Verfügbarkeit: Discrete and continuous fourier transforms

Discrete and continuous fourier transforms: analysis, applications and fast algorithms

"Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms presents the fundamental of Fourier analysis and their deployment in signal process using DFT and FFT algorithms. This book provides meaningful interpretations of essential formulas in the context of applica...

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Bibliographische Detailangaben
1. Verfasser:	Chu, Eleanor 1950- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Boca Raton, Fla. [u.a.] Chapman & Hall/CRC Press 2008
Schlagworte:	Fourier transformations Fourier analysis Fourier-Transformation
Online-Zugang:	Inhaltsverzeichnis
Zusammenfassung:	"Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms presents the fundamental of Fourier analysis and their deployment in signal process using DFT and FFT algorithms. This book provides meaningful interpretations of essential formulas in the context of applications, building a solid foundation for the application of Fourier analysis in the many diverging and continuously evolving areas in digital signal processing enterprises."--BOOK JACKET.
Beschreibung:	Includes bibliographical references and index
Beschreibung:	XXIII, 400 S. graph. Darst.
ISBN:	9781420063639

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245	1	0	\|a Discrete and continuous fourier transforms \|b analysis, applications and fast algorithms \|c Eleanor Chu
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500			\|a Includes bibliographical references and index
520	1		\|a "Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms presents the fundamental of Fourier analysis and their deployment in signal process using DFT and FFT algorithms. This book provides meaningful interpretations of essential formulas in the context of applications, building a solid foundation for the application of Fourier analysis in the many diverging and continuously evolving areas in digital signal processing enterprises."--BOOK JACKET.
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Datensatz im Suchindex

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adam_text	Contents List of Figures xi List of Tables xv Preface xvii Acknowledgments xxi About the Author xxiii I Fundamentals, Analysis and Applications 1 1 Analytical and Graphical Representation of Function Contents 3 1.1 Time and Frequency Contents of a Function ............... 3 1.2 The Frequency-Domain Plots as Graphical Tools ............ 4 1.3 Identifying the Cosine and Sine Modes .................. 6 1.4 Using Complex Exponential Modes .................... 7 1.5 Using Cosine Modes with Phase or Time Shifts ............. 9 1.6 Periodicity and Commensurate Frequencies ............... 12 1.7 Review of Results and Techniques ..................... 13 1.7.1 Practicing the techniques ..................... 15 1.8 Expressing Single Component Signals ................... 19 1.9 General Form of a Sinusoid in Signal Application ............ 20 1.9.1 Expressing sequences of discrete-time samples .......... 21 1.9.2 Periodicity of sinusoidal sequences ................ 22 1.10 Fourier Series: A Topic to Come ..................... 23 1.11 Terminology ................................. 25 2 Sampling and Reconstruction of Functions-Part I 27 2.1 DFT and Band-Limited Periodic Signal ................. 27 2.2 Frequencies Aliased by Sampling ..................... 32 2.3 Connection: Anti-Aliasing Filter ..................... 35 2.4 Alternate Notations and Formulas .................... 36 2.5 Sampling Period and Alternate Forms of DFT .............. 37 2.6 Sample Size and Alternate Forms of DFT ................ 40 vi CONTENTS The Fourier Series 45 3.1 Formal Expansions ............................. 45 3.1.1 Examples .............................. 48 3.2 Time-Limited Functions .......................... 51 3.3 Even and Odd Functions .......................... 51 3.4 Half-Range Expansions ........................... 53 3.5 Fourier Series Using Complex Exponential Modes ............ 