Transform Techniques in Chemistry

The application of the Fourier transform is being seen to an increasing extent in all branches of chemistry, but it is in the area of chemical analysis that the greatest activity is taking place. Fourier transform infrared and nuclear magnetic resonance spectrometry are already routine methods for obtaining high-sensitivity IR and NMR spectra. Analogous methods are now being developed for mass spectrometry (Fourier transform ion cyclo­ tron resonance spectrometry) and microwave spectroscopy, and Fourier transform techniques have been successfully applied in several areas of electrochemistry. In addition the fast Fourier transform algorithm has been used for smoothing, interpolation, and more efficient storage of data, and has been studied as a potential method for more efficient identification of samples using pattern recognition techniques. Linear transforms have also been shown to be useful in analytical chemistry. Probably the most important of these is the Hadamard transform, which has been applied in alternative methods for obtaining IR and NMR data at high sensitivity. Even though measurements involving this algorithm will probably not be applied as universally as their Fourier transform ana­ logs, in the area of pattern recognition application of the Hadamard trans­ form will in all probability prove more important than application of the Fourier transform.

1 Transform Techniques in Chemistry: Past, Present, and Future.- 1.1. The Past.- 1.1.1. Optical Spectroscopy.- 1.1.2. NMR Spectroscopy.- 1.1.3. Data Processing.- 1.2. The Present.- 1.3. The Future.- References.- 2 The Fourier Transform and Related Concepts: A First Look.- 2.1. Introduction: Guitar Tuning.- 2.2. Differences in Space and Time: Phase.- 2.3. Sums, Integrals, and Orthogonality.- 2.4. Various Expressions of Fourier Transform Relationships.- 2.5. Concepts and Corollaries for Fourier Transforms.- 2.6. More on Phase and Phase Correction.- 2.7. Apodization and Resolution Enhancement.- 2.8. The Discrete Fourier Transform.- 2.9. Walsh and Hadamard Transforms.- 2.10. Summary.- References.- 3 Multichannel Methods in Spectroscopy.- 3.1. Introduction.- 3.2. Spectrometer Sources and Detectors.- 3.2.1. Terminology.- 3.2.2. Single-Channel (Scanning-Type) Spectrometer.- 3.2.3. Multidetector Spectrometer.- 3.3. Weights on a Balance: The Multichannel Advantage. Multiplex Methods.- 3.3.1. One-at-a-Time Weighing: The Scanning Spectrometer.- 3.3.2. Many Balances: The Multidetector Spectrometer.- 3.3.3. Half the Weights on the Balance at Once: Hadamard Multiplexing.- 3.3.4. All the Weights on the Balance at Once: The Fourier Advantage.- 3.4. Hadamard Multiplexing of Spatially Dispersed Spectra.- 3.5. Advantages of Coherent Radiation in Spectrometer Detection.- 3.6. Fourier Methods.- 3.6.1. Fourier Multiplexing: The Multichannel Advantage.- 3.6.2. Fourier Analysis of Detector Response: Spectral Line Shape.- 3.6.3. Pulsed Monochromatic Coherent Radiation as a Broad-Band Radiation Source.- 3.7. Summary: Relations Between Different Spectrometers.- 3.8. Appendix. Noise Considerations for Multichannel Spectrometers.- 3.8.1. NB ? (signal)1/2: “Source-Limited” Noise.- 3.8.2. NA = constant: “Detector-Limited” Noise.- 3.8.3. NC ? signal: “Fluctuation” Noise.- References and Notes.- 4 Data Handling in Fourier Transform Spectroscopy.- 4.1. The Computer System.- 4.1.1. Introduction to Computers.- 4.1.2. Data Acquisition.- 4.1.3. Timing in Data Acquisition.- 4.1.4. The Sampling Theorem.- 4.1.5. Digital Phase Correction.- 4.1.6. Signal Averaging.- 4.1.7. Signals Having High Dynamic Range.