Z Transform Pairs and Properties

Z Transform Pairs and Properties Z Transform Pairs Time Domain * [k] (unit impulse) γ[k] † (unit step) ak e-bTk k sin(bk) cos(bk) aksin(bk) akcos(bk)

Z Domain z-1 1

z 1

(z) 

z z 1

z za z z  e bT z  z  12

1 1  z 1 1 1  z 1a 1 1  z 1e bT z 1

(z) 

1  z 1 2

z sin(b) 2 z  2z cos(b)  1

z 1 sin(b) 1  2z 1 cos(b)  z 2

z  z  cos(b)  2 z  2z cos(b)  1 az sin(b) 2 z  2az cos(b)  a 2

1  z 1 cos(b) 1  2z 1 cos(b)  z 2 az 1 sin(b) 1  2az 1 cos(b)  a 2 z 2

z  z  a cos(b)  2 z  2az cos(b)  a 2

1  az 1 cos(b) 1  2az 1 cos(b)  a 2 z 2

*All time domain functions are implicitly=0 for k<0 (i.e. they are multiplied by unit step, γ[k]). †u[k] is more commonly used for the step, but is also used for other things. γ[k] is chosen to avoid confusion (and because in the Z domain it looks a little like a step function, Γ(z)).

Z Transform Properties Property Name Linearity

Illustration Z af1[k]  bf 2 [k]   aF1 (z)  bF2 (z)

Left Shift by 1

Z f[k  1]   zF(z)  zf[0]

Left Shift by 2

Z f[k  2]   z 2 F(z)  z 2 f[0]  zf[1] n 1

Z f[k  n]   z n F(z)  z n  f[k]z  k

Left Shift by n

Right Shift by n Multiplication by time Scale in z Scale in time

k 0

   z n  F(z)   f[k]z  k  k 0   Z n f[k  n]   z F(z) dF(z) Z kf[k]  z dz z Z a k f[k]   F  a n is an integer k Z f     F  zn  ; n 1 n  n 1

Convolution

Z f1[k]  f 2 [k]   F1 (z)F2 (z)

Initial Value Theorem

f[0]  lim F(z)

Final Value Theorem

lim f[k]  lim(z  1)F(z)

(if final value exists)

z 

k 

z 1