Showing papers on "Fourier transform published in 1987"
01 Jan 1987
TL;DR: In this article, a broad perspective of spectral estimation techniques and their implementation is provided, focusing on spectral estimation of discretespace sequences derived by sampling continuous space signals, including parametric methods, minimum variance method, eigenanalysis-based estimators, multichannel methods, and twodimensional methods.
Abstract: This new book provides a broad perspective of spectral estimation techniques and their implementation. It concerned with spectral estimation of discretespace sequences derived by sampling continuousspace signals. Among its key features, the book: · Emphasizes the behavior of each spectral estimator for short data records. · Provides 35 computer programs, including fast algorithms. · Provides the theoretical background and review material in linear systems, Fourier transforms matrix algebra, random processes, and statics. · Summarizes classical spectral estimation as it is practiced today. · Covers Prony’s method, parametric methods, the minimum variance method, eigenanalysis-based estimators, multichannel methods, and twodimensional methods. · Includes problems. · Contains appendices that cover Sunspot Numbers, Complex Test Data, Temperature Data, and Program Conversion for Complex-to-Real Case. Of Special Interest A disk is included that has a double-sides 360kB format readable by any personal computer with an MS-DOS 2 or 3 operating system, such as the IBM XT or AT.
1,975 citations
1,250 citations
Proceedings Article•
01 Jan 1987
TL;DR: The problem of optimal detection of orientation in arbitrary neighborhoods is solved in the least squares sense and it is shown that this corresponds to fitting an axis in the Fourier domain of the n-dimensional neighborhood, the solution of which is a well known solution of a matrix eigenvalue problem.
Abstract: The problem of optimal detection of orientation in arbitrary neighborhoods is solved in the least squares sense. It is shown that this corresponds to fitting an axis in the Fourier domain of the n-dimensional neighborhood, the solution of which is a well known solution of a matrix eigenvalue problem. The eigenvalues are the variance or inertia with respect to the axes given by their respective eigen vectors. The orientation is taken as the axis given by the least eigenvalue. Moreover it is shown that the necessary computations can be pursued in the spatial domain without doing a Fourier transformation. An implementation for 2-D is presented. Two certainty measures are given corresponding to the orientation estimate. These are the relative or the absolute distances between the two eigenvalues, revealing whether the fitted axis is much better than an axis orthogonal to it. The result of the implementation is verified by experiments which confirm an accurate orientation estimation and reliable certainty measure in the presence of additive noise at high level as well as low levels.
558 citations
TL;DR: It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.
Abstract: This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.
426 citations
TL;DR: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane which produces a correlation output formed by an inverse Fourier transform of this binarized Fouriers transform interference intensity.
Abstract: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane. The reference and input images are illuminated by a coherent light at the object plane of a Fourier transform lens system. A image detection device, such as a charge coupled device, is disposed at the Fourier plane of this Fourier transform lens system. A thresholding network detects the median intensity level of the imaging cells of the charge coupled device at the Fourier plane and binarizes the Fourier transform interference intensity. The correlation output is formed by an inverse Fourier transform of this binarized Fourier transform interference intensity. In the preferred embodiment this is achieved via a second Fourier transform lens system. This binary data is then applied to spatial light modulator device operating in a binary mode located at the object plane of a second Fourier transform lens system. This binary mode spatial light modulator device is illuminated by coherent light producing the correlation output at the Fourier plane of the second Fourier transform lens system. The inverse Fourier transform may also be formed via a computer. In an alternative embodiment, the Fourier transform interference intensity is thresholded into one of three ranges. An inverse Fourier transform of this trinary Fourier transform interference intensity produces the correlation output.
333 citations
TL;DR: A Gerchberg-type iterative technique is used to eliminate edge effects in several types of interferogram and results are presented for seeing measurements and interferometric tests.
Abstract: Fourier transform techniques have been used to map the complex fringe visibility in several types of interferogram. A Gerchberg-type iterative technique is used to eliminate edge effects. Results are presented for two specific cases: seeing measurements and interferometric tests.
293 citations
TL;DR: In this article, the wave function at various times during the propagation was split into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction area, and the latter by a single application of a free particle propagator.
