Digital Signal Processing Applications(310253) UNIT-III Z-Transform Prof. Vina M. Lomte RMDSSOE,Warje 3/18/2016

Digital Signal Processing Applications Teaching Scheme: Examination Scheme: Theory: 3 Hrs/Week In Semester Assessment: 30 Marks End Semester Assessment: 70 Marks 3/18/2016

Course Objectives: · Study and understanding of representation of signals and systems. · To learn and understand different Transforms for Digital Signal Processing · Design and analysis of Discrete Time signals and systems · To Generate foundation for understanding of DSP and its applications like audio, Image, telecommunication and real world 3/18/2016

Syllabus Definition of Z-Transform, ZT and FT, ROC, ZT properties, pole-zero plot, Inverse Z-Transform, Methods, System function H(Z), Analysis of DT LTI systems in Z-domain: DT system representation in time and Z domain. Relationship of FT and ZT 3/18/2016

Teaching Plan Sr. No.TopicLectures Required References 01Definition of Z-Transform, ZT and FT Steven W. Smith, “The Scientist and Engineer's Guide to Digital Signal Processing” 2. P. Ramesh Babu, Fourth Edition,” Digital Signal Processing” 02ROC, ZT properties & Examples03(1 + 2 Extra ) 03Pole-zero plot & Examples03(1 + 2 Extra ) 04Inverse Z-Transform, Methods, System function H(Z) & Examples 02(1 + 1 Extra ) 05Analysis of DT LTI systems in Z-domain: DT system representation in time and Z domain 02(1 + 1 Extra ) 06Relationship of FT and ZT01 3/18/2016

Session 1 Introduction Why z-Transform? Definition of Z-Transform, Relationship ZT and FT 3/18/2016

What is ZT What is Z ? It is Z= x + iy 3/18/2016 Real

Why z-Transform? It is very simple method for analyzing system(by ROC properties) ex. LTI system A generalization of Fourier transform Why generalize it? FT does not converge on all sequence Notation good for analysis Bring the power of complex variable theory deal with the discrete-time signals and systems The z-transform is a very important tool in describing and analyzing digital systems. It offers the techniques for digital filter design and frequency analysis of digital signals.

A signal can be characterized with its Z-transform (poles, final value …) In an LTI system, Z-transform of Y(z) is the multiplication of Z-transform of U(z) and the transfer function The LTI system can be characterized by the transfer function, or the Z-transform of the unit impulse response

Definition The z-transform of sequence x(n) is defined by Let z = e  j . Fourier Transform Frequency Domain Time Domain Convert

Relationship Between FT and ZT 3/18/2016 The following Eq.(1) and (2) are FT and ZT, respectively. Replacing Z with, ZT will become FT

Session 2 ROC ZT properties, pole-zero plot 3/18/2016

Definition of ROC The region in which Z is valid Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|< , is called the region of convergence. ROC is centered on origin and consists of a set of rings.

Example: Region of Convergence Re Im ROC is annual ring centered an on the origin. r

Stable Systems A stable system requires that its Fourier transform is uniformly convergent. Re Im 1 Fact: Fourier transform is to evaluate z- transform on a unit circle. A stable system requires the ROC of z- transform to include the unit circle.

Example: A right sided Sequence n x(n)x(n)... All positive values A right hand sequence x(n) is one for which x(n)>=0 for all n =0 the resulting sequence is causal sequence. For such type of sequence ROC is entire z-plane except at z=0

Example: A right sided Sequence For convergence of X(z), we require that

a aa Example: A right sided Sequence ROC for x(n)=a n u(n) Re Im 1 a aa Re Im 1 Which one is stable? ROC includes unit circle

Example: A left sided Sequence n x(n)x(n)... Negative Values A left hand sequence x(n) is one for which x(n)>=0 for all n<no where no is +ve or –ve but finite. If n0<=0 the resulting sequence is anticausal sequence. For such type of sequence ROC is entire z-plane except at z=∞

Example: A left sided Sequence For convergence of X(z), we require that

a aa Example: A left sided Sequence ROC for x(n)=  a n u(  n  1) Re Im 1 a aa Re Im 1 Which one is stable?

