Chapter # 3 : The z-transform and Its Application to the ...site.iugaza.edu.ps/musbahshaat/files/chapter-3_B_handout.pdfEELE 4310: Digital Signal Processing (DSP) Chapter # 3 : The

EELE 4310: Digital Signal Processing (DSP)

Chapter # 3 : The z-transform and Its Application to the

Analysis of LTI Systems(Part Two)

Spring, 2012/2013

EELE 4310: Digital Signal Processing (DSP) - Ch.3 Dr. Musbah Shaat 1 / 15

Outline

1 Rational z-Transform


Rational z-TransformPoles and Zeros ... 1

An important family of z-transforms are those for which X (z) is arational function (a ratio of two polynomials in z−1 (or z)).

The zeros of a z-transform X (z) are the values of z for whichX (z) = 0.

The poles of a z-transform X (z) are the values of z for whichX (z) =∞.

If X (z) is a rational function,

X (z) = B(z)A(z)

= b0+b1z−1+···+bMz−M

a0+a1z−1+···+aNz−N =∑M

k=0 bkz−k∑N

k=0 bkz−k

If a0 6= 0 and b0 6= 0, the negative powers can be avoided as follows:

X (z) = B(z)A(z)

= b0z−M

a0z−N

zM+b1b0

zM−1+···+ bMb0

zN+a1a0zN−1+···+ aN

a0



B(z) and A(z) are polynomials in z ⇒ can be expressed in factoredform as

X (z) = B(z)A(z) = b0

a0z−M+N (z−z1)(z−z2)···(z−zM)

(z−p1)(z−p2)···(z−pN)

X (z) = GzN−M∏M

k=1(z−zk )∏Nk=1(z−pk )

where G ≡ b0a0

.

X (z) has M finite zeros at z = z1, z2, · · · , zM (the roots of thenumerator polynomial).

X (z) has N finite poles at z = p1, p2, · · · , pN (the roots of thedenominator polynomial).

X (z) has |N −M| zeros (if N > M) or poles (if N < M) at theorigin z = 0.

X (z) has exactly the same number of poles and zeros (countingthose at zero and infinity).



A pole-zero plot (or pattern)

Its a graphical representation of X (z).

The cross (x) represents a location of pole while the circle (o)represents the location of the zeros.

The multiplicity of multiple-order zeros or poles is indicated by anumber close to the corresponding circle or cross.

The ROC of a z-transform should not contain any poles.

Ex. 3.3.1: Determine the pole-zero plot for the signalx(n) = anu(n), a > 0

X (z) = 11−az−1 = z

z−a , ROC:|Z | > aThus, has one zero at z = 0 and one poleat p1 = a



Ex. 3.3.2: Determine the pole-zero plot for the signalx(n) = an[u(n)− u(n −M)], a > 0

From the definition of the z-transform,X (z) =

∑M−1n=0 (az−1)n

= 1−(az−1)M

1−az−1

= zM−aM

zM−1(z−a)

The equation ZM = aM has M roots at

zk = ae j2πk/M , k = 0, 1, · · · ,M − 1

The zero (z0 = a) cancels the pole at(z = a)

⇒ X (z) =(z−z1)(z−z2)···(z−zM−1)

zM−1

which has M − 1 zeros and M − 1 poles.



Ex. 3.3.3: Determine the z-transform and the signal thatcorresponds to the following pole-zero plot

There are two zeros (M = 2) atz1 = 0, z2 = r cosw0 and two poles(N = 2) at p1 = re jw0 , p2 = re−jw0

Thus

X (z) = G (z−z1)(z−z2)(z−p1)(z−p2)

= G z(z−r cosw0)

(z−re jw0 )(z−re−jw0 ), ROC:|Z | > r

After some manipulations, we obtain

X (z) = G 1−rz−1 cosw01−2rz−1 cosw0+r2z−2 ,ROC:|Z | > r

⇒ x(n) = G (rn cosw0n)u(n)



If a polynomial has real coefficients, its root are either real or occur incomplex-conjugate pair.

|X (z)|, the magnitude of the z-transform X (z), is a two dimensionalfunction describe a surface (Always positive and real).

Ex. For X (z) = z−1−z−2

1−1.2732z−1+0.81z−2


Rational z-TransformPole Location and Time-Domain Behavior for Causal Signals ... 1

Consider only real and causal signals.

The characteristic behavior of causal signals depends on whether thepoles of the transform are contained in the region |z | < 1 (unitcircle) or in the region |z | > 1, or on the circle |z | = 1.

Case #1: Real signal with one pole

This pole has to be real and the only such a signal is the realexponential.

x(n) = anu(n)z↔X (z) = 1

1−az−1 , ROC: |z | > |a|Has one zero at z1 = 0 and one pole at p1 = a on the real axis.





Case #1: Real signal with one pole (summary)

The pole is inside the unit circle ⇒ the signal is decaying.

The pole is on the unit circle ⇒ the signal is fixed.

The pole is outside the unit circle ⇒ the signal is growing(unbounded and should be avoided in real systems).

The pole is negative ⇒ the signal is alternating in sign.

Case #2: Real signal with a double real pole

nanu(n)↔ az−1

(1−az−1)2,ROC: |z | > |a|

A double real pole (also its the case for complex-conjugate multiplepoles) on the unit circle results in unbounded signal.





Case #3: Pair of a Complex-conjugate Poles

Results in an exponentially weighted sinusoidal signal.

rn(cosω0n)u(n)z↔ 1−rz−1 cosω0

1−2rz−1 cosω0+r2z−2 ,ROC: |z | > |r |

The amplitude is growing if r > 1, constant if r = 1 and decaying ifr < 1.





Summary

Causal real signals with simple real poles or simplecomplex-conjugate pair of poles are always bounded if the poles areinside or on the time circle.

Signals with multiple poles (real or complex-conjugate) on the unitcircle are unbounded.

The signal with a pole near the origin decays more rapidly than onehas a pole near (but inside) the unit circle.

The time behavior of a signal depends strongly on location of itspoles relative to the unit circle.

What is applied to causal signal is also valid for causal LTI system(the impulse response is unbounded if it has a pole outside the unitcircle and hence the system is unstable).


Documents

Chapter # 3 : The z-transform and Its Application to the ...site.iugaza.edu.ps/musbahshaat/files/chapter-3_B_handout.pdfEELE 4310: Digital Signal Processing (DSP) Chapter # 3 : The