Foreword ..................................................... xiii
Preface ........................................................ xv
Acknowledgments ............................................... xix
Abbreviations ................................................. xxi
1 Fourier Analysis Techniques for Signal Processing .......... 1
1.1 Review of Basic Signals and Systems ........................ 2
1.1.1 Basic Continuous-Time Signals ....................... 2
1.1.2 Basic Discrete-Time Signals ......................... 3
1.1.3 System and Its Properties ........................... 4
1.1.4 LTI Systems ......................................... 5
1.2 Continuous-Time Fourier Transform .......................... 6
1.2.1 Properties of the CTFT .............................. 7
1.2.2 Examples of CTFT .................................... 9
1.3 Discrete-Time Fourier Transform ........................... 17
1.3.1 Properties of DTFT ................................. 19
1.3.2 Examples of DTFT ................................... 22
1.4 Z-Transform ............................................... 30
1.4.1 Examples of z-Transform ............................ 34
1.5 Discrete Fourier Transform ................................ 36
1.5.1 Properties of the DFT .............................. 37
1.5.2 Examples of DFT .................................... 40
1.6 Fast Fourier Transform .................................... 50
1.6.1 Decimation-in-Time FFT (DIT-FFT) ................... 50
1.6.1.1 Computational Savings ..................... 53
1.6.1.2 In-Place Computation ...................... 54
1.6.2 Decimation-in-Frequency FFT (DIF-FFT) .............. 54
1.6.3 Inverse DFT from FFT ............................... 56
1.6.4 Linear Convolution Using DIT-FFT and DIF-FFT ....... 57
References ................................................ 58
2 Pitfalls in the Computation of DFT ........................ 59
2.1 Sampling, Reconstruction, and Aliasing .................... 59
2.1.1 WKS Sampling Theorem ............................... 63
2.1.2 Reconstruction of Continuous-Time Signals from
Discrete-Time Samples .............................. 67
2.2 Frequency Leakage Effect .................................. 69
2.2.1 Zero Leakage Case .................................. 71
2.2.2 Maximum Leakage Case ............................... 73
2.3 DFT as a Filter Bank ...................................... 76
2.4 Picket-Fence Effect or Scalloping Loss .................... 78
2.5 Zero-Padding and Frequency Resolution ..................... 80
2.5.1 Zero-Padding ....................................... 80
2.5.2 Frequency Resolution ............................... 84
References ................................................ 85
3 Review of Window Functions ................................ 87
3.1 Introduction .............................................. 87
3.2 Characteristics of a Window Function ...................... 87
3.3 List of Windows ........................................... 88
3.3.1 Rectangular (Box Car) Window ....................... 89
3.3.2 Triangular (Bartlett) Window ....................... 89
3.3.3 Cos(x) Window ...................................... 91
3.3.4 Hann (Raised-Cosine) Window ........................ 91
3.3.5 Truncated Taylor Family ............................ 94
3.3.6 Hamming Window ..................................... 96
3.3.7 Cos3(x) Window ..................................... 96
3.3.8 Sum-Cosine Window .................................. 97
3.3.9 Cos4(x) Window .................................... 101
3.3.10 Raised-Cosine Family .............................. 102
3.3.11 Blackman Window ................................... 106
3.3.12 Optimized Blackman Window ......................... 107
3.3.13 Blackman-Harris Window ............................ 109
3.3.14 Parabolic Window .................................. 110
3.3.15 Papoulis Window ................................... 111
3.3.16 Tukey Window ...................................... 111
3.3.17 Parzen (Jackson) Window ........................... 113
3.3.18 Dolph-Chebyshev Window ............................ 114
3.3.19 Kaiser's Modified Zeroth-Order Bessel Window
Function Family ................................... 115
3.3.20 Kaiser's Modified First-Order Bessel Window
Function Family ................................... 116
3.4 Rate of Fall-Off Side-Lobe Level ......................... 118
3.4.1 Theorem ........................................... 119
3.4.2 Side-Lobe Fall-Off Rate in the Time-Domain ........ 121
