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In many situations, it is convenient to analyse continuous systems using Laplace space. This approach is also applicable for discrete systems, but it is more convenient to use the discrete Laplace and z-transforms. Linear time-invariant systems are the core of signal filtering algorithms. Discrete linear systems can be fully described by appropriate difference equations or their integral characteristics: impulse response in the time domain, and frequency response and transfer function in the frequency and z-domain respectively.

Determination of the residue at a prime pole If f (z) is a rational function, f (z) = P (z) Q(z) where P (z) and Q(z) are exponential polynomials. 53) Q (z) z=zk 2. Determination of the residue at the “m” multiple pole If for the same value of z = zk function f (z) has “m” multiple poles, then resk = d m−1 1 {f (z)(z − zk )m }z=zk (m − 1)! 4 Examples of Inverse z -Transform Calculations 1. x(z) = zn z ; f (z) = z−a z−a There is one prime pole at the point z = a. 53), we obtain x(n) = 2. 55) z=a z zn ; f (z) = (z − 1)2 (z − 1)2 In this case, the pole is at the point z = 1 with multiplicity m = 2.

1) k=0 where x(n) and y(n) are input and output signals respectively; n = 0, 1, . . corresponds to the time instant nT, where T is the clock or sampling period; and ak (n) and bk (n) are time-varying coefficients and a0 (n) = 0 for any n [5, 8]. Coefficients ak (n) correspond to a recursive part of the system, and bk (n) correspond to a non-recursive (transversal) part of the system. For K1 > 0, a system is called a recursive or infinite impulse response (IIR) system of the K1 order, whereas for K1 = 0, it is called a non-recursive or finite impulse response (FIR) system.

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