Folding band sharpening of the cascaded-integrator-comb filter results in a high passband droop. In applications requiring low passband deviation, the droop is reduced by connecting a low-order FIR compensator in cascade with the filter. In this Letter, the authors present a method for optimum design of compensators whose coefficients are expressed as sums of signed-power-of-two terms. The method is based on the minimisation of passband deviation, with specified total number of terms.

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In this paper a class of system transfer functions based on the impulse response symmetry criterion is presented. The class is obtained using nonlinear optimization procedure. Optimization of the second to tenth order system is executed. The time and frequency domain properties of obtained system or filter class are given and compared to commonly known filter approximations. INTRODUCTIONLinear phase systems are used when a small time domain signal distortion is required. For this purpose filter transfer function approximating constant group delay with equiripple or maximally flat behavior is used. However, this is an indirect requirement because it is given in the frequency domain. One should expect better results from optimum systems based on the time domain criteria. The general property of the ideal linear phase system is a symmetric impulse response with which we expect minimum time domain error. This property was used earlier as an additional criterion in the antialiasing filter design[l]. In this paper the optimization of causal systems with the maximum impulse response symmetry is executed.We expect that such a class of systems shall approximate a constant group delay in the frequency domain but not necessarily with equiripple or maximally flat behavior.Integral Eo represents energy of the impulse response to which the symmetry error will be normalized. So, the error of the impulse response symmetry, will be given by:(4) E E = 1 -2 3 . EO E will be used as a goal function in the optimization procedure.For the n-th order system, transfer function is given by: m k=O where ai and bk are polynomial coefficients. If the transfer function poles are simple and m a , the impulse response can be written as: n r = l h(t) = x K r e P r t , where the pole residues Kr, r=1,2, 41, are: Kr = 2. IMPULSE RESPONSE SYMMETRYFor the impulse response symmetry error the following integral is used:tm where h is the filter impulse response and tm is the abscissa of symmetry line. Arranging the above equation, we obtain t m 00 Es = Jh2(t)dt-2 Jh(tm + t)h(tm -t)dt ( 2 ) ( 3 ) 0 0 Es = Eo -2Ea m i=O x a i s i 5 kbkSk-l k = l S=Pr (7) 0-7803-4455-3/98/%10.00 0 1998 EEE 111-485The integrals Eo and Ea, now can be expressed as function of polynomial coefficients and poles:

In multistage decimation filters some stages operate at the low sampling rate, thus reducing the required number of additions per output sample (APOS). State-of-the-art methods for the design of these filters usually consider only two stages. In addition, these methods do not optimize the number of APOS explicitly. This Letter proposes a framework for the design of multistage decimation filters with minimum number of APOS. To accomplish this task, the structures which contain maximum number of stages allowed by specified decimation factor are optimized. As a case study, multistage filters whose stages are cascaded-integrator-comb filters are considered. It is illustrated by several examples that the proposed optimum filters ensure smaller number of APOS and higher folding-band attenuation than those obtained by recent methods for multistage filter design.