Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1 – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2 – – δ0(n-k) 1 n = k 0 n ≠ k z-k 3 s 1 1(t)
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Table of Laplace and Z-transforms
X(s) x(t) x(kT) or x(k) X(z)
1. - -
Kronecker delta
(k)1 k = 0
0 k 0
2. - -
(n-k)1 n = k
0 n k
1(t) 1(k)
-at -akT t kT (kT) 112zzT (kT) 2113
zzzT ass 1 - e -at 1 - e -akT zez bsas -at - e -bt -akT - e -bkT zeze zee bTaT bTaT 10. -at kTe -akT zTe 11. (1 - at)e -at (1 - akT)e -akT zeaT 12. -at (kT) -akT 112
zzeeT aTaT 13. ass at - 1 + e -at akT - 1 + e -akT zez zzaTeeeaT aTaTaT 14. sin t sin kT cos21 sin zTz 15. cos t cos kT cos21 cos1 zTz 16. -at sin t e -akT sin kT 221
cos21 sin zeTze Tze aTaT 17. -at cos t e -akT cos kT 221
cos21 cos1 zeTze Tze aTaT
18. - - a
19. - -
k-1 k = 1, 2, 3, ...20. - - ka
k-121. - - k
k-1 azz22. - - k
k-1 2211zaazz
23. - - k
k-1332211
11111zazaazz
24. - - a
cos k x(t) = 0 for t < 0 x(kT) = x(k) = 0 for k < 0Unless otherwise noted, k = 0, 1, 2, 3, ...
Definition of the Z-transform
Z{x(k)}
zkxzX Important properties and theorems of the Z-transform x(t) or x(k) Z{x(t)} or Z {x(k)}1. )(tax )(zaX
2. )t(bx)t(ax
zbXzaX3. )Tt(x or )k(x1 )(zx)z(zX0
4. )Tt(x2
)T(zx)(xz)z(Xz05. )k(x2
)(zx)(xz)z(Xz106. )kTt(x
)TkT(zx)T(xz)(xz)z(Xz kkk7. )kTt(x
)z(Xz8. )kn(x
)k(zx)(xz)(xz)z(Xz kkk 11109. )kn(x
)z(Xz10. )t(tx
)z(X11. )k(kx
)z(X12. )t(xe
)ze(X13. )k(xe
)ze(X14. )k(xa
15. )k(xka
16. )(x0
)(limzX if the limit exists17. )(x )(1lim
zXz if )z(Xz is analytic on and outside the unit circle18. )k(x)k(x)k(x1
)z(Xz19. )k(x)k(x)k(x1
)(zx)z(Xz01 20. )k(x )z(X21. )a,t(x
)a,z(X22. )k(xk
)z(X 23.)kTnT(y)kT(x )z(Y)z(X 24.
)k(xquotesdbs_dbs2.pdfusesText_4