EEL205 : Signals and Systems

Kushal K. ShahAsst. Prof. @ EE, IIT Delhi

Email : kkshah@ee.iitd.ac.inWeb : http://web.iitd.ac.in/~kkshah

Z Transform

Z Transform

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

=∞

∑n=−∞

{x [n] r−n

}e−jωn

= F{x [n] r−n

}X (jω) = X (z)

∣∣∣∣∣r=1

ROC : range of r for which X (z) converges

Z Transform

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

=∞

∑n=−∞

{x [n] r−n

}e−jωn

= F{x [n] r−n

}X (jω) = X (z)

∣∣∣∣∣r=1

ROC : range of r for which X (z) converges

Z Transform

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

=∞

∑n=−∞

{x [n] r−n

}e−jωn

= F{x [n] r−n

}X (jω) = X (z)

∣∣∣∣∣r=1

ROC : range of r for which X (z) converges

Z Transform

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

=∞

∑n=−∞

{x [n] r−n

}e−jωn

= F{x [n] r−n

}X (jω) = X (z)

∣∣∣∣∣r=1

ROC : range of r for which X (z) converges

Z Transform

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

=∞

∑n=−∞

{x [n] r−n

}e−jωn

= F{x [n] r−n

}X (jω) = X (z)

∣∣∣∣∣r=1

ROC : range of r for which X (z) converges

Z-Plane

X (z) =∞

∑n=−∞

x [n]z−n z = re jω

ROC : range of r for which X (z) convergesIf ROC includes the unit circle, then the DTFT of x [n] also exists

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

=∞

∑n=−∞

anu [n]z−n

=∞

∑n=0

(az−1

)n=

1

1−az−1if

∣∣az−1∣∣ < 1

=z

z−aif |z |> |a|

Z-Transform : Example 1

x [n] = anu [n]

⇒ X (z) =z

z−aif |z |> |a|

If a = 0.5,

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =∞

∑n=−∞

x [n]z−n

= −∞

∑n=−∞

anu [−n−1]z−n

= −−1

∑n=−∞

(az−1

)n=−

∞

∑n=1

(a−1z

)n=

z

z−aif |z |< |a|

Z-Transform : Example 2

x [n] = −anu [−n−1]

⇒ X (z) =z

z−aif |z |< |a|

If a = 0.5,

Z-Transform

x [n] = anu [n]

⇒ X (z) =z

z−a

ROC : |z |> |a|

a = 0.5

x [n] = −anu [−n−1]

⇒ X (z) =z

z−a

ROC : |z |< |a|

a = 0.5

Z-Transform : Example 3

x [n] = anu [n]−bnu [−n−1]

⇒ X (z) =z

z−a+

z

z−b

ROC : |a|< |z |< |b| iff |b|> |a|

If a = 0.5 and b = 0.75,

R

I

1

1

0.5

0.75

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Z-Transform : Example 4

x [n] = b|n| b > 0

= bnu [n]−b−nu [−n−1]

⇒ X (z) =z

z−b+

z

z−1/b

ROC1 : |z |> b

ROC2 : |z |< 1/b

ROC : b < |z |< 1

biff 0 < b < 1

Poles and ZerosPole : Value of |z | for which X (z) is not finiteZero : Value of |z | for which X (z) is zero

Properties of ROC

Property 1ROC of Z-transform consists of a ring in the z-plane centered aboutthe origin

X (z) =∞

∑n=−∞

x [n]z−n

∞

∑n=−∞

|x [n]| r−n < ∞

R

I

1

1

0.5

0.75

Property 2ROC does not contain any poles

Property 3

If x [n] is of finite duration, then the ROC is the entire z-plane exceptpossibly z = 0 and/or z = ∞

δ [n] ←→ 1 |z | ≥ 0

δ [n−1] ←→ 1

z|z |> 0

δ [n+1] ←→ z 0≤ |z |< ∞

Property 4

If x [n] is a right-handed signal, and if the circle |z |= r0 is in the ROC,then all finite values of z for which |z |> r0 will also be in the ROC.

∞

∑n=N1

|x [n]| r−n0

< ∞

⇒∞

∑n=N1

|x [n]| r−n < ∞ if r > r0

Property 5

If x [n] is a left-handed signal, and if the circle |z |= r0 is in the ROC,then all finite values of z for which |z |< r0 will also be in the ROC.

N2

∑n=−∞

|x [n]| r−n0

< ∞

⇒N2

∑n=−∞

|x [n]| r−n < ∞ if r < r0

Property 6

If x [n] is a two-handed signal, and if the circle |z |= r0 is in the ROC,then the ROC will consist of a ring in the z-plane that includes thecircle |z |= r0

Property 7

If X (z) is rational, then the ROC is bounded by poles or extends toinfinity

Property 8

If X (z) is rational, and if x [n] is right-sided, then the ROC is theregion in z-plane outside the outermost pole

If x [n] is causal, then the ROC of X (z) extends to infinity

Property 9

If X (z) is rational, and if x [n] is left-sided, then the ROC is theregion in z-plane inside the innermost pole

