oscillations

  • oscillations are everywhere
  • sustainable dynamic systems are always oscillatory
  • things that don’t move in circles don’t last
    • bombs
    • rockets
    • human beings (only partially oscillatory)

clip: gyroscopic stability of a disc player in zero-gravity - OFF vs. ON

  • an oscillation is cyclic, it goes around in circles
    • all oscillations can be described with a combination of cosine and sine functions

representing an oscillation

  • a complex reference system centered on the origin of the oscillation is used for this
  • a complex exponential is used to describe the position on the plane of oscillation
  • \(x(t) = \exp(j\omega t)\), where \( \exp(j\omega t)\) is the complex exponential
    • \(j = \sqrt{-1}\)
    • \(\omega \): rotation/oscillation frequency
    • \(t\): physical domain time

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fig: complex exponential in complex plane with sin and cos functions

  • trigonometric expansion of complex exponential:
    • \(x(t) = \exp(j\omega t) = \cos(\omega t) + j \sin(\omega t)\)
    • called euler’s formula

sample notation with complex exponential

  • \(x[n]\): discretized sample notation
  • complex exponential sample notation
    • \( x[n] = A\exp(j(\omega n + \phi)) \)
  • trigonometric complex sample notation
    • \( x[n] = A [ \cos(\omega n + \phi) + j\sin(\omega n + \phi)] \)
  • where:
    • \( \omega \): frequency (radians)
    • \( \phi \): initial phase (radians)
    • \( A \): amplitude
  • note that phase and frequency are both in radians when using the complex exponential paradigm

complex exponential notation

justification for use in dsp

  • every sinusoid can be written as a sum of the sine and cosine
    • sine and cosine live together
  • trigonometry becomes algebra, so notation is simpler
    • helps avoiding the use of trigonometric identities
      • significantly reduces the number of terms in equations
    • phase can be managed with exponent summation and split rule
    • phase shifts are simple complex multiplications
  • complex numbers can be used in digital systems without obstacles

anatomy of the complex exponential

  • polar form of complex exponential:
    • \( e^{j \alpha} = \cos \alpha + j \sin \alpha \)

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fig: \( e^{j \alpha}\) on the complex plane (argand diagram)

  • \( e^{j \alpha} \) is a unit circle on the complex plane
    • unit circle: \( \lvert e^{j \alpha} \rvert = 1\)

rotation about the origin complex plane

  • say \( z \) is a point on the complex plane
    • by multiplying it with \( e^{j \alpha} \)
    • \( z \) is rotated about the origin
      • in the anti-clockwise direction by amount \(\alpha\)
      • about the origin
      • with radius of rotation \( = \lvert z \rvert \)

comp-exp-2

fig: rotation by multiplication with \( e^{j \alpha}\)

  • this rotation operation is the basis of the complex exponential generating algorithm
    • used to synthesize signals
      • \(x[n] = e^{j \omega} \)
      • \(x[n+1] = e^{j \omega}x[n] \)
      • \(x[0] = 1\)
    • with an initial phase:
      • \(x[n] = e^{j \omega + \phi} \)
      • \(x[n+1] = e^{j \omega}x[n] \)
      • \(x[0] = e^{j \phi}\)
  • in discrete time, a sinusoid \( e^{j\omega n} \) is periodic only if:
    • \(\omega = \frac{M}{N}2\pi\); \(M,N \in \mathbb{N}\)
    • i.e. only if:
      • frequency, \(\omega \), is a rational multiple of \( 2\pi \)
      • or equivalently, if \(e^{j\omega N} = 1\)
    • so not every sinusoid is periodic in discrete time

aliasing

  • the same point in the unit circle may have many names:
    • the point at \(e^{j\alpha}\) can
      • \(e^{2\pi + j\alpha}\)
      • \(e^{6\pi + j\alpha}\)
      • \(e^{-2\pi + j\alpha}\)
  • this is called aliasing
    • natural property of complex exponential

  • in discrete time, this limits how fast we can go around the unit circle with a discrete-time signal

  • the frequency of the discrete-time machine is limited
    • \(0 \leq \omega < 2\pi\)
    • when it is faster than \(2\pi\), due to the periodicity of the complex exponential,
      • we fall back via a modulo operation
  • even within the range, care must be taken between backwards and forwards motion
    • when \(\omega = \pi \),
      • the point simply oscillates between \( 1 \) and \(-1\) on the unit circle
    • when \(\omega > \pi\)
      • it can also be view as rotation backwards to get that point
      • it is shorter to get to that point in the backwards direction
    • so anytime \(\omega > \pi\),
      • it appears like a smaller step in the clockwise direction
    • this reverse effect aliasing is even more pronounced when \(\omega\) is close to \(2\pi\)
    • if \(\omega » \pi \), the rotating body appears stationary due to aliasing
  • so at different frequencies, i.e. \(\omega \) values, aliasing introduces different artifacts of illusion

eigenvalues and eigenvectors

  • almost all vectors change direction when they are multiplied by a matrix
  • however, certain vectors are in the same direction even after multiplication with that matrix
    • those certain vectors are eigenvectors
  • since they are the in the same direction, multiplication with the matrix is like scaling the original vector
    • the equivalent scaling factors are called eigenvalues
  • consider equation: \( Ax = \lambda x \)
    • \(A\): square matrix
    • \(\lambda\): eigenvalues of \(A\)
    • \(x\): eigenvectors of \(A\)
  • rearranging this, we get: \( A - \lambda I = 0\)
    • from this we get A’s characteristic equation:
      • \(\lvert A - \lambda I \rvert = 0\)
      • where \(\lvert A \rvert\): determinant of matrix \(A\)
    • the roots of this equation are eigenvalues \(\lambda\)

  • having computed the eigenvalues \(\lambda\), the eigenvectors \(x\) can be found using \( Ax = \lambda x \)

  • if eigenvector elements can take on arbitrary values based on a relationship between them, make sure to normalize them with the square root of the sum of squares of all elements
    • i.e. normalize it with the length (first modulus) of the vector

references