Digital Signal Processing

Lecture Notes

Rein van den Boomgaard

with contributions from

Jose Lagerberg

Taco Walstra

Maxim van den Berg

• 1. Systems and Signals
  • 1.1. Systems
  • 1.2. Signals
  • 1.3. Decibels
  • 1.4. Exercises
• 2. Linear Time Invariant Systems
  • 2.1. Definition and Properties
    • 2.1.1. Linearity and Time Invariance
    • 2.1.2. Stable Systems
    • 2.1.3. Causal Systems
  • 2.2. Convolutions
    • 2.2.1. Linearity + Translation Invariance = Convolution
    • 2.2.2. Properties and Recipe
  • 2.3. Linear Discrete Time Systems
  • 2.4. Eigenfunctions
  • 2.5. Exercises
• 3. The Frequency Domain
  • 3.1. Continuous Time Fourier Series
    • 3.1.1. The Complex Exponential Functions
    • 3.1.2. The Fourier Series
    • 3.1.3. Properties of the CT Fourier Series
      • 3.1.3.1. Real Valued Signal
      • 3.1.3.2. Even Signal
      • 3.1.3.3. Odd Signal
      • 3.1.3.4. Differentiation
      • 3.1.3.5. Convolution
      • 3.1.3.6. Fourier Series Examples
        • 3.1.3.6.1. Block Wave Function
        • 3.1.3.6.2. From Time Domain to Frequency Domain
        • 3.1.3.6.3. Triangular Signal
    • 3.1.4. Exercises
  • 3.2. Continuous Time Fourier Transform
    • 3.2.1. The Fourier Transform
    • 3.2.2. Properties of the CT Fourier Transform
      • 3.2.2.1. Real Signals
      • 3.2.2.2. Even and Odd Signals
      • 3.2.2.3. Derivatives
      • 3.2.2.4. Pulse Function
      • 3.2.2.5. Time Shifts
      • 3.2.2.6. Complex Exponential
      • 3.2.2.7. Periodic Signal
      • 3.2.2.8. Pulse Train
      • 3.2.2.9. Convolution
      • 3.2.2.10. Duality
    • 3.2.3. Fourier Transform Pairs
      • 3.2.3.1. Complex Exponential \(\FTright\) Pulse
      • 3.2.3.2. Pulse \(\FTright\) Complex Exponential
    • 3.2.4. Bode Plots
    • 3.2.5. Exercises
  • 3.3. The Discrete Time Fourier Transform
    • 3.3.1. DTFT and IDTFT
    • 3.3.2. Properties of the DTFT
    • 3.3.3. Fourier Transform Pairs
    • 3.3.4. Introducing Time
    • 3.3.5. Exercises
  • 3.4. Discrete Time Fourier Series
    • 3.4.1. Synthesis and Analysis Equations
    • 3.4.2. Properties of Discrete Time Fourier Series
    • 3.4.3. Time and Frequency
    • 3.4.4. DTFS in Numpy
    • 3.4.5. DTFS in Linear Algebra Disguise
    • 3.4.6. The Fast Fourier Transform
    • 3.4.7. Exercises
• 4. The Complex Domain
  • 4.1. The S-Domain
    • 4.1.1. The Laplace Transform
      • 4.1.1.1. Definition
      • 4.1.1.2. Eigenfunctions of an LTI system
      • 4.1.1.3. The Laplace Transform and the Fourier Transform
    • 4.1.2. Properties of the Unilateral Laplace transform
    • 4.1.3. Pairs of the (Unilateral) Laplace Transform
    • 4.1.4. Differential Equations and the Laplace Transform
      • 4.1.4.1. Poles and Zeros in the S-plane
      • 4.1.4.2. An Electronic Example
      • 4.1.4.3. A Mechanical Example
    • 4.1.5. Exercises
  • 4.2. The Z-Domain
    • 4.2.1. The \(\ZT\)-Transform
      • 4.2.1.1. Definition
      • 4.2.1.2. Eigenfunctions of an LTI System
      • 4.2.1.3. Finite and Infinite Signals
      • 4.2.1.4. The Z-Transform and the Fourier Transform
    • 4.2.2. \(\ZT\)-Transform Properties
    • 4.2.3. Pairs of Z-Transforms
    • 4.2.4. Difference Equations in the Z-domain
      • 4.2.4.1. From Difference Equation to Transfer Function
      • 4.2.4.2. Poles and Zeros
    • 4.2.5. The Z-Operator
    • 4.2.6. Exercises
• 5. Applications
  • 5.1. Sound and Sound Processing
    • 5.1.1. What is sound?
    • 5.1.2. Human Perception of Sound
      • 5.1.2.1. High Frequency Limit
      • 5.1.2.2. Absolute Threshold of Hearing
      • 5.1.2.3. Simultaneous Masking
      • 5.1.2.4. Temporal Masking
    • 5.1.3. Sound Level Measurement
    • 5.1.4. A word of warning
    • 5.1.5. Sound Recording
    • 5.1.6. Sound Compression
  • 5.2. Analog Electronics
  • 5.3. Sampling
  • 5.4. Transforming Analog Filters into Digital Filters
    • 5.4.1. Analog Filters
      • 5.4.1.1. Ideal Filters
      • 5.4.1.2. The Canonical (Low Pass) First Order Filter and its Transformations
        • 5.4.1.2.1. Selecting the Corner Frequency
        • 5.4.1.2.2. From Low Pass to High Pass
        • 5.4.1.2.3. From Low Pass to Bandpass
      • 5.4.1.3. Second Order Filters
        • 5.4.1.3.1. From Low Pass to High Pass
        • 5.4.1.3.2. Second Order Filters from the (Audio) Cookbook
      • 5.4.1.4. Higher Order Filters
    • 5.4.2. From Analog to Digital through the Bilinear Transform
      • 5.4.2.1. The Bilinear Transform
      • 5.4.2.2. Pre-Warping
    • 5.4.3. IIR Filters in Python
    • 5.4.4. Exercises
  • 5.5. Classical Control Theory
• 6. Mathematical Tools
  • 6.1. Geometric Sequences
  • 6.2. Complex Numbers and Complex Functions
    • 6.2.1. Algebra
    • 6.2.2. Complex Functions
    • 6.2.3. Geometrical Interpretation
    • 6.2.4. Complex Numbers in Python/Numpy
    • 6.2.5. Exercises
• 7. Python for Sound Signal Processing
  • 7.1. Read, Write and Play Sound
  • 7.2. Visualize Sound Signals
  • 7.3. Real Time Processing Sound Signals
  • 7.4. Exercises