Abstract: Various methods using fast Fourier transform algorithms or other ‘‘grid’’ methods for solving the time‐dependent Schrodinger equation are very efficient if the wave function remains spatially localized throughout its evolution. Here we present and test an extension of these methods which is efficient even if the wave function spreads out, provided that the potential remains localized. The idea is to split the wave function at various times during the propagation into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction region, and the latter by a single application of a free particle propagator. This splitting is performed whenever the interaction region wave function comes close to the end of the grid. The total asymptotic wave function at a given time t is reconstructed by adding coherently all the asymptotic wave function pieces which were split at earlier times, after they have been propagated to the common time t. The method is tested by studying the wave function of a diatomic molecule dissociated by a strong laser field. We compute the rate of energy absorption and dissociation and the momentum distribution of the fragments.
261 citations
TL;DR: In this paper, a truncated series expansion of the inverse operator that maps object opacity function to hologram intensity was proposed, which is shown to be equivalent to conventional (optical) reconstruction, with successive terms increasingly supressing the twin image.
Abstract: Digitally sampled in-line holograms may be linearly filtered to reconstruct a representation of the original object distribution, thereby decoding the information contained in the hologram The decoding process is performed by digital computation rather than optically Substitution of digital for optical decoding has several advantages, including selective suppression of the twin-image artifact, elimination of the far-field requirement, and automation of the data reduction and analysis process The proposed filter is a truncated series expansion of the inverse of that operator that maps object opacity function to hologram intensity The first term of the expansion is shown to be equivalent to conventional (optical) reconstruction, with successive terms increasingly sup-pressing the twin image The algorithm is computationally efficient, requiring only a single fast Fourier transform pair
223 citations
TL;DR: The dependence of the calculated Ruderman-Kittel-Kasuya-Yosida range function of a one-dimensional free-electron metal on the order of the integrations over the occupied states, and over the Fourier components of the perturbation is discussed and clarified.
Abstract: The dependence of the calculated Ruderman-Kittel-Kasuya-Yosida range function of a one-dimensional free-electron metal on the order of the integrations over the occupied states, and over the Fourier components of the perturbation is discussed and clarified. The correct result is derived. The range function of a magnetized layer (e.g., a magnetic layer in a multilayer material) is also calculated and compared with the range functions of point sources in the one-dimensional and in the three-dimensional cases.
213 citations
TL;DR: A new implementation of the real-time joint transform correlator architecture using inexpensive LCTVs will be discussed and preliminary experimental results are presented to verify the usefulness of the technique.
Abstract: Recently, many applications of the liquid crystal television (LCTV) to real-time signal processing have been reported. A basic description of the application of the LCTV to realtime pattern recognition was first reported by Liu et al. Gregory later also showed a successful space-invariant correlation using a LCTV as a spatial light modulator. Both methods were based on the use of a complex matched (VanderLugt) spatial filter. The joint transform architecture is an alternative approach to optical pattern recognition. The joint transform method has proved to be suitable for adaptive programmable correlation because no matched spatial filter is required. The reference pattern may simply be generated by a computer and input to a low space-bandwidth product (SBP), electronically addressed, spatial light modulator. Conversely, it is very difficult to generate a dynamic matched spatial filter in the Fourier plane. A microcomputer-based programmable optical correlator using a magnetooptical device (MOD) and a liquid crystal light valve (LCLV) was proposed recently by Yu and Ludman. In this Letter, a new implementation of the real-time joint transform correlator architecture using inexpensive LCTVs will be discussed. Preliminary experimental results are presented to verify the usefulness of the technique. There are three major objections to using commercially available liquid crystal TVs for optical processing applications: (1) low contrast ratio, (2) phase nonuniformity, and (3) low resolution and low SBP. These problems must be minimized if a LCTV is to be used as a spatial light modulator in a coherent optical system. The contrast uniformity
TL;DR: Prony's method as discussed by the authors provides a way of extracting the locations (projected on the path of propagation) and weighting coefficients of scattering centers from the backscattered field as a fuction of frequency.
