z-transforms Definition of the z-transform Chapter
|
|
- Marian Arnold
- 5 years ago
- Views:
Transcription
1 z-transforms Chapter 7 In the study of discrete-time signal and systems, we have thus far considered the time-domain and the frequency domain. The z- domain gives us a third representation. All three domains are related to each other. A special feature of the z-transform is that for the signals and system of interest to us, all of the analysis will be in terms of ratios of polynomials. Working with these polynomials is relatively straight forward. Definition of the z-transform Given a finite length signal as xn [ ], the z-transform is defined N Xz ( ) xk [ ]z k k 0 k 0 xk [ ]( z ) k (7.) where the sequence support interval is [0, N], and z is any complex number This transformation produces a new representation of denoted Xz ( ) Returning to the original sequence (inverse z-transform) xn [ ] requires finding the coefficient associated with the nth power of z N xn [ ] 7
2 Definition of the z-transform Formally transforming from the time/sequence/n-domain to the z-domain is represented as A sequence and its z-transform are said to form a z-transform pair and are denoted (7.2) In the sequence or n-domain the independent variable is n In the z-domain the independent variable is z Example: Using the definition Thus, N n-domain z-domain xn [ ] xk [ ]δ[ n k] Xz ( ) k 0 xn [ ] δ[ n n 0 ] N z z xn [ ] Xz ( ) z N k 0 xk [ ]z k Xz ( ) xk [ ]z k δ [ k n ]z k 0 z n 0 k 0 N k 0 δ[ n n 0 ] z n 0 z 7 2
3 The z-transform and Linear Systems Example: xn [ ] 2δ[ n] + 3δ[ n ] + 5δ[ n 2] + 2δ[ n 3] By inspection we find that Example: Xz ( ) 4 5z 2 + Xz ( ) 2 + 3z + 5z 2 + By inspection we find that z 3 2z 4 2z 3 xn [ ] 4δ[ n] 5δ[ n 2] + δ[ n 3] 2δ[ n 4] What can we do with the z-transform that is useful? The z-transform and Linear Systems The z-transform is particularly useful in the analysis and design of LTI systems The z-transform of an FIR Filter We know that for any LTI system with input impulse response hn [ ], the output is yn [ ] xn [ ]*h[ n] xn [ ] and (7.3) We are interested in the z-transform of FIR filter hn [ ] b k δ[ n k] k 0 hn [ ], where for an (7.4) 7 3
4 To motivate this, consider the input The output is The z-transform and Linear Systems The term in parenthesis is the z-transform of known as the system function of the FIR filter Like ( ) function as yn [ ] xn [ ] z n, < n < yn [ ] b k xn [ k ] b k z n k He jω k 0 k 0 b k z n z k k 0 k 0 b k z k n z (7.5) (7.6) hn [ ], also was defined in Lecture 5, we define the system Hz ( ) b k z k k 0 k 0 hk [ ]z k The z-transform pair we have just established is z hn [ ] Hz ( ) (7.7) k 0 b k δ[ n k] z k 0 b k z k Another result, similar to the frequency response result, is yn [ ] hn [ ]*z n Hz ( )z n (7.8) 7 4
5 The z-transform and Linear Systems Note if z, we in fact have the frequency response result of Chapter 6 e jωˆ The system function is an th degree polynomial in complex variable z As with any polynomial, it will have roots or zeros, that is there are values z 0 such that Hz ( 0 ) 0 These zeros completely define the polynomial to within a gain constant (scale factor), i.e., Hz ( ) b 0 + b z + + b z where ( z z )( z 2 z ) ( z z ) ( z z )( z z 2 ) ( z z ) z z k, k,, denote the zeros Example: Find the Zeros of hn [ ] δ[ n] + The z-transform is --δ [ n ] 6 --δ [ n 2] 6 Hz ( ) + --z z z --z 2 3 z z -- 2 z 3 7 5
6 The zeros of Hz ( ) are -/2 and +/3 The difference equation yn [ ] 6x[ n] + xn [ ] xn [ 2] has the same zeros, but a different scale factor; proof: Properties of the z-transform Properties of the z-transform The z-transform has a few very useful properties, and its definition extends to infinite signals/impulse responses The Superposition (Linearity) Property ax [ n] + bx 2 [ n] ax ( z) + bx 2 ( z) z (7.9) proof Xz ( ) ( ax [ n] + bx 2 [ n] )z n 0 N a x [ n ]z + b x 2 n n 0 ax ( z) + bx 2 ( z) N n 0 [ ]z 7 6