- 4.1.8. Other Computer Requirements.- 4.1.9. Disk-Based Data Acquisition.- 4.1.10. Comparison of Data System Requirements in NMR and IR.- 4.2. The Fourier Transform.- 4.2.1. Introduction.- 4.2.2. The Cooley-Tukey Algorithm.- 4.2.3. The Signal Flow Graph.- 4.2.4. In-Place Transforms.- 4.3. Writing a Fourier Transform for a Minicomputer.- 4.3.1. Introduction.- 4.3.2. The Form of W.- 4.3.3. The Fundamental Operations.- 4.3.4. The Sine Look-Up Table.- 4.3.5. Binary Fractions.- 4.3.6. The Sine Look-Up Routine.- 4.3.7. Scaling during the Transform.- 4.3.8. Forward and Inverse Transforms.- 4.3.9. Forward Transforms of Real Data.- 4.3.10. Inverse Real Transforms.- 4.3.11. Baseline Correction.- 4.3.12. A Fourier Transform Routine.- 4.3.13. Correlation.- 4.3.14. Disk-Based Fourier Transforms.- 4.3.15. Hardware Fourier Processors.- 4.4. Noise in the Fourier Transform Process.- 4.4.1. Round-Off Errors.- 4.4.2. Block Averaging.- 4.4.3. Double-Precision Fourier Transforms.- 4.5. Summary.- References.- 5 Fourier Transform Infrared Spectrometry: Theory and Instrumentation.- 5.1. Introduction.- 5.2. The Michelson Interferometer.- 5.3. Resolution and Apodization.- 5.4. Effect of Beam Divergence.- 5.5. Mirror Drive Tolerance.- 5.6. Dynamic Range.- 5.7. Scan Speed and Spectral Modulation.- 5.8. Data Acquisition.- 5.9. Beamsplitters.- 5.10. Lamellar Grating Interferometers.- 5.11. Detectors for FT-IR.- 5.11.1. Far-Infrared Detectors.- 5.11.2. Mid- and Near-Infrared Detectors.- 5.11.3. Ultraviolet-Visible Spectroscopy.- 5.12. Auxiliary Optics.- 5.12.1. Source Optics.- 5.12.2. Absorption Spectroscopy.- 5.12.3. Reflection Spectroscopy.- 5.13. Data Systems.- 5.13.1. Far-Infrared Spectroscopy.- 5.13.2. Mid-Infrared Spectroscopy.- 5.13.3. Ultra-High-Resolution Spectroscopy.- 5.14. Dual-Beam Fourier Transform Spectroscopy.- References.- 6 Infrared Fourier Transform Spectrometry: Applications to Analytical Chemistry.- 6.1. FT-IR versus Grating Spectrophotometers.- 6.1.1. Fellgett’s Advantage.- 6.1.2. Jacquinot’s Advantage.- 6.1.3. Effect of Detector Performance.- 6.1.4. Other Differences.- 6.1.5. Implications.- 6.2. Spectra of Transient Species.- 6.2.1. GC-IR.- 6.2.2. LC-IR.- 6.2.3. Reaction Kinetics.- 6.3. Low-Energy Absorption Spectrometry.- 6.3.1. Far-Infrared Spectrometry.- 6.3.2. Mid-Infrared Absorption Spectrometry.- 6.4. Difference Spectroscopy.- 6.5. Reflection Spectrometry.- 6.6. Emission Spectrometry.- 6.7. Atomic Spectrometry.- References.- 7 Hadamard Transform Analytical Systems.- 7.1. Introduction.- 7.2. Weighing Designs and Optical Multiplexing.- 7.3. Historical Background of Multiplexing by Means of Masks.- 7.4. Mathematical Development.- 7.5. Varieties of Encoded Spectrometers.- 7.6. Limitations: HTS Instruments and Interferometers.- 7.7. Imagers and Spectrometric Imagers.- 7.8. Signal and Noise Limitations.- 7.9. Special Optical Systems.- 7.10. Some Future Applications.- References.- 8 Pulsed and Fourier Transform NMR Spectroscopy.- 8.1. Introduction.- 8.2. Basic Concepts of FT-NMR.- 8.3. Basic Instrumentation.- 8.3.1. The Spectrometer.- 8.3.2. The Sample Probe.- 8.4. Recent Instrumental Improvements.- 8.4.1. Coherent Broad-Band Decoupling.- 8.4.2. Gated Decoupling Methods and Quantitative Measurements.- 8.4.3. Microsample Techniques.- 8.4.4. Selective Population Transfer.- 8.4.5. Studies of Chemical Dynamics.- 8.4.6. High-Resolution 13C NMR in Solid Materials.- 8.4.7. FT-NMR at High Fields.- References.- 9 Advanced Techniques in Fourier Transform NMR.- 9.1. Introduction.- 9.2. Systematic Noise Reduction.- 9.2.1. Noise Reduction Methods.- 9.2.2. Relaxation Times and Spin Echoes.