14.4 Characteristic Seyueiices arid System Functions of Discrete LTI-Systernd341 14.3.2 Solving Linear Difference Equations The are two possible methods for solving the linear difference equation (14.6): rrurrierical aiitl aria1ytic;il 14.3.2.1 Numerical Solution Ky rearranging the diffcrciice equation (14.6), we obtain the expression (14.7) cl, begirirririg with X = , For k = 1, the that can be numerically eval calculation of the initial ns starts with the known inpul signal z [ k : a i d , U[-N I] For k > the values already with the initial d u e s g[O], known, yjk - 11, ,y[k 21, , are entered into the right-lmid side This mrthod has practical importaricr, BY the right siclc of (14.7) can be calculated quickly arid very accurately using a digital cwmpiiter A delay by n can hc iniplciiicnted by shifting the location of the values in the memory of the computer ~ 14.3.2-2 AnaXytical Solution Therc is also an analytical solution which is possible in mut-11the sanie way as with t lie linear differential c.quation The fundamental priiiciple is agairi splitting thc solution into an external and internal part (14.8) Here, y[k] rrJpcresents the solution of the diflererice etlitation iiiider the initial conditions, yext[k] describes the response to an input signal if the sy icst aiid ylllt[k]describes the honlogenoiis solution of the iuitial colidition problern without an input signal The effect of the combined terrnh is shown in Figurr 14.2, which corresponds to Figure 7.1 for corrtiriuons initial value problems The exkraal &lid internal tclrms C i i D be calciilated in a similar way u5ing the z-transform, just as the Laplace transform was used with coiitinuoucl systems (we Chapter 7) P me~i~~~sequen~es In Chapter 13.1.1and 13.5 we rnentioned that the inverse z-transform,oan be interpreted as superimposing eigensequences of diwrete LTT-systems The broperties of