7 Properties of the z-transform The Time-Delay Property and xn [ ] z z Xz ( ) z xn [ n 0 ] z n 0 Xz ( ) (7.0) (7.) proof: Consider then Let so Similarly Xz ( ) α 0 + α z + + α z N N N xn [ ] α k δ[ n k] k 0 α 0 δ[ n] + α δ[ n ] + + α N δ[ n N] Yz ( ) z Xz ( ) α 0 z + α z α N z N yn [ ] α 0 δ[ n ] + α δ[ n 2] + + α N δ[ n N ] xn [ ] Yz ( ) z n 0 Xz ( ) yn [ ] xn [ n 0 ] 7 7
8 A General z-transform Formula having support inter- We have seen that for a sequence val 0 n N the z-transform is The z-transform as an Operator (7.2) This definition extends for doubly infinite sequences having support interval n to (7.3) There will be discussion of this case in next Lecture when we deal with infinite impulse response (IIR) filters The z-transform as an Operator The z-transform can be considered as an operator. Unit-Delay Operator N Xz ( ) xn [ ]z n n 0 xn [ ] Xz ( ) xn [ ]z n n xn [ ] xn [ ] Unit Delay z yn [ ] xn [ ] 7 8
9 The z-transform as an Operator In the case of the unit delay, we observe that yn [ ] z { xn [ ]} xn [ ] unit delay operator which is motivated by the fact that Yz ( ) z Xz ( ) Similarly, the filter yn [ ] xn [ ] xn [ ] can be viewed as the operator yn [ ] ( z ){ xn [ ]} xn [ ] xn [ ] since Yz ( ) Xz ( ) z Xz ( ) ( z )Xz ( ) (7.4) Example: Two-Tap FIR Using the operator convention, we can write by inspection that Yz ( ) b 0 Xz ( ) + b z Xz ( ) yn [ ] b 0 xn [ ] + b xn [ ] 7 9
10 Convolution and the z-transform Convolution and the z-transform The impulse response of the unity delay system is hn [ ] δ[ n ] and the system output written in terms of a convolution is yn [ ] xn [ ]*δ[ n ] xn [ ] The system function (z-transform of hn [ ]) is Hz ( ) z and by the previous unit delay analysis, Yz ( ) z Xz ( ) We observe that Yz ( ) Hz ( )Xz ( ) (7.5) proof: yn [ ] xn [ ]*h[ n] hk [ ]xn [ k] k 0 (7.6) We now take the z-transform of both sides of (7.6) using superposition and the general delay property Yz ( ) hk [ ]( z k Xz ( )) k 0 k 0 hk [ ]z k Xz ( ) Hz ( )Xz ( ) (7.7) 7 0
11 Convolution and the z-transform Note: For the case of xn [ ] a finite duration sequence, Xz ( ) is a polynomial, and Hz ( )Xz ( ) is a product of polynomials in z Example: Convolving Finite Duration Sequences Suppose that We wish to find yn [ ] by first finding Yz ( ) We begin by z-transforming each of the sequences We find We find Yz ( ) xn [ ] 2δ[ n] 3δ[ n 2] + 4δ[ n 3] hn [ ] δ[ n] + 2δ[ n ] + δ[ n 2] Yz ( ) Convolve directly? Xz ( ) 2 3z 2 + 4z 3 z 2 Hz ( ) + 2z + by direct multiplication Yz ( ) ( 2 3z 2 + 4z 3 )( + 2z + z 2 ) yn [ ] 2 + 4z z 2 2z 3 + 5z 4 + 4z 5 using the delay property on each of the terms of yn [ ] 2δ[ n] + 4δ[ n ] δ[ n 2] 2δ[ n 3] + 5δ[ n 4] + 4δ[ n 5] 7
12 Convolution and the z-transform This section has established the very important result that polynomial multiplication can be used to replace sequence convolution, when we work in the z-domain, i.e., z-transform Convolution Theorem yn [ ] hn [ ]*x[ n] Hz ( )Xz ( ) z Yz ( ) Cascading Systems We have seen cascading of systems in the time-domain and the frequency domain, we now consider the z-domain We know from the convolution theorem that Wz ( ) H ( z)xz ( ) It also follows that Yz ( ) H 2 ( z)wz ( ) so by substitution Yz ( ) [ H 2 ( z)h ( z) ]Xz ( ) [ H ( z)h 2 ( z) ]Xz ( ) (7.8) 7 2
