DOCSLIB.ORG

Digital Signal Processing Lecture 8

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Digital Signal Processing Lecture 8 Lecture 8 Recap Introduction CT->DT Digital Signal Processing Impulse Invariance Lecture 8 - Filter Design - IIR Bilinear Trans. Example Electrical Engineering and Computer Science University of Tennessee, Knoxville Overview Lecture 8 1 Recap Recap Introduction CT->DT 2 Introduction Impulse Invariance 3 Bilinear Trans. CT->DT Example 4 Impulse Invariance 5 Bilinear Trans. 6 Example Roadmap Lecture 8 Introduction Discrete-time signals and systems - LTI systems Unit sample response h[n]: uniquely characterizes an Recap LTI system Introduction Linear constant-coefficient difference equation CT->DT Frequency response: H(ej!) Impulse Invariance Complex exponential being eigenfunction of an LTI j! j! Bilinear Trans. system: y[n] = H(e )x[n] and H(e ) as eigenvalue. Example z transform P1 −n The z-transform, X(z) = n=−∞ x[n]z Region of convergence - the z-plane System function, H(z) Properties of the z-transform The significance of zeros 1 H n−1 The inverse z-transform, x[n] = 2πj C X(z)z dz: inspection, power series, partial fraction expansion Sampling and Reconstruction Transform domain analysis - nwz Review - Design structures Lecture 8 Different representations of causal LTI systems LCDE with initial rest condition H(z) with jzj > R+ and starts at n = 0 Recap Block diagram vs. Signal flow graph and how to determine Introduction system function (or unit sample response) from the graphs CT->DT Design structures Impulse Invariance Direct form I (zeros first) Bilinear Trans. Direct form II (poles first) - Canonic structure Example Transposed form (zeros first) IIR: cascade form, parallel form, feedback in IIR (computable vs. noncomputable) FIR: direct form, cascade form, parallel form, linear phase Metric: computational resource and precision Sources of errors: coefficient quantization error, input quantization error, product quantization error, limit cycles Pole sensitivity of 2nd-order structures: coupled form Coefficient quantization examples: direct form vs. cascade form Rationale Lecture 8 Recap Review of complex exponentials as eigenfunctions of Introduction the LTI CT->DT x[n] = ej!0n ! y[n] = H(ej!0 )ej!0n Impulse or x[n] = cos ! n ! y[n] = jH(ej!0 )j cos(! n + θ) Invariance 0 0 Bilinear Trans. Separation of signal when they occupate different Example frequency bands — choose the system function that is unity at selective frequencies Given a set of specifications, design a rational transfer function that approximates the ideal filter maintaining specifications of δp, δs, and the transition band. Stages of digital filter design Lecture 8 Recap Introduction The specification of the desired properties of the CT->DT system Impulse application-dependent Invariance usually done in the frequency domain Bilinear Trans. Example The approximation of the specifications using a causal discrete-time system The realization of the system e.g., DSP board Practical frequency-selective filters Lecture 8 Approximate ideal filters by a rational function or LCDE Recap Introduction CT->DT Impulse Invariance Bilinear Trans. Example Factors that affect the filter performance the maximum tolerable passband ripple, 20 log10 δp the maximum tolerable stopband ripple, 20 log10 δs the passband edge frequency !p the stopband edge frequency !s M and N: order of the LCDE Example Lecture 8 Design a discrete-time lowpass filter to filter a Recap continuous-time signal with the following specs (with a Introduction sampling rate of 104 samples/s): CT->DT The gain should be within ±0:01 of unity in the Impulse Invariance frequency band 0 ≤ Ω ≤ 2π(2000) Bilinear Trans. The gain should be no greater than 0.001 in the Example frequency band 2π(3000) ≤ Ω Parameter setup δp1 = δp2 = 0:01, δs = 0:001 4 4 !p = 2π(2000)=10 , !s = 2π(3000)=10 Ideal passband gain in decibels? maximum passband gain in decibels? maximum stopband gain in decibels? Design techniques for IIR filters Lecture 8 Recap Introduction CT->DT Impulse Invariance Analytical — closed-form solution of transfer function Bilinear Trans. Continuous-time ! Discrete-time Example Algorithmic General guidelines for CT->DT Lecture 8 Recap Introduction CT->DT continuous ! discrete Impulse Invariance Ha(s) ! H(z) Bilinear Trans. ha(t) ! h[n] Example jΩ-axis (s-plane) ! unit circle (z-plane) if Ha(s) is stable ! H(z) is