Common z-Transform Pairs
Signal, x(n) z-Transform, X(z) ROC
1 (n) 1 All z2 u(n) 1
1z1 |z| > 13 anu(n) 1
1az1 |z| > |a|4 nanu(n) az
1(1az1)2 |z| > |a|
5 anu(n 1) 11az1 |z| < |a|
6 nanu(n 1) az1(1az1)2 |z| < |a|
7 cos(0n)u(n)1z1 cos0
12z1 cos0+z2 |z| > 18 sin(0n)u(n)
z1 sin012z1 cos0+z2 |z| > 1
9 an cos(0n)u(n)1az1 cos0
12az1 cos0+a2z2 |z| > |a|10 an sin(0n)u(n)
1az1 sin012az1 cos0+a2z2 |z| > |a|
z-Transform Properties
Property Time Domain z-Domain ROC
Notation: x(n) X(z) ROC: r2 < |z| < r1x1(n) X1(z)
ROC1x2(n) X2(z) ROC2
Linearity: a1x1(n) + a2x2(n) a1X1(z) + a2X2(z) At least ROC1
ROC2Time shifting: x(n k) zkX(z) At least ROC, except
z = 0 (if k > 0)and z = (if k < 0)
z-Scaling: anx(n) X(a1z) |a|r2 < |z| < |a|r1Time reversal
x(n) X(z1) 1r1 < |z| < 1r2Conjugation: x(n) X(z)
ROCz-Differentiation: n x(n) z dX(z)dz r2 < |z| <
r1Convolution: x1(n) x2(n) X1(z)X2(z) At least ROC1 ROC2
1
DTFT Theorems and Properties
Property Time Domain Frequency Domain
Notation: x(n) X()x1(n) X1()x2(n) X1()
Linearity: a1x1(n) + a2x2(n) a1X1() + a2X2()Time shifting: x(n
k) ejkX()Time reversal x(n) X()Convolution: x1(n) x2(n)
X1()X2()Multiplication: x1(n)x2(n)
12pi
2piX1()X2( )d
Correlation: rx1x2(l) = x1(l) x2(l) Sx1x2() = X1()X2()=
X1()X
2 () [if x2(n) real]
Frequency Differentiation: nx(n) j dX()dWiener-Khintchine:
rxx(l) = x(l) x(l) Sxx() = |X()|2
DTFT Symmetry Properties
Time Sequence DTFT
x(n) X()x(n) X()x(n) X()x(n) X()xR(n) Xe() =
12 [X() +X
()]jxI(n) Xo() =
12 [X()X()]
X() = X()XR() = XR()
x(n) real XI() = XI()|X()| = |X()|X() = X()
xe(n) =12 [x(n) + x
(n)] XR()xo(n) =
12 [x(n) x(n)] jXI()
2
