Z Transform

October 25, 2017 | Author: crux_123 | Category: Algebra, Electrical Engineering, Analysis, Signal Processing, Applied Mathematics
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Z Transform...

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Ch3.4: Z-Transform Information source and input transducer

Source Coding

• Ch3.4: z-Transform

Channel Coding

Modulator

Questions to be answered: – z-Transform: A general version of DTFT – Region of Convergence: Where the transform exists.

Channel

– Inverse z-Transform: Back to timedomain – LTI System in Transform Domain: A general version of the frequency response function Information sink and output transducer

Elec3100 Chapter 3.4

Source Decoding

Channel Decoding

Demodulator (Matched Filter) 1

Ch3.4: z-Transform • Definition of z-Transform • Region of Convergence • Inverse z-Transform • Z-Transform Properties • LTI System in Transform Domain

Elec3100 Chapter 3.4

2

Generalizing DTFT •

The DTFT provides a frequency-domain representation of discretetime signals and LTI discrete-time systems.



However, because of the convergence condition, in many cases, the DTFT of a sequence may not exist.



As a result, it is not possible to make use of such frequency-domain characterization in these cases.



Question: Can we generalize DTFT?

Elec3100 Chapter 3.4

3

z-Transform •

The z-transform X(z) of a sequence x[n] is defined as

where z is a continuous complex variable. Generally, we can where r = express the complex variable z in polar form as |z| > 0 is the magnitude and is the angle of z. •

z-transform gives ∑



In particular, when r = 1, then becomes ∑ |

Elec3100 Chapter 3.4

and the above expression

4

z-Transform • The relationship between and can also be illustrated in the z-plane (a real-imaginary plane for complex number).

The z-transform evaluated on the unit circle (as ω varies from 0 to 2π) corresponds to the DTFT of x[n].

Elec3100 Chapter 3.4

5

z-Transform • Main advantages of using z-transform over DTFT: – z-transform can encompass a broader class of signals than DTFT – Recall that a sufficient condition for convergence of the DTFT is: |

|



– Example: The DTFT of 0.5 1 does not exist since it is unbounded in the negative direction: -5 -4 -3 -2 -1 0 -8 -16

-4 -2

n

-32 Elec3100 Chapter 3.4

6

z-Transform •

On the other hand, the z-transform exists if ∞ We can choose region of convergence (ROC) for z such that the z-transform converges.



Example: z-transform of



Proof:

∑ ∑ ∑

0.5

0.5 0.5 0.5

1 exists if 0.5

1

1

. .

Elec3100 Chapter 3.4

.

7

z-Transform • Other advantages of using z-transform: – More convenient notation than DTFT when dealing with analytical problems

– Convenient block diagram representation for implementation of practical systems, Eg., a unit delay system is expressed as: x[n]

z-1

y[ n] = x[n − 1]

– Useful for determining and solving difference equations for discrete-time systems

Elec3100 Chapter 3.4

8

Application: Filter Design •

The convolution sum description of an LTI discrete-time system with ∑ . an impulse response is given by



The output to an input ∑ ∑ ∑



For any given input sequence , the output will be indicating that is an eigen-function for any LTI system.



How can we utilize this property?

Elec3100 Chapter 3.4

is given by

,

9

Use of z-Transform: Filter Design • Can you figure out what kind of filter has the following ztransform and magnitude spectrum?

Elec3100 Chapter 3.4

10

Ch3.4: z-Transform • Definition of z-Transform • Region of Convergence • Inverse z-Transform • Z-Transform Properties • LTI System in Transform Domain

Elec3100 Chapter 3.4

11

Region of Convergence (ROC) •

Since z can be any point on the z-plane, generally there exists some z which makes X(z) not converge |



|

→∞

The set of z values for which X(z) converges ∞ is called the region of convergence (ROC).



The ROC must be specified along with X(z) in order for the ztransform to be completely defined.

Elec3100 Chapter 3.4

12

Poles and Zeros in z-Transform •

In many applications, X(z) can be expressed as a rational function: M

M

P( z ) X ( z) = = Q( z )

∑b z

k

∑a z

k

k

k =0 N

k

b ( z − c1 )(z − c2 )L (z − cM ) b0 = 0 = a0 ( z − d1 )( z − d 2 )L (z − d N ) a0

k =0

∏ (z − c ) k

k =1 N

∏ (z − d )

a0 ≠ 0, b0 ≠ 0

k

k =1



P(z) and Q(z) are numerator and denominator polynomials in z



The values of z for which X(z) = 0, i.e. roots of P(z), are called zeros of X(z)



The values of z for which X(z) = ∞, i.e. roots of Q(z), are called poles of X(z)

Elec3100 Chapter 3.4

13

Properties of the ROC •

The ROC is ring or disk in the z-plane centered at the origin, i.e. 0 | | ∞



The DTFT of converges absolutely iff the ROC of the ztransform includes the unit circle.



