Preface ............................................................................................................................................ 17
Acknowledgments ................................................................................................................... 17
Book Series Overview ............................................................................................................ 18
Chapter 1 Introduction and Overview ..................................................................................... 20
1.1 Organization ............................................................................................................ 20
1.2 Overview ................................................................................................................. 21
1.2.1 Elementary Spectrum Analysis ....................................................................... 21
1.2.2 The Short-Time Fourier Transform (STFT) and Time-Frequency Displays ... 22
1.2.3 Short-Time Analysis, Modification, and Resynthesis ..................................... 22
1.2.4 Applications .................................................................................................... 22
1.2.5 Multirate Polyphase Filter and Wavelet Banks ............................................... 23
1.2.6 Appendices ...................................................................................................... 23
Chapter 2 Fourier Transforms for Continuous/Discrete Time/Frequency ............................... 24
2.1 Discrete Time Fourier Transform (DTFT) .............................................................. 25
2.2 Fourier Transform (FT) and Inverse ........................................................................ 26
2.2.1 Existence of the Fourier Transform ................................................................. 26
2.3 Fourier Theorems for the DTFT .............................................................................. 27
2.3.1 Linearity of the DTFT ..................................................................................... 27
2.3.2 Time Reversal ................................................................................................. 28
2.3.3 Symmetry of the DTFT for Real Signals ........................................................ 28
2.3.3.1 Real Even (or Odd) Signals ..................................................................... 29
2.3.4 Shift Theorem for the DTFT ........................................................................... 30
2.3.5 Convolution Theorem for the DTFT ............................................................... 31
2.3.6 Correlation Theorem for the DTFT ................................................................. 32
2.3.7 Autocorrelation ............................................................................................... 33
2.3.8 Power Theorem for the DTFT ......................................................................... 33
2.3.9 Stretch Operator .............................................................................................. 34
2.3.10 Repeat (Scaling) Operator ............................................................................... 35
2.3.11 Stretch/Repeat (Scaling) Theorem .................................................................. 36
2.3.12 Downsampling and Aliasing ........................................................................... 37
2.3.12.1 Proof of Aliasing Theorem .............................................................. 38
2.3.13 Differentiation Theorem Dual ......................................................................... 39
2.4 Continuous-Time Fourier Theorems ....................................................................... 40
2.4.1 Scaling Theorem ............................................................................................. 40
2.4.2 Spectral Roll-Off ............................................................................................. 41
2.5 Spectral Interpolation .............................................................................................. 42
2.5.1 Ideal Spectral Interpolation ............................................................................. 42
2.5.2 Interpolating a DFT ......................................................................................... 43
2.5.3 Zero Padding in the Time Domain .................................................................. 43
2.5.3.1 Practical Zero Padding ............................................................................ 44
2.5.4 Zero-Phase Zero Padding ................................................................................ 47
2.5.4.1 Matlab/Octave fftshift utility ................................................................... 49
Chapter 3 Spectrum Analysis Windows ................................................................................... 52
3.1 Rectangular Window ............................................................................................... 52
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