System Function of discrete-time systems

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System Function of discrete-time systems

• System representation
• Where the input signal is x(n) of z-transform
  X(z)
• The output is y(n) of z-transform Y(z).

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System Function of discrete-time systems

• The system has an impulse response h(n)
• Define so that
• and hence
• Clearly when X(z) 1 then G(z) Y(z) i.e. G(z)
  is the z-transform of the impulse response h(n)

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Frequency Response

• Let the input be
• Then the output is
• Where
• and

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System Function of discrete-time systems

• However
• While the amplitude phase responses are
• And hence

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System Functions-Amplitude response

• Evidently
• And hence
• Thus
• ie for real systems amplitude is an even
  function, and phase an odd function of frequency

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System Functions-Amplitude response

• Moreover from
• Since at is finite we
  obtain

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System Functions-Phase response

• From
• At we have
• Thus for real systems the amplitude response must
  approach zero frequency with zero slope, while
  the phase rsponse must be zero at the origin

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System Functions-Phase response

• For ,
• Hence

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System Functions-Group delay

• Thus
• where

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Suppression of a frequency band

• A real rational transfer function H(z) cannot
  suppress a band of frequencies completely.
• i.e. cannot be identically zero for
  
• in
• This may be demonstrated as follows

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System Function of discrete-time systems

• To produce a zero at say we must
  have in the numerator of H(z) a factor of the
  form
• Therefore for one zero in the band
  the factor is
• and since there are an infinite number of points
  in the band we need factors in the numerator as

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System Function of discrete-time systems

• Clearly the result is not a rational function
• Hence it cannot be the transfer function of a
  digital signal processing system.

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Stability Test

• For stability a DSP transfer function must have
  poles inside the unit circle on the z-plane.
• We need to have a means of determining whether
  the denominator of a given transfer function has
  all its zeros inside the unit circle.
• The procedures for doing so are called stability
  tests.

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Stability Test

• Let the transfer function to be tested be
• where n is the order of the transfer function.
  Set A 1.
• For stability Dn(z) must have no zeros in
  the region

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Stability Test

• Consider the simple case of a quadratic
  denominator
• Rewrite as (ignore the factor )
• If the roots are complex, say
• then

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Stability Test

• Thus and
• For stability and thus
• For real roots
• If choose root with largest
  absolute value and make less than 1

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Stability Test

• Thus
• And since quantities are positive we obtain
• Similarly for
• Thus jointly we have

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Stability Test

• These conditions form the Stability Triangle

Stability region inside triangle
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Stability Test

• For higher order functions most tests rely on an
  iterative precedure that involves
• reduction of the polynomial degree by unity
• a simple test
• Jury-Marden Test We write Dn(z) as
• where is a constant chosen to make of
  degree (n - 1)

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Stability Test

• Repeat equation
• Hence
• And thus
• Set
• so that
• is of degree (n-1) when

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Stability Test

• Rouches Theorem If the polynomials and
  are such that in the same region
• then has the same number of
  zeros in that region as

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Stability Test

• we observe that Dn(z) has as many zeros as
  either or
  depending on whether
• or
• Ie or

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Stability Test

• Thus if then is
  unstable as it has as many zeros as
  which has at most (n - 1) zeros within z lt 1.
• If then can have as
  many zeros within z lt 1 as
• The zero at z 0 can be removed and the
  procedure repeated for the remaining polynomial

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Stability Test

• An alternative test Consider
• So that
• For this equation to be a polynomial we require
  the constant term in the numerator to be zero so
  as to be able to cancel through a factor z

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Stability Test

• Thus or
• The rest of the argument is similar to the
  previous case

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Further Stability Test

• Given that and
• show that on the unit circle for any real
• Construct
• Repeat the previous arguments

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Digital Two-Pairs

• The LTI discrete-time systems considered so far
  are single-input, single-output
• Often such systems can be efficiently realised by
  interconnecting two-input, two-output structures,
  known as two-pairs

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Digital Two-Pairs

• Figures below show two commonly used block
  diagram representations of a two-pair
• Here and denote the two outputs, and
  and denote the two inputs, where the
  dependencies on the variable z has been omitted
  for simplicity

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Digital Two-Pairs

• The input-output relation of a digital two-pair
  is given by
• In the above relation the matrix t given by
• is called the transfer matrix of the two-pair

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Digital Two-Pairs

• An alternate characterisation of the two-pair is
  in terms of its chain parameters as
• where the matrix G given by
• is called the chain matrix of the two-pair

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Digital Two-Pairs

• The transfer and chain parameters are related as

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Two-Pair Interconnections

• Cascade Connection - G-cascade
• Here

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Two-Pair Interconnections

• But from figure, and
• Substituting the above relations in the first
  equation on the previous slide and combining the
  two equations we get
• Hence,

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Two-Pair Interconnections

• Cascade Connection - t-cascade
• Here

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Two-Pair Interconnections

• But from figure, and
• Substituting the above relations in the first
  equation on the previous slide and combining the
  two equations we get
• Hence,

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Two-Pair Interconnections

• Constrained Two-Pair
• It can be shown that

H(z)