13 Convolution and the z-transform In summary, when we cascade two LTI systems, we arrive at the cascade impulse response as a cascade of impulse responses in the time-domain and a product of the z-transforms in the z-domain hn [ ] h [ n]*h 2 [ n] H ( z)h 2 ( z) z Hz ( ) Factoring z-polynomials ultiplying z-transforms creates a cascade system, so factoring must create subsystems Example: Hz ( ) + 3z 2z 2 + z 3 Since Hz ( ) is a third-order polynomial, we should be able to factor it into a first degree and second degree polynomial We can use the ATLAB function roots() to assist us >> p roots([ 3-2 ]) p i i >> conv([ -p(2)],[ -p(3)]) ans i With one real root, the logical factoring is to create two polynomials as follows 7 3
14 Convolution and the z-transform H ( z) z H 2 ( z) ( ( j0.42)z ) The cascade system is thus: ( ( j0.42)z ) z z 2 xn [ ] z wn [ ] z z 2 yn [ ] Xz ( ) Wz ( ) Yz ( ) H ( z) H 2 ( z) As a check we can multiply the polynomials >> conv([ -p()],conv([ -p(2)],[ -p(3)])) ans.0000, , i, i The difference equations for each subsystem are wn [ ] xn [ ] x[ n ] yn [ ] wn [ ] w[ n ] w[ n 2] Deconvolution/Inverse Filtering In a two subsystems cascade can the second system undo the action of the first subsystem? For the output to equal the input we need Hz ( ) We thus desire H ( z)h 2 ( z) or H 2 ( z) H ( z) 7 4
15 Convolution and the z-transform Example: H ( z) az, a < The inverse filter is H 2 ( z) This is no longer an FIR filter, it is an infinite impulse response (IIR) filter, which is the topic of next Chapter We can approximate az H ( z) H 2 ( z) az as an FIR filter via long division + az + a 2 z 2 + az az az a 2 z 2 a 2 z 2 a 2 z 2 a 3 z 3 a 3 z 3 An + term approximation is H 2 ( z) a k z k k 0 7 5
16 Relationship Between the z-domain and the Frequency Domain Relationship Between the z-domain and the Frequency Domain He jωˆ ωˆ - Domain ( ) b k e jωˆ k k 0 versus z - Domain Hz ( ) b k z k k 0 Comparing the above we see that the connection is setting z e jωˆ in Hz ( ), i.e., He jωˆ ( ) Hz ( ) z e jωˆ (7.9) The z-plane and the Unit Circle If we consider the z-plane, we see that evaluating Hz ( ) on the unit circle z-plane Im z ωˆ j π -- 2 z He jωˆ ( ) e jωˆ corresponds to ωˆ ± π z unit circle ωˆ Re ωˆ 0 z z ωˆ j π -- 2 ECE 260 Signals and Systems 7 6
17 Relationship Between the z-domain and the Frequency Domain From this interpretation we also can see why He ( jωˆ ) is periodic with period 2π As ωˆ increases it continues to sweep around the unit circle over and over again The Zeros and Poles of H(z) Consider Hz ( ) + b z + b 2 z 2 + b 3 z 3 (7.20) where we have assumed that Factoring Hz ( ) results in b 0 Hz ( ) ( z z )( z 2 z )( z 3 z ) (7.2) ultiplying by z 3 z 3 allows to write Hz ( ) in terms of positive powers of z Hz ( ) z 3 + b z 2 + b 2 z + b 3 z (7.22) The zeros are the locations where Hz ( ) 0, i.e., z, z 2, z 3 The poles are where Hz ( ), i.e., z 0 Note that the poles and zeros only determine constant; recall the example on page 7-5 z 3 ( z z )( z z 2 )( z z 3 ) z 3 Hz ( ) to within a 7 7
18 Relationship Between the z-domain and the Frequency Domain A pole-zero plot displays the pole and zero locations in the z- plane z-plane Im z 2 Three poles at z 0 z 3 Re z 3 Example: Hz ( ) + 2z + 2z 2 + ATLAB has a function that supports the creation of a polezero plot given the system function coefficients >> zplane([ 2 2 ],) z 3 Imaginary Part Real Part 3 7 8
