Some basic Applications of Digital filters :

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Some basic Applications of Digital filters

• Least-square fitting of polynomials
• Given M data points (tm, um), m1,2,,M wish
  to fit (approximate) these data, in some sense,
  by a polynomial uu(t) of degree N where M?N1
• Principle of least square
• the sum of the squares of the residuals is
  the least

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• Ex consider a set of 5 equally spaced data
  points.
• tmm for m-2, -1, 0, 1, 2
• um unspecified
• fit the above data by a straight line
  uABt
• in least-square sense.

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• From the frequency point of view
• Spose the input function is one of the
  eigenfunctions in the complex form eiwt of
  frequency w.
• Since the system is linear, the same function
  will emerge at the output except that it is
  multiplied by its eigenvalue H(w).

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transfer function
In general
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the more terms used , the more rapid are the
wiggles of H(w), and the more the envelope of the
wiggles is squeezed toward the frequency axis.
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• Least-squares Quadratics and Quartics
• Instead of a straight line, fit by a
  quardratic (or a cubic) u(t)ABtCt2

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• The effect of the higher-degree polynomial is a
  higher degree of tangency at w0
• The use of more terms in the smoothing formula
  makes the curve come down sooner.

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• Modified least squares
• Smoothing Window for 2N1 points
• modified Smoothing Windows for 2N1 points
• reduce the two end values to one-half their
  assigned values

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• The curve will come down more rapidly
• The main lobe is slightly wider.

? the end constants can be chosen as a parameter
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• Differences and Derivatives
• The difference operator
• the operator annihilates a polynomial
  Pn(x) of degree n in X that is

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• from frequency point of view

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• Since , so the
  amplification at frequency w is contained in the
  factor
• ?decreases the amplitude of
  any frequency
• there is an amplification

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high-pass behavior of the difference operator ?k
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• Differences are also used to approximate
  derivatives.
• Central difference formula
• from the formula (with h1)

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• The ratio of the calculated to the true answer
  (which is iw) is
• w0, R1
• w?0, Rlt1 ? the formula underestimates the
  value of the derivatives for all other freqs.
• For the 2nd derivate

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not informative

• Spencers smoothing formula
• 15-point
• 21-point

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• plot the logs of the numbers H(w)
• 20 logratio decibel units (dB)
• 20 dB factor of 10

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• Missing Data and Interpolation
• The reasons or situations for the occurrence of
  missing data in a long record of data
• the measurements may never have been made they
  may have been misrecorded and thus later removed
• the formula used to compute successive values of
  the function may have involved an indeterminate
  form, such as at x0, and the computer
  refused to divide by zero.

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• Usually, an interpolation formula based on the
  assumption that the data locally is a polynomial
  of some odd degree.
• This is equivalent to the assumption that the
  next higher-order difference is zero.

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• For instance, k4
• note the sum of the sequences of the binomial
  coeffs. of order N is . Hence for the 2k
  th difference formula, the noise amplification is
  

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• If set
• then

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• H(0)1 However, for high freqs, the value is not
  too
• good, particularly for very high
  freqs.
• Negative values on the graph of the transfer
  function imply a change in sign.
• The above figure points out the damager of
  interpolating a missing value when the data is
  noisy, which means that the data has numerous
  high frequencies.

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• Interpolation midpoint values linear
  interpolation gives
• if 4 adjacent points are used

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• A Class of Nonre cursive Smoothing Filters
• Design of filters
• H(0)1 exact at dc(lowest freq.)
• H(?)0 no highest freq. get through

L.P.
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• Since H(?)0 the factor cos w1 had to occur
• Now one can select a filter that approximately
  meets the requirement.

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• for Ex.1, if we require

Ex.2 if we set
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• Ex. 3. Spose that we try to do as well as
  possible in the neighborhood of zero freq.

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• Ex we pick our filter form as

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• Integration Recursive Filters
• Trapezoid Rule (using y00)

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• The true answer for integration of is

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• Simpsons Rule
• w0 , R1
• and has a tangency through the 3rd derivative

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• Midpoint integration formula (using y00)

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• Leo Tick Formula

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• Simpsons formula amplifies the upper part of
  Nyquist interval (the higher frequencies) where
  as the trapezoid rule damps them out.
• In the presence of noise. Simpsons formula is
  more dangerous to use than are the trapezoid or
  midpoint formulas. But when there is relatively
  little high freq. In the function being
  integrated, then the flatness of Simpsons
  formula for low freqs. show why it is superior.