60 3.6 Complex-Valued Functions ......................... 60 3.7 Fourier Series in Other Variables ..................... 61 3.8 Truncated Fourier Series and Least Squares ............... 61 3.9 Orthogonal Projections and Fourier Series ................ 63 3.9.1 The Cauchy-Schwarz inequality .................. 68 3.9.2 The Minkowski inequality ..................... 71 3.9.3 Projections ............................. 72 3.9.4 Least-squares approximation ................... 74 3.9.5 Bessel s inequality and Riemann s lemma ............ 77 3.10 Convergence of the Fourier Series ..................... 80 3.10.1 Starting with a concrete example ................. 80 3.10.2 Pointwise convergence — a local property ............. 83 3.10.3 The rate of convergence—a global property ........... 88 3.10.4 The Gibbs phenomenon ...................... 90 3.10.5 The Dirichlet kernel perspective .................. 92 3.10.6 Eliminating the Gibbs effect by the Cesaro sum ......... 96 3.10.7 Reducing the Gibbs effect by Lanczos smoothing ........ 100 3.10.8 The modification of Fourier series coefficients .......... 101 3.11 Accounting for Aliased Frequencies in DFT ............... 103 3.11.1 Sampling functions with jump discontinuities .......... 105 DFT and Sampled Signals 109 4.1 Deriving the DFT and IDFT Formulas .................. 109 4.2 Direct Conversion Between Alternate Forms ............... 114 4.3 DFT of Concatenated Sample Sequences ................. 116 4.4 DFT Coefficients of a Commensurate Sum ................ 117 4.4.1 DFT coefficients of single-component signals ........... 117 4.4.2 Making direct use of the digital frequencies ........... 121 4.4.3 Common period of sampled composite signals .......... 123 4.5 Frequency Distortion by Leakage ..................... 126 4.5.1 Fourier series expansion of a nonharmonic component ..... 128 4.5.2 Aliased DFT coefficients of a nonharmonic component ..... 129 4.5.3 Demonstrating leakage by numerical experiments ........ 131 4.5.4 Mismatching periodic extensions ................. 131 4.5.5 Minimizing leakage in practice .................. 134 4.6 The Effects of Zero Padding ........................ 134 4.6.1 Zero padding the signal ...................... 134 CONTENTS vii 4.6.2 Zero padding the DFT ....................... 138 4.7 Computing DFT Defining Formulas Per Se ................ 148 4.7.1 Programming DFT in MATLAB® ................ 148 Sampling and Reconstruction of Functions-Part II 159 5.1 Sampling Nonperiodic Band-Limited Functions .............160 5.1.1 Fourier series of frequency-limited X( ƒ).............161 5.1.2 Inverse Fourier transform of frequency-limited X{ ƒ) ......161 5.1.3 Recovering the signal analytically .................162 5.1.4 Further discussion of the sampling theorem ...........163 5.2 Deriving the Fourier Transform Pair ...................165 5.3 The Sine and Cosine Frequency Contents ................166 5.4 Tabulating Two Sets of Fundamental Formulas .............167 5.5 Connections with Time/Frequency Restrictions .............169 5.5.1 Examples of Fourier transform pair ................169 5.6 Fourier Transform Properties .......................175 5.6.1 Deriving the properties .......................175 5.6.2 Utilities of the properties .....................178 5.7 Alternate Form of the Fourier Transform .................180 5.8 Computing the Fourier Transform from Discrete-Time Samples .... 181 5.8.1 Almost time-limited and band-limited functions .........182 5.9 Computing the Fourier Coefficients from Discrete-Time Samples . . . 