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The second approach that may be used is to design an analog low-pass filter, map it into a digital filter using a suitable s-plane to z-plane mapping, and then apply an appropriate frequency transformation in the discrete-time domain to produce the desired frequency selective digital filter. Table 9-6 provides a list of some digital-to-digital transformations. The two approaches do not always result in the same design. For example, although the second approach could be used to design a high-pass filter using the impulse invariance technique, with the first approach the design would be unacceptable due to the aliasing that would occur when sampling the analog high-pass filter.
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The design techniques described in the previous section are based on converting an analog filter into a digital filter. It is also possible to perform the design directly in the time domain without any reference to an analog filter. This section describes several methods for designing a digital filter directly.
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Table 9-6 The Transformation of a Digital Low-Pass Filter with a Cutoff Frequency w, to Other Frequency Selective Filters
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Design Parameters sin[(w,. - w:.)/2] sinl(w,. q ' . ) / 2 ] w:. = desired cutoff frequency
High-pass
--2-'
+ cr ul
cos[(w,. w:.)/21 cos[(w, - w3/21 w: = desired cutoff frequency
Bandpass
wCl= desired lower cutoff frequency = desired upper cutoff frequency
Bandstop
w,,~= desired lower cutoff frequency w,2 = desired upper cutoff frequency
9.5.1 Pad4 Approximation
Let h d ( n )be the unit sample response of an ideal filter that is to be approximated by a causal filter that has a unit sample response, h ( n ) ,and a rational system function,
Because H ( z ) has p q 1 free parameters, it is generally possible to find values for the coefficients a ( k ) and h(k) so that h ( n ) = h d ( n )for n = 0, 1 , . . . , p q . The procedure that is used to find these coefficients is to write H ( z ) = B ( z ) / A ( z )as follows,
and note that, in the time domain, the left-hand side corresponds to a convolution
(note that b(n) is a finite-length sequence that is equal to zero for n .c0 and n z q). Setting h ( n ) = hd(n)for n = 0 1, . . . , p q results in a set of p q + 1 linear equations in p q + 1 unknowns, .
hd(n)
+ &o(k)hd(n
~ = I
-k ) =
n = 0. I . . . . , q {E(n) n=q+l, ...,q+p
FILTER DESIGN
[CHAP. 9
that may be solved using a two-step approach. In the first step, the coefficients a ( k ) are found using the last p equations in Eq. (9.14),which may be written in matrix form as
Assuming that these equations are linearly independent, the coefficients may be uniquely determined. In the second step, the coefficients b ( k ) are found from the first 9 1 equations in Eq. (9.14) as follows:
Although PadC's method produces an exact match of h ( n ) to h d ( n )for n = 0. 1. . . . , p 9, because h ( n ) is unconstrained for n > p q , the PadC method does not generally produce a good approximation to h d ( n )for n > p+q.
9.5.2 Prony ' Method s
With a least-squares approach to filter design, the problem is to find the coefficients a ( k ) and b ( k )that minimize the least-squares error
where U is some preselected upper limit. Because E is a nonlinear function of the coefficients a ( k ) and b(k), solving this minimization problem is, in general, difficult. With Prony's method, however, an approximate least-squares solution may be found using a two-step procedure as follows. Ideally, because [see Eq. (9.14)]
the first step is to find the coefficients a ( k ) that minimize
where Once the coefficients a ( k ) have been determined, the coefficients h ( k ) are found using the PadC approach of' forcing h ( n ) = h d ( n )for n = 0, 1. . . . , 9 :
The coefficients a ( k ) that minimize E may be found by setting the partial derivatives of E equal to zero,
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