Window Functions and Their Applications in Signal Processing

Window Functions and Their Applications in Signal Processing

by K. M. Prabhu
Released September 2018
Publisher(s): CRC Press
ISBN: 9781351832274

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Book description

Window functions—otherwise known as weighting functions, tapering functions, or apodization functions—are mathematical functions that are zero-valued outside the chosen interval. They are well established as a vital part of digital signal processing. Window Functions and their Applications in Signal Processing presents an exhaustive and detailed account of window functions and their applications in signal processing, focusing on the areas of digital spectral analysis, design of FIR filters, pulse compression radar, and speech signal processing.

Comprehensively reviewing previous research and recent developments, this book:

  • Provides suggestions on how to choose a window function for particular applications
  • Discusses Fourier analysis techniques and pitfalls in the computation of the DFT
  • Introduces window functions in the continuous-time and discrete-time domains
  • Considers two implementation strategies of window functions in the time- and frequency domain
  • Explores well-known applications of window functions in the fields of radar, sonar, biomedical signal analysis, audio processing, and synthetic aperture radar

Table of contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Table of Contents
  7. Foreword
  8. Preface
  9. Acknowledgments
  10. Abbreviations
  11. 1. Fourier Analysis Techniques for Signal Processing
    1. 1.1 Review of Basic Signals and Systems
      1. 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
    2. 1.2 Continuous-Time Fourier Transform
      1. 1.2.1 Properties of the CTFT
      2. 1.2.2 Examples of CTFT
    3. 1.3 Discrete-Time Fourier Transform
      1. 1.3.1 Properties of DTFT
      2. 1.3.2 Examples of DTFT
    4. 1.4 Z-Transform
      1. 1.4.1 Examples of z-Transform
    5. 1.5 Discrete Fourier Transform
      1. 1.5.1 Properties of the DFT
      2. 1.5.2 Examples of DFT
    6. 1.6 Fast Fourier Transform
      1. 1.6.1 Decimation-in-Time FFT (DIT-FFT)
        1. 1.6.1.1 Computational Savings
        2. 1.6.1.2 In-Place Computation
      2. 1.6.2 Decimation-in-Frequency FFT (DIF-FFT)
      3. 1.6.3 Inverse DFT from FFT
      4. 1.6.4 Linear Convolution Using DIT-FFT and DIF-FFT
    7. References
  12. 2. Pitfalls in the Computation of DFT
    1. 2.1 Sampling, Reconstruction, and Aliasing
      1. 2.1.1 WKS Sampling Theorem
      2. 2.1.2 Reconstruction of Continuous-Time Signals from Discrete-Time Samples
    2. 2.2 Frequency Leakage Effect
      1. 2.2.1 Zero Leakage Case
      2. 2.2.2 Maximum Leakage Case
    3. 2.3 DFT as a Filter Bank
    4. 2.4 Picket-Fence Effect or Scalloping Loss
    5. 2.5 Zero-Padding and Frequency Resolution
      1. 2.5.1 Zero-Padding
      2. 2.5.2 Frequency Resolution
    6. References
  13. 3. Review of Window Functions
    1. 3.1 Introduction
    2. 3.2 Characteristics of a Window Function
    3. 3.3 List of Windows
      1. 3.3.1 Rectangular (Box Car) Window
      2. 3.3.2 Triangular (Bartlett) Window
