Basic Tasks of Signal Processing in Spectroscopy Challenges with quantification of time signals The quantum-mechanical concept of resonances in scattering and spectroscopy Resonance profiles Why is this topic relevant to biomedical researchers and clinical practitioners? The Role of Quantum Mechanics in Signal Processing Direct link of quantum-mechanical spectral analysis with rational response functions Expansion methods for signal processing Recurrent time signals and their generating fractions as spectra with no recourse to Fourier integrals Fast Padé transform (FPT) for quantum-mechanical spectral analysis and signal processing Padé acceleration and analytical continuation of time series Description of the background contribution by the off-diagonal FPT Diagonal and para-diagonal FPT Froissart doublets and the exact number of resonances Harmonic Transients in Time Signals Rational response function to generic external perturbations The exact solution for the general harmonic inversion problem General time series Response or Green function The key prior knowledge: internal structure of time signals The Rutishauser quotient-difference recursive algorithm The Gordon product-difference recursive algorithm The Lanczos continued fractions The Padé-Lanczos approximant FPT(−)outside the unit circle FPT(+)inside the unit circle Signal-Noise Separation via Froissart Doublets Critical importance of poles and zeros in generic spectra Spectral representations via Padé poles and zeros: pFPT(±)and zFPT(±) Padé canonical spectra Signal-noise separation: exclusive reliance upon resonant frequencies Model reduction problem via Padé canonical spectra Denoising Froissart filter Signal-noise separation: exclusive reliance upon resonant amplitudes Padé partial fraction spectra Model reduction problem via Padé partial fraction spectra Disentangling genuine from spurious resonances Padé Processing for Magnetic Resonance (MR) Total Shape Spectra from in vivoFree Induction Decays (FIDs)Description of the background contribution by the off-diagonal FPT Diagonal and para-diagonal FPT Froissart doublets and the exact number of resonances Harmonic Transients in Time Signals Rational response function to generic external perturbations The exact solution for the general harmonic inversion problem General time series Response or Green function The key prior knowledge: internal structure of time signals The Rutishauser quotient-difference recursive algorithm The Gordon product-difference recursive algorithm The Lanczos continued fractions The Padé-Lanczos approximant FPT(−)outside the unit circle FPT(+)inside the unit circle Signal-Noise Separation via Froissart Doublets Critical importance of poles and zeros in generic spectra Spectral representations via Padé poles and zeros: pFPT(±)and zFPT(±) Padé canonical spectra Signal-noise separation: exclusive reliance upon resonant frequencies Model reduction problem via Padé canonical spectra Denoising Froissart filter Signal-noise separation: exclusive reliance upon resonant amplitudes Padé partial fraction spectra Model reduction problem via Padé partial fraction spectra Disentangling genuine from spurious resonances Padé Processing for Magnetic Resonance (MR) Total Shape Spectra from in vivoFree Induction Decays (FIDs)inside the unit circle Signal-Noise Separation via Froissart Doublets Critical importance of poles and zeros in generic spectra Spectral representations via Padé poles and zeros: pFPT(±)and zFPT(±) Padé canonical spectra Signal-noise separation: exclusive reliance upon resonant frequencies Model reduction problem via Padé canonical spectra Denoising Froissart filter Signal-noise separation: exclusive reliance upon resonant amplitudes Padé partial fraction spectra Model reduction problem via Padé partial fraction spectra Disentangling genuine from spurious resonances Padé Processing for Magnetic Resonance (MR) Total Shape Spectra from in vivoFree Induction Decays (FIDs)Free Induction Decays (FIDs) Comparison of the performances of the FPT and fast Fourier transform (FFT) for total shape spectra The FIDs, convergence regions, and absorption spectra at full signal length for 4T and 7T Convergence patterns of the FPT(−)and FFT for absorption spectra at 4T and 7T Error analysis Prospects for comprehensive applications of the FPT to in vivoMR time signals for brain tumor diagnostics Exact Reconstructions of Spectral Parameters by FPT Tabular data Absorption