184 5.9.1 Periodic and almost band-limited function ............184 Sampling and Reconstruction of Functions-Part III 187 6.1 Impulse Functions and Their Properties .................187 6.2 Generating the Fourier Transform Pairs .................190 6.3 Convolution and Fourier Transform ....................191 6.4 Periodic Convolution and Fourier Series .................194 6.5 Convolution with the Impulse Function ..................196 6.6 Impulse Train as a Generalized Function .................197 6.7 Impulse Sampling of Continuous-Time Signals ..............204 6.8 Nyquist Sampling Rate Rediscovered ...................205 6.9 Sampling Theorem for Band-Limited Signal ...............209 6.10 Sampling of Band-Pass Signals ......................211 Fourier Transform of a Sequence 213 7.1 Deriving the Fourier Transform of a Sequence ..............213 7.2 Properties of the Fourier Transform of a Sequence ...........217 7.3 Generating the Fourier Transform Pairs .................219 7.3.1 The Kronecker delta sequence ...................219 7.3.2 Representing signals by Kronecker delta .............220 7.3.3 Fourier transform pairs .......................221 7.4 Duality in Connection with the Fourier Series ..............228 viii CONTENTS 7.4.1 Periodic convolution and discrete convolution .......... 229 7.5 The Fourier Transform of a Periodic Sequence .............. 231 7.6 The DFT Interpretation .......................... 234 7.6.1 The interpreted DFT and the Fourier transform ......... 236 7.6.2 Time-limited case .......................... 237 7.6.3 Band-limited case .......................... 238 7.6.4 Periodic and band-limited case .................. 239 8 The Discrete Fourier Transform of a Windowed Sequence 241 8.1 A Rectangular Window of Infinite Width ................241 8.2 A Rectangular Window of Appropriate Finite Width ..........243 8.3 Frequency Distortion by Improper Truncation ..............245 8.4 Windowing a General Nonperiodic Sequence ...............246 8.5 Frequency-Domain Properties of Windows ................247 8.5.1 The rectangular window ......................248 8.5.2 The triangular window .......................249 8.5.3 The von Hann window .......................250 8.5.4 The Hamming window .......................252 8.5.5 The Blackman window .......................253 8.6 Applications of the Windowed DFT ...................254 8.6.1 Several scenarios ..........................254 8.6.2 Selecting the length of DFT in practice .............263 9 Discrete Convolution and the DFT 269 9.1 Linear Discrete Convolution ........................ 269 9.1.1 Linear convolution of two finite sequences ............ 269 9.1.2 Sectioning a long sequence for linear convolution ........ 275 9.2 Periodic Discrete Convolution ....................... 276 9.2.1 Definition based on two periodic sequences ........... 276 9.2.2 Converting linear to periodic convolution ............ 278 9.2.3 Defining the equivalent cyclic convolution ............ 278 9.2.4 The cyclic convolution in matrix form .............. 281 9.2.5 Converting linear to cyclic convolution .............. 282 9.2.6 Two cyclic convolution theorems ................. 282 9.2.7 Implementing sectioned linear convolution ............ 285 9.3 The Chirp Fourier Transform ....................... 285 9.3.1 The scenario ............................ 285 9.3.2 The equivalent partial linear convolution ............. 287 9.3.3 The equivalent partial cyclic convolution ............. 288 10 Applications of the DFT in Digital Filtering and Filters 293 10.1 The Background ..............................293 10.2 Application-Oriented Terminology ....................294 10.3 Revisit Gibbs Phenomenon from the Filtering Viewpoint ........297 CONTENTS ix 10.4 Experimenting with Digital Filtering ала Filter Design ......... 298 II Fast Algorithms 305 11 Index Mapping and Mixed-Radix FFTs 307 11.1 Algebraic DFT versus FFT-Computed DFT ............... 307 11.2 The Role of Index Mapping ........................ 308 11.2.1 The decoupling process — Stage I ................. 309 11.2.2 The decoupling process — Stage II ................. 311 11.2.3 The decoupling process—Stage III ................ 313 11.3 The Recursive Equation Approach .................... 315 11.3.1 Counting short DFT or DFT-like transforms .......... 315 11.3.2 The recursive equation for arbitrary composite N........ 315 11.3.3 Specialization to the radix-2 DIT FFT for N = 2 ....... 317 11.4 Other Forms by Alternate Index Splitting ................ 319 11.4.1 The recursive equation for arbitrary composite N........ 320 11.4.2 Specialization to the radix-2 DIF FFT for N - 2 ....... 