      3. 3.3.3 Cos(x) Window
      4. 3.3.4 Hann (Raised-Cosine) Window
      5. 3.3.5 Truncated Taylor Family
      6. 3.3.6 Hamming Window
      7. 3.3.7 Cos3(x) Window
      8. 3.3.8 Sum-Cosine Window
      9. 3.3.9 Cos4(x) Window
      10. 3.3.10 Raised-Cosine Family
      11. 3.3.11 Blackman Window
      12. 3.3.12 Optimized Blackman Window
      13. 3.3.13 Blackman–Harris Window
      14. 3.3.14 Parabolic Window
      15. 3.3.15 Papoulis Window
      16. 3.3.16 Tukey Window
      17. 3.3.17 Parzen (Jackson) Window
      18. 3.3.18 Dolph–Chebyshev Window
      19. 3.3.19 Kaiser’s Modified Zeroth-Order Bessel Window Function Family
      20. 3.3.20 Kaiser’s Modified First-Order Bessel Window Function Family
    4. 3.4 Rate of Fall-Off Side-Lobe Level
      1. 3.4.1 Theorem
      2. 3.4.2 Side-Lobe Fall-Off Rate in the Time-Domain
    5. 3.5 Comparison of Windows
    6. References
  14. 4. Performance Comparison of Data Windows
    1. 4.1 Definition of Window Parameters
    2. 4.2 Computation of Window Parameters
    3. 4.3 Discussion on Window Selection
    4. References
  15. 5. Discrete-Time Windows and Their Figures of Merit
    1. 5.1 Different Classes of Windows
    2. 5.2 Discrete-Time Windows
      1. 5.2.1 Rectangular (Box Car) Window
      2. 5.2.2 Triangular (Bartlett) Window
      3. 5.2.3 Cosαx Window Family
      4. 5.2.4 Hann Window
      5. 5.2.5 Truncated Taylor Family of Windows
      6. 5.2.6 Hamming Window
      7. 5.2.7 Sum-Cosine Window
      8. 5.2.8 Raised-Cosine Window Family
      9. 5.2.9 Blackman Window
      10. 5.2.10 Optimized Blackman Window
      11. 5.2.11 Tukey Window
      12. 5.2.12 Blackman–Harris Window
      13. 5.2.13 Nuttall Window Family
      14. 5.2.14 Flat-Top Window
      15. 5.2.15 Parabolic Window
      16. 5.2.16 Riemann Window
      17. 5.2.17 Poisson Window
      18. 5.2.18 Gaussian Window
      19. 5.2.19 Cauchy Window
      20. 5.2.20 Hann–Poisson Window
      21. 5.2.21 Papoulis (Bohman) Window
      22. 5.2.22 Jackson (Parzen) Window
      23. 5.2.23 Dolph–Chebyshev Window
      24. 5.2.24 Modified Zeroth-Order Kaiser–Bessel Window Family
      25. 5.2.25 Modified First-Order Kaiser–Bessel Window Family
      26. 5.2.26 Saramäki Window Family
      27. 5.2.27 Ultraspherical Window
      28. 5.2.28 Odd and Even-Length Windows
    3. 5.3 Figures of Merit
    4. 5.4 Time–Bandwidth Product
    5. 5.5 Applications of Windows
      1. 5.5.1 FIR Filter Design Using Windows
      2. 5.5.2 Spectral Analysis
      3. 5.5.3 Window Selection for Spectral Analysis
    6. References
  16. 6. Time-Domain and Frequency-Domain Implementations of Windows
    1. 6.1 Time-Domain Implementation
    2. 6.2 A Programmable Windowing Technique
    3. 6.3 Computational Error in Time and Frequency-Domains
    4. 6.4 Canonic Signed Digit Windowing
      1. 6.4.1 Window 1
      2. 6.4.2 Window 2
      3. 6.4.3 Window 3
      4. 6.4.4 Window 4
      5. 6.4.5 Window 5
      6. 6.4.6 Window 6
      7. 6.4.7 Window 7
      8. 6.4.8 Window 8
      9. 6.4.9 Window 9
      10. 6.4.10 Window 10
      11. 6.4.11 Window 11
      12. 6.4.12 Window 12
      13. 6.4.13 Window 13
      14. 6.4.14 Window 14
    5. 6.5 Modified Zeroth-Order Kaiser–Bessel Window Family
