DiSCRETE AND CONTINUOUS

FOURIER TRANSFORMS

ANALTSIS, APPLICATIONS AND FASTALGORITHMS

Eleanor Chu University of Guelph

Guelph, Ontario, Canada

CRC Press Taylor & Francis Group BocaRaton London New York

CRC Press i« in imprimofthe Taylor & Francis Group, an informa business

A CHAPMAN & HALL BOOK

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

v

vi CONTENTS

3 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

4 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 vü

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

5 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(f) 161 5.1.2 Inverse Fourier transform of frequency-limited X{f) 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 CoefRcients from Discrete-Time Samples . . . 184

5.9.1 Periodic and almost band-limited function 184

6 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

7 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 triangulär 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 D F T 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 and 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

x 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 j/-dimensional CRT index map 368 13.5.5 The i/-dimensional Ruritanian index map 368 13.5.6 Organizing the i/-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 Emcient Implementation of the PFA 374

14 On Computing the DFT 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