19 Relationship Between the z-domain and the Frequency Domain The Significance of the Zeros of H(z) The difference equation is the actual time domain means for calculating the filter output for a given filter input The difference equation coefficients are the polynomial coefficients in Hz ( ) n For xn [ ] z 0 we know that n yn [ ] Hz ( 0 )z 0, (7.23) so in particular if z 0 is one of the zeros of Hz ( ), Hz ( 0 ) 0 and the output yn [ ] 0 If a zero lies on the unit circle then the output will be zero for a sinusoidal input of the form n xn [ ] z 0 ( e jωˆ 0 n ) e jωˆ 0n (7.24) where ωˆ 0 is the angle of the zero relative to the real axis, which is also the frequency of the corresponding complex sinusoid; why? yn [ ] Hz ( ) jωˆ e 0n z e jωˆ 0 0 (7.25) Nulling Filters The special case of zeros on the unit circle allows a filter to null/block/annihilate complex sinusoids that enter the filter at frequencies corresponding to the angles the zeros make with respect to the real axis in the z-plane 7 9
20 Relationship Between the z-domain and the Frequency Domain The nulling property extends to real sinusoids since they are composed of two complex sinusoids at ±ωˆ 0, and zeros not on the real axis will always occur in conjugate pairs if the filter coefficients are real This nulling/annihilating property is useful in rejecting unwanted jamming and interference signals in communications and radar applications Example: Hz ( ) 2cos( ωˆ 0 )z + z 2, xn [ ] cos( ωˆ 0n) Factoring Hz ( ) we find that Expanding Hz ( ) e jωˆ 0 z e jωˆ 0 xn [ ] we see that xn [ ] The nulling action of Hz ( ) at ± ωˆ 0 will remove the signal from the filter output We can set up a simple simulation in ATLAB to verify this >> n 0:00; >> w0 pi/4; >> x cos(w0*n); >> y filter([ -2*cos(w0) ],,x); >> stem(n,x,'filled') >> hold Current plot held >> stem(n,y,'filled','r') >> axis([ ]); grid z z 2 --e jωˆ 0n e jωˆ 0n 2 z 7 20
21 Relationship Between the z-domain and the Frequency Domain 0.8 Input (blue) ωˆ 0 π Amplitude of x[n] and y[n] Output (red) Time Index n Since the input is applied at n 0, we see a small transient while the filter settles to the final output, which in this case is zero >> zplane([ -2*cos(w0) ],)% check the pole-zero plot 0.8 Imaginary Part ωˆ 0 π Real Part 7 2
22 Relationship Between the z-domain and the Frequency Domain Graphical Relation Between z and When we make the substitution z in Hz ( ) we know that we are evaluating the z-transform on the unit circle and thus obtain the frequency response If we plot say Hz ( ) over the entire z-plane we can visualize how cutting out the response on just the unit circle, gives us the frequency response magnitude ωˆ e jωˆ Example: L 9 Here we have oving Average Filter (9 taps/8th-order) Hz ( ) 9 -- z k 9 k ( e j2πk 9 z ) 9 k Im 8 Poles at z 0 create the tree trunk H e jωˆ z-plane agnitude Surface Re 7 22
23 Relationship Between the z-domain and the Frequency Domain >> zplane([ones(,9)]/9,) 9-Tap oving Average FIlter Imaginary Part Real Part >> w -pi:(pi/500):pi; >> H freqz([ones(,9)/9],,w); H(e jω ) H(e jω ) hat(ω) 7 23
24 Useful Filters Useful Filters The L-Point oving Average Filter The L-point moving average (running sum) filter has yn [ ] and system function (z-transform of the impulse response) Hz ( ) (7.26) (7.27) The sum in (7.27) can be simplified using the geometric series sum formula Hz ( ) Notice that the zeros of the equation L -- xn [ k] L k 0 L -- z k L k 0 L -- z k -- z L L L z k 0 Hz ( ) z L 0 z L (7.28) are determined by the roots of (7.29) The roots of this equation can be found by noting that e j2πk for k any integer, thus the roots of (7.29) (zeros of (7.28)) are z k e j2πk L These roots are referred to as the L roots of unity -- L z L ( z ) z L, k 02,,,, L (7.30) ECE 260 Signals and Systems 7 24
25 Useful Filters One of the zeros sits at z, but there is also a pole at z, so there is a pole-zero cancellation, meaning that the pole-zero plot of Hz ( ) corresponds to the L-roots of unity, less the root at z 0 z-plane 2π L L- Pole-zero cancellation occurs here L 8 shown We have seen the frequency response of this filter before The first null occurs at frequency H e jωˆ ωˆ 0 2π L π L 4π L π ωˆ ECE 260 Signals and Systems 7 25
