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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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
Chapman & Hall/CRC Press
2008
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "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. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXIII, 400 S. graph. Darst. |
| ISBN: | 9781420063639 |
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| 008 | 080603s2008 xxud||| |||| 00||| eng d | ||
| 010 | |a 2008007070 | ||
| 020 | |a 9781420063639 |c hardback : alk. paper |9 978-1-4200-6363-9 | ||
| 035 | |a (OCoLC)193909740 | ||
| 035 | |a (DE-599)BVBBV023325257 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-703 |a DE-83 |a DE-11 |a DE-384 |a DE-634 |a DE-824 | ||
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| 082 | 0 | |a 515/.723 | |
| 084 | |a SK 450 |0 (DE-625)143240: |2 rvk | ||
| 100 | 1 | |a Chu, Eleanor |d 1950- |e Verfasser |0 (DE-588)13820764X |4 aut | |
| 245 | 1 | 0 | |a Discrete and continuous fourier transforms |b analysis, applications and fast algorithms |c Eleanor Chu |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b Chapman & Hall/CRC Press |c 2008 | |
| 300 | |a XXIII, 400 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 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. | |
| 650 | 4 | |a Fourier transformations | |
| 650 | 4 | |a Fourier analysis | |
| 650 | 0 | 7 | |a Fourier-Transformation |0 (DE-588)4018014-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Fourier-Transformation |0 (DE-588)4018014-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016509283 | ||
Record in the Search Index
| _version_ | 1804137665989181440 |
|---|---|
| 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 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Chu, Eleanor 1950- |
| author_GND | (DE-588)13820764X |
| author_facet | Chu, Eleanor 1950- |
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| bvnumber | BV023325257 |
| callnumber-first | Q - Science |
| callnumber-label | QA403 |
| callnumber-raw | QA403.5 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)193909740 (DE-599)BVBBV023325257 |
| dewey-full | 515/.723 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.723 |
| dewey-search | 515/.723 |
| dewey-sort | 3515 3723 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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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 |
| open_access_boolean | |
| 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 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Chapman & Hall/CRC Press |
| record_format | marc |
| 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 |