3.5 Comparison of Windows .................................... 123
References ............................................... 126
4 Performance Comparison of Data Windows ................... 129
4.1 Definition of Window Parameters .......................... 130
4.2 Computation of Window Parameters ......................... 132
4.3 Discussion on Window Selection ........................... 139
References ............................................... 142
5 Discrete-Time Windows and Their Figures of Merit ......... 145
5.1 Different Classes of Windows ............................. 145
5.2 Discrete-Time Windows .................................... 148
5.2.1 Rectangular (Box Car) Window ...................... 148
5.2.2 Triangular (Bartlett) Window ...................... 148
5.2.3 Cosαx Window Family ............................... 148
5.2.4 Hann Window ....................................... 148
5.2.5 Truncated Taylor Family of Windows ................ 149
5.2.6 Hamming Window .................................... 149
5.2.7 Sum-Cosine Window ................................. 149
5.2.8 Raised-Cosine Window Family ....................... 149
5.2.9 Blackman Window ................................... 150
5.2.10 Optimized Blackman Window ......................... 150
5.2.11 Tukey Window ...................................... 150
5.2.12 Blackman-Harris Window ............................ 151
5.2.13 Nuttall Window Family ............................. 151
5.2.14 Flat-Top Window ................................... 152
5.2.15 Parabolic Window .................................. 153
5.2.16 Riemann Window .................................... 153
5.2.17 Poisson Window .................................... 153
5.2.18 Gaussian Window ................................... 153
5.2.19 Cauchy Window ..................................... 154
5.2.20 Hann-Poisson Window ............................... 154
5.2.21 Papoulis (Bohman) Window .......................... 154
5.2.22 Jackson (Parzen) Window ........................... 155
5.2.23 Dolph-Chebyshev Window ............................ 155
5.2.24 Modified Zeroth-Order Kaiser-Bessel Window
Family ............................................ 155
5.2.25 Modified First-Order Kaiser-Bessel Window Family .. 156
5.2.26 Saramäki Window Family ............................ 156
5.2.27 Ultraspherical Window ............................. 157
5.2.28 Odd and Even-Length Windows ....................... 158
5.3 Figures of Merit ......................................... 159
5.4 Time-Bandwidth Product ................................... 171
5.5 Applications of Windows .................................. 172
5.5.1 FIR Filter Design Using Windows ................... 172
5.5.2 Spectral Analysis ................................. 177
5.5.3 Window Selection for Spectral Analysis ............ 181
References ............................................... 188
6 Time-Domain and Frequency-Domain Implementations
of Windows ............................................... 189
6.1 Time-Domain Implementation ............................... 189
6.2 A Programmable Windowing Technique ....................... 190
6.3 Computational Error in Time and Frequency-Domains ........ 192
6.4 Canonic Signed Digit Windowing ........................... 195
6.4.1 Window 1 .......................................... 197
6.4.2 Window 2 .......................................... 197
6.4.3 Window 3 .......................................... 198
6.4.4 Window 4 .......................................... 202
6.4.5 Window 5 .......................................... 202
6.4.6 Window 6 .......................................... 202
6.4.7 Window 7 .......................................... 203
6.4.8 Window 8 .......................................... 205
6.4.9 Window 9 .......................................... 207
6.4.10 Window 10 ......................................... 209
6.4.11 Window 11 ......................................... 209
6.4.12 Window 12 ......................................... 209
6.4.13 Window 13 ......................................... 210
6.4.14 Window 14 ......................................... 211
6.5 Modified Zeroth-Order Kaiser-Bessel Window Family ........ 213
6.6 Summary .................................................. 216
References ............................................... 217
7 FIR Filter Design Using Windows .......................... 219
7.1 Ideal Filters ............................................ 219
7.1.1 Lowpass Filter .................................... 219
