Analog and digital filters; design and realization
Optimizing Elliptic Filter Selectivity. This is a preview of subscription content, log in to check access. Lindquist, C. Google Scholar. Calahan, D. Hayden Publishers, Bronshtein, I. Van Nostrand, Spenceley, G. Smithsonian Institution, Corral, C. Circuits, Devices and Syst. EDN , pp. Vlcek, M. Circuits Syst. Ying-Fai Lam.
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The text appears unmarked. The binding is solid. Bookseller: R. Ships with Tracking Number! May not contain Access Codes or Supplements. May be ex-library. Buy with confidence, excellent customer service! In the approximation problem for VDFs, the required task is to describe an input-output relationship e. This section introduces research topics on VDFs from the viewpoints of the approximation problem and the realization problem.
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Two methods have been widely used for approximation and realization of VDFs: one is based on the variable transformation of transfer functions and the other is based on the multi-dimensional M-D polynomial approximation of filter coefficients. In the sequel details of these two methods are reviewed and some recent results on these two methods are introduced. Next, we apply a variable transformation to this prototype transfer function and obtain a desired VDF, where the variable transformation makes use of a function which includes variable parameters that are associated with the components to be changed in frequency characteristics.
Many approaches exist for variable transformations, and the most famous approach is the frequency transformation [ 3 ].
The frequency transformation makes use of all-pass functions for the variable transformation. Although details of the frequency transformation are well reviewed in [ 1 ], this chapter will also review this topic with some additional discussions. If frequency transformation is used to obtain this VLPF, the first step is to prepare the transfer function of a prototype low-pass filter.
Such a transfer function is denoted by H p z. We next discuss the realization problem for this VLPF. From the realization point of view, Eq. To explain this problem, consider a second-order IIR prototype filter with the transfer function given by. It is now clear that Figure 4 b includes delay-free loops, and hence it is impossible to implement this block diagram.
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It is well known that delay-free loops can be avoided by means of mathematical manipulations of transfer function or difference equation. However, such manipulations are not good solutions in the case of VDF realization.
In particular, the filter coefficients in Eq. One of the popular methods to overcome this problem is the Taylor approximation-based description [ 4 ]. This method applies the first-order Taylor series approximation to all of the rational polynomials of filter coefficients in VDFs, under the assumption that the absolute values of all variable parameters are small. For example, in the case of Eq.
These new coefficients do not require divisions, and hence the VLPF can be realized in terms of additions and multiplications, as shown in Figure 5. Although the VLPFs based on the Taylor approximation provide an effective realization method, they have a serious drawback that the range of variable cutoff frequency is quite limited. In order to overcome these problems, some alternative methods are proposed [ 4 , 5 , 6 ].
All of these methods make use of low sensitivity structures for realization of block diagrams for the prototype filter.
Analog and digital filters ; design and realization
Although the methods given by [ 4 , 5 , 6 ] can be applied to the limited classes of transfer functions, the Taylor approximation error becomes smaller than the standard VDFs based on the direct form. This approach is also extended to the 2-D VDFs [ 7 ]. There are some other approaches for the reduction of the Taylor approximation error. In [ 8 ], the approach based on wave digital filters is presented. Although this approach requires the knowledge of analog filter theory, very high precision is attained in the resultant VDFs, and hence the variable cutoff frequency can be controlled in relatively wide range.
In [ 9 ], state-space representation is used for construction of the block diagram of the prototype filter, and series approximations are applied to avoid the significant increase of the implementation cost of frequency transformation-based VDFs. This approach does not need any restriction that appeared in the conventional methods, and hence the method of [ 9 ] can be applied to arbitrary transfer functions and arbitrary state-space structures.
Furthermore, in [ 10 ], the VDFs based on the combination of frequency transformation and coefficient decimation are proposed, and it is shown through FPGA implementation and performance evaluation that the proposed method attains very low cost for hardware implementation. It should be noted that, however, this problem does not always happen. In general, this problem happens if the all-pass function in the frequency transformation has a nonzero constant term in the numerator.
This case corresponds to the VDFs with variable bandwidth. In other words, the problem of delay-free loops does not happen when the VDFs have fixed bandwidth, as shown in Figure 2. We conclude this subsection with a summary of the merits and the drawbacks of the frequency transformation-based VDFs. The merits are as follows: Variable characteristics can be easily obtained because the theory of controlling cutoff frequency is based on the simple variable transformations.
If Taylor approximation is not carried out, the frequency transformation preserves many useful properties on the shape of magnitude responses. For example, when a prototype low-pass filter is the Butterworth filter that possesses the monotonic and maximally flat magnitude response, the VDFs given by applying frequency transformations to this prototype filter also possess the monotonic and maximally flat magnitude responses.
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The aforementioned merit facilitates the design of adaptive band-pass or band-stop filters because the cost function for adaptive filtering becomes unimodal, leading to an adaptive algorithm that converges to the globally optimal solution. Details will be discussed in the next section. Compared with the VDFs based on the M-D polynomial approximation, the frequency transformation-based VDFs require much less computational cost in the filtering.
Next, the drawbacks are summarized as follows: As stated earlier, if the bandwidth needs to be variable in VDFs, the frequency transformation causes delay-free loops and this problem must be appropriately solved. If one wishes to obtain VDFs with multiple passbands or stopbands such as VBPFs, VBSFs, and variable multi-band filters, it is necessary to use high-order all-pass functions for the frequency transformation.
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As a result, the order of VDFs becomes higher than that of the prototype filter. Linear-phase VDFs cannot be obtained because the all-pass functions to be used in the frequency transformation are IIR filters. Realization of variable characteristics is quite limited. To be specific, the frequency transformation can provide only the VDFs with variable cutoff frequencies. In other words, other components such as the transition bandwidth and the stopband attenuation cannot be controlled.
One of the significant benefits of this approach over the frequency transformation-based VDFs is that many kinds of variable characteristics as well as variable cutoff frequencies can be attained. For example, this approach can provide VLPFs with variable transition bandwidth and variable stopband attenuation, as shown in Figure 6. Such variable parameters are referred to as spectral parameters.
After this step, filter coefficients of the desired VDFs are described as M-D polynomials with respect to these variable parameters. In realization of the VDFs given as above, Farrow structure [ 24 ] is widely used. The transfer function of this VDF is given by. A drawback of the M-D polynomial approximation-based VDFs is the high computational cost in the filtering because the filter coefficients are described by M-D polynomials.
In addition, this approach limits the range of variable characteristics. As in the case of frequency transformation with Taylor approximation, this limitation comes from the M-D polynomial approximation. Furthermore, since this approach requires a number of filters with fixed coefficients, their hardware implementation may cause an increase of characteristic degradations that comes from finite wordlength effects such as coefficient sensitivity and roundoff noise. However, such degradations can be suppressed by using high accuracy filter structures, and this approach has been recently proposed by the authors [ 23 ].
In addition to the aforementioned two approaches, many other methods have also been presented in the literature.