Biomedical Signal Processing

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Biomedical Signal Processing
Chapter 5. Design of Non-recursive Digital Filters
Dept. of Biomedical Engineering Yonsei
University Prof. Young Ro Yoon yoon_at_yonsei.ac.kr 0
10-2007-2440, 033-760-2440,2501
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The Von Hann Hamming Windows (7)

• A disadvantage of the Von Hann Hamming filters
  
• Their main lobes are a lot wider than the 10
  specified.
• ?To some extent we could narrow them by
  requesting a smaller bandwidth in the first place.

H(W) dB
Frequency response of a 101-term low-pass filter
based upon the Hamming window (abscissa 320
samples)
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The Kaiser Window (1)
Specifying the design of a Kaiser-window filter.
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The Kaiser Window (2)

• The parameter a depends on the allowable ripple
  value d. This is because a controls the taper of
  the window, and hence its sidelobe levels. For a
  given value of ripple(and hence a), the
  transition width D is related to the window
  length. Hence if D is specified, we can find the
  parameter M.
• Ripple level
• Given the value of A and the transition width D,
  M is found using

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Equiripple Filters (1)

• As we move away from the transition region, the
  error between desired and actual responses
  becomes smaller.
• This raises the interesting possibility that, if
  the error can be distributed more equally over
  the range.
• This raises the interesting possibility that, if
  the error can be distributed more equally over
  the range 0 ? ? ? ?, we may be able to achieve a
  better overall compromise between ripple levels,
  transition bandwidth, and filter order.
• Equiripple filter The aim is to find an
  approximation giving acceptable levels of ripple
  throughout the passband and stopband - rather
  than just meeting the specification at one
  frequency, and greatly exceeding it elsewhere.

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Equiripple Filters(2)
Specifying the design of a Kaiser-window filter.
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Equiripple Filters(3)

• Hermann and Schuessler specified the parameter M,
  d1 and d2, allowing WP and WS to vary. They
  showed that the equiripple behavior of Figure
  5.21 could be expressed by a set of nonlinear
  equations.
• The difficulty of solving the equations for large
  values of M led Hofstetter, Oppenheim, and Siegel
  to develop an iterative algorithm for finding a
  trigonometric polynomial with the required
  properties.
• Parks and McClellan chose to specify M, WP and
  WS, and the ripple ratio d1/d2, while allowing
  the actual value of d1 to vary. Their approach
  has the advantage that the transition bandwidth
  an important feature of most practical designs
  is properly controlled. The design problem was
  shown to reduce to a so-called Chebyshev
  approximation over disjoint sets.

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Digital differentiators (1)
The first-order difference(FOD) of an input signal
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Digital differentiators (2)
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Digital differentiators (3)
(b)
(a)
Frequency responses of digital differentiators
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Example 5.2 (1)

Finding the impulse response hn corresponding
to the frequency response H(W) jB(W), where B(W)
is as shown in figure below. Sketch the form of
hn for a causal differentiating filter
truncated to 21 terms.
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Example 5.2 (2)
Solution
We have
The inverse Fourier Transform (see
equation(5.10)) is
Giving
Integrating by parts, we obtain
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Example 5.2 (3)
If n is odd, exp(jnp) exp(-jnp) -1. If n is
even, exp(jnp) exp(jnp)1 .
and
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Example 5.2 (4)

• The case n0 is a little awkward because of the
  denominators n and n2 in the above expressions.
  However if we put n0 in equation (5.40), we
  readily find that h00. These results show that
  hn is antisymmetrical about n0, in theory the
  tails are infinitely long.
• Figure 5.24 shows the impulse response truncated
  to 21 terms, and shifted to begin at n0.
  (Strictly, perhaps we should say 20 terms,
  because the middle one is zero!) This causal
  version of the differentiator will give a
  best-fit approximation to the desired frequency
  response in the least-squares sense, and will
  introduce a pure delay of ten sampling intervals

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Example 5.2 (5)
(a)
(b)
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Three point central difference filter
System is
Having zeros at z1 and z-1.
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Least-squares polynomial derivative approximation
System is
Have zeros at z1 and z-1, z-0.25j0.968,
z-0.25-j0.968
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Derivatives. (a) Amplitude response. (b) Phase
response. Solide lineTwo-point. Circles
Three-point central difference. Dashed line
Least-squares parabolic approximation for L2
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Second derivative filter
System is
Second derivative. (a) Signal-flow graph. (b)
Unit-circle diagram.