stable Different CT->DT approaches Lecture 8 Recap Introduction Mapping differentials to differences CT->DT Impulse z = 1 + sT Invariance the jΩ-axis is NOT mapped to the unit circle Bilinear Trans. stable poles might not be mapped to inside the unit Example circle Impulse invariance Bilinear transformation Impulse invariance Lecture 8 h[n] = Td hc(nTd ) (1) Recap 1 Introduction j! X j! j2πk H(e ) = Hc[ + ] (2) CT->DT Td Td k=−∞ Impulse Invariance Bilinear Trans. Preserve good time-domain characteristics Example Linear scaling of frequency axis, ! = ΩT Existence of aliasing Impulse invariance doesn’t imply step invariance Impulse invariance (cont’) Lecture 8 N N Recap X A X T A ( ) = k ! ( ) = d k Introduction Hc s H z s T −1 s − sk 1 − e k d z CT->DT k=1 k=1 Impulse Invariance Mapping poles Bilinear Trans. Example sk Td s = sk ! z = e Preserve residues s = jΩ ! z = ejΩTd = ej!, the unit circle if sk is stable, i.e., region of sk is less than 0, ! jzk j < 1 ! digital filter is stable Impulse invariance - An example Lecture 8 Recap Introduction Find the system function of the digital filter mapped CT->DT from the analog filter with a system function s+a Impulse Hc(s) = 2 2 . Compare magnitude of the Invariance (s+a) +b Bilinear Trans. frequency response and pole-zero distributions in the s- Example and z-plane −aT −1 Sol: H(z) = 1−(e cos bT )z (1−e−(a+jb)T z−1)(1−e−(a−jb)T z−1) Note that zeros are not mapped. Also note that jHs(jΩ)j is not periodic but jH(ej!)j is. Bilinear transformation Lecture 8 Mapping from s-plane to z-plane by relating s and z according to a bilinear transformation. Hc(s) ! H(z) Recap −1 sT 2 1 − z 1 + 2 Introduction s = ( ); or z = T 1 + z−1 1 − sT CT->DT 2 Impulse Two guidelines Invariance Preserves the frequency characteristics? I.e., maps the Bilinear Trans. jΩ-axis to the unit circle? Example Stable analog filter mapped to stable digital filter? Important properties of bilinear transformation Left-side of the s-plane ! interior of the unit circle; Right-side of the s-plane ! exterior of the unit circle. Therefore, stable analog filters ! stable digital filters. The jΩ-axis gets mapped exactly once around the unit circle. No aliasing The jΩ-axis is infinitely long but the unit circle isn’t ! nonlinear distortion of the frequency axis Blinear transformation - Mappings Lecture 8 Recap Introduction CT->DT Impulse Invariance Bilinear Trans. Example Bilinear transformation - How to tolerate distortions? Lecture 8 Prewarp the digital cutoff frequency to an analog cutoff 2 ! frequency through Ω = T tan 2 Recap Better used to approximate piecewise constant filters which Introduction will be mapped as constant as well CT->DT Can’t be used to obtain digital lowpass filter with linear-phase Impulse Invariance Bilinear Trans. Example Avoid aliasing at the price of distortion of the frequency axis The class of analog filters Lecture 8 Butterworth filter jH (jΩ)j2 = 1 c 1+( jΩ )2N Recap jΩc Note about the butterworth circle with radius Ω Introduction c Ωc is also called the 3dB-cutoff frequency when CT->DT 2 −10log10jHc(jΩ)j jΩ=Ω ≈ 3 Impulse c Invariance Monotonic function in both passband and stopband Bilinear Trans. Matlab functions: buttord, butter Example Chebyshev filter Type I Chebyshev has an equiripple freq response in the passband and varies monotonicaly in the stopband, 2 1 jHc(jΩ)j = 2 2 1+ TN (Ω=Ωp) Type II Chebyshev is monotonic in the passband and 2 1 equiripple in the stopband, jHc(jΩ)j = T (Ω =Ω ) 1+2[ N s p ]2 TN (Ωs=Ω) Matlab functions: cheb1ord, cheby1, cheb2ord, cheby2 The class of analog filters (cont’d) Lecture 8 Recap Introduction Elliptic filter CT->DT 2 1 jHc(jΩ) = 2 2 where RN (Ω) is a rational Impulse 1+ RN (Ω=Ωp) Invariance function of order N satisfying the perperty Bilinear Trans. RN (1=Ω) = 1=RN (Ω) with the roots of its numerator Example lying within the interval 0 < Ω < 1 and the roots of its denominator lying in the interval 1 < Ω < 1 Equiripple in both the passband and the stopband Matlab functions: ellipord, ellip Example Lecture 8 Recap Specs of the discrete-time filter: passband gain Introduction between 0dB and -1dB, and stopband attenuation of at CT->DT least -15dB. Impulse Invariance 1 − δp ≥ −1dB; δs ≤ −15dB Bilinear Trans. Example j0:2π j0:2π −0:05 20 log10 jH(e )j ≥ −1 ! jH(e )j ≥ 10 = 0:89130 ≤ j!j ≤ 0:2π (3) j0:3π j0:3π −0:75 20 log10 jH(e )j ≤ −15 ! jH(e )j ≤ 10 = 0:17780:3π ≤ j!j ≤ π (4) Example (cont’d) Lecture 8 Impulse invariance Round up to the next integer of N Due to aliasing problem, meet the passband exactly Recap with exceeded stopband Introduction j 0:2π CT->DT 1 + ( T )2N = 100:1 (5) Impulse jΩc Invariance j 0:3π Bilinear Trans. 1 + ( T )2N = 101:5 (6) Example jΩc Bilinear transformation Round up to the next integer of N By convention, choose to meet the stopband exactly with exceeded passband j2 tan(0:1π) 1 + ( )2N = 100:1 (7) jΩc j2 tan(0:15π) 1 + ( )2N = 101:5 (8) jΩc Example - Comparison Lecture 8 Recap Introduction CT->DT Impulse Invariance Bilinear Trans. Example.