total shape spectra Residual spectra and consecutive difference spectra Absorption component shape spectra of individual resonances Distributions of reconstructed spectral parameters in the complex plane Discussion Relevance of exact quantification in brain tumor diagnostics Machine Accurate Padé Quantification and Exact Signal-Noise Separation Numerical presentation of the spectral parameters Direct comparison of the performance of the FFT and the FPT Convergence of total shape spectra versus component spectra in the FPT Signal-noise separation through the concept of Froissart doublets/pole-zero cancellation Diagnostic significance of the Froissart filter for exact signal-noise separation Magnetic Resonance Spectroscopy (MRS) and Magnetic Resonance SpectroscopicImaging (MRSI) in Neuro-Oncology: Achievements and Challenges MRS and MRSI as a key non-invasive diagnostic modality for neuro-oncology Major limitations and dilemmas with MRS and MRSI in neuro-oncology related to reliance upon conventional Fourier-based data analysis Accurate extraction of clinically relevant metabolite concentrations for neurodiagnostics via MRS Padé Quantification of Malignant and Benign Ovarian MRS Data Studies to date using in vivoproton MRS to evaluate benign and malignant ovarian lesions Insights for ovarian cancer diagnostics from in vitroMRS Performance of the FPT for in vitroMRS data derived from benign and malignant ovarian cyst fluid, and comparisons with the FFT Prospects for Padé-optimized MRS for ovarian cancer diagnostics Breast Cancer and Non-Malignant Breast Data: Quantification by FPT Current challenges in breast cancer diagnostics In vivoMR-based modalities for breast cancer diagnostics and clinical assessment Insights for breast cancer diagnostics from in vitroMRS Performance of the FPT for MRS data from breast tissue Prospects for Padé-optimized MRS for breast cancer diagnostics Multiplet Resonances in MRS Data from Normal and Cancerous Prostate Dilemmas in prostate cancer diagnostics and screening Insights for prostate cancer diagnostics by means of 2D in vivoMRS and in vitroMRS Performance of the FPT for MRS data from prostate tissue Prospects for Padé-optimized MRSI within prostate cancer diagnostics General Discussion Why the FPT for signal processing? The two variants of the FPT converging inside and outside the unit circle Computation of the complex frequencies and amplitudes by FPT Interpolation and extrapolation by the FPT Determination of the exact number of metabolites Lorentzian and non-Lorentzian spectra both computed by FPT Validity assessment of the FPT Error analysis Clinical ramifications of implementing Padé-based in vivoMRS: special importance for cancer diagnostics Conclusions and Outlooks Prediction and extrapolation for resolution improvement lt;/P> Relevance of exact quantification in brain tumor diagnostics Machine Accurate Padé Quantification and Exact Signal-Noise Separation Numerical presentation of the spectral parameters Direct comparison of the performance of the FFT and the FPT Convergence of total shape spectra versus component spectra in the FPT Signal-noise separation through the concept of Froissart doublets/pole-zero cancellation Diagnostic significance of the Froissart filter for exact signal-noise separation Magnetic Resonance Spectroscopy (MRS) and Magnetic Resonance SpectroscopicImaging (MRSI) in Neuro-Oncology: Achievements and Challenges MRS and MRSI as a key non-invasive diagnostic modality for neuro-oncology Major limitations and dilemmas with MRS and MRSI in neuro-oncology related to reliance upon conventional Fourier-based data analysis Accurate extraction of clinically relevant metabolite concentrations for neurodiagnostics via MRS Padé Quantification of Malignant and Benign Ovarian MRS Data Studies to date using in vivoproton MRS to evaluate benign and malignant ovarian lesions Insights for ovarian cancer diagnostics from in vitroMRS Performance of the FPT for in vitroMRS data derived from benign and malignant ovarian cyst fluid, and comparisons with the FFT Prospects for Padé-optimized MRS for ovarian cancer diagnostics Breast Cancer and Non-Malignant Breast Data: Quantification by FPT Current challenges in breast cancer diagnostics In vivoMR-based modalities for breast cancer diagnostics and clinical assessment Insights for breast cancer diagnostics from in vitroMRS Performance of the FPT for MRS data from breast tissue Prospects for Padé-opti.