    6. 6.6 Summary
    7. References
  17. 7. FIR Filter Design Using Windows
    1. 7.1 Ideal Filters
      1. 7.1.1 Lowpass Filter
      2. 7.1.2 Highpass Filter
      3. 7.1.3 Bandpass Filter
      4. 7.1.4 Bandstop Filter
    2. 7.2 Linear Time Invariant Systems
    3. 7.3 FIR Filters
      1. 7.3.1 Advantages of FIR Filters
    4. 7.4 IIR Filters
      1. 7.4.1 Properties of IIR Filters
    5. 7.5 Structure of an FIR Filter
      1. 7.5.1 Filter Specifications
    6. 7.6 FIR Filter Design
      1. 7.6.1 Linear-Phase Filters
      2. 7.6.2 Types of FIR Filters
      3. 7.6.3 Frequency Response of Type FIR Filter
      4. 7.6.4 Design Procedure for Filters
    7. 7.7 Kaiser–Bessel Windows for FIR Filter Design
      1. 7.7.1 Filter Design Using Kaiser–Bessel Zeroth-Order (I0–Sinh) Window
      2. 7.7.2 Filter Design Using Kaiser–Bessel First-Order (I1-Cosh) Window
    8. 7.8 Design of Differentiator by Impulse Response Truncation
    9. 7.9 Design of Hilbert Transformer Using Impulse Response Truncation
    10. References
  18. 8. Application of Windows in Spectral Analysis
    1. 8.1 Nonparametric Methods
      1. 8.1.1 Periodogram PSD Estimator
      2. 8.1.2 Modified Periodogram PSD Estimator
      3. 8.1.3 Spectral Analysis Using Kaiser–Bessel Window
      4. 8.1.4 Bartlett Periodogram
      5. 8.1.5 Welch Periodogram Method
      6. 8.1.6 Blackman–Tukey Method
      7. 8.1.7 Daniel Periodogram
      8. 8.1.8 Application of the FFT to the Computation of a Periodogram
      9. 8.1.9 Short-Time Fourier Transform
      10. 8.1.10 Conclusions
    2. References
  19. 9. Applications of Windows
    1. 9.1 Windows in High Range Resolution Radars
      1. 9.1.1 HRR Target Profiling
      2. 9.1.2 Simulation Results
    2. 9.2 Effect of Range Side Lobe Reduction on SNR
      1. 9.2.1 Introduction
      2. 9.2.2 Loss Factor
      3. 9.2.3 Weighting Function
      4. 9.2.4 Results and Discussions
    3. 9.3 Window Functions in Stretch Processing
    4. 9.4 Application of Window Functions in Biomedical Signal Processing
      1. 9.4.1 Biomedical Signal Processing
      2. 9.4.2 FIR Filtering of Biomedical Signals
      3. 9.4.3 Moving Average Filtering of Biomedical Signals
      4. 9.4.4 QRS Detection in ECG Based on STFT
    5. 9.5 Audio Denoising Using the Time–Frequency Plane
      1. 9.5.1 Time–Frequency Plane
      2. 9.5.2 Audio Denoising Using Time–Frequency Plane
      3. 9.5.3 Block Thresholding
      4. 9.5.4 Effect of Windows
    6. 9.6 Effect of Windows on Linear Prediction of Speech
      1. 9.6.1 Linear Prediction Coder
      2. 9.6.2 Line Spectral Frequencies
      3. 9.6.3 LSF Variation due to Windows
    7. 9.7 Application of Windows in Image Processing
      1. 9.7.1 Windows for ISAR Images
      2. 9.7.2 Experimental Analysis
      3. 9.7.3 Results and Conclusions
    8. 9.8 Windows to Improve Contrast Ratio in Imaging Systems
      1. 9.8.1 Experimental Analysis
      2. 9.8.2 Results and Conclusions
    9. References
  20. Index

Product information

  • Title: Window Functions and Their Applications in Signal Processing
  • Author(s): K. M. Prabhu
  • Release date: September 2018
  • Publisher(s): CRC Press
  • ISBN: 9781351832274