Properties of ROC A ring or disk in the z-plane centered at the origin. The Fourier Transform of x(n) is converge absolutely iff the ROC includes the unit circle. The ROC cannot include any poles Finite Duration Sequences: The ROC is the entire z-plane except possibly z=0 or z= . Right sided sequences: The ROC extends outward from the outermost finite pole in X(z) to z= . Left sided sequences: The ROC extends inward from the innermost nonzero pole in X(z) to z=0.

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 if you need stability then the ROC must contain the unit circle.  If you need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence.  If you need an anticausal system then the ROC must contain the origin and the system function will be a left- sided sequence.  If you need both, stability and causality, all the poles of the system function must be inside the unit circle.

Represent z-transform as a Rational Function where P(z) and Q(z) are polynomials in z. Zeros: The values of z’s such that X(z) = 0 Poles: The values of z’s such that X(z) =  Pole and Zeros

Example: A right sided Sequence Re Im a ROC is bounded by the pole and is the exterior of a circle.

Example: A left sided Sequence Re Im a ROC is bounded by the pole and is the interior of a circle.

Example: Sum of Two Right Sided Sequences Re Im 1/2  1/3 1/12 ROC is bounded by poles and is the exterior of a circle. ROC does not include any pole.

Example: A Two Sided Sequence Re Im 1/2  1/3 1/12 ROC is bounded by poles and is a ring. ROC does not include any pole.

Example: A Finite Sequence Re Im ROC: 0 < z <  ROC does not include any pole. N-1 poles N-1 zeros Always Stable

BIBO Stability Bounded Input Bounded Output Stability  If the input is bounded, we want the output is bounded too  For limited input sequence its output should respectively limited 3/18/2016

Z-Transform Pairs Sequencez-TransformROC All z All z except 0 (if m>0) or  (if m<0)

Z-Transform Pairs Sequencez-TransformROC

Signal TypeROC Finite-Duration Signals Infinite-Duration Signals Causal Anticausal Two-sided Causa l Anticausal Two-sided Entire z-plane Except z = 0 Entire z-plane Except z = infinity Entire z-plane Except z = 0 And z = infinity |z| < r 1 |z| > r 2 r 2 < |z| < r 1

Some Common z-Transform Pairs Sequence Transform ROC 1.  [n] 1 all z 2. u[n] z/(z-1) |z|>1 3. -u[-n-1] z/(z-1) |z|<1 4.  [n-m] z -m all z except 0 if m>0 or ฅ  if m 0 or ฅ  if m<0 5. a n u[n] z/(z-a) |z|>|a| 6. -a n u[-n-1] z/(z-a) |z|<|a| 7. na n u[n] az/(z-a) 2 |z|>|a| 8. -na n u[-n-1] az/(z-a) 2 |z|<|a| 9. [cos  0 n]u[n] (z 2 -[cos  0 ]z)/(z 2 -[2cos  0 ]z+1) |z|>1 10. [sin  0 n]u[n] [sin  0 ]z)/(z 2 -[2cos  0 ]z+1) |z|>1 11. [r n cos  0 n]u[n] (z 2 -[rcos  0 ]z)/(z 2 -[2rcos  0 ]z+r 2 ) |z|>r 12. [r n sin  0 n]u[n] [rsin  0 ]z)/(z 2 -[2rcos  0 ]z+r 2 ) |z|>r 13. a n u[n] - a n u[n-N] (z N -a N )/z N-1 (z-a) |z|>0

Z-Transform Properties: Notation Linearity – Note that the ROC of combined sequence may be larger than either ROC – This would happen if some pole/zero cancellation occurs – Example: 1.Linearity

Proof: According to defination of ZT Here x(n)=a1x1(n) + a2x2(n) Writing two terms separately we get, Here a1 & a2 are constants se we can take it outside the summation sign By comparing eqn 1 & 3 we get X(z) =a1X1(z)+a2X2(z) Hence proved ROC : the combined ROC is overlapped or intersection of the individual ROCs of X1(z) & X2(z) 3/18/2016