321 12 Kronecker Product Factorization and FFTs 323 12.1 Reformulating the Two-Factor Mixed-Radix FFT ............324 12.2 From Two-Factor to Multi-Factor Mixed-Radix FFT ..........330 12.2.1 Selected properties and rules for Kronecker products ......331 12.2.2 Complete factorization of the DFT matrix ............333 12.3 Other Forms by Alternate Index Splitting ................335 12.4 Factorization Results by Alternate Expansion ..............337 12.4.1 Unordered mixed-radix DIT FFT .................337 12.4.2 Unordered mixed-radix DIF FFT .................339 12.5 Unordered FFT for Scrambled Input ...................339 12.6 Utilities of the Kronecker Product Factorization .............341 13 The Family of Prime Factor FFT Algorithms 343 13.1 Connecting the Relevant Ideas .......................344 13.2 Deriving the Two-Factor PFA .......................345 13.2.1 Stage I: Nonstandard index mapping schemes ..........346 13.2.2 Stage II: Decoupling the DFT computation ...........347 13.2.3 Organizing the PFA computation-Part 1.............348 13.3 Matrix Formulation of the Two-Factor PFA ...............350 13.3.1 Stage III: The Kronecker product factorization .........350 13.3.2 Stage IV: Defining permutation matrices .............350 13.3.3 Stage V: Completing the matrix factorization ..........352 13.4 Matrix Formulation of the Multi-Factor PFA ..............352 13.4.1 Organizing the PFA computation — Part 2............354 13.5 Number Theory and Index Mapping by Permutations .........355 CONTENTS 13.5.1 Some fundamental properties of integers ............. 356 13.5.2 A simple case of index mapping by permutation ......... 365 13.5.3 The Chinese remainder theorem .................. 366 13.5.4 The iz-dimensional CRT index map ................ 368 13.5.5 The v-dimensional Ruritanian index map ............ 368 13.5.6 Organizing the ľ-factor PFA computation — Part 3....... 370 13.6 The In-Place and In-Order PFA ...................... 370 13.6.1 The implementation-related concepts ............... 370 13.6.2 The in-order algorithm based on Ruritanian map ........ 373 13.6.3 The in-order algorithm based on CRT map ........... 374 13.7 Efficient Implementation of the PFA ................... 374 14 On Computing the ĐFT of Large Prime Length 377 14.1 Performance of FFT for Prime N..................... 378 14.2 Fast Algorithm I: Approximating the FFT ................ 380 14.2.1 Array-smart implementation in MATLAB® ........... 381 14.2.2 Numerical results .......................... 383 14.3 Fast Algorithm II: Using Bluestein s FFT ................ 384 14.3.1 Bluestein s FFT and the chirp Fourier transform ........ 384 14.3.2 The equivalent partial linear convolution ............. 385 14.3.3 The equivalent partial cyclic convolution ............. 386 14.3.4 The algorithm ............................ 387 14.3.5 Array-smart implementation in MATLAB® ........... 388 14.3.6 Numerical results .......................... 390 Bibliography 391 Index 395
adam_txt	Contents List of Figures xi List of Tables xv Preface xvii Acknowledgments xxi About the Author xxiii I Fundamentals, Analysis and Applications 1 1 Analytical and Graphical Representation of Function Contents 3 1.1 Time and Frequency Contents of a Function . 3 1.2 The Frequency-Domain Plots as Graphical Tools . 4 1.3 Identifying the Cosine and Sine Modes . 6 1.4 Using Complex Exponential Modes . 7 1.5 Using Cosine Modes with Phase or Time Shifts . 9 1.6 Periodicity and Commensurate Frequencies . 12 1.7 Review of Results and Techniques . 13 1.7.1 Practicing the techniques . 15 1.8 Expressing Single Component Signals . 19 1.9 General Form of a Sinusoid in Signal Application . 20 1.9.1 Expressing sequences of discrete-time samples . 21 1.9.2 Periodicity of sinusoidal sequences . 22 1.10 Fourier Series: A Topic to Come . 23 1.11 Terminology . 25 2 Sampling and Reconstruction of Functions-Part I 27 2.1 DFT and Band-Limited Periodic Signal . 27 2.2 Frequencies Aliased by Sampling . 32 2.3 Connection: Anti-Aliasing Filter . 35 2.4 Alternate Notations and Formulas . 