26 Practical Filter Design A Complex Bandpass Filter see text A Bandpass Filter with Real Coefficients see text Practical Filter Design Here we will use fdatool from the ATLAB signal processing toolbox to design an FIR filter Properties of Linear-Phase Filters A class of FIR filters having symmetrical coefficients, i.e., b k b k for k 0,,, has the property of linear phase The Linear Phase Condition For a filter with symmetrical coefficients we can show that ( ) is of the form He jωˆ He ( jωˆ ) Re ( jωˆ )e jω 2 (7.3) where Re ( jωˆ ) is a real function The fact that Re ( jωˆ ) is real means that the phase of He ( jωˆ ) is a linear function of frequency plus the possibility of ±π phase jumps whenever Re ( jωˆ ) passes through zero 7 26
27 Properties of Linear-Phase Filters Example: Hz ( ) b 0 + b z + b 2 z 2 + b z 3 + b 0 z 4 By factoring out z 2 we can write Hz ( ) [ b 0 ( z 2 + z 2 ) + b ( z + z ) + b 2 ]z 2 We now move to the frequency response by letting z ( ) [ 2b 0 cos( 2ωˆ ) + 2b cos( ωˆ ) + b 2 ]e jωˆ 4 2 He jωˆ e jωˆ Note that here we have 4, so we see that the linear phase term is indeed of the form e jωˆ 2 and the real function Re ( jωˆ ) is of the form Re jωˆ ( ) b 2 + 2[ b 0 cos( 2ωˆ ) + b cos( ωˆ )] Locations of the Zeros of FIR Linear-Phase Systems Further study of Hz ( ) for the case of symmetric coefficients reveals that H( z) z Hz ( ) (7.32) A consequence of this condition is that for Hz ( ) having a zero at z 0 it will also have a zero at z 0 Assuming the filter has real coefficients, complex zeros occur in conjugate pairs, so the even symmetry condition further implies that the zeros occur as quadruplets z 0 z*, 0, ----, z * z
28 Properties of Linear-Phase Filters z-plane z ---- * 0 z 0 z 0 * Quadruplet Zeros for Linear Phase FIlters ---- z 0 Example: Hz ( ) 2z + 4z 2 2z 3 + z 4 >> zplane([ ],).5 Imaginary Part Real Part 7 28
Ch. 7: Z-transform Reading
c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient
More informationLecture 19 IIR Filters
Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class
More informationLECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)
LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS
More informationDiscrete-Time Systems
FIR Filters With this chapter we turn to systems as opposed to signals. The systems discussed in this chapter are finite impulse response (FIR) digital filters. The term digital filter arises because these
More informationDigital Signal Processing Lecture 4
Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir E-mail:
More informationDiscrete Time Systems
Discrete Time Systems Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz 1 LTI systems In this course, we work only with linear and time-invariant systems. We talked about
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 6: January 30, 2018 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial
More informationE : Lecture 1 Introduction
E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation
More information# FIR. [ ] = b k. # [ ]x[ n " k] [ ] = h k. x[ n] = Ae j" e j# ˆ n Complex exponential input. [ ]Ae j" e j ˆ. ˆ )Ae j# e j ˆ. y n. y n.
[ ] = h k M [ ] = b k x[ n " k] FIR k= M [ ]x[ n " k] convolution k= x[ n] = Ae j" e j ˆ n Complex exponential input [ ] = h k M % k= [ ]Ae j" e j ˆ % M = ' h[ k]e " j ˆ & k= k = H (" ˆ )Ae j e j ˆ ( )
More informationLecture 8 Finite Impulse Response Filters
Lecture 8 Finite Impulse Response Filters Outline 8. Finite Impulse Response Filters.......................... 8. oving Average Filter............................... 8.. Phase response...............................
