Lecture, week 03

The z-transformation — Part II

Week 03, INF3190/4190

Andreas Austeng

Department of Informatics, University of Oslo

September 4, 2019

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 1 / 11

Outline

Outline

1 The system function of a LTI system

Properties

System function algebra

Pole-zero locations

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 2 / 11

The system function of a LTI system

Outline

1 The system function of a LTI system

Properties

System function algebra

Pole-zero locations

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 3 / 11

The system function of a LTI system

The impulse response & the system function

An LTI system is completely characterized in time domain by its

impulse response:

y [n] =X1

k=�1h[k ] x [n � k ].

From the convolution property of the z-transform, we get

Y (z) = H(z) X (z).

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 4 / 11

The system function of a LTI system

The system function of a LTI system

Y (z) = H(z)X (z).

H(z) =P1

n=�1 h[n]z�1 (and ROC) and h[n] are equivalent

descriptions of a system in the two domains.

H(z) is called the system function.

Linear constant-coefficient difference equations:

y [n] = �P

N

k=1aky [n � k ] +

PM

k=0bkx [n � k ] gives

H(z) =P

M

k=0bk z�k

1+P

N

k=1ak z�k

.

If ak = 0 for k = 1..N; all-zero system/ FIR system / MA system.

If bk = 0 for k = 1..M; all-pole system/ IIR system.

General form; pole-zero system/ IIR system.

The invers system function: H�1(z) = HI(z) = 1/H(z).Poles becomes zeros and v.v.

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 5 / 11

The system function of a LTI system Properties

Properties

Causuality

I A causal system has h[n] = 0 for n < 0, i.e. H(z) =P1

n=0h[n]

I I.e. no positive powers of z ) ROC given as |z| > r .

Stability

I LTI system stable if X1

n=�1|h[n]| < 1

I Equivalent condition:

|H(z)| =X1

n=�1|h[n] z

�n| < 1

for |z| = 1 ) ROC must include the unite circle.

Causal and stable:

I All poles inside the unit circle, and ROC extending outward

including the unit circle!

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 6 / 11

The system function of a LTI system System function algebra

System function algebra

h[n] = h1[n] + h2[n] ) H(z) = H1(z) + H2(z).

h[n] = h1[n] ⇤ h2[n] ) H(z) = H1(z)H2(z).

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 7 / 11

The system function of a LTI system Pole-zero locations

Pole-zero location and time-domain behavior

Causal first-order system:

y [n] = ay [n � 1] + bx [n], |a| < 1.

System function:

H(z) =Y (z)

X (z)=

b

1 � az�1=

bz

z � a.

Since system is causal ) h[n] = banu[n].

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 8 / 11

The system function of a LTI system Pole-zero locations

Pole-zero location and time-domain behavior ...

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 9 / 11

The system function of a LTI system Pole-zero locations

Pole-zero location and time-domain behavior

Causal second-order system:

y [n] = �a1y [n � 1]� a2y [n � 2] + b0x [n] + b1x [n � 1].

System function:

H(z) =Y (z)

X (z)=

b0 + b1z�1

1 + a1z�1 + a2z�2=

z(b0z + b1)

z2 + a1z + a2

.

I ) zeros for z1 = 0 and z2 = b1/b0.

I ) poles given as p1,2 = � a1

2±

pa2

1�4a2

2

1 Real and distinct; a2

1 > 4a2.

2 Real and equal; a2

1 = 4a2.

3 Complex conjugate; a2

1 > 4a2.

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 10 / 11

The system function of a LTI system Pole-zero locations

Pole-zero location and time-domain behavior ...

Impulse response, pair of complex conjugate poles.

AA, IN3190/4190 (Ifi/UiO) Lecture, week 03 Sept. 2019 11 / 11