2. Time Shifting Here n o is an integer – If positive the sequence is shifted right – If negative the sequence is shifted left The ROC can change the new term may – Add or remove poles at z=0 or z=  Example Here x(n-no) indicates that the sequence is shifted in the time domain by (-no) samples corresponds to multiplication by in the frequency domain

Proof Statement : if X(n) z Z(z) Then x(n-k) ) z Z -k X(z) Then Z{x(n-k)} = Now Z -n can be written as Z- (n-k) Z(x(n-k) = Since the limits of summation are in terms of n we can take Z-k outside of the summation Z(x(n-k) = Now put n-k=m on RHS the limit will change as follows At n=-∞, -∞-k = m m=-∞ At n= +∞, ∞-k=m, m= ∞ Z{x(n-k)} = /18/2016

Compare eqn 1 & 4 Z{x(n-k) = Z -k X(z) hence X(n) z Z(z) Similarly it can be shown that x(n-k) z Z -k X(z) = x(n-k) z z +k X(z) Here x(n-k) indicates that the sequence is shifted in time domain by (-k) samples corresponding to multiplication by z -k in the frequency domain ROC of z -k is same as that X(Z) except z=0 if k>0 and z=∞ if k>0 3/18/2016

Example Find ZT of x( n) = (n-k) That means (n) Z 1 x(n-k) ) Z Z -1 X(z) hence (n-k) Z Z -k. 1 3/18/2016

3. Scaling in Z-domain (Multiplication by Exponential) ROC is scaled by |z o | All pole/zero locations are scaled If z o is a positive real number: z-plane shrinks or expands If z o is a complex number with unit magnitude it rotates Example: We know the z-transform pair Let’s find the z-transform of

4. Differentiation Example: We want the inverse z-transform of Let’s differentiate to obtain rational expression Making use of z-transform properties and ROC Multiplying the sequence in time domain by n is equivalent to multiplying the sequence the derivation of its ZT by –Z in the Z-domain

5. Conjugation Example

6. Time Reversal ROC is inverted Example: Time reversed version of Here x(-n) is the folded version of x(n) so,x(-n) is the time reverse signal thus the folding of signal in time domain is equivalent to replacing z by z-1 in the z-domain Replacing z by z-1 in the z-domain is called as inversion hence folding in the time domain is equivalent to the inversion in z-domain

7. Convolution Convolution in time domain is multiplication in z-domain Example:Let’s calculate the convolution of Multiplications of z-transforms is ROC: if |a| 1 if |a|>1 ROC is |z|>|a| Partial fractional expansion of Y(z)

Linearity Overlay of the above two ROC’s

Shift

Multiplication by an Exponential Sequence

Differentiation of X(z)

Conjugation

Reversal

Initial Value Theorem Initial Value

Convolution of Sequences

3/18/2016 Z- Transform Properties Examples: a and b are arbitrary constants. Example Problem: Find z- transform of Using z- transform table: Linearity: Therefore, we get

3/18/2016 Shift Theorem: Verification: Since x(n) is assumed to be causal: Then we achieve,

3/18/2016 Example Problem: Find z- transform of Solution: Using shift theorem, Using z- transform table

3/18/2016 Convolution In time domain Eq. (1) In z- transform domain, Verification: Using z- transform in Eq. (1)

3/18/2016 Example Problem: Given the sequences, Find the z-transform of their convolution. Solution: Applying z-transform on the two sequences, From the table Therefore we get,

System Function x(n) y(n) By using convolution property & system function Y(z)=H(z)X(z) ROC: at least the intersection of the ROCs of H(z) & X(z) Proof: x(n) * h(n) =y(n) X(z)H(z)=Y(z) H(z) = Y(z) / x(z) 3/18/2016 h(n)

