EE513 Audio Signals and Systems

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EE513Audio Signals and Systems

• Complex Oscillator
• Kevin D. DonohueElectrical and Computer
  EngineeringUniversity of Kentucky

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Oscillator Design

• Marginally Stable approach Design a system by
  placing poles so that a marginally stable system
  results, which oscillates with a fundamental
  frequency of f0 when excited by a unit impulse.
• Show that TF and difference equation of
  oscillator system are given by
• where K scales the input and relates to the
  amplitude of oscillations, and ? relates to the
  frequency of oscillation fo and sampling
  frequency fs by

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Oscillator Design

• Trig-Identity Approach Design a system by
  selecting values of A and B in the trig-identity
  below so that yn can substitute out the
  cos(nT?0) function (T is sampling period) and
  result in a second order autoregressive
  difference equation.
• Show that difference equation of oscillator
  system is given by
• Oscillator is initiated with non-zero initial
  conditions. For n0, let

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Analyze Design

• Consider Z transform of general second order
  system
• Show that Z transform can be expressed as

Input term
Initial condition term
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Multiple Frequency Oscillator

• Excite a bank of oscillators (in parallel) tuned
  to different frequencies
• Each term represents a separate difference
  equation (second order system) where their
  outputs can be added together

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Multiple-Frequency Oscillator

• To obtain a direct form implementation for use
  with the filter function in Matlab, each parallel
  term must be combined to obtain a higher order,
  but single term, transfer function
• The numerator and denominator coefficients can
  be used directly in the direct-form I or II
  implementation of a complex (multiple-frequency)
  oscillator. The left hand side represents a
  parallel implementation.
• See Matlab functions residuez, filter, fdatool,
  dfilt

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Direct Form I Implementations

• Direct form I implementation of an IIR filter.
  The square blocks represent unit delays, the
  triangles represent multiplies, and the circles
  represent accumulators. The variable wn is an
  intermediate value output from the all-zero
  component and the input to the all-pole component
  of the filter, as suggested by the factorization
  in the equation.

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Direct Form II Implementation

• Direct form II implementation of an IIR filter.
  Note
• 1. The difference from direct form I is that the
  input and wn are associated with the all-pole
  component of the filter while the output and wn
  are associated with the all-zero part as
  suggested by the equation below.
• 2. The filter coefficients are the same for
  either direct form I or II implementations.

? ?
? ?
? ?
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Cascade Implementation

• From a direct form implementation a cascaded
  series of second order filters can be also be
  derived for another implementation of second
  order systems. Note in this case the coefficient
  are no longer the same as for the direct form
  implementations.
• In Matlab see method convert and sos and filter
  for dfilt, and tf2sos, sos2tf.

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Parallel Implementation

• A parallel bank of second order filters can be
  also be derived (obtained directly from the
  oscillator design procedures described in these
  notes). Note in this case the coefficient are not
  the same as in the direct form or cascade
  implementations.
• In Matlab see method convert and parallel for
  dfilt.