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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

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)

The Kaiser Window (1)

Specifying the design of a Kaiser-window filter.

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

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.

Equiripple Filters(2)

Specifying the design of a Kaiser-window filter.

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.

Digital differentiators (1)

The first-order difference(FOD) of an input signal

Digital differentiators (2)

Digital differentiators (3)

(b)

(a)

Frequency responses of digital differentiators

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.

Example 5.2 (2)

Solution

We have

The inverse Fourier Transform (see

equation(5.10)) is

Giving

Integrating by parts, we obtain

Example 5.2 (3)

If n is odd, exp(jnp) exp(-jnp) -1. If n is

even, exp(jnp) exp(jnp)1 .

and

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

Example 5.2 (5)

(a)

(b)

Three point central difference filter

System is

Having zeros at z1 and z-1.

Least-squares polynomial derivative approximation

System is

Have zeros at z1 and z-1, z-0.25j0.968,

z-0.25-j0.968

Derivatives. (a) Amplitude response. (b) Phase

response. Solide lineTwo-point. Circles

Three-point central difference. Dashed line

Least-squares parabolic approximation for L2

Second derivative filter

System is

Second derivative. (a) Signal-flow graph. (b)

Unit-circle diagram.