Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 59

Using the computed h ( n ) values, the filter response can be obtained in
z -transform using the relationship

N− 1

h ( n ) z −n

H ( z )= (2.160)

n =0

The frequency sampling method is therefore a direct method compared to
the window technique because it avoids transformations from the time domain
to the frequency domain. The large overshoots that usually occur at the atten-
uation or transition band of the filter response can be minimized by leaving
some unconstrained terms in the response function. It is now known that this
technique yields more ecient filters than the window method (Tompkins,
1993).

2.6.5 Integer Filters
Integer filters are another form of digital filter that are primarily deployed in
environments requiring fast online processing. The previous digital filters can
be implemented on computer software, however, the floating point operations
performed on the real coe cients of the transfer function limit somewhat
the speed of computation. In integer filters, these coecients are replaced by
integers making the computations faster and more ecient by using integer
arithmetic operations. These operations require only bit shifting operations
rather than the slower floating point unit (FPU) for computations. Such filter-
ing is especially desirable for high-frequency digital signals or when computers
have slow microprocessors. The major limitation of the integer filter is that
it becomes dicult to obtain sharp cutoff frequencies by using only integer
coecients in the filter transfer function.

2.6.5.1 Design of Integer Filters

The design of integer filters revolves around placement of zeroes on the unit
circle in the z -plane followed by positioning of poles to cancel the zeroes,
such that frequencies defined by these specific locations are allowed through
the passband. Since each point on the unit circle corresponds to a particular
frequency, the frequency response of the integer filter can be determined. The
transfer function commonly used (Tompkins, 1993) is

z −m ] p [1 −

H I ( z )= (2.161)

[1 − 2 cos θz − 1 + z − 2 ] q

where m represents the number of zeroes around the unit circle, p and q
are integers, which give the order of magnitude for the filter and θ is the
angular location of the poles. Raising the order of magnitude of the filter
corresponds to cascading these filters. The first step is to place the zeroes
using the numerator of the transfer function in Equation 2.161. One may use