A class of systems with symmetric impulse response

Abstract

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:

Systems with constant group delay and symmetric impulse response (CGDSIR)

“…Working with Vucic and Babic [1] approach, we noticed that the error function proposed works only with systems with a specific t m . So, we first need to know the abscissa of the symmetry line of the systems.…”

“…where θ is the phase angle and ω is the frequency In the same way as Vucic and Babic [1], optimum system parameters were determined by searching the minimum of the error function, in our case the CGDSE, because we search for maximally flat delay in t m seconds.…”

“…There is a direct relationship between the group delay of a filter and its impulse response. To be more precise, the group delay of a filter function is essentially equal to t out -t in , where t in is the time when the input attains its steady-state value, and t out is the time when the output settles down to its steady state condition [1]. Vucic and Babic [1] mention that if we finally look for systems with small time domain signal distortion is better to work in the same domain, and then propose, for this kind of systems, the general form of its transfer function, and the minimization procedure to find its coefficients.…”

Filter families with minimum time-bandwidth products

“…The imaginary parts of poles are equidistant. Such properties are typical for linear phase and symmetric impulse response systems [5].…”

Filter Designs for VDSL Analog Front End

“…Three different all-pole configurations are designed using Butterworth (B), Chebyshev (C) and Symmetric impulse response (S) [16] approximations and one elliptic filter is designed with elliptic (E) transfer function approximation.…”