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transform

# transform - 330 APPENDIX 1 Table 1.1 Some Properties of the...

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Unformatted text preview: 330 APPENDIX 1 Table 1.1 Some Properties of the z-Transform SEQUENCE l.f,, n=0,l,2,... 41> 2. afn +‘bg,. 3. a"f,, 4. fun; n = 0, k, 2k, .. 5- fn+1 @ka k>0 7- fn—l 8. f,,_k k > o 9.nf,, 10. n(n—1)(n—2),...,(n-m+1)f,, 11. fn @gn 12- fn _fn—-1 7| 11ka ”=011a29"‘ k=0 a 14, 3—11 fn (a is a parameter of fn) 15. Series sum property 16. Alternating sum property 17. Initial value theorem 18. Intermediate value theorem 19. Final value theorem Z-TRANSFORM Fe) = 20M aF(z) + bG(z) F(az) F (2") 1. g [1%) ~51 F(2) k . , _ _ z'_k_1f;._] Zak 1': zF(z) z"F(z) d z E F (z) 2'" d7: F (2) F (2)0(2) (1 - Z)F(z) F (2) l —- z a a F(z) F<1>=§ofn F(—1) =n§o(‘l)"f" F (0) =fo 1 d"F(z) __ — n n! dz" 2:0 lim (1 — z)F(z) =fm 241 1.2. THE z-TRANSFORM 331 Table 1.2 Some‘z-Transform Pairs ________________-—é————————-——-——-- . SEQUENCE Z-TRANSFORM , , co l.f,, n =0,'l,2,... 41> F(z) = Efnz" n=0 2 1 ‘ n = 0 l ‘ "" - o n 9'5 o 3 up; 2" . l 4.6,,=1 n=o,1,2,.. 1-2 ZR: 5. 6H 1 -2 1 . A 6. Au” ‘ - 1 -— ocz 7 n uz: . not —-—(1 _¥ ”)2 8 z ' . n , (1 _‘z)2 , - uz(l + (12) 2 __ 9. not“ (1 _ “2): “ ' - z(l +2) 2 10. n 1 (1 ’_ 2):, 11 ( +1); _1__ . n a (1 _ ”)2 ' _ , ‘ 1 I?" 1‘" +1); (1 - z)? t 13 1 ' 1 1 n 1 . ;n_!(n + m)(n + m — ) (117+ )0: p (1 _ “2W“ _ , 1 ’14‘. — e” , We; n! In words, then, we have that the z-transform of the convolution of two sequences is equal to the product of the z-transform of each of the sequences themselves. - k g In Table 1.1 we list a number of important properties of the z-transform, and following that in Table 1.2 we provide a list of important common z-transforms. Table 1.3 " Some Properties of the Laplace Transform ‘ k _ . FUNCTION ‘ ‘ ' TRANSFORM ' * W 1. f(t) t 2 0 c» , F*(s) = f(t)e““dt o— 2. af(t) + bg(t) aF’,“(s) + bG*(s)r 3. f6!) (a > 0) a_F*(a;) 4. f(t — a) e"“’F*(s) 5. e’“‘f’(t) F*(s -+ a) ' LdF*(s) 6. tf(t) — ds ' ’ '_ , ”F* 7. 17(1) 1 (- 1) '13:” 8. I? f F *(sl) ds1 9.1%) i f (131 f dsz . - - f dsnF*(s,,) sl=s 32=31 sn=s,._1 10- f0) ® 5’“) ~ ' » ~ F*(S)G*(S) ~ df(t) ' 11.1' 7 ’ sF*(s) 12.’r d2? )1 s”F*(s) t __ * 13.? f f(t)dt _ F3“) t' t " a: 14.1 f --- f f(t)(dt)" stf) —ao —ao ntimes a a f (t) [a is a parameter] a F (s) w 16. Integral property F*(0) = _ f(t) dt 17. Initial value theorem lim sF*(s) = lim f(t) . _ a—wo t_.o 18. Final value theorem lin‘i) sF*(s) = lim f(t) _ if sF*(s) is analytic for Re (3) 2 0 _' m '1 * To be complete, we wish to show the form of the transform for entries 11— 14 m , the case when f(t) may have nonzero values for t < 0 also: ' (gigofr snF*(s) — s”‘lf(0‘) " "'2f(1’(0‘)— ' ' ' ~_f(n—1)(o—) . ‘. * t F* (—1) o— “2’ 0‘ 0'”) 0- I, f... ”NWO" s_(_s)+ r S( ) +f \$151+) ”+f s( 1) —: times 346 1.3. THE LAPLACE TRANSFORM 347 Table 1.4 'Some Laplace Transform Pairs _____________________._—————————-—— FUNCTION TRANSFORM / l. f(t) t z 0 a» F*(s) = J- f(t)e—“dt o- 2. uo(t) (unit impulse) 1 3. uo(t — a) e‘” 4 A E- t n "140) = dt un—1( ) S L 1 s u 1(t) é do) (unlt step) 3 e—as 6 u_1(t — a) ‘—s— A tn—l 1 7 “—"m ‘ (n w s7 A —at 8 Ae 6(t) s + a 1 —at 9. re 6(t) (s + (1)2 10 it" ‘at 6 1 ' n! (t) (s + a)"+1 __-____'_____________.———————————‘—' gives the effect of a parameter shift in the 6 and 7 show the eﬂ'ect of multiplication by t differentiation in the transform whereas Property 5, its dual, transform domain. Properties (to some power), which corresponds to domain; similarly, Properties 8. and 9 show the effect of division by t (to some I E power), which corresponds to integration. Property 10, a most important property (derived earlier), shows the effect of convolution in the time domain going over to simple multiplication in the transform domain. Properties 11 and 12 give the effect of time differentiation; it should be noted that this E corresponds to multiplication by s (to a power equal to the number of E diﬂ‘erentiations in time) times the original transform. In a similar way , Properties 13 and 14 show the effect Of time integration going over to division ; by s in the transform domain. Property 15 shows that differentiation with E respect to a parameter of f (t') corresponds to differentiation in the transform domain as well. Property 16, the integral property, shows the simple way in which the transform may be evaluated at the origin to give the total integral ...
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