Abstract: High-frequency scattering can often be described in terms of scattering centers, and an understanding of the geometries which give rise to these centers is important in the area of radar cross section modification. Certain canonical geometries have been treated theoretically with asymptotic methods, but, in general, one must study the behavior of scattering centers empirically. Prony's method provides a way of extracting the locations (projected on the path of propagation) and weighting coefficients of scattering centers from the backscattered field as a fuction of frequency. It has been found to be superior to the conventional Fourier transform technique in resolution and dynamic range.
TL;DR: Experiments with synthetic and real boundaries show that estimates closer to the true values of Fourier descriptors of complete boundaries are obtained and classification experiments performed using real boundaries indicate that reasonable classification accuracies are obtained even when 20-30 percent of the data is missing.
Abstract: We present a method for the classification of 2-D partial shapes using Fourier descriptors. We formulate the problem as one of estimating the Fourier descriptors of the unknown complete shape from the observations derived from an arbitrarily rotated and scaled shape with missing segments. The method used for obtaining the estimates of the Fourier descriptors minimizes a sum of two terms; the first term of which is a least square fit to the given data subject to the condition that the number of missing boundary points is not known and the second term is the perimeter2/area of the unknown shape. Experiments with synthetic and real boundaries show that estimates closer to the true values of Fourier descriptors of complete boundaries are obtained. Also, classification experiments performed using real boundaries indicate that reasonable classification accuracies are obtained even when 20-30 percent of the data is missing.
TL;DR: In this article, the authors considered a system consisting of an arbitrary number of multiconductor transmission lines joined and terminated by arbitrary linear networks, and analyzed the system in the frequency domain, and the inverse Fourier transform was used to obtain the time-domain response.
Abstract: Systems are considered consisting of an arbitrary number of multiconductor transmission lines joined and terminated by arbitrary linear networks. The fines are assumed to be Iossy, with frequency-dependent parameters. The system is analyzed in the frequency domain, and the inverse Fourier transform is used to obtain the time-domain response.
TL;DR: In this article, a positive Radon measure with compact support in the euclidean n-space ℝn is introduced, where the Fourier transform and the average over the spheres are introduced.
Abstract: Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transformand the averages over the spheres we can write the α-energy, 0 < α < n, of μ aswhere the positive constants c1 and c2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .
TL;DR: In this paper, the Fourier transform of a K-dimensional function is zero on continuous surfaces (here called zero sheets) of dimension (2K − 2) in a space that effectively has 2K dimensions.
Abstract: A multiple convolution (e.g., an image formed by convolving several individual components) is automatically deconvolvable, provided that its dimension (i.e., the number of variables of which it is a function) is greater than unity. This follows because the Fourier transform of a K-dimensional function (having compact support) is zero on continuous surfaces (here called zero sheets) of dimension (2K − 2) in a space that effectively has 2K dimensions. A number of important practical applications are transfigured by the concept of the zero sheet. Image restoration can be effected without prior knowledge of the point-spread function, i.e., blind deconvolution is possible even when only a single blurred image is given. It is in principle possible to remove some of the additive noise when the form of the point-spread function is known. Fourier phase can be retrieved directly, and, unlike for readily implementable iterative techniques, complex images can be handled as straightforwardly as real images.
TL;DR: An experimental demonstration of a holographic associative memory that utilizes an array of classic VanderLugt correlators to implement in parallel the inner product between an input and a set of stored reference images.
Abstract: An experimental demonstration of a holographic associative memory is presented. The system utilizes an array of classic VanderLugt correlators to implement in parallel the inner product between an input and a set of stored reference images. Each inner product is used to read out an associated image. Theoretical analysis of the system is given, and experimental results are shown.
TL;DR: The effect of multiexponential transverse relaxation of degenerate transitions on coherence transfer phenomena in high-resoln. NMR is discussed in this paper, and the implications for multiple-quantum-filtered 2-dimensional correlation spectra of large mols are analyzed.
TL;DR: In this article, the adaptive least mean square (LMS) algorithm was used for the calculation of the digital Fourier transform (DFT) using a set of N periodic complex phasors whose frequencies are equally spaced from dc to the sampling frequency.
