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7 summarises the recommended design process for deriving G(S) for a Chebyshev lowpass continuous filter. 2 Impulse-invariant Design Method 2 •3 •4 For a continuous ftlter the impulse response,g(t), is defmed as £- 1 [G(S)]. Similarly, for a digital ftlter the impulse response,g(n)T) is defmed as z- 1 [G(Z)]; n being an integer corresponding to the nth sampling instant, that is n = 0,1 ,2, .... 12) e-Stt At each sampling instant we wantg(n)Tto equalg(t), that isg(n)T= g(t) for t =0, T, 2T, '· ..

2. 11). (b) The two poles of the filter are inside the unit-circle in the Z-plane, therefore the filter is stable. (c) Substituting ejwT for Z in G(Z) we obtain G(eiwT) = e 0 2 jwT 0 08 - . - . 25) + j(sin 2wT) In this example w = w 5 /4, but w 5 = 2rr/T, therefore wT= rr/2, and 2wT= rr. 077. 5 65°. 49° Hence it is seen that the values obtained by both methods in part (c) are identical, which therefore demonstrates the validity of both methods. 6, that is in the form G(Z) = i; g(i)TZ- 1 i=O Let us now suppose that the sampled-data signal (or system) hasN distinct values, that isg(O)T,g(1)T,g(2)T, ..

For example, if N = 1024, the DFT has 1 048 576 complex multiplications, and in contrast the FFT has only 5120, which clearly is a very considerable reduction. 20. 18. Comment on the result and the method. TakeN to be equal to 8. 19. Furthermore, in the above calculations no complex multiplications are involved so the FFT requires far less computer time than the DFT. 9 CONCLUDING REMARKS This chapter has attempted to provide an introduction to some important basic concepts of sampled-data signals and digital ftlters.

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