- 9.3. Sideband Filters and Quadrature Detection NMR.- 9.3.1. The Crystal Sideband Filter.- 9.3.2. Quadrature Detection Spectroscopy.- 9.3.3. Operational Details in Quadrature NMR.- 9.3.4. Comparison between Crystal Sideband Filter and Quadrature Detection.- 9.4. Rapid-Scan (Correlation) NMR.- 9.4.1. General Description.- 9.4.2. Data Processing Methods.- 9.5. Noise Excitation Methods.- 9.5.1. Stochastic Resonance Spectroscopy.- 9.5.2. Hadamard Transform NMR.- 9.5.3. Tailored Excitation.- 9.6. Measure of the Spin-Lattice Relaxation Time T1.- 9.6.1. General Description.- 9.6.2. Reasons for Measuring T1.- 9.6.3. Methods of Measuring T1.- 9.6.4. Progressive Saturation.- 9.6.5. Homospoil-T1 Methods.- 9.6.6. Experimental Techniques in the Measurement of T1.- 9.7. Techniques for the Suppression of Strong Solvent Peaks.- 9.7.1. Introduction.- 9.7.2. Block Averaging.- References.- 10 Fourier Transform Ion Cyclotron Resonance Spectroscopy.- 10.1. Introduction.- 10.2. Fundamental Equations for ICR Linewidth and Resolution.- 10.3. Fourier Transform Ion Cyclotron Resonance (FT-ICR) Spectroscopy.- 10.4. Analytical FT-ICR Linewidth and Mass Resolution.- 10.5. FT-ICR Mass Range, Computer Data Size, and Sampling Rate.- 10.6. Discussion.- References.- 11 Fourier Domain Processing of General Data Arrays.- 11.1. Introduction.- 11.2. Fourier Transformation and a General Data Array.- 11.3. Amplitude and Phase Arrays.- 11.4. Transformation as a Reversible Operation.- 11.5. Specific Manipulations of Data in the Fourier Domain.- 11.5.1. Fourier Domain Manipulations without Using Weighting Functions.- 11.5.1.1. Zero Filling.- 11.5.1.2. Contrast Enhancement.- 11.5.2. Fourier Domain Manipulations Using Weighting Functions.- 11.5.2.1. Smoothing.- 11.5.2.2. Elimination of Low-Frequency Interferences.- 11.5.2.3. Differentiation and Integration.- 11.5.2.4. Resolution Enhancement and Functional Isolation.- 11.6. Summary.- References.- 12 Fourier and Hadamard Transforms in Pattern Recognition.- 12.1. Introduction.- 12.1.1. Basic Pattern Recognition System.- 12.1.2. Preprocessor-Feature Extractor.- 12.1.3. Classifier.- 12.2. Binary Pattern Classifiers.- 12.2.1. Pattern Vectors.- 12.2.2. Similarity and Clustering.- 12.2.3. K-Nearest-Neighbor Classification.- 12.2.4. Decision Surfaces.- 12.2.5. TLUs as Binary Pattern Classifiers.- 12.2.5.1. Training of TLUs Using Error Correction Feedback.- 12.2.5.2. Properties of TLUs.- 12.2.6. Preprocessing and Transformations.- 12.3. Fourier and Hadamard Transforms in Pattern Recognition.- 12.3.1. Feature Reduction.- 12.3.2. Pattern Recognition Analysis of NMR Data.- 12.3.2.1. Simulated Free-Induction Decay Analysis.- 12.3.2.2. Hadamard-Transformed Data Analysis.- 12.3.2.3. Autocorrelation Transforms.- 12.4. Conclusions.- References.- 13 Spectral Representations for Quantized Chemical Signals.- 13.1. Introduction.- 13.2. 13C FID Signals and Their Spectra.- 13.3. Orthogonal Expansions and Spectral Representations.- 13.4. Clipped Signals and Their Spectral Representations.- 13.5. Random Real-Zero Signals.- 13.6. Zero-Based Product Representations for Band-Limited Signals.- 13.7. Spectra of Clipped FID Signals.- 13.8. Summary, Implications, and Open Questions.- Notation.- Appendix. Intermodulation Distortion in the CFID.- References.- 14 Applications of the FFT in Electrochemistry.- 14.1. Introduction.- 14.2. Faradaic Admittance Measurements―Basic Principles.- 14.3. Instrumentation.- 14.4. Kinetics of Electrode Processes.- 14.5. Relevant Properties of the FFT for Electrochemical Relaxation Measurements.- 14.6. Published and Future Applications of the FFT in Electrochemistry.- References.