Abstract: The digital Fourier transform (DFT) and the adaptive least mean square (LMS) algorithm have existed for some time. This paper establishes a connection between them. The result is the "LMS spectrum analyzer," a new means for the calculation of the DFT. The method uses a set of N periodic complex phasors whose frequencies are equally spaced from dc to the sampling frequency. The phasors are weighted and then are summed to generate a reconstructed signal. Weights are adapted to realize a best least squares fit between this reconstructed signal and the input signal whose spectrum is to be estimated. The magnitude squares of the weights correspond to the power spectrum. For a proper choice of adaptation speed, the LMS spectrum analyzer will provide an exact N -sample DFT. New DFT outputs will be available in steady flow after the introduction of each new data sample.
TL;DR: In this paper, the estimation of a local empirical covariance function from a set of observations was done in the Faeroe Islands region using both the space domain method and the frequency domain method.
Abstract: The estimation of a local empirical covariance function from a set of observations was done in the Faeroe Islands region. Gravity and adjusted Seasat altimeter data relative to theGPM2 spherical harmonic approximation were selected holding one value in celles of1/8°×1/4° covering the area. In order to center the observations they were transformed into a locally best fitting reference system having a semimajor axis1.8 m smaller than the one ofGRS80. The variance of the data then was273 mgal2 and0.12 m2 respectively. In the calculations both the space domain method and the frequency domain method were used. Using the space domain method the auto-covariances for gravity anomalies and geoid heights and the cross-covariances between the quantities were estimated. Furthermore an empirical error estimate was derived. Using the frequency domain method the auto-covariances of gridded gravity anomalies was estimated. The gridding procedure was found to have a considerable smoothing effect, but a deconvolution made the results of the two methods to agree.The local covariance function model was represented by a Tscherning/Rapp degree-variance model,A/((i−1)(i−2)(i+24))(RB/RE)2i+2, and the error degree-variances related to the potential coefficient setGPM2. This covariance function was adjusted to fit the empirical values using an iterative least squares inversion procedure adjusting the factor A, the depth to the Bjerhammar sphere(RE−RB), and a scale factor associated with the error degree-variances. Three different combinations of the empirical covariance values were used. The scale factor was not well determined from the gravity anomaly covariance values, and the depth to the Bjerhammar sphere was not well determined from geoid height covariance values only. A combination of the two types of auto-covariance values resulted in a well determined model.
Journal Article•
TL;DR: In this paper, a general approach to obtaining smoothed even-order derivative spectra using Fourier transforms is developed, which is easier to use and yields higher signal-to-noise ratios than do techniques which involve the use of convolution functions.
Abstract: A general approach to obtaining smoothed even-order derivative spectra using Fourier transforms is developed. The method is easier to use and yields higher signal-to-noise ratios than do techniques which involve the use of convolution functions.
TL;DR: In this paper, the authors used deconvolved and second derivative Fourier transform infrared spectra of the proteins flavodoxin and triosephosphate isomerase to obtain the first experimental infrared data on proteins with parallel β-chains.
TL;DR: In this paper, the eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)1/2)exp[(2πi/n)jk], i=(−1) 1/2, is determined.
Abstract: The eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)1/2)exp[(2πi/n)jk], i=(−1)1/2, is determined. In doing so, a multitude of identities, some of which may be new, are encountered. A conjecture is advanced.
TL;DR: In this paper, a Fourier transform spectrometer designed to operate at high resolution at wavelengths down to 170 nm is described, and the principal instrumental parameters are: mirror travel, 200 mm; resolving limit, 0.025 cm-1; collimator aperture ratio, f/25; overall dimensions of vacuum tank, 1.5 m*0.25 m *0.
Abstract: A Fourier transform spectrometer designed to operate at high resolution at wavelengths down to 170 nm is described. The principal instrumental parameters are: mirror travel, 200 mm; resolving limit, 0.025 cm-1; collimator aperture ratio, f/25; overall dimensions of vacuum tank, 1.5 m*0.25 m*0.25 m. Test results show (i) a signal-to-noise ratio in the transformed spectrum at 200 nm better than 1000:1 for an iron-neon hollow cathode lamp at a resolving power of quarter of a million, (ii) fully resolved line profiles in the same source at a resolving limit of 0.03 cm-1 (resolving power 1.5*106), (iii) relative wavenumbers of Fe II emission lines reproducible to +or-0.0006 cm-1 (3.4 fm), and (iv) a significant luminosity gain over grating spectrometers operating in the same region.
