Dot Product

Lecture 3: Digital Signal What?

• Differentiation is Convolution
• Integration is Convolution
• A linear system with same impulse response (shifted) is Convolution [§ 2.3.3]
  • Called Linear Shift Invariant system
  • For time-based signals, Linear Time Invariant system [pp. 63]
  • Convolution as a matrix
• Convolution is associative [§ 2.3.4]
  • Passing a signal through two cascaded LSI systems is equivalent to passing through one resultant LSI system.
  • The impulse response (filter kernel) of the combined system is the convolution of the two impulse responses
  • (Note the difference between this and associativity of linear systems.)
• Energy of a signal/vector
  • Sum of squares of sample values for discrete signal [pp. 47]
  • Integral of squared signal for continuous signal
  • Square of pythagorean length
• Fitting a basis vector (u) to another vector (x)
  • Scaling the basis such that remaining vector has minimum energy
  • The scale is given by dot product of basis and given vector divided by energy of basis vector
  • After this has been subtracted, the remnant is orthogonal to the basis vector, i.e.,
  • If you take all you can out of someone, you cannot take any more!