More informationDSP-I DSP-I DSP-I DSP-I
DSP-I DSP-I DSP-I DSP-I Digital Signal Processing I (8-79) Fall Semester, 005 OTES FOR 8-79 LECTURE 9: PROPERTIES AD EXAPLES OF Z-TRASFORS Distributed: September 7, 005 otes: This handout contains in outline
More informationLinear Convolution Using FFT
Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular
More informationHow to manipulate Frequencies in Discrete-time Domain? Two Main Approaches
How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous
More informationZ-Transform. x (n) Sampler
Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal
More informationLet H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )
Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:
More informationZ Transform (Part - II)
Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence
More informationVU Signal and Image Processing
052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/18s/
More informationSignals and Systems. Spring Room 324, Geology Palace, ,
Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 6: January 31, 2017 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial
More informationLecture 11 FIR Filters
Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges
More informationDiscrete-time signals and systems
Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the
More informationECE503: Digital Signal Processing Lecture 6
ECE503: Digital Signal Processing Lecture 6 D. Richard Brown III WPI 20-February-2012 WPI D. Richard Brown III 20-February-2012 1 / 28 Lecture 6 Topics 1. Filter structures overview 2. FIR filter structures
More informationStability Condition in Terms of the Pole Locations
Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability
More informationz-transform Chapter 6
z-transform Chapter 6 Dr. Iyad djafar Outline 2 Definition Relation Between z-transform and DTFT Region of Convergence Common z-transform Pairs The Rational z-transform The Inverse z-transform z-transform
More informationSignals and Systems. Problem Set: The z-transform and DT Fourier Transform
Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the
More informationPS403 - Digital Signal processing
PS403 - Digital Signal processing 6. DSP - Recursive (IIR) Digital Filters Key Text: Digital Signal Processing with Computer Applications (2 nd Ed.) Paul A Lynn and Wolfgang Fuerst, (Publisher: John Wiley
More informationUse: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}
1 VI. Z Transform Ch 24 Use: Analysis of systems, simple convolution, shorthand for e jw, stability. A. Definition: X(z) = x(n) z z - transforms Motivation easier to write Or Note if X(z) = Z {x(n)} z
More informationECE503: Digital Signal Processing Lecture 5
ECE53: Digital Signal Processing Lecture 5 D. Richard Brown III WPI 3-February-22 WPI D. Richard Brown III 3-February-22 / 32 Lecture 5 Topics. Magnitude and phase characterization of transfer functions
More informationECE4270 Fundamentals of DSP Lecture 20. Fixed-Point Arithmetic in FIR and IIR Filters (part I) Overview of Lecture. Overflow. FIR Digital Filter
ECE4270 Fundamentals of DSP Lecture 20 Fixed-Point Arithmetic in FIR and IIR Filters (part I) School of ECE Center for Signal and Information Processing Georgia Institute of Technology Overview of Lecture
More informationDigital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant
Digital Filter Chapter 8 Digital Filter Block Diagram Representation Equivalent Basic FIR Digital Filter Basic IIR Digital Filter. Block Diagram Representation In the time domain, the input-output relations
More informationAnalog vs. discrete signals
Analog vs. discrete signals Continuous-time signals are also known as analog signals because their amplitude is analogous (i.e., proportional) to the physical quantity they represent. Discrete-time signals
More informationEEL3135: Homework #4
EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]
More informationLecture 7 - IIR Filters
Lecture 7 - IIR Filters James Barnes (James.Barnes@colostate.edu) Spring 204 Colorado State University Dept of Electrical and Computer Engineering ECE423 / 2 Outline. IIR Filter Representations Difference
More informationMultidimensional digital signal processing
PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,
More informationLecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE
OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE EEE 43 DIGITAL SIGNAL PROCESSING (DSP) 2 DIFFERENCE EQUATIONS AND THE Z- TRANSFORM FALL 22 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr
More information2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.
. Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7) .1. All-Pass Systems An all-pass system is defined as a system which has
More informationECE503: Digital Signal Processing Lecture 4
ECE503: Digital Signal Processing Lecture 4 D. Richard Brown III WPI 06-February-2012 WPI D. Richard Brown III 06-February-2012 1 / 29 Lecture 4 Topics 1. Motivation for the z-transform. 2. Definition
More informationProblem Value Score No/Wrong Rec 3
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING QUIZ #3 DATE: 21-Nov-11 COURSE: ECE-2025 NAME: GT username: LAST, FIRST (ex: gpburdell3) 3 points 3 points 3 points Recitation
More informationVery useful for designing and analyzing signal processing systems
z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing
More informationDetailed Solutions to Exercises
Detailed Solutions to Exercises Digital Signal Processing Mikael Swartling Nedelko Grbic rev. 205 Department of Electrical and Information Technology Lund University Detailed solution to problem E3.4 A
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #21 Friday, October 24, 2003 Types of causal FIR (generalized) linear-phase filters: Type I: Symmetric impulse response: with order M an even
More informationZ-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1
Chapter 6 Z-Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 z-transform The DTFT provides a frequency-domain representation of discrete-time
More informationDiscrete-Time David Johns and Ken Martin University of Toronto
Discrete-Time David Johns and Ken Martin University of Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University of Toronto 1 of 40 Overview of Some Signal Spectra x c () t st () x s () t xn
More informationIII. Time Domain Analysis of systems
1 III. Time Domain Analysis of systems Here, we adapt properties of continuous time systems to discrete time systems Section 2.2-2.5, pp 17-39 System Notation y(n) = T[ x(n) ] A. Types of Systems Memoryless
More informationENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin
ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency
More informationDiscrete-Time Fourier Transform (DTFT)
Discrete-Time Fourier Transform (DTFT) 1 Preliminaries Definition: The Discrete-Time Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]
More information( ) John A. Quinn Lecture. ESE 531: Digital Signal Processing. Lecture Outline. Frequency Response of LTI System. Example: Zero on Real Axis
John A. Quinn Lecture ESE 531: Digital Signal Processing Lec 15: March 21, 2017 Review, Generalized Linear Phase Systems Penn ESE 531 Spring 2017 Khanna Lecture Outline!!! 2 Frequency Response of LTI System
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 19: Lattice Filters Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E. Barner (Univ. of Delaware) ELEG
More informationDiscrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections
Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More informationUNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).
Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the -
More informationChapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals
z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties
More information! z-transform. " Tie up loose ends. " Regions of convergence properties. ! Inverse z-transform. " Inspection. " Partial fraction
Lecture Outline ESE 53: Digital Signal Processing Lec 6: January 3, 207 Inverse z-transform! z-transform " Tie up loose ends " gions of convergence properties! Inverse z-transform " Inspection " Partial
More informationTransform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write
Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp. 4 9. For LTI systems we can write yœn D xœn hœn D X kd xœkhœn Alternatively, this relationship can be expressed in the z-transform
More informationLecture 5 Frequency Response of FIR Systems (III)
EE3054 Signal and Sytem Lecture 5 Frequency Repone of FIR Sytem (III Yao Wang Polytechnic Univerity Mot of the lide included are extracted from lecture preentation prepared by McClellan and Schafer Licene
More information8. z-domain Analysis of Discrete-Time Signals and Systems
8. z-domain Analysis of Discrete-Time Signals and Systems 8.. Definition of z-transform (0.0-0.3) 8.2. Properties of z-transform (0.5) 8.3. System Function (0.7) 8.4. Classification of a Linear Time-Invariant
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Solutions for Problem Set 2
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Issued: Tuesday, September 5. 6.: Discrete-Time Signal Processing Fall 5 Solutions for Problem Set Problem.
More informationDiscrete Time Systems
1 Discrete Time Systems {x[0], x[1], x[2], } H {y[0], y[1], y[2], } Example: y[n] = 2x[n] + 3x[n-1] + 4x[n-2] 2 FIR and IIR Systems FIR: Finite Impulse Response -- non-recursive y[n] = 2x[n] + 3x[n-1]
More informationDigital Signal Processing:
Digital Signal Processing: Mathematical and algorithmic manipulation of discretized and quantized or naturally digital signals in order to extract the most relevant and pertinent information that is carried
More informationDigital Signal Processing Lecture 3 - Discrete-Time Systems
Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of
More informationCMPT 889: Lecture 5 Filters
CMPT 889: Lecture 5 Filters Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 7, 2009 1 Digital Filters Any medium through which a signal passes may be regarded
More informationCMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation
CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is
More informationDigital Filters. Linearity and Time Invariance. Linear Time-Invariant (LTI) Filters: CMPT 889: Lecture 5 Filters
Digital Filters CMPT 889: Lecture 5 Filters Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 7, 29 Any medium through which a signal passes may be regarded as
More informationAPPLIED SIGNAL PROCESSING
APPLIED SIGNAL PROCESSING DIGITAL FILTERS Digital filters are discrete-time linear systems { x[n] } G { y[n] } Impulse response: y[n] = h[0]x[n] + h[1]x[n 1] + 2 DIGITAL FILTER TYPES FIR (Finite Impulse
More information(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform
z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand
More informationDiscrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function
Discrete-Time Signals and s Frequency Domain Analysis of LTI s Dr. Deepa Kundur University of Toronto Reference: Sections 5., 5.2-5.5 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More informationEE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley
EE123 Digital Signal Processing Today Last time: DTFT - Ch 2 Today: Continue DTFT Z-Transform Ch. 3 Properties of the DTFT cont. Time-Freq Shifting/modulation: M. Lustig, EE123 UCB M. Lustig, EE123 UCB
More informationReview of Discrete-Time System
Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.
More informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More informationModule 4 : Laplace and Z Transform Problem Set 4
Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential
More informationDigital Signal Processing Lecture 9 - Design of Digital Filters - FIR
Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time
More informationy[n] = = h[k]x[n k] h[k]z n k k= 0 h[k]z k ) = H(z)z n h[k]z h (7.1)
7. The Z-transform 7. Definition of the Z-transform We saw earlier that complex exponential of the from {e jwn } is an eigen function of for a LTI System. We can generalize this for signals of the form
More informationDigital Signal Processing Lecture 10 - Discrete Fourier Transform
Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015 Overview 1 2 3 4 Review - 1 Introduction Discrete-time
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-04 COURSE: ECE-2025 NAME: GT #: LAST, FIRST Recitation Section: Circle the date & time when your Recitation
More informationModule 1: Signals & System
Module 1: Signals & System Lecture 6: Basic Signals in Detail Basic Signals in detail We now introduce formally some of the basic signals namely 1) The Unit Impulse function. 2) The Unit Step function
More informationIntroduction to DSP Time Domain Representation of Signals and Systems
Introduction to DSP Time Domain Representation of Signals and Systems Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407)
More informationChapter 7: The z-transform
Chapter 7: The -Transform ECE352 1 The -Transform - definition Continuous-time systems: e st H(s) y(t) = e st H(s) e st is an eigenfunction of the LTI system h(t), and H(s) is the corresponding eigenvalue.