7.1.2 Highpass Filter ................................... 220
7.1.3 Bandpass Filter ................................... 221
7.1.4 Bandstop Filter ................................... 221
7.2 Linear Time Invariant Systems ............................ 222
7.3 FIR Filters .............................................. 224
7.3.1 Advantages of FIR Filters ......................... 225
7.4 IIR Filters .............................................. 226
7.4.1 Properties of IIR Filters ......................... 226
7.5 Structure of an FIR Filter ............................... 226
7.5.1 Filter Specifications ............................. 227
7.6 FIR Filter Design ........................................ 228
7.6.1 Linear-Phase Filters .............................. 229
7.6.2 Types of FIR Filters .............................. 234
7.6.3 Frequency Response of Type 1 FIR Filter ........... 236
7.6.4 Design Procedure for Filters ...................... 239
7.7 Kaiser-Bessel Windows for FIR Filter Design .............. 243
7.7.1 Filter Design Using Kaiser-Bessel Zeroth-Order
(I0-Sinh) Window .................................. 243
7.7.2 Filter Design Using Kaiser-Bessel First-Order
(I1-Cosh) Window .................................. 249
7.8 Design of Differentiator by Impulse Response Truncation .. 252
7.9 Design of Hilbert Transformer Using Impulse Response
Truncation ............................................... 253
References ............................................... 256
8 Application of Windows in Spectral Analysis .............. 259
8.1 Nonparametric Methods .................................... 260
8.1.1 Periodogram PSD Estimator ......................... 260
8.1.2 Modified Periodogram PSD Estimator ................ 265
8.1.3 Spectral Analysis Using Kaiser-Bessel Window ...... 271
8.1.4 Bartlett Periodogram .............................. 278
8.1.5 Welch Periodogram Method .......................... 280
8.1.6 Blackman-Tukey Method ............................. 283
8.1.7 Daniel Periodogram ................................ 292
8.1.8 Application of the FFT to the Computation of
a Periodogram ..................................... 293
8.1.9 Short-Time Fourier Transform ...................... 293
8.1.10 Conclusions ....................................... 297
References ............................................... 301
9 Applications of Windows .................................. 303
9.1 Windows in High Range Resolution Radars .................. 303
9.1.1 HRR Target Profiling .............................. 303
9.1.2 Simulation Results ................................ 305
9.2 Effect of Range Side Lobe Reduction on SNR ............... 306
9.2.1 Introduction ...................................... 307
9.2.2 Loss Factor ....................................... 309
9.2.3 Weighting Function ................................ 314
9.2.4 Results and Discussions ........................... 316
9.3 Window Functions in Stretch Processing ................... 317
9.4 Application of Window Functions in Biomedical Signal
Processing ............................................... 322
9.4.1 Biomedical Signal Processing ...................... 323
9.4.2 FIR Filtering of Biomedical Signals ............... 324
9.4.3 Moving Average Filtering of Biomedical Signals .... 328
9.4.4 QRS Detection in ECG Based on STFT ................ 333
9.5 Audio Denoising Using the Time-Frequency Plane ........... 336
9.5.1 Time-Frequency Plane .............................. 336
9.5.2 Audio Denoising Using Time-Frequency Plane ........ 340
9.5.3 Block Thresholding ................................ 341
9.5.4 Effect of Windows ................................. 342
9.6 Effect of Windows on Linear Prediction of Speech ......... 343
9.6.1 Linear Prediction Coder ........................... 344
9.6.2 Line Spectral Frequencies ......................... 346
9.6.3 LSF Variation due to Windows ...................... 347
9.7 Application of Windows in Image Processing ............... 349
9.7.1 Windows for ISAR Images ........................... 350
9.7.2 Experimental Analysis ............................. 352
9.7.3 Results and Conclusions ........................... 356
9.8 Windows to Improve Contrast Ratio in Imaging Systems ..... 357
9.8.1 Experimental Analysis ............................. 361
9.8.2 Results and Conclusions ........................... 361
References ............................................... 361
Index ......................................................... 365
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