Signal Processing
1.1
  • 1. Systems and Signals
    • 1.1. Systems
    • 1.2. Signals
      • 1.2.1. Plotting Signals
        • 1.2.1.1. Plotting Real Valued Signals
        • 1.2.1.2. Plotting Complex Valued Signals
      • 1.2.2. Signal Properties
      • 1.2.3. Signal Transformations
      • 1.2.4. Basic Signals
        • 1.2.4.1. Constant Signal
        • 1.2.4.2. Step Function
        • 1.2.4.3. Pulse Function
        • 1.2.4.4. Complex Exponential Functions
        • 1.2.4.5. Chirp Signal
    • 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
      • 3.1.4. 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
      • 5.2.1. Ohm’s Law
      • 5.2.2. Serial Circuits / Voltage Divider
      • 5.2.3. Parallel Circuits
      • 5.2.4. Kirchhoffs Laws
        • 5.2.4.1. Kirchhoff’s Current Law
        • 5.2.4.2. Kirchhoff Voltage Law
      • 5.2.5. Highway to Hell
      • 5.2.6. Power
      • 5.2.7. Inductors
      • 5.2.8. Capacitors
      • 5.2.9. Low Pass Filter
      • 5.2.10. High Pass Filter
      • 5.2.11. Operational Amplifiers
      • 5.2.12. Active Analog Filters
      • 5.2.13. Measurements
        • 5.2.13.1. Multi-meter
        • 5.2.13.2. Oscilloscope
      • 5.2.14. Excercises
        • 5.2.14.1. Equivalent Circuits
        • 5.2.14.2. Complex Impedance
        • 5.2.14.3. Control Sound Level
        • 5.2.14.4. Bode Diagram
        • 5.2.14.5. Design 2-way X-over
        • 5.2.14.6. RIAA Correction
        • 5.2.14.7. Non Inverting OpAmp
        • 5.2.14.8. Sallen Key opamp filter
        • 5.2.14.9. Audio Equalizer
    • 5.3. Sampling
      • 5.3.1. The Sampling Theorem
      • 5.3.2. Interpolation
        • 5.3.2.1. Sinc Interpolation
      • 5.3.3. Exercises
    • 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
        • 5.4.1.3. Second Order Filters
        • 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
      • 5.5.1. Physical Modelling of Dynamic Systems
        • 5.5.1.1. Warming Up
        • 5.5.1.2. A Bumpy Road
        • 5.5.1.3. The Inverted Pendulum
      • 5.5.2. Block Diagrams
        • 5.5.2.1. Cascade of Systems
        • 5.5.2.2. Addition of System outputs
        • 5.5.2.3. Feedback Systems
      • 5.5.3. Canonical Systems
        • 5.5.3.1. First Order Systems
        • 5.5.3.2. Second Order Systems
      • 5.5.4. Control Systems
        • 5.5.4.1. Feedback Control System
        • 5.5.4.2. Stability
        • 5.5.4.3. PID Controller
        • 5.5.4.4. Root Locus Analysis
  • 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
Signal Processing
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  • 4. The Complex Domain »
  • 4.2. The Z-Domain
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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

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