Linear Systems

Lecture 2: Digital Signal What?

• Convolution [§ 2.3.3]
  • Examples
    • Room response equalization
    • Room response reproduction (filtering)
    • Figuring out what is the convolution kernel
      • Seismic ranging
      • Medical sonography
  • Representing "live" time signals as "dead" marks on a blackboard
  • Causal filters - physical systems doing time domain convolution [§ 2.3.5]
  • Non-causal filters
    • Offline processing [pp. 69]
    • Or where there is no time involved
      • e.g. image domain [pp. 69]
  • How to find the filter kernel - Impulse response [§ 2.3.2]
  • Finite impulse response filters [§ 2.3.7]
    • Implemented on a computer
      • Using latches to delay the input
      • Using gather convolution
      • Called direct form realization, or tapped delay line [§ 7.2.1]
  • Convolution is commutative [§ 2.3.4]
  • Convolution is linear [pp. 65]
    • Convolution exhibits scaling property
    • Convolution exhibits superposition property
• Vectors
  • Stuff that can be added...
    • Closure
    • Commutativity
    • Associativity
    • Existence of identity (zero vector)
  • ...and scaled (by a real number)
    • Closure
    • Distribution
    • Multiplication by 1
    • Multiplication by 0
  • Examples
    • 3-sample signal
    • 5-sample signal
    • infinite-sample signal
    • continuous signal
• Linear Systems
  • A function from a vector space to a vector space which follows superposition and scaling
  • Takes vector zero to vector zero
  • If you know it's response to every unit vector (impulses), then you know everything about it
  • Is a "linear network" between elements of the input and elements of the output
    • Think of it as scatter and gather
  • Cascaded give linear systems
    • An edge of the resultant system is got by "multiply-adding" the corresponding scatter response of the first system and the corresponding gather response of the second system
    • Cascading is associative
  • The "linear networks", tidied, are called matrices
    • Scatter responses in columns
    • Gather responses in rows
    • Application of linear system becomes matrix-vector multiplication
    • Cascading linear systems becomes matrix-matrix multiplication