Session 3 Inverse Z-Transform, Methods to find IZT 3/18/2016

Inverse z- Transform: The Procedure of obtaining x(n) from its ZT X(Z) is called Inverse ZT Methods to find Inverse z- Transform: 1.Power series expansion 2.Partial fraction expansion 3.Residue method

1. Inverse Z-Transform by Power Series Expansion The z-transform is power series In expanded form Z-transforms of this form can generally be inversed easily Especially useful for finite-length series Example X(z)=a0+a1z -1 +a2Z an Z -n If the sequence is causal then the limits of n will be n=0 to n=∞ Expanding the above expression we get, X(Z) = x(0) z 0 + x(1) z -1+x(2) z -2+ ……..x(n) z -n = x(0) + x(1) z -1+x(2) z -2 ……..x(n) z -n

3/18/2016 Expansion of ZT for matching standard pair of ZT to get original sequence back By comparing two equs of X(z) we can write X(0) =a0 X(1)= a1 X(2)=a2. X(n)=an Thus the general expression of discrete time causal sequence x(n) is, X(n) = an n>=0

Assume that a given z-transform can be expressed as Apply partial fractional expansion First term exist only if M>N – B r is obtained by long division Second term represents all first order poles Third term represents an order s pole – There will be a similar term for every high-order pole Each term can be inverse transformed by inspection It is expressed as ratios of two polynomials X(z) = N(z)/D(z) = bo+b1z -1 +b1z bMz -M / a0+a1z -1 +a2z aNz -N N(z)- numerator polynomial, D(z) - denominator polynomial bo…,bM –coefficient numerator polynomial, ao..aN - coefficient denominator polynomial, M – Degree of numerator & N- Degree of denominator 2. Inverse z-Transform by Partial Fraction

Session 4 System function H(Z) 3/18/2016

System Function

Signal Characteristics from Z-Transform If U(z) is a rational function, and Then Y(z) is a rational function, too Poles are more important – determine key characteristics of y(k) zeros poles

Why are poles important? Z -1 Z domain Time domain poles components

Shift-Invariant System h(n)h(n) h(n)h(n) x(n)x(n) y(n)=x(n)*h(n) X(z)X(z)Y(z)=X(z)H(z) H(z)H(z)

Shift-Invariant System H(z)H(z) H(z)H(z) X(z)X(z) Y(z)Y(z)

Analysis of DT LTI systems in Time domain 3/18/2016 Session 5

Time-Domain Representation Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range x[n] defined only for integer values of n and undefined for non integer values of n Discrete-time signal represented by {x[n]} 3/18/2016

Discrete-time signal may also be written as a sequence of numbers inside braces: { [x n]} ={K,− 0.2,2.2,1.1,0.2,− 3.7,2.9,K} ↑ In the above, x[−1] = −0.2, x[0] = 2.2, x[1] =1.1, etc. The arrow is placed under the sample at time index n = 0 3/18/2016

Graphical representation of a discrete-time signal with real-valued samples is as shown below: 3/18/2016

In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal x a ( t ) at uniform intervals of time 3/18/2016

Here, n-th sample is given by x[n] = xa (t) t=nT = xa (nT), n =K,− 2,−1,0,1,K The spacing T between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T, denoted as FT, is called the sampling frequency: FT= 1/T 3/18/2016

Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete-valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation 3/18/2016

A discrete-time signal may be a finitelength or an infinite-length sequence Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval:N1 ≤ n ≤ N2 where −∞ < N1 and N2 < ∞ with N1 ≤ N2 Length or duration of the above finitelength sequence is N = N2 − N1 +1 3/18/2016

Analysis of DT LTI systems in Z- domain 3/18/2016 Session 6

LTI System description Previous basis function: unit sample or DT impulse  The input sequence is represented as a linear combination of shifted DT impulses.  The response is given by a convolution sum of the input and the reflected and shifted impulse response Now, use eigenfunctions of all LTI systems as basis function

Relation between DFT & ZT This means if Z-T is evaluated on the unit circle at evenly spaced points only; then it become DFT 3/18/2016