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Biomedical Signal Processing

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Title: Biomedical Signal Processing


1
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
2
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)
3
The Kaiser Window (1)
Specifying the design of a Kaiser-window filter.
4
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

5
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.

6
Equiripple Filters(2)
Specifying the design of a Kaiser-window filter.
7
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.

8
Digital differentiators (1)
The first-order difference(FOD) of an input signal
9
Digital differentiators (2)
10
Digital differentiators (3)
(b)
(a)
Frequency responses of digital differentiators
11
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.
12
Example 5.2 (2)
Solution
We have
The inverse Fourier Transform (see
equation(5.10)) is
Giving
Integrating by parts, we obtain
13
Example 5.2 (3)
If n is odd, exp(jnp) exp(-jnp) -1. If n is
even, exp(jnp) exp(jnp)1 .
and
14
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

15
Example 5.2 (5)
(a)
(b)
16
Three point central difference filter
System is
Having zeros at z1 and z-1.
17
Least-squares polynomial derivative approximation
System is
Have zeros at z1 and z-1, z-0.25j0.968,
z-0.25-j0.968
18
Derivatives. (a) Amplitude response. (b) Phase
response. Solide lineTwo-point. Circles
Three-point central difference. Dashed line
Least-squares parabolic approximation for L2
19
Second derivative filter
System is
Second derivative. (a) Signal-flow graph. (b)
Unit-circle diagram.
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