The ROC cannot contain any poles.



Example: The z-transform is given by .

,

of the sequence

0.6

0.6

Here, the ROC is just outside the circle going through the point z 0.6.

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14

ROC of Finite-Length Sequence • If is a finite-duration sequence, i.e. a sequence that is ∞, then zero except over a finite interval ∞ the ROC is the entire z-plane, except z = 0. • Example: Consider a finite-length sequence defined for −M ≤ n ≤ N, where M and N are non-negative integers and | ∞ | • Its z-transform is given by

• Note: has M poles at ∞ and N poles at 0. • Thus, the ROC is the entire z-plane except possibly at and/or at ∞ Elec3100 Chapter 3.4

N1

N2

n

0

15

ROC of Right-Sided Sequence •

A right-sided sequence is a sequence with nonzero sample values for . A right-sided sequence with nonzero sample values for 0 is called a causal sequence.



Consider a causal sequence



converges exterior to a circle , including the point ∞. On the other hand, a right-sided sequence with nonzero sample values only for with M nonnegative has a ztransform with M poles at ∞. is exterior to a circle , excluding the point The ROC of ∞.





Elec3100 Chapter 3.4

, whose z-transform is given by

16

Example 5.1 Right-Sided Sequence • Determine the z-transform of



converges if ∑ 1⇒

|

|

∞. This requires

| |, and gives

Note: If |a| < 1, the ROC contains the unit circle and hence the DTFT of this sequence x[n] exists. Shaded area = ROC



Remark: u[n] has z-transform, but not DTFT

Elec3100 Chapter 3.4

x denotes as the pole of X(z) o denotes as the zero of X(z) 17

ROC of Left-Sided Sequence •



A left-sided sequence is a sequence with nonzero sample values for . A left-sided sequence with nonzero sample values for 0 is called an anticausal sequence. Consider an anticausal sequence , whose z-transform is given by



It can be shown that converges interior to a circle , including the point 0.



On the other hand, a left-sided sequence with nonzero sample values only for with N nonnegative has a z-transform with N poles at 0. is interior to a circle , excluding the point The ROC of 0.



Elec3100 Chapter 3.4

18

Example 5.2 Left-Sided Sequence •

Determine the z-transform of another signal

1

1 •

converges if ∑ 1⇒

|

|

∞. This requires

| |, and gives

Note: If |a| < 1, the ROC does not contain the unit circle and the DTFT of this sequence x[n] does not exist. x denotes as the pole of X(z) o denotes as the zero of X(z)

Elec3100 Chapter 3.4

19

Observation from Example 5.1 and 5.2 •

Different signals can give same z-transform, although the ROCs will differ

x[n] = a n u[n]

x[n] = −a nu[−n − 1]

1 X ( z) = −1 if z > a 1 − az

X ( z) =

( ) exists if a < 1 ( ) does not exist if a > 1

X e jω X e jω Elec3100 Chapter 3.4

1 if z < a −1 1 − az

( ) exists if a > 1 ( ) does not exist if

X e jω X e jω

a

if

1 2

n

1 ⎛ −1 ⎞ Z x2 [n] = ⎜ ⎟ u[n] ←⎯→ X 2 ( z) = 1 + 13 z −1 ⎝ 3 ⎠

X ( z) =

1

1

+

1 − 12 z −1 1 + 13 z −1

(

)

1 2 z z − 12 = z − 12 z + 13

(

Elec3100 Chapter 3.4

)(

)

if

if

z >

1 3

1⎞ ⎛ 1⎞ 1 ⎛ ⎜ z > ⎟ I⎜ z > ⎟ ⇒ z > 3⎠ ⎝ 2⎠ 2 ⎝ The overall ROC is the region of overlap of the ROC for X1(z) and the ROC for X2(z); unless there is pole-zero cancellation. 23

Table 3.4.1: Common z-transform pairs

Elec3100 Chapter 3.4

24

Ch3.4: z-Transform • Definition of z-Transform • Region of Convergence • Inverse z-Transform • Z-Transform Properties • LTI System in Transform Domain

Elec3100 Chapter 3.4

25

The Inverse z-Transform •

There are 4 commonly used techniques to evaluate the inverse ztransform. They are – 1. Inspection Method – 2. Partial Fraction Expansion – 3. Power Series Expansion – 4. Cauchy Integral Theorem • This is the formal inverse z-transform expression based on Contour integral (an integration techniques together with right-hand rule) onto z-plane. • We will not cover this method in this course.