Load more

Recommended publications
  • Dsp Notes Prepared
    DSP NOTES PREPARED BY Ch.Ganapathy Reddy Professor & HOD, ECE Shaikpet, Hyderabad-08 Ch Ganapathy Reddy, Prof and HOD, ECE, GNITS id:[email protected],9052344333 1 DIGITAL SIGNAL PROCESSING A signal is defined as any physical quantity that varies with time, space or another independent variable. A system is defined as a physical device that performs an operation on a signal. System is characterized by the type of operation that performs on the signal. Such operations are referred to as signal processing. Advantages of DSP 1. A digital programmable system allows flexibility in reconfiguring the digital signal processing operations by changing the program. In analog redesign of hardware is required. 2. In digital accuracy depends on word length, floating Vs fixed point arithmetic etc. In analog depends on components. 3. Can be stored on disk. 4. It is very difficult to perform precise mathematical operations on signals in analog form but these operations can be routinely implemented on a digital computer using software. 5. Cheaper to implement. 6. Small size. 7. Several filters need several boards in analog, whereas in digital same DSP processor is used for many filters. Disadvantages of DSP 1. When analog signal is changing very fast, it is difficult to convert digital form .(beyond 100KHz range) 2. w=1/2 Sampling rate. 3. Finite word length problems. 4. When the signal is weak, within a few tenths of millivolts, we cannot amplify the signal after it is digitized. 5. DSP hardware is more expensive than general purpose microprocessors & micro controllers. Ch Ganapathy Reddy, Prof and HOD, ECE, GNITS id:[email protected],9052344333 2 6.
    [Show full text]
  • Noise Will Be Noise: Or Phase Optimized Recursive Filters for Interference Suppression, Signal Differentiation and State Estimation (Extended Version) Hugh L
    Available online at https://arxiv.org/ 1 Noise will be noise: Or phase optimized recursive filters for interference suppression, signal differentiation and state estimation (extended version) Hugh L. Kennedy Abstract— The increased temporal and spectral resolution of oversampled systems allows many sensor-signal analysis tasks to be performed (e.g. detection, classification and tracking) using a filterbank of low-pass digital differentiators. Such filters are readily designed via flatness constraints on the derivatives of the complex frequency response at dc, pi and at the centre frequencies of narrowband interferers, i.e. using maximally-flat (MaxFlat) designs. Infinite-impulse-response (IIR) filters are ideal in embedded online systems with high data-rates because computational complexity is independent of their (fading) ‘memory’. A novel procedure for the design of MaxFlat IIR filterbanks with improved passband phase linearity is presented in this paper, as a possible alternative to Kalman and Wiener filters in a class of derivative-state estimation problems with uncertain signal models. Butterworth poles are used for configurable bandwidth and guaranteed stability. Flatness constraints of arbitrary order are derived for temporal derivatives of arbitrary order and a prescribed group delay. As longer lags (in samples) are readily accommodated in oversampled systems, an expression for the optimal group delay that minimizes the white-noise gain (i.e. the error variance of the derivative estimate at steady state) is derived. Filter zeros are optimally placed for the required passband phase response and the cancellation of narrowband interferers in the stopband, by solving a linear system of equations. Low complexity filterbank realizations are discussed then their behaviour is analysed in a Teager-Kaiser operator to detect pulsed signals and in a state observer to track manoeuvring targets in simulated scenarios.
