Malaysia COMPUTER-BASED STUDY OF MCP-COUPLED ACTIVE FILTERS FOR STRUCTURES LIKE FLF , SCF , CBQ , AND LF

This paper deals with the problem of the realization of low sensitivity narrow BP active filter implementing multiple critical pole approximation methods. Coupled active filter structures are also studied. The position of a MCP (multiple critical pole) transfer function containing multiple poles is realized as FLF (Follow the Leader Feedback) and and remaining part as cascade. Further MCP transfer functions are combined with CBQ (Coupled Biquads), SCF (Shifted Companion Form), and LF (Leap Frog) structure.


I. INTRODUCTION
Several authors -3 have contributed to the reduction of sensitivities of cascaded filter structures at the aproximating stage by reducing the critical pole Q value at the expense of increasing the total degree of the approximating function. In general, the sensitivity depend on the Q-factors of the poles and the highest Q-factor will be most relevant. This method gives a more complicated network with more active and passive elements for its realization. Premoli  J. Tow 7-8 suggested a GELF configuration to cascade general second-order blocks with feedback from the output of each section to the input of the first one. This is applicable to any voltage transfer functions, whereas the FLF is restricted to the design of symmetrical BP and BR filter.
In this paper, given first is the synthesis procedure for realization of the MCP transfer function. Then the position of the MCP transfer function containing multiple poles is realized as FLF topology and the remaining part as cascade. Further, it has been shown that the lower sensitivities of MCP transfer functions in comparison with standard approximating functions can be further decreased by applying a FLF configuration instead of cascade. 29 The following three realizations for sensitivity have been compared" a) The cascaded realization of a transfer function without MCPs. b) The cascaded realization of the MCP transfer. c) The multiloop realization of the MCP transfer function.

II. MULTIPLE CRITICAL POLE FOLLOW THE LEADER FEEDBACK FILTER
In a low-pass filter, the locations of poles in the complex frequency plane is given as follows (Fig. 1).

FIGURE
Pole locations of the corresponding MCP transfer function.
The quality factor, Q, of a pole can be found as follows. For a given pole location, as in Fig. 2 The pole pair closest to the jw-axis, i.e., with the highest Q-factor, is critical from sensitivity point of view of the overall transfer function.
For an nth order transfer function, the critical part can be separated out from   (2), when realized as an FLF network, will give the following diagram: (

S -{-"ii
In order to retain the modularity of the cascaded approach, all the second-order blocks will be considered equal Eqn (6) (16)  The resulting network has a form similar to the one in Fig. 3. The second order blocks must be replaced by fourth order BP blocks with transfer functions.  (21) A second-order LP-to-BP two second-order cascade of BP will be given as After determining Q0 value, the remaining problem is to maximize the signal swing within the filter. This means that the voltage maxima at all the OPAMPS outputs should be equal. The following formula is used" The single parameter-relative sensitivity can be defined as the relative variation of a network function F due to small change in a component X. The variation in amplitude response is minimum for MUCROER FLE MUCROER CASCADE has less sensitivity compared to Chebyshev cascade but MUCROER CASCADE is the best so far as sensitivity is concerned.
The following example illustrates this: a) First we consider cascade realization of an 8th order Chebyshev filter with reflection co-efficient P 10%. (Fig. 4) b) Next we consider realization of a 10th order MUCROER filter with P 10%. (Fig. 5) c) Finally, a FLF realization of a 10th order MUCROER filter with P 10%. (Fig. 6) From the reflection co-effecient, one finds the single factor constant using the formula P V't2/(1 + t2).

IV. APPLICATIONS TO THE CBQ SCF AND LF CONFIGURATIONS
The MCP TF given in the example was tried for CBQ, SCF and LF (Fig. 5). These were compared to the corresponding 8th order Chebyshev filter realized by the same structure. The calculation for CBQ is better so far as sensitivity is concerned in stop band. For SCF realization, the sensitivity curve is almost equal to FLF case. There are slight differences within pass band, i.e., the SCF design has a little better sensitivity in a very narrow region around the center frequency, whereas FLF is better in the rest of the pass band. For both, sensitivities increase in the region outside the pass band. The CBQ design has very good sensitivity behavior in the pass band as well as in the stop band. It is much better than the cascade and somewhat better than the FLF and SCF designs. Outside the pass band, sensitivities increase but not as much as in some others. The sensitivity of LF structure is the lowest within the pass band. It has also the lowest ripple in the pass band, i.e., it is almost constant in that region. At the pass band edges, the sensitivity increases.

V. COMPUTER BASED STUDY OF COUPLED ACTIVE FILTERS FOR STRUCTURES LIKE FLF, SCF, LF AND CBQ
The formulae for TFs for the structure shown in Fig. 4 and Fig  The TF for individual second-order blocks have been calculated using the data given in tables.
From the graphs, it can be considered that the sensitivity of the MCP part of TF can be reduced implementing coupling topologies such as FLF, SCF, CBQ, and LE It has been found that the LF structure is superior to all from the sensitivity point of view.

IV. CONCLUSION
In Narrow BP filter realization using multiple critical-pole approximation and coupled-filter structure methods, the internal interaction between different biquads of the filter by coupling them resulted in lesser sensitivity. There is striking correspondence between pole quality factor and pole frequencies of one biquad to the other.
Introducing a multiplicity of the critical pole and then coupling each secondorder blocks of low-pass sections results in a large reduction of pole-quality factors, which resulted in not only sensitivity minimization but also further stability of the filter if Hurwitz polynomial criterion is consulted.
The cascade design of SCF, CBQ, and LF requires a little less effort than FLF structure and, even though the quality factors are not very much reduced, it still adds to sensitivity minimization. It can be concluded that once the multiple critical portion has been separated and then coupling topology is applied, sensitivity minimization results. The LF structure is by far the best, but its realization is difficult. The infinite pole-Q of the second-order section is a bit of nuisance. LC prototype methods are used. RL methods are used for the rest. CBQ structure is the simplest, but has more sensitivity compared to SCF and FLF topology.
The essential points are: (1) Coupling topology results in reduction of pole-quality factors of individual LP prototype filters compared to those of Chebyshev filters. (2) MUCROER examples have lower second-order section Q factors compared to Chebyshev filters. CBQ, LF, and SCF configurations of MUCROER polynomial give less sensitivity within the pass band than a cascade design. CBQ is even better in stop band.
(3) The pole frequencies and quality factors are reduced by a factor of /(1 + /3K) compared to the critical pole.
(4) This is only applicable to narrow band pass filters. So it is slightly restrictive in application.