Elec3100 Chapter 3.4

26

Inverse z-transform: Inspection Method •

Inspection Method: Familiar with, or recognizing “by inspection”, certain transform pairs.



Example: – Find the inverse z-transform of

– This gives

X ( z) =

1

z X ( z) = z − ( 12 )

z>

1 2

1 z> 2

1 − (12 )z −1

– By making use of the transform pair in Table 3.4.1, n

⎛1⎞ x[n] = ⎜ ⎟ u[n] ⎝ 2⎠ Elec3100 Chapter 3.4

27

Partial Fraction Expansion Method •

A rational



If

can be expressed as ∑ ∑

, then

can be re-expressed as

where the degree of

is less than N. The rational function

is

called a proper fraction.

Elec3100 Chapter 3.4

28

Inverse Transform: Partial-Fraction Expansion •

Simple Poles: In most cases, the rational z-transform of interest is a proper fraction with simple poles. Let the poles of be at ,1 . A partial-fraction expansion of is then of the form 1



The constants



Each term of the sum in partial-fraction expansion has an ROC given by and, thus has an inverse transform of the form . Therefore, the inverse transform of G(z) is given by ∑ .



Reminder: There are cases with multiple poles.

Elec3100 Chapter 3.4

are called the residues and given by 1 |

29

Inverse Transform: Partial-Fraction Expansion •

Example 5.5: Determine whose z-transform is given by 2 1 2 1 0.2 1 0.6 0.2 0.6



Solution: A partial-fraction expansion of 1



0.2

1

is of the form 0.6

Given

we have

Elec3100 Chapter 3.4

. .

. .

. Thus,

30

Inverse Transform by Long Division •

The z-transform G(z) of a sequence can be expanded in a power series in . In the series expansion, the coefficient multiplying the term is then the n-th sample of .



For a rational z-transform expressed as a ratio of polynomials in the power series expansion can be obtained by long division.



Example – Consider 1



2 0.12

Long division of the numerator by the denominator yields 1



1 0.4

,

1.6

0.52

0.4

1 1.6 0.52 0.4 As a result: result with that of Example 5.5.

Elec3100 Chapter 3.4

0.2224



0.2224 … . Compare this 31

Ch3.4: z-Transform • Definition of z-Transform • Region of Convergence • Inverse z-Transform • Z-Transform Properties • LTI System in Transform Domain

Elec3100 Chapter 3.4

32

z-Transform Properties •

Linearity Z Z If x1[n]←⎯→ X 1 ( z ), ROC = Rx1 and x2 [n ]←⎯→ X 2 ( z ), ROC = Rx Z then ax1 [n] + bx2 [n]←⎯→ aX 1 ( z ) + bX 2 ( z ), ROC contains Rx I Rx 1

• •

Example: Consider the two-sided sequence 1. and Two sequences , | | and transforms



With linearity property, we arrives at

• •

1 ROC is given by the overlap regions of | |? Question: What happens if

Elec3100 Chapter 3.4

1

2

2

1 have z, | | 1 1 | | and

| |.

33

z-Transform Properties •

Time Shifting Z x[n − n0 ]←⎯→ z − n0 X ( z ),

ROC = Rx

ROC is unchanged, except for the possible addition or deletion of −n the points z = 0 or z = ∞. It is because the factor z 0 can alter the number of poles at z = 0 or z =∞. •

Multiplication by an Exponential Sequence Z z 0n x[n]←⎯→ X ( z z0 ),

ROC = z 0 Rx

where ROC | | denotes that the ROC is scaled by | |; ie. if is the set of values of z such that rR < |z| < rL, then the new ROC is the set of values of z such that |z0|rR < |z| < |z0| rL

