MICROPROCESSORS

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MICROPROCESSORS

• Dr. Hugh Blanton
• ENTC 4337/ENTC 5337

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Z Transform, Sampling and Definition

• Most real signals are analog and in order to
  utilize the processing power of modern digital
  processors it is necessary to convert these
  analog signals into some form which can be stored
  and processed by digital devices.

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Z Transform, Sampling and Definition

• The standard method is to sample the signal
  periodically and digitize it with an A to D
  converter using a standard number of bits 8, 16
  etc.
• Digital signal processing is primarily concerned
  with the processing of these sampled signals.

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Z Transform, Sampling and Definition

• The diagram below illustrates the situation.
• The blue line shows the analog signal while the
  red lines shows the samples arising from periodic
  sampling at intervals T.

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• A mathematical representation of the sampled
  signal is shown below.

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• This is equivalent to modulating a train of delta
  functions by the analog signal.
• The delta function effectively "filters" out the
  values of the signal at times corresponding to
  the zeros in the argument of the delta function.

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• This process is also referred to as "ideal"
  sampling since it results in sampled signals of
• "zero" width,
• "infinite" height,
• magnitude x(t) and
• whose spectrum is perfectly periodic.

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• The previous equation is equivalent to the
  following since the delta function has the effect
  of making x(t) nonzero only at times t kT.

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• Taking the Laplace transform of the sampled
  signal using the integral definition and the
  properties of the delta function results in the
  following

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• The Laplace transform has the Laplace variable s
  occurring in the exponent and can be awkward to
  handle.

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• A much simpler expression results if the
  following substitutions are made

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• The definition of the Z Transform is

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• If the sampling time T is fixed then the Z
  Transform can also be written

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• If the sampling time T is fixed then the Z
  Transform can also be written

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• The final result is a polynomial in Z.
• The Z Transform plays a similar role in the
  processing of sampled signals as the Laplace
  transform does in the processing of continuous
  signals.

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Z Transform, Step and Related Functions

• The definition of the Z transform is shown below.

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• The step function is defined as
• and is shown graphically below.
• A continuous step function shown above is plotted
  in blue and the sampled step in red.

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• When a step function is sampled, each sample has
  a constant value of 1.
• The Z Transform can be written as a sum of terms
  as indicated below.
• The expression for X(z) is a geometric series
  which converges if z gt 1 to-

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• A Step function delayed by 1 sampling interval.
• The Z transform is

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• This can be summed to give the Z transform of the
  delayed step.
• The Z transform of x(k-1) can be written as
  z-1X(z) where X(z) is the Z transform of x(k).

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• For a kT interval delay of the step function the
  Z transform is multiplied by z-k

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• The equation for a ramp and its samples are shown
  below

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• The Z transform of the ramp is given by

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• Multiplying by z-1 gives
• Subtracting the last 2 equations give

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• Rearranging the expression for the Z transform
  gives the final expression for the Z transform of
  the ramp as

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Z Transform Table
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Z Transform Table
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Z Transform Table
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Z Transform Table