance of the estimators and to carry out an analysis of estimation performance and a comparison with the case of a linear chirp with a rectangular and a Gaussian amplitude modulation. The analysis and the calculations of the CRLB are also extended to a train of hyperbolic chirps. Results corroborate that at narrowband the hyperbolic chirp is less Doppler tolerant than the linear chirp and show that the hyperbolic chirp provides a comparable measurement accuracy to the linear chirp. Results at wideband corroborate the superior Doppler tolerance of the hyperbolic chirp with respect to that of the linear chirp.

Hyperbolic FRFT‐OFDM system BER analysis over high Doppler Rician fading channels

Hyperbolic fractional Fourier transform (HFRFT) based on hyperbolic chirps utilisation in FRFT is introduced. The exact BER of the HFRFT in a Rician fading channel is derived mathematically. In high speed vehicle applications, specially in LEO orbit satellite communications that have an inherent high Doppler frequency shift, with the HFRFT-OFDM system better BERs than other OFDM systems can be achieved.

Estimation of the instantaneous amplitude and the instantaneous frequency of audio signals using complex wavelets

This work presents a method of analyzing and synthesizing audio signals that uses complex wavelets. In the method, the input signal is filtered by a complex bandpass filter bank through a discrete version of the complex continuous wavelet transform. A general theoretical signal with time-dependent amplitude and phase has been analyzed. The analysis of the information provided by the modulus and the phase of the complex continuous wavelet transform was used to develop an additive analysis/synthesis algorithm for audio signals called complex wavelet additive synthesis. To illustrate the method, this new algorithm has been used to analyze the following four synthetic, non-stationary signals that have different frequency variation laws: a linear chirp, a quadratic chirp, a hyperbolic chirp and an FM signal with a sinusoidal excursion. Finally, the validity of the algorithm has been tested by comparing the results of an analysis of two real sounds with the proposed algorithm and with one existing technique, namely spectral modelling synthesis (SMS).