Elec3100 Chapter 3.4

34

z-Transform Properties •

Differentiation

dX ( z ) , nx[n]←⎯→ − z dz Z

ROC = Rx

ROC is unchanged except for the possible addition or deletion of z = 0. •

Conjugation of a Complex Sequence Z x ∗ [n ]←⎯→ X ∗ ( z ∗ ),



Z Time Reversal x[− n ]←⎯→ X (1 z ),

ROC = Rx ROC = 1 Rx

where ROC 1/ denotes that the ROC is inverted; i.e. if is the set of values of z such that rR < |z| < rL, then the ROC is the set of values of z such that 1/rL < |z| < 1/rR

Elec3100 Chapter 3.4

35

Example 5.6: Time-Reversal • Find the z-transform of x[ n] = a − nu[− n] • Solution:

1 , a u[ n] ←⎯→ z>a −1 1 − az 1 − a −1 z −1 Z −n = a u[ − n] ←⎯→ , −1 −1 1 − az 1 − a z n

Elec3100 Chapter 3.4

Z

z < a −1

36

z-Transform Properties •

Convolution of Sequences Z Z If x1 [n]←⎯→ X 1 ( z ), ROC = Rx1 and x2 [n ]←⎯→ X 2 ( z ), Z then x1 [n]∗ x2 [n]←⎯→ X 1 ( z ) X 2 ( z ),

ROC = Rx2

ROC contains Rx1 I Rx2

The resulting ROC contains at least the intersection of Rx1 and Rx2 – If there is no pole-zero cancellation in the linear combination, the ROC is exactly equal to Rx1 I Rx2 . – If there is pole-zero cancellation in the linear combination, the ROC may be larger. •

Initial-Value Theorem If x[n] is zero for n < 0 (i.e. if x[n] is causal), then x[0] = lim X ( z ) z →∞

Elec3100 Chapter 3.4

37

Example 5.7: Convolution •

Find the convolution sum of 1.

and

, where



Solution: – Method 1: direct convolution Z −1 – Method 2: Y ( z ) = X 1 ( z ) X 2 ( z ) ⎯⎯→ y[ n] = x1 [n ]∗ x2 [n ] 1 1 X 1 ( z) = , z > a and X ( z ) = , z >1 2 −1 −1 1 − az 1− z This gives Y ( z) =

1

1 −1

−1

z >1

,

1 − az 1 − z 1 ⎛ 1 a ⎞ = − ⎜⎜ ⎟⎟ − 1 − 1 1− a ⎝1− z 1 − az ⎠

y[n] = Elec3100 Chapter 3.4

(

)

1 u[n] − a n+1u[n] 1− a

38

Table 3.4.2: z-transform properties

Elec3100 Chapter 3.4

39

Ch3.4: z-Transform • Definition of z-Transform • Region of Convergence • Inverse z-Transform • Z-Transform Properties • LTI System in Transform Domain

Elec3100 Chapter 3.4

40

LTI Systems in the Transform Domain •

An LTI discrete-time system is completely characterized in the timedomain by its impulse response sequence .



Each LTI discrete-time representation.



Such transform-domain representations provide additional insight into the behavior of such systems. It is easier to design and implement these systems in the transform-domain for certain applications.



We consider now the use of the DTFT and the z-transform in developing the transform domain representation of an LTI system.

Elec3100 Chapter 3.4

system

has

a

transform-domain

41

The Transfer Function •

Transfer Function: A generalization of the frequency response function.



The convolution sum description of an LTI discrete-time system with ∑ . an impulse response is given by



Taking the z-transform of both sides, we get ∑ ∑ ∑ ∑ ∑ ∑ ∑



∑ ∑

/ is the z-transform of Here, transfer function or the system function.

Elec3100 Chapter 3.4

and is called the 42

FIR vs. IIR Filter •

A finite-duration impulse response (FIR) filter has a impulse response ⎧bn 0 ≤ n ≤ M − 1

h ( n) = ⎨ ⎩0



else

The difference equation representation is

y[ n] = b0 x[ n] + b1x[ n − 1] + L + bM −1 x[ n − M + 1] which is a linear convolution of finite support. • •

IIR filters are characterized by infinite-duration impulse response. The difference equation representation of an IIR filter is expressed as

y[n] =

Elec3100 Chapter 3.4

M −1

N −1

m=0

m =1

∑ bm x[n − m] − ∑ am y[n − m] 43

LTI Systems in the Transform Domain • Consider an LTI discrete-time system characterized by linear constant coefficient difference equations of the form: ∑ ∑ . • Applying the z-transform to both sides of the difference equation, we arrive at ∑ ∑ where and denote the z-transforms of and with associated ROCs, respectively. • A more convenient form of the z-domain representation of the difference equation is given by Elec3100 Chapter 3.4

