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By Prof. Dr.-Ing. Rolf Isermann (auth.)

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Srn6 w(k)+Srn_ 1 6 (3. 1-4) In the following, another method for obtaining difference equations which is also valid for large sample times T0 is described. 2 Impulse Trains An expedient mathematical treatment of discrete-time functions is obtained if the pulse train x (t) is approximated by an impulse train, p where an impulse is defined by o(tl ={ 0 t'fo ( 3. 1-5) t=O and has the area I a

6-4) and Eel. 0 x, (k+1) 0 x 2 (k+1) 0 0 0 0 -a xn (k+1) n -a 0 x, (k) 0 x 2 (k) -a, xn(k) 0 + 0 u(k) (3. 6-7) xn(k) or, after introduction of a state vector ~I a system's matrix a con- ~I trol vector b and an output vector c: ~(k+1) y(k) ~ ~(k) (3. 6-9) ~ ~(k). Setting bn = 1 and b 0 , ... ,bn-l = 0, Eq. y(z) = n z +a 1 z If, however, bn f n-l + ... 6-2) and Eq. = x 1 (z). 1 and b 0 , ... 6-3) lead to ( 3. 6-10) 0, then Eq. 6-2) and Eq. 6-11) From Eq. 6-12) is also valid. Here xn(k+1) comes from Eq.

1a). This band-limited signal is sampled with sam~le time T0 and approximated by the impulse train x*(t). If T0 is sufficiently small, the Fourier-transform then exists in a "basic spectrum" -w max ,;; w ,;; w max and periodically with w0 repeating "complementary spectra" (sidebands) v=±1,±2, ... as shown in Fig. 1b. In comparison to the continuous signal, additional higher frequency components arise from the sampled signal. If the originally continuous signal has to be reconstructed from the sampled signal by filtering with the ideal bandpass filter I G(iw) I I G(iw) I -w 0 max w < -w ,;; w ,;; w max max and w > w max (see Fig.

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