TL;DR: A shift-invariant all-optical holographic associative memory implemented using phase conjugate mirrors and Fourier transform holograms with large storage capacity obtained through the use of nonlinearities in the correlation domain is described.
Abstract: We describe a shift-invariant all-optical holographic associative memory implemented using phase conjugate mirrors and Fourier transform holograms. A key feature of our system is the large storage capacity obtained through the use of nonlinearities in the correlation domain. The use of angularly multiplexed plane wave reference beams allows access to the correlation domain where nonlinearities in the phase conjugate mirrors can be used to reduce greatly crosstalk and correlation noise.
TL;DR: In this article, the second-order constant was shown to reflect the polarity of the sample and the third order one was positive being consistent with cooperative dipoles, and the ferroelectric hysteresis loops obtained at very high fields were analyzed in terms of the higher-order Fourier coefficients.
Abstract: Experimental procedures and exemplifying results are described in relation to a nonlinear dielectric investigation of ferroelectric polymers. The apparatus developed here was used to apply a sinusoidal electric field and to detect fundamental and higher harmonic electric displacements with a 0.01% accuracy by means of digital sampling and Fourier transform techniques. Three types of experiments were made for PVDF. Measurements at relatively low fields determined the nonlinear dielectric constants in that the second-order constant was shown to reflect the polarity of the sample and the third order one was positive being consistent with cooperative dipoles. The ferroelectric hysteresis loops obtained at very high fields were analyzed in terms of the higher-order Fourier coefficients. The use of a double frequency wave determined the field-dependent linear dielectric constants which showed a maximum during polarization reversal.
Book•
01 Nov 1987TL;DR: The Geometric Series--An Important Relationship: Difference Equations for Nth-Order Systems and the Discrete Fourier Transform examines the relationships between Linear Time-Invariant Systems, the Inversion Formula, and the DFT.
Abstract: 1. Introduction. Preview. Processing of Speech Signals. Processing of Seismic Signals. Radar Signal Processing. Image Processing. Kalman Filtering and Estimators. Review. References and Other Sources of Information 2. Signals and Systems. Preview. Types of Signals. Sequences Some Basic Sequences. Shifted and Special Sequences. Exponential and Sinusoidal Sequences. General Periodic Sequences. Sampling Continuous-Time Sinusoids and the Sampling Theorem. Systems and Their Properties. Linearity. Time-Invariance. Linear-Time Invariant (LTI) Systems. Stability. Causality. Approximation of Continuous-Time Processes by Discrete Models. Discrete Approximation of Integration. Discrete Approximation of Differentiation. Review. Vocabulary and Important Relations. Problems. References and Other Sources of Information. 3. Linear Time-Invariant Systems. Preview. Linear Constant-Coefficient Difference Equations. The Geometric Series--An Important Relationship. Difference Equations for Nth-Order Systems. Computer Solution of Difference Equations. System Diagrams or Realizations. Unit Sample Response. Convolution. A General Way to Find System Response. Computer Evaluation of Convolution. Analytical Evaluation of Convolution. An Application: Stability and the Unit Sample Response. Interconnected Systems. Cascade Connection. Parallel Connection. Initial Condition Response and Stability of LTI Systems. Forced and Total Response of LTI Systems. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 4. Frequency Response and Filters. Preview. Sinusoidal Steady-State Response of LTI Systems. Frequency Response. Sinusoidal Steady--State Response--General Statement. The Nature of H(Ejq). Computer Evaluation of Frequency Response. Frequency Response from the System Difference Equations. Filters. A Typical Filtering Problem. Comparison of Two Filters. Ideal Filters. Interconnected Systems. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 5. Frequency Response--A Graphical Method. Preview. Graphical Concepts. Geometric Algorithms for Sketching the Frequency Response. Graphical Design of Filters. Stability. Effects of Poles and Zeros on the Frequency Response. Correspondence Between Analog and Digital Frequencies. Some Design Problems. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 6. Z-Transforms. Preview. Definitions. Right-Sided Sequences--Some Transform Pairs. Sample (Impulse) Sequence. Step Sequence. Real Exponential Sequence. Complex Exponential Sequence. General Oscillatory Sequence. Cosine Sequence. Properties and Relations. Linearity. Shifting Property. Multiplication by n and Derivatives In z. Convolution. Transfer Functions. Stability. Frequency Response Revisited. The Evaluation of Inverse Transforms. Inverse Transforms from the Definition. Inverse Transforms from Long Division. Inverse Transforms from Partial Fraction Expansions and Table Look-Up. Partial Fraction Expansion--General Statement. Checking Partial Fraction Expansions and Inverse Transforms. Solution of Difference Equations. Connections Between the Time Domain and the z-Domain. Poles and Zeros and Time Response. System Response to Some Special Inputs. General Results and Miscellany. Noncausal Systems. Convergence and Stability. The Inversion Formula. Review. Vocabulary and Important Relations. Problems. References and Other Sources of Information. 7. Discrete Fourier Transform. Preview. Periodic Sequences. Complex Exponentials. Discrete Fourier Series. Finite Duration Sequences and the Discrete Fourier Transform. Some Important Relationships. DFTs and the Fourier Transform. Relationships Among Record Length, Frequency Resolution, and Sampling Frequency. Properties of the DFT. Linearity. Circular Shift of a Sequence. Symmetry Properties. Alternative Inversion Formula. Duality and the DFT. Computer Evaluation of DFTs and Inverse DFTs. Another Look at Convolution. Periodic Convolution. Circular Convolution. Frequency Convolution. Correlation. Some Properties of Correlation Sequences. Circular Correlation. Computer Evaluation of Correlation. Block Filtering or Sectioned Convolution. Spectrum Analysis. Periodogram Methods for Spectrum Estimation. Use of Windows In Spectrum Analysis. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 8. The Fast Fourier Transform. Preview. Decomposition in Time. Development of the Basic Algorithm. Computer Evaluation of the Algorithm. Decomposition in Frequency.Variations of the Basic Algorithms. Fast Convolution. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 9. Nonrecursive Filter Design. Preview. Design by Fourier Series. Fourier Coefficients. Lowpass Design. Highpass, Bandpass and Bandstop Design. Gibbs' Phenomenon. Windows in the Fourier design. Design of a Differentiator. Linear Phase Characteristics. Comb Filters. Design by Frequency Sampling. Design Using the Inverse Discrete Fourier Transform. Frequency Sampling Filters. Computer-Aided Design (CAD) of Linear Phase Filter. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 10. Recursive Filter Design. Preview. Analog Filter Characteristics Sinusoidal Steady-State. Frequency Response, Graphical Method. Computer Evaluation of Frequency Response. Determination of Filter Transfer Function from Frequency Response. Analog Filter Design. Butterworth Lowpass Prototype Design. Chebyshev Lowpass Prototype Design. Elliptic Lowpass Prototype Design. Analog Frequency Transformations. Design of Lowpass, Highpass, Bandpass, and Bandstop Filters. Digital Filter Design. Matched z-Transform Design. Impulse and Step-Invariant Design. Bilinear Transform Design. Digital Frequency Transformations. Direct Design of Digital Lowpass, Highpass, Bandpass, and Bandstop Filters. Optimization. Some Comments on Recursive and Nonrecursive Filters. Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. 11. Structures, State Equations, and Applications. Preview. System Implementations. Direct Structure. Second-Order Substructures. Cascade Realization. Parallel Realization (Partial Fraction Expansion). Lattice Filters. Mason's Gain Rule. State Difference Equations. Writing State Equations. Solution of State Equations. Computer Solution of State Equations. Two Different Systems. Digital Control of a Continuous-Time System. Deconvolution Review. Vocabulary and Important Relations Problems. References and Other Sources of Information. Appendix A: Complex Numbers. Appendix B: Fourier Series. Appendix C: Laplace Transform. Appendix D: Frequency Response of Continuous-Time (Analog) Systems. Appendix E: .A Summary of Fourier Paris. Appendix F: Matrices and Determinants. Appendix G: Continuous-Time Systems with a Piecewise Constant Input. Answers to Selected Problems. Index.