36 2.5 Sampling Period and Alternate Forms of DFT . 37 2.6 Sample Size and Alternate Forms of DFT . 40 vi CONTENTS The Fourier Series 45 3.1 Formal Expansions . 45 3.1.1 Examples . 48 3.2 Time-Limited Functions . 51 3.3 Even and Odd Functions . 51 3.4 Half-Range Expansions . 53 3.5 Fourier Series Using Complex Exponential Modes . 60 3.6 Complex-Valued Functions . 60 3.7 Fourier Series in Other Variables . 61 3.8 Truncated Fourier Series and Least Squares . 61 3.9 Orthogonal Projections and Fourier Series . 63 3.9.1 The Cauchy-Schwarz inequality . 68 3.9.2 The Minkowski inequality . 71 3.9.3 Projections . 72 3.9.4 Least-squares approximation . 74 3.9.5 Bessel's inequality and Riemann's lemma . 77 3.10 Convergence of the Fourier Series . 80 3.10.1 Starting with a concrete example . 80 3.10.2 Pointwise convergence — a local property . 83 3.10.3 The rate of convergence—a global property . 88 3.10.4 The Gibbs phenomenon . 90 3.10.5 The Dirichlet kernel perspective . 92 3.10.6 Eliminating the Gibbs effect by the Cesaro sum . 96 3.10.7 Reducing the Gibbs effect by Lanczos smoothing . 100 3.10.8 The modification of Fourier series coefficients . 101 3.11 Accounting for Aliased Frequencies in DFT . 103 3.11.1 Sampling functions with jump discontinuities . 105 DFT and Sampled Signals 109 4.1 Deriving the DFT and IDFT Formulas . 109 4.2 Direct Conversion Between Alternate Forms . 114 4.3 DFT of Concatenated Sample Sequences . 116 4.4 DFT Coefficients of a Commensurate Sum . 117 4.4.1 DFT coefficients of single-component signals . 117 4.4.2 Making direct use of the digital frequencies . 121 4.4.3 Common period of sampled composite signals . 123 4.5 Frequency Distortion by Leakage . 126 4.5.1 Fourier series expansion of a nonharmonic component . 128 4.5.2 Aliased DFT coefficients of a nonharmonic component . 129 4.5.3 Demonstrating leakage by numerical experiments . 131 4.5.4 Mismatching periodic extensions . 131 4.5.5 Minimizing leakage in practice . 134 4.6 The Effects of Zero Padding . 134 4.6.1 Zero padding the signal . 134 CONTENTS vii 4.6.2 Zero padding the DFT . 138 4.7 Computing DFT Defining Formulas Per Se . 148 4.7.1 Programming DFT in MATLAB® . 148 Sampling and Reconstruction of Functions-Part II 159 5.1 Sampling Nonperiodic Band-Limited Functions .160 5.1.1 Fourier series of frequency-limited X( ƒ).161 5.1.2 Inverse Fourier transform of frequency-limited X{ ƒ) .161 5.1.3 Recovering the signal analytically .162 5.1.4 Further discussion of the sampling theorem .163 5.2 Deriving the Fourier Transform Pair .165 5.3 The Sine and Cosine Frequency Contents .166 5.4 Tabulating Two Sets of Fundamental Formulas .167 5.5 Connections with Time/Frequency Restrictions .169 5.5.1 Examples of Fourier transform pair .169 5.6 Fourier Transform Properties .175 5.6.1 Deriving the properties .175 5.6.2 Utilities of the properties .178 5.7 Alternate Form of the Fourier Transform .180 5.8 Computing the Fourier Transform from Discrete-Time Samples . 181 5.8.1 Almost time-limited and band-limited functions .182 5.9 Computing the Fourier Coefficients from Discrete-Time Samples . . . 184 5.9.1 Periodic and almost band-limited function .184 Sampling and Reconstruction of Functions-Part III 187 6.1 Impulse Functions and Their Properties .187 6.2 Generating the Fourier Transform Pairs .190 6.3 Convolution and Fourier Transform .191 6.4 Periodic Convolution and Fourier Series .194 6.5 Convolution with the Impulse Function .196 6.6 Impulse Train as a Generalized Function .197 6.7 Impulse Sampling of Continuous-Time Signals .204 6.8 Nyquist Sampling Rate Rediscovered .205 6.9 Sampling Theorem for Band-Limited Signal .209 6.10 Sampling of Band-Pass Signals .211 Fourier Transform of a Sequence 213 7.1 Deriving the Fourier Transform of a Sequence .213 7.2 Properties of the Fourier Transform of a Sequence .217 7.3 Generating the Fourier Transform Pairs .219 7.3.1 The Kronecker delta sequence .219 7.3.2 Representing signals by Kronecker delta .220 7.3.3 Fourier transform pairs .221 7.4 Duality in Connection with the Fourier Series .228 viii CONTENTS 7.4.1 Periodic convolution and discrete convolution . 