If x [n] is anti-causal, then the ROC of X (z) also includes z = 0

Property 10

The ROC must be a connected region

Properties of Z-Transform

Z-Transform : PropertiesI Linearity

ax1 +bx2 ←→ aX1 +bX2 ROC contains R1∩R2

x1 = x2 b =−a

I Time shifting

x [n−n0]←→ z−n0X (z) ROC=R±{0,∞}

I Scaling in the z-domain

zn0 x [n]←→ X

(z

z0

)ROC= |z0|R

If |z0|= 1,

e jω0nx [n]←→ X(e−jω0nz

)ROC=R

Z-Transform : PropertiesI Linearity

ax1 +bx2 ←→ aX1 +bX2 ROC contains R1∩R2

x1 = x2 b =−a

I Time shifting

x [n−n0]←→ z−n0X (z) ROC=R±{0,∞}

I Scaling in the z-domain

zn0 x [n]←→ X

(z

z0

)ROC= |z0|R

If |z0|= 1,

e jω0nx [n]←→ X(e−jω0nz

)ROC=R

Z-Transform : PropertiesI Linearity

ax1 +bx2 ←→ aX1 +bX2 ROC contains R1∩R2

x1 = x2 b =−a

I Time shifting

x [n−n0]←→ z−n0X (z) ROC=R±{0,∞}

I Scaling in the z-domain

zn0 x [n]←→ X

(z

z0

)ROC= |z0|R

If |z0|= 1,

e jω0nx [n]←→ X(e−jω0nz

)ROC=R

Z-Transform : Properties

I Linearity

ax1 +bx2 ←→ aX1 +bX2 ROC contains R1∩R2

x1 = x2 b =−a

I Time shifting

x [n−n0]←→ z−n0X (z) ROC=R±{0,∞}

I Scaling in the z-domain

zn0 x [n]←→ X

(z

z0

)ROC= |z0|R

If |z0|= 1,

e jω0nx [n]←→ X(e−jω0nz

)ROC=R

Z-Transform : PropertiesI Time Reversal

x [−n]←→ X

(1

z

)ROC=

1

R

I Time Expansion

x(k) [n]←→ X(zk

)ROC=R1

/k

I Conjugation

x∗ [n]←→ X ∗ (z∗) ROC=R

I Convolution

x1 [n]∗ x2 [n]←→ X1 (z)X2 (z) ROC contains R1∩R2

Z-Transform : PropertiesI Time Reversal

x [−n]←→ X

(1

z

)ROC=

1

R

I Time Expansion

x(k) [n]←→ X(zk

)ROC=R1

/k

I Conjugation

x∗ [n]←→ X ∗ (z∗) ROC=R

I Convolution

x1 [n]∗ x2 [n]←→ X1 (z)X2 (z) ROC contains R1∩R2

Z-Transform : Properties

I Time Reversal

x [−n]←→ X

(1

z

)ROC=

1

R

I Time Expansion

x(k) [n]←→ X(zk

)ROC=R1

/k

I Conjugation

x∗ [n]←→ X ∗ (z∗) ROC=R

I Convolution

x1 [n]∗ x2 [n]←→ X1 (z)X2 (z) ROC contains R1∩R2

Z-Transform : Properties

I Time Reversal

x [−n]←→ X

(1

z

)ROC=

1

R

I Time Expansion

x(k) [n]←→ X(zk

)ROC=R1

/k

I Conjugation

x∗ [n]←→ X ∗ (z∗) ROC=R

I Convolution

x1 [n]∗ x2 [n]←→ X1 (z)X2 (z) ROC contains R1∩R2

Z-Transform : Properties

I Differentiation in z-domain

nx [n]←→−z dXdz

ROC=R

I Initial Value Theorem

x [0] = limz→∞

X (z) if x [n] = x [n]u [n]

X (z) =∞

∑n=0

x [n]z−n

Z-Transform : Properties

I Differentiation in z-domain

nx [n]←→−z dXdz

ROC=R

I Initial Value Theorem

x [0] = limz→∞

X (z) if x [n] = x [n]u [n]

X (z) =∞

∑n=0

x [n]z−n

Causality

I A discrete-time LTI system is causal iff the ROC of its systemfunction is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

R

I

1

1

0.5

0.75

CausalityI A discrete-time LTI system is causal iff the ROC of its system

function is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

CausalityI A discrete-time LTI system is causal iff the ROC of its system

function is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

CausalityI A discrete-time LTI system is causal iff the ROC of its system

function is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

Causality

I A discrete-time LTI system is causal iff the ROC of its systemfunction is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

Causality

I A discrete-time LTI system is causal iff the ROC of its systemfunction is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

Causality

I A discrete-time LTI system is causal iff the ROC of its systemfunction is the exterior of a circle, including infinity

H (z) =∞

∑n=0

h [n]z−n

I A discrete-time LTI system with rational system function H (z)is causal iff

I ROC is the exterior of a circle outside the outermost poleI Order of the numerator cannot be greater than the order of the

denominator

H (z) =z3−2z

2 + z

z2 + z +1Not causal

I Stable : ROC must include |z |= 1

I Causal+Stable : All poles must lie within the unit circle

Inverse Z-Transform

I Inspection MethodI Partial Fraction MethodI Power Series Method

X (z) =∞

∑n=−∞

x [n]z−n

Inverse Z-Transform

I Inspection MethodI Partial Fraction MethodI Power Series Method

X (z) =∞

∑n=−∞

x [n]z−n

Inverse Z-Transform

I Inspection MethodI Partial Fraction MethodI Power Series Method

X (z) =∞

∑n=−∞

x [n]z−n