44

LTI Systems in the Transform Domain •

Thus, ∑ ∑ ∑ ∑ ∏ ∏

• , ,…, are finite zeros, and , , … , are finite poles. • If , there are additional zeros at 0. • If , there are additional poles at 0. • For a causal digital filter, the impulse response is a causal sequence. The ROC of the causal transfer function is exterior to a circle going through the pole furthest from the origin with max | | Elec3100 Chapter 3.4

45

Example 5.8: Moving Average Filter •



Consider the M-point moving-average FIR filter with an impulse ,0 1 response, 0, otherwise Its transfer function is given by 1

1

1 1

• • •

1

The transfer function has M zeros on the unit circle at / ,0 1. There is an 1)th order pole at 0 and a single pole at 1. The ROC is the entire z-plane except 0

Elec3100 Chapter 3.4

46

Example 5.8: Moving Average Filter

Elec3100 Chapter 3.4

47

Frequency Response from Transfer Function •

If the ROC of the transfer function H(z) includes the unit circle, then the frequency response of the LTI digital filter can be obtained | simply as follows:



For a stable rational transfer function in the form ∏ ∏ the frequency response is given by ∏ ∏



It is convenient to visualize the contributions of the zero factor and the pole factor from the factored form of the frequency response.

Elec3100 Chapter 3.4

48

Frequency Response from Transfer Function •



The magnitude response function is given by: ∏ | | | | || | ∏ | ∏ | | | | ∏ | |

| |

The phase response is of the form arg arg



arg

How can we use this?

Elec3100 Chapter 3.4

49

Geometric Interpretation •

The factored form of frequency response ∏ ∏ is convenient to develop a geometric interpretation of the frequency response computation from the pole-zero plot as ω varies from 0 to 2π on the unit circle.



A typical factor in the factored form of the frequency response is where is a zero or pole. given by



It can be observed from the right figure that represents a vector starting at z

and ending on the unit circle .

Elec3100 Chapter 3.4

50

Geometric Interpretation •

As indicated by |

|

|



|∏

| |

| |

, the magnitude

response at a specific frequency is given by the product of the magnitudes of all zero vectors divided by the product of the magnitudes of all pole vectors. • • •

Question: When can we obtain the smallest magnitude? A zero (pole) vector has the smallest magnitude when To attenuate signal components in a specified frequency range, we need to place zeros very close to or on the unit circle in this range. To emphasize signal components in a specified frequency range, we need to place poles very close to or on the unit circle in this range.

Elec3100 Chapter 3.4

51

Can we build an ideal bandpass filter?

Elec3100 Chapter 3.4

52

Can we build an ideal bandpass filter? •

The z-transform of an FIR impulse response can be expressed as a simple polynomial P(z) of degree L - 1 where L is the number of nonzero taps of the filter.



The fundamental theorem of algebra states that such a polynomial has at most L - 1 roots. Thus, the frequency response of an FIR filter can never be identically zero over a frequency interval since, if it were, its z-transform would have an infinite number of roots.



The argument can be easily extended to rational transfer functions, confirming the impossibility of a realizable filter whose characteristic is piecewise perfectly flat.



Then, how can we design a bandpass filter? (To be continued…)

Elec3100 Chapter 3.4

53

Stability Condition •

A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., S ∑ | | ∞.



Thus, an FIR digital filter with bounded impulse response is always stable.



Question: What is the condition in terms of pole locations of ?



The ROC of the for the impulse response sequence is defined by values of where ∑ | | ∞. Thus, if the digital filter is stable, the ROC includes the unit circle.

Elec3100 Chapter 3.4

54

Stability Condition in Terms of Pole Position •

Consider the causal IIR digital filter with a rational transfer function H(z) given by



• •

∑ ∑

. Its impulse response is right-sided

sequence. The ROC of the is exterior to a circle going through the pole furthest from z 0. But stability requires ROC includes the unit cycle. Conclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle. The stability region (shown shaded) in the z-plane is shown below

Elec3100 Chapter 3.4

z-Transform

55

Example 5.9: Stability Condition IIR Filter

Elec3100 Chapter 3.4

56

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