229 7.5 The Fourier Transform of a Periodic Sequence . 231 7.6 The DFT Interpretation . 234 7.6.1 The interpreted DFT and the Fourier transform . 236 7.6.2 Time-limited case . 237 7.6.3 Band-limited case . 238 7.6.4 Periodic and band-limited case . 239 8 The Discrete Fourier Transform of a Windowed Sequence 241 8.1 A Rectangular Window of Infinite Width .241 8.2 A Rectangular Window of Appropriate Finite Width .243 8.3 Frequency Distortion by Improper Truncation .245 8.4 Windowing a General Nonperiodic Sequence .246 8.5 Frequency-Domain Properties of Windows .247 8.5.1 The rectangular window .248 8.5.2 The triangular window .249 8.5.3 The von Hann window .250 8.5.4 The Hamming window .252 8.5.5 The Blackman window .253 8.6 Applications of the Windowed DFT .254 8.6.1 Several scenarios .254 8.6.2 Selecting the length of DFT in practice .263 9 Discrete Convolution and the DFT 269 9.1 Linear Discrete Convolution . 269 9.1.1 Linear convolution of two finite sequences . 269 9.1.2 Sectioning a long sequence for linear convolution . 275 9.2 Periodic Discrete Convolution . 276 9.2.1 Definition based on two periodic sequences . 276 9.2.2 Converting linear to periodic convolution . 278 9.2.3 Defining the equivalent cyclic convolution . 278 9.2.4 The cyclic convolution in matrix form . 281 9.2.5 Converting linear to cyclic convolution . 282 9.2.6 Two cyclic convolution theorems . 282 9.2.7 Implementing sectioned linear convolution . 285 9.3 The Chirp Fourier Transform . 285 9.3.1 The scenario . 285 9.3.2 The equivalent partial linear convolution . 287 9.3.3 The equivalent partial cyclic convolution . 288 10 Applications of the DFT in Digital Filtering and Filters 293 10.1 The Background .293 10.2 Application-Oriented Terminology .294 10.3 Revisit Gibbs Phenomenon from the Filtering Viewpoint .297 CONTENTS ix 10.4 Experimenting with Digital Filtering ала Filter Design . 298 II Fast Algorithms 305 11 Index Mapping and Mixed-Radix FFTs 307 11.1 Algebraic DFT versus FFT-Computed DFT . 307 11.2 The Role of Index Mapping . 308 11.2.1 The decoupling process — Stage I . 309 11.2.2 The decoupling process — Stage II . 311 11.2.3 The decoupling process—Stage III . 313 11.3 The Recursive Equation Approach . 315 11.3.1 Counting short DFT or DFT-like transforms . 315 11.3.2 The recursive equation for arbitrary composite N. 315 11.3.3 Specialization to the radix-2 DIT FFT for N = 2". 317 11.4 Other Forms by Alternate Index Splitting . 319 11.4.1 The recursive equation for arbitrary composite N. 320 11.4.2 Specialization to the radix-2 DIF FFT for N - 2" . 321 12 Kronecker Product Factorization and FFTs 323 12.1 Reformulating the Two-Factor Mixed-Radix FFT .324 12.2 From Two-Factor to Multi-Factor Mixed-Radix FFT .330 12.2.1 Selected properties and rules for Kronecker products .331 12.2.2 Complete factorization of the DFT matrix .333 12.3 Other Forms by Alternate Index Splitting .335 12.4 Factorization Results by Alternate Expansion .337 12.4.1 Unordered mixed-radix DIT FFT .337 12.4.2 Unordered mixed-radix DIF FFT .339 12.5 Unordered FFT for Scrambled Input .339 12.6 Utilities of the Kronecker Product Factorization .341 13 The Family of Prime Factor FFT Algorithms 343 13.1 Connecting the Relevant Ideas .344 13.2 Deriving the Two-Factor PFA .345 13.2.1 Stage I: Nonstandard index mapping schemes .346 13.2.2 Stage II: Decoupling the DFT computation .347 13.2.3 Organizing the PFA computation-Part 1.348 13.3 Matrix Formulation of the Two-Factor PFA .350 13.3.1 Stage III: The Kronecker product factorization .350 13.3.2 Stage IV: Defining permutation matrices .350 13.3.3 Stage V: Completing the matrix factorization .352 13.4 Matrix Formulation of the Multi-Factor PFA .352 13.4.1 Organizing the PFA computation — Part 2.354 13.5 Number Theory and Index Mapping by Permutations .355 CONTENTS 13.5.1 Some fundamental properties of integers . 356 13.5.2 A simple case of index mapping by permutation . 365 13.5.3 The Chinese remainder theorem . 366 13.5.4 The iz-dimensional CRT index map . 368 13.5.5 The v-dimensional Ruritanian index map . 