Digital Fourier Analysis: Fundamentals

✍ Scribed by Ken’iti Kido

• Publisher: Springer
• Year: 2015
• Tongue: English
• Leaves: 211
• Series: Undergraduate Lecture Notes in Physics
• Category: Library

✦ Table of Contents

Preface
Contents
1 Sine and Cosine Waves
1.1…Synthesis of an Impulse by Cosine Waves
1.2…Synthesis of Rectangular Waveforms by Sine and Cosine Waves
1.3…Time Period
1.4…Harmonics and Waveforms
1.5…Fourier Series
1.6…Shift of Harmonics on the Time Axis
1.7…Complex Exponential Functions and Sine and Cosine Functions
1.8…Phases of Cosine and Sine Functions
1.9…Synthesis of Sine Wave with Arbitrary Phase
1.10…Instantaneous Phase and Frequency
1.11…Exercise
2 Fourier Series Expansion
2.1…Integrals of Sine and Cosine Functions
2.2…Calculations of Fourier Coefficients
2.3…Expressing Waveforms by Even Functions
2.4…Expressing Waveforms by Odd Functions
2.5…Expressing Waveforms by Complex Exponential Functions
2.6…Fourier Transform
2.7…Gibbs’ Phenomenon
2.8…Exercises
3 Numerical (Digitized) Waveforms
3.1…Fourier Series Expansion of Spectrum
3.2…Reproduction of the Continuous Waveform from the Sequence of Sample Values
3.3…Frequency Bandwidth and Sampling Frequency
3.4…Smoothing of Sample Sequence by Low-Pass Filtering
3.5…Sampling Theorem
3.6…Smoothing of a Sample Sequence Using the Sampling Theorem
3.7…Folding (Aliasing) of the Spectrum
3.8…Sampling Frequency Conversion I (Application of the Fourier transform)
3.9…Sampling Frequency Conversion II (Application of LPF)
3.10…Exercises
4 Discrete Fourier Transform
4.1…Fourier Transform of Discrete Sequence of Numbers
4.2…Inverse Discrete Fourier Transform (IDFT)
4.3…The DFT and the Fourier Transform
4.4…Waveform and Its DFT
4.4.1 Sine and Cosine Waves
4.4.2 Phase and Spectrum
4.4.3 Harmonics
4.4.4 Symmetric and Antisymmetric Waveforms
4.4.5 Sine Waveforms with Noninteger Frequencies
4.4.6 Too Wide Sample Spacing
4.4.7 Rectangular Wave
4.5…Discrete Cosine Transform (DCT)
4.6…Extension of the Discrete Cosine Transform
4.7…Exercises
5 Fast Fourier Transform
5.1…Decimation in Time Algorithm
5.2…Decimation in Frequency Algorithm
5.3…2m-Point FFT by Decimation in Time Algorithm
5.4…2m-Point FFT by Decimation in Frequency Algorithm
5.5…Rearrangement of the Bit-Reversed Order
5.6…Speed-Up Technique by Parallel Computation
5.7…Exercises
6 DFT and Spectrum
6.1…Periodogram
6.2…Uncertainty Principle
6.3…Spreading of the Spectrum
6.4…Analysis of Short Waves
6.5…DFT of Sine Waves
6.6…Removal of Discontinuity by the Adjustment of the Sampling Frequency
6.7…Removal of Discontinuity by the Weighting of Sample Sequences
6.8…Exercise
7 Time Window
7.1…Fourier Transform of a Product of Two Time Functions
7.2…Spectra of Tapered Functions
7.3…DFT of Short Waveforms
7.4…Various Time Windows
7.4.1 Rectangular Window
7.4.2 Hanning Window (Von Hann Window)
7.4.3 Hamming Window
7.4.4 Blackman-Harris Window
7.4.5 Half-Sine Window and Riesz Window
7.4.6 Flat-Top Window
7.4.7 Bartlett Window
7.4.8 Gaussian Window
7.5…Comparison of Windows by the Results of Frequency Analysis
7.6…Exercise
Appendix
References
Index