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 02 DSP Fundamentals 14/01/21 http://www.ee.unlv.edu/~b1morris/ee482/
More information2. CONVOLUTION. Convolution sum. Response of d.t. LTI systems at a certain input signal
2. CONVOLUTION Convolution sum. Response of d.t. LTI systems at a certain input signal Any signal multiplied by the unit impulse = the unit impulse weighted by the value of the signal in 0: xn [ ] δ [
More informationAn Fir-Filter Example: Hanning Filter
An Fir-Filter Example: Hanning Filter Josef Goette Bern University of Applied Sciences, Biel Institute of Human Centered Engineering - microlab Josef.Goette@bfh.ch February 7, 2018 Contents 1 Mathematical
More informationPoles, Zeros and System Response
Time Response After the engineer obtains a mathematical representation of a subsystem, the subsystem is analyzed for its transient and steady state responses to see if these characteristics yield the desired
More informationFourier Series Representation of
Fourier Series Representation of Periodic Signals Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline The response of LIT system
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More information7.17. Determine the z-transform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the z-plane. X(z) = x[n]z n.
Solutions to Additional Problems 7.7. Determine the -transform and ROC for the following time signals: Sketch the ROC, poles, and eros in the -plane. (a) x[n] δ[n k], k > 0 X() x[n] n n k, 0 Im k multiple
More informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-04 COURSE: ECE-2025 NAME: GT #: LAST, FIRST Recitation Section: Circle the date & time when your Recitation
More informationSolution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet
Solution 10 July 2015 ECE301 Signals and Systems: Midterm Cover Sheet Test Duration: 60 minutes Coverage: Chap. 1,2,3,4 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes are not allowed.
More informationEE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.
EE : Signals, Systems, and Transforms Spring 7. A causal discrete-time LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discrete-time
More informationSolutions: Homework Set # 5
Signal Processing for Communications EPFL Winter Semester 2007/2008 Prof. Suhas Diggavi Handout # 22, Tuesday, November, 2007 Solutions: Homework Set # 5 Problem (a) Since h [n] = 0, we have (b) We can
More informationIT DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING IT6502 - DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A 1. What is a continuous and discrete time signal? Continuous
More informationUNIVERSITY OF OSLO. Faculty of mathematics and natural sciences. Forslag til fasit, versjon-01: Problem 1 Signals and systems.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 1th, 016 Examination hours: 14:30 18.30 This problem set
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 20) Final Examination December 9, 20 Name: Kerberos Username: Please circle your section number: Section Time 2 am pm 4 2 pm Grades will be determined by the correctness of your answers (explanations
More information6.003: Signals and Systems
6.003: Signals and Systems Z Transform September 22, 2011 1 2 Concept Map: Discrete-Time Systems Multiple representations of DT systems. Delay R Block Diagram System Functional X + + Y Y Delay Delay X
More informationTransform Analysis of Linear Time-Invariant Systems
Transform Analysis of Linear Time-Invariant Systems Discrete-Time Signal Processing Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-Sen University Kaohsiung, Taiwan ROC Transform
More informationLecture 5 - Assembly Programming(II), Intro to Digital Filters
GoBack Lecture 5 - Assembly Programming(II), Intro to Digital Filters James Barnes (James.Barnes@colostate.edu) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 18: Applications of FFT Algorithms & Linear Filtering DFT Computation; Implementation of Discrete Time Systems Kenneth E. Barner Department of Electrical and
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 05 IIR Design 14/03/04 http://www.ee.unlv.edu/~b1morris/ee482/
More informationDHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A
DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete
More informationAnalog LTI system Digital LTI system
Sampling Decimation Seismometer Amplifier AAA filter DAA filter Analog LTI system Digital LTI system Filtering (Digital Systems) input output filter xn [ ] X ~ [ k] Convolution of Sequences hn [ ] yn [
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationMulti-rate Signal Processing 3. The Polyphase Representation
Multi-rate Signal Processing 3. The Polyphase Representation Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by
More information