368 13.5.6 Organizing the ľ-factor PFA computation — Part 3. 370 13.6 The In-Place and In-Order PFA . 370 13.6.1 The implementation-related concepts . 370 13.6.2 The in-order algorithm based on Ruritanian map . 373 13.6.3 The in-order algorithm based on CRT map . 374 13.7 Efficient Implementation of the PFA . 374 14 On Computing the ĐFT of Large Prime Length 377 14.1 Performance of FFT for Prime N. 378 14.2 Fast Algorithm I: Approximating the FFT . 380 14.2.1 Array-smart implementation in MATLAB® . 381 14.2.2 Numerical results . 383 14.3 Fast Algorithm II: Using Bluestein's FFT . 384 14.3.1 Bluestein's FFT and the chirp Fourier transform . 384 14.3.2 The equivalent partial linear convolution . 385 14.3.3 The equivalent partial cyclic convolution . 386 14.3.4 The algorithm . 387 14.3.5 Array-smart implementation in MATLAB® . 388 14.3.6 Numerical results . 390 Bibliography 391 Index 395
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id	DE-604.BV023325257
illustrated	Illustrated
index_date	2024-07-02T20:55:21Z
indexdate	2024-07-09T21:15:55Z
institution	BVB
isbn	9781420063639
language	English
lccn	2008007070
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016509283
oclc_num	193909740
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owner	DE-703 DE-83 DE-11 DE-384 DE-634 DE-824
owner_facet	DE-703 DE-83 DE-11 DE-384 DE-634 DE-824
physical	XXIII, 400 S. graph. Darst.
publishDate	2008
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publisher	Chapman & Hall/CRC Press
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spelling	Chu, Eleanor 1950- Verfasser (DE-588)13820764X aut Discrete and continuous fourier transforms analysis, applications and fast algorithms Eleanor Chu Boca Raton, Fla. [u.a.] Chapman & Hall/CRC Press 2008 XXIII, 400 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index "Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms presents the fundamental of Fourier analysis and their deployment in signal process using DFT and FFT algorithms. This book provides meaningful interpretations of essential formulas in the context of applications, building a solid foundation for the application of Fourier analysis in the many diverging and continuously evolving areas in digital signal processing enterprises."--BOOK JACKET. Fourier transformations Fourier analysis Fourier-Transformation (DE-588)4018014-1 gnd rswk-swf Fourier-Transformation (DE-588)4018014-1 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016509283&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Chu, Eleanor 1950- Discrete and continuous fourier transforms analysis, applications and fast algorithms Fourier transformations Fourier analysis Fourier-Transformation (DE-588)4018014-1 gnd
subject_GND	(DE-588)4018014-1
title	Discrete and continuous fourier transforms analysis, applications and fast algorithms
title_auth	Discrete and continuous fourier transforms analysis, applications and fast algorithms
title_exact_search	Discrete and continuous fourier transforms analysis, applications and fast algorithms
title_exact_search_txtP	Discrete and continuous fourier transforms analysis, applications and fast algorithms
title_full	Discrete and continuous fourier transforms analysis, applications and fast algorithms Eleanor Chu
title_fullStr	Discrete and continuous fourier transforms analysis, applications and fast algorithms Eleanor Chu
title_full_unstemmed	Discrete and continuous fourier transforms analysis, applications and fast algorithms Eleanor Chu
title_short	Discrete and continuous fourier transforms
title_sort	discrete and continuous fourier transforms analysis applications and fast algorithms
title_sub	analysis, applications and fast algorithms
topic	Fourier transformations Fourier analysis Fourier-Transformation (DE-588)4018014-1 gnd
topic_facet	Fourier transformations Fourier analysis Fourier-Transformation
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016509283&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT chueleanor discreteandcontinuousfouriertransformsanalysisapplicationsandfastalgorithms

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