By E.A. BOLTZMANN. GUGGENHEIM
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24. =0 L: vnz- n , n=O L: hnz- n . 00 00 V(z) = H(z) = n=O Determine the transfer function H(z) of the digital filter (ai+l _bi+l) b a_ Un-i , where a =I- b. 25. Let be an input signal, where the ai's are positive constants and 0 ~ Wo < ... < < 211". Design a simple FIR digital filter that passes the frequency Wo without changing its magnitude, but stops the frequencies WI, ... ,Wk. 26. Consider the transfer function 00 H(z) = L: hnz- n n=O of a (stable) ARMA digital filter. By using partial fractions, demonstrate that the radius of convergence of this power series is strictly less than 1.
E-jWO - (aelnwo . e- jW1 (aei(n-1)wo einWo(e-jwo _ e-jw1) . e-jWO - . + bel nw1 ) . e-jW1 +b + bei(n-1)wl) ei(n-1)wl _ ei(n-1)wl . e-jWO - . e- jW1 = aei nwo . Let us now return to the transfer function =L 00 H(z) hnz- n . n=O If the digital filter is an IIR filter, then H(z) cannot be a polynomial in Z-l. The simplest non-polynomial function that is analytic at z = 00 is a rational function in z-l. We will see that indeed a rational function which is analytic at 00 provides a realizable IIR digital filter.
On the other hand, if the transfer matrix of a linear system is given, it is important to be able to find a state-space description of the system. It will be clear, however, that there is no unique way to describe the system. A very important problem is to find a system with the minimum dimension. This is called the problem of minimal (state-space) realizl,tion, a topic that will be discussed in Sects. 2. 1 Transfer Matrices of Linear Time-Invariant Systems We first need the notion of the one-sided z-transform of a sequence of r x s matrices, Eo, E I , E 2 , .
Boltzmann's distribution law by E.A. BOLTZMANN. GUGGENHEIM