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Low-pass filters realizable as all-pass sums: design via a new flat delay filter

Low-pass filters realizable as all-pass sums: design via a new flat delay filter,10.1109/82.749080,IEEE Transactions on Circuits and Systems Ii: Analo

Low-pass filters realizable as all-pass sums: design via a new flat delay filter   (Citations: 19)
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This paper describes a new class of maximally flat low-pass recursive digital filters. The filters are realizable as a parallel sum of two all-pass filters, a structure for which low-complexity low-noise implementations exist. Note that, with the classical Butterworth filter of degree N which is retrieved as a special case, it is not possible to adjust the delay (or phase linearity). However, with the more general class of filters described in this paper, the adjustment of the delay becomes possible, and the tradeoff between the delay and the phase linearity can be chosen. The construction of these low-pass filters depends upon a new maximally flat delay all-pole filter, for which the degrees of flatness at ω=0 and ω=π are not necessarily equal. For the coefficients of this flat delay filter, an explicit solution is introduced, which also specializes to a previously known result
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    • ...In a similar way, the method given in [6] designs IIR filters with flat magnitude responses...

    Alfonso Fernandez-Vazquezet al. DESIGN OF IIR FILTERS USING NEW THREE ALLPASS FILTERS STRUCTURE

    • ...In [10], IIR lowpass filters with MF magnitude responses, whose numerator is a mirror image polynomial, have been presented by using the above-mentioned all-pole filters, where the flat group delay response can be also obtained by appropriately adjusting the delay of the filter, resulting in a parallel structure of a delay and an allpass filter...
    • ...It should be noted that unlike [9,10], the numerator will contribute to the group delay also, since it is not restricted to be a mirror image polynomial...

    Xi Zhang. Design of maximally flat IIR filters with flat group delay responses

    • ...B.3. The phase value of D(z) at ωp, φp = φD(ωp), is given by (10)...

    Alfonso Fernandez Vazquezet al. Design of Linear Phase IIR Filters with Flat Magnitude Response using ...

    • ...Design of an IIR filter based on an allpass filter controlling only the central frequency (the frequency at 3 dB) is proposed in [3]...
    • ...In the following, we demonstrate that the result of the design [3] and the method proposed here are equal for one special case, otherwise they are different...
    • ...Therefore, the filter coefficients in (2) are reduced to ones proposed in [3], where the number of constraints assigned to and are equal...
    • ...In this special case, both methods, the method [3] and the proposed one, can not control either the passband or the stopband, as shown in Fig. 3...
    • ...where is the paraconjugate of . The group delay of is given as [3]...
    • ...Similarly, the phase of can be expressed as [3]...
    • ...We recall the result from [3] where a relationship between an...
    • ...Using similar equations to (25) and (26) in [3], it can be shown that in both cases, real and complex, the total number of null derivatives at and is . Thus, according with the definition in [3], the filters designed here are not maximally flat...
    • ...Using similar equations to (25) and (26) in [3], it can be shown that in both cases, real and complex, the total number of null derivatives at and is . Thus, according with the definition in [3], the filters designed here are not maximally flat...
    • ...The proposed method may seem to some reader to be an extension of method [3], which includes both, real and complex allpass filters...
    • ...The method [3] does not relate the desired filter specifications with the design parameters: cutoff frequency , degree of delay , and the number of constraints assigned to and , and , respectively...
    • ...From the other side, the parameter in [3] allows a trade-off between phase-linearity and delay...
    • ...The method [3] also allows for flatness at and to be different...
    • ...Consequently, by changing parameters in [3], there are a number of designs with different magnitude and phase responses...
    • ...In following, we compare Example 6 with the design [3]...
    • ...DESIGN PARAMETERS AND IIR FILTER ORDER FOR THE METHOD [3]...
    • ...Generally, only in the case where the order of the IIR filter is odd, and for , , the method [3] and the proposed method will result in the same design...
    • ...Fig. 10. Magnitude responses and group delays using the method [3]...
    • ...As mentioned before, this filter can not be obtained by method [3], because is even...
    • ...According to [3], using the proposed method, the designed filters are not maximally flat...

    Alfonso Fernandez-Vazquezet al. A new method for the design of IIR filters with flat magnitude respons...

    • ...Thedirect methodproposed in[2]also improves the thenonlinearity ofthegroupdelay...
    • ...Equation (2)indicates that wecanobtain acausal andstable...
    • ...A linear phaseIIRfilter, whichsatisfy (2)andisrelated withtwocomplex allpass filters, isgiven by[9],...
    • ...Therefore, forthisexamples thebestpolynomial isE]2 (z)...
    • ...2)ForeachzeroZminside, andeachzerozloutside ofthe unitcircle, wehave,...
    • ...4)Thefilter coefficients ofthetransfer function Hc(z) are showninTable IV. asing Thegroupdelay ofthedesigned filter isshowninFig. 2(b), while Figs...
    • ...2(c) and2(d) present themagnitude response...
    • ...Figure 2(b)and2(c)showthegroupdelays andmagnitude...

    Alfonso Femandez-Vazquezet al. New Method for Flat IIR Filter Design with Improved Group Delay

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