Course Outline - Rhea

Course Outline

I. Introduction

• Review of course policies
• Why linear systems theory is important

II. Signals [OW 1.0-1.4]

• Types - continuous time, discrete time, and digital [OW 1.0-1.1]
• Transformations of the independent variable [OW 1.2]
  • Time reversal
  • Time delay
  • Time scaling
• Signal Properties
  • Periodic signals
  • Even/odd signals
  • Energy
  • Power
  • Average value
• Exponential Signals [OW 1.3]
  • Continuous time
  • Discrete time
• Impulse and step functions [OW 1.4]
  • Discrete time
    • Relationship between impulse and step functions
    • Representation of DT signals with DT impulses (Sifting Prop.)
  • Continuous time
    • Definition of CT impulse [OW 2.5]
    • Relationship between impulse and step functions
    • Representation of CT signals with CT impulses (Sifting Prop.)

III. Systems [OW 1.5-1.6]

• Input/output models for systems [OW 1.5]
• System Properties [OW 1.6]
  • Review of formal logic [From notes/handouts]
  • Continuous time and Discrete time systems
  • Causal and noncausal systems
  • Memory and memoryless systems
  • Linear and nonlinear systems
  • Time varying and time invariant systems
  • Stable and unstable systems
  • Formal definitions of system properties

IV. Linear Time-Invariant Systems [OW 2.0-2.4]

• Time domain analysis of linear systems [OW 2.0]
  • Discrete time systems [OW 2.1]
    • impulse function and impulse response
    • discrete time convolution
  • Continuous time [OW 2.2]
    • impulse function and impulse response
    • continuous time convolution
• Properties for LTI systems [OW 2.3]
  • Memoryless
  • Causal and anticausal
  • Stable
• LTI analysis of linear differential equations [OW 2.4]]
• Complex exponential inputs to LTI systems [OW 3.2]

V. Frequency Analysis

• Orthonormal Tranforms [From notes]
  • General analysis of orthonormal transformations
  • Functions as vectors
  • Innerproducts on functions
  • Parseval’s theorem for orthonormal transforms
• Continuous time Fourier series (CTFS) [OW 3.0-3.3,3.5,3.8-3.9]
  • Derivation as orthogonal transform [OW 3.0-3.3]
  • CTFS examples
  • Properties of CTFS [OW 3.5]
  • LTI system analysis using CTFS [OW 3.8,3.9]
• Overview of transforms we will cover [From notes and handout]
• Continuous time Fourier transform (CTFT) [OW 4.0-4.8]
  • Derivation of tranform [OW 4.0-4.1]
  • The convolution property and LTI systems [OW 4.4]
  • CTFT properties [OW 4.3]
  • Transform pairs for aperiodic signals [See OW 4.6]
  • CTFT of periodic functions [OW 4.2]
  • Transform pairs for periodic signals [See OW 4.6]
  • Impulse train sampling [OW 7.1.1]
  • Systems characterized by linear differential equations [OW 4.7]
• The DFT [OW 3.6-3.7]
  • Derivation as orthogonal transform [From notes and OW 3.6]
  • Example transforms
  • DFT properties and circular convolution [OW 3.7]
• Discrete time Fourier transform (DTFT) [OW 5.0-5.1,5.3-5.6,5.8]
  • Tranform definition [OW 5.0,5.1]
  • DTFT properties [OW 5.3]
  • Transform pairs [See OW 5.6]
  • The convolution property and LTI systems [OW 5.4]
  • Systems characterized by linear difference equations [OW 5.8]

VI. Sampling and reconstruction [From Notes, OW Chapter 7]

• Overview of sampling systems [OW 7.0]
• Sampling
  • Relationship between CTFT and DTFT
  • Aliasing and the Nyquist frequency
• Reconstruction
  • Relationship between DTFT and CTFT
  • Aliasing and reconstruction filters
  • Zero order sample and holds

VII. (Didn’t get to this) The Z-Transform [OW 10.0-10.7]

• Definition of Z-transform
• Region of convergence
• The inverse Z-transform
• More on the Z-transform
  • Left and right hand signals
  • Stable and unstable signals
  • Causal and anticausal signals
  • Z-transform properties
• Analysis of DT systems
  • FIR systems
  • IIR systems
  • Stability analysis

[OW ] - Refers to Oppenheim and Willsky text