Norm Estimates for Selfadjoint Toeplitz Operators on the Fock Space

An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes the fact that the norm of such operator is strictly less than the supremum norm of the symbol. Consequences for time-frequency localization operators are also given.

Underwater Target Localization Using Opportunistic Ship Noise Recorded on a Compact Hydrophone Array

In this research, a new application using broadband ship noise as a source-of-opportunity to estimate the scattering field from the underwater targets is reported. For this purpose, a field trial was conducted in collaboration with JASCO Applied Sciences at Duncan’s Cove, Canada in September 2020. A hydrophone array was deployed in the outbound shipping lane at a depth of approximately 71 m to collect broadband noise data from different ship types and effectively localize the underwater targets. In this experiment, a target was installed at a distance (93 m) from the hydrophone array at a depth of 25 m. In this study, a matched field processing (MFP) algorithm is utilized for localization. Different propagation models are presented using Green’s function to generate the replica signal; this includes normal modes in a shallow water waveguide, the Lloyd-mirror pattern for deep water, as well as the image model. We use the MFP algorithm with different types of underwater environment models and a proposed estimator to find the best match between the received signal and the replica signal. Finally, by applying the scatter function on the proposed multi-channel cross correlation coefficient time-frequency localization algorithm, the location of target is detected.

Fourier Interpolation and Time-Frequency Localization

We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

Time-Frequency Signal Synthesis and Its Application in Multimedia Watermark Detection

A novel approach is proposed in this thesis to synthesize the time domain chirp signal from the joint time-frequency distribution (TFD). The objective is to reconstruct the original signal from its corrupted version. The new signal synthesis technique is based on the Discrete Polynomial Phase Transform (DPPT) and the TFD of the signal to be synthesized. The TFD is used to separate the mono-component signals from a multi-component signal. The DPPT is then applied on the estimated mono-components to have a final synthesized version of the individual time domain signals. The candidate TFD to be used in the synthesis technique is chosen from a group of common TFDs based on their performance with different types of signals. The criteria for the comparison are joint time-frequency localization, low susceptibility to noise, cross-term suppression and the precision of the instantaneous frequency estimated from these distributions. Smoothed Psuedo Wigner-Ville Distribution is chosen as the processing TDFD in the proposed signal synthesis technique. The proposed chirp synthesis technique is applied to detect the presence of the chirp signal embedded as a watermark message in multimedia security applications. The technique can detect the presence of chirp signals from a corrupted chirp with a bit error rate up to signal synthesis is proved to be less than that of the detection method based on the Hough Radon Transform and the proposed signal synthesis technique may also be used as an error correction tool in other applications.

Time-Frequency Signal Synthesis and Its Application in Multimedia Watermark Detection

A novel approach is proposed in this thesis to synthesize the time domain chirp signal from the joint time-frequency distribution (TFD). The objective is to reconstruct the original signal from its corrupted version. The new signal synthesis technique is based on the Discrete Polynomial Phase Transform (DPPT) and the TFD of the signal to be synthesized. The TFD is used to separate the mono-component signals from a multi-component signal. The DPPT is then applied on the estimated mono-components to have a final synthesized version of the individual time domain signals. The candidate TFD to be used in the synthesis technique is chosen from a group of common TFDs based on their performance with different types of signals. The criteria for the comparison are joint time-frequency localization, low susceptibility to noise, cross-term suppression and the precision of the instantaneous frequency estimated from these distributions. Smoothed Psuedo Wigner-Ville Distribution is chosen as the processing TDFD in the proposed signal synthesis technique. The proposed chirp synthesis technique is applied to detect the presence of the chirp signal embedded as a watermark message in multimedia security applications. The technique can detect the presence of chirp signals from a corrupted chirp with a bit error rate up to signal synthesis is proved to be less than that of the detection method based on the Hough Radon Transform and the proposed signal synthesis technique may also be used as an error correction tool in other applications.

A prediction framework with time-frequency localization feature for detecting the onset of seismic events

Onset detection of P-wave in seismic signals is of vital importance to seismologists because it is not only crucial to the development of early warning systems but it also aids in estimating the seismic source parameters. All the existing P-wave onset detection methods are based on a combination of statistical signal processing and time-series modeling ideas. However, these methods do not adequately accommodate some advanced ideas that exist in fault detection literature, especially those based on predictive analytics. When combined with a time-frequency (t-f) / temporal-spectral localization method, the effectiveness of such methods is enhanced significantly. This work proposes a novel real-time automatic P-wave detector and picker in the prediction framework with a time-frequency localization feature. The proposed approach brings a diverse set of capabilities in accurately detecting the P-wave onset, especially in low signal-to-noise ratio (SNR) conditions that all the existing methods fail to attain. The core idea is to monitor the difference in squared magnitudes of one-step-ahead predictions and measurements in the time-frequency bands with a statistically determined threshold. The proposed framework essentially accommodates any suitable prediction methodology and time-frequency transformation. We demonstrate the proposed framework by deploying auto-regressive integrated moving average (ARIMA) models for predictions and the well-known maximal overlap discrete wavelet packet transform (MODWPT) for the t-f projection of measurements. The ability and efficacy of the proposed method, especially in detecting P-waves embedded in low SNR measurements, is illustrated on a synthetic data set and 200 real-time data sets spanning four different geographical regions. A comparison with three prominently used detectors, namely, STA/LTA, AIC, and DWT-AIC, shows improved detection rate for low SNR events, better accuracy of detection and picking, decreased false alarm rate, and robustness to outliers in data. Specifically, the proposed method yields a detection rate of 89% and a false alarm rate of 11.11%, which are significantly better than those of existing methods.

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators $$A_a^{\varphi _1,\varphi _2}$$ A a φ 1 , φ 2 with symbols a in the wide weighted modulation space $$ M^\infty _{w}({\mathbb {R}^{2d}})$$ M w ∞ ( R 2 d ) , and windows $$ \varphi _1, \varphi _2 $$ φ 1 , φ 2 in the Gelfand–Shilov space $$\mathcal {S}^{\left( 1\right) }(\mathbb {R}^d)$$ S 1 ( R d ) . If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $$A_a^{\varphi _1,\varphi _2}$$ A a φ 1 , φ 2 have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space $$ \mathcal {S}^{(\gamma )} (\mathbb {R^{d}}) $$ S ( γ ) ( R d ) , where the parameter $$\gamma \ge 1 $$ γ ≥ 1 is related to the growth of the considered weight. An important role is played by $$\tau $$ τ -pseudodifferential operators $$Op_{\tau } (\sigma )$$ O p τ ( σ ) . In that direction we show convenient continuity properties of $$Op_{\tau } (\sigma )$$ O p τ ( σ ) when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $$Op_{\tau } (\sigma )$$ O p τ ( σ ) when the symbol $$\sigma $$ σ belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.