    [Show full text]
  • Overview Textbook Prerequisite Course Outline
    ESE 337 Digital Signal Processing: Theory Fall 2018 Instructor: Yue Zhao Time and Location: Tuesday, Thursday 7:00pm - 8:20pm, Javits Lecture Center 101 Contact: Email: [email protected], Office: 261 Light Engineering Office Hours: Tuesday, Thursday 1:30pm - 3:00pm, or by appointment Teaching Assistants, and Office Hours: • Jiaming Li ([email protected]): THU 12:30pm - 2:00pm, or by appointment, Location: 208 Light Engineering (Changes of hours, if any, will be updated on Blackboard.) Overview Digital Signal Processing (DSP) lies at the heart of modern information technology in many fields including digital communications, audio/image/video compression, speech recognition, medical imaging, sensing for health, touch screens, space exploration, etc. This class covers the basic principles of digital signal processing and digital filtering. Skills for analyzing and synthesizing algorithms and systems that process discrete time signals will be developed. 3 credits. Textbook • A.V. Oppenheim and R.W. Schafer, Discrete Time Signal Processing, Prentice Hall, Third Edition, 2009 Prerequisite • ESE 305, Deterministic Signals and Systems Course Outline Week 1 DT signals, DT systems and properties, LTI systems Week 2 Convolution, properties of LTI systems Week 3 Examples of LTI systems, eigen functions, frequency response of DT systems, DTFT 1 ESE 337 Syllabus Week 4 Convergence and properties of DTFT Week 5 Theorems of DTFT, useful DTFT pairs, examples, Z-transform Week 6 Examples of Z-transform, properties of ROC Week 7 Inverse Z-transform,
    [Show full text]
  • Understanding Digital Signal Processing
    Understanding Digital Signal Processing Richard G. Lyons PRENTICE HALL PTR PRENTICE HALL Professional Technical Reference Upper Saddle River, New Jersey 07458 www.photr,com Contents Preface xi 1 DISCRETE SEQUENCES AND SYSTEMS 1 1.1 Discrete Sequences and Their Notation 2 1.2 Signal Amplitude, Magnitude, Power 8 1.3 Signal Processing Operational Symbols 9 1.4 Introduction to Discrete Linear Time-Invariant Systems 12 1.5 Discrete Linear Systems 12 1.6 Time-Invariant Systems 17 1.7 The Commutative Property of Linear Time-Invariant Systems 18 1.8 Analyzing Linear Time-Invariant Systems 19 2 PERIODIC SAMPLING 21 2.1 Aliasing: Signal Ambiquity in the Frequency Domain 21 2.2 Sampling Low-Pass Signals 26 2.3 Sampling Bandpass Signals 30 2.4 Spectral Inversion in Bandpass Sampling 39 3 THE DISCRETE FOURIER TRANSFORM 45 3.1 Understanding the DFT Equation 46 3.2 DFT Symmetry 58 v vi Contents 3.3 DFT Linearity 60 3.4 DFT Magnitudes 61 3.5 DFT Frequency Axis 62 3.6 DFT Shifting Theorem 63 3.7 Inverse DFT 65 3.8 DFT Leakage 66 3.9 Windows 74 3.10 DFT Scalloping Loss 82 3.11 DFT Resolution, Zero Padding, and Frequency-Domain Sampling 83 3.12 DFT Processing Gain 88 3.13 The DFT of Rectangular Functions 91 3.14 The DFT Frequency Response to a Complex Input 112 3.15 The DFT Frequency Response to a Real Cosine Input 116 3.16 The DFT Single-Bin Frequency Response to a Real Cosine Input 117 3.17 Interpreting the DFT 120 4 THE FAST FOURIER TRANSFORM 125 4.1 Relationship of the FFT to the DFT 126 4.2 Hints an Using FFTs in Practice 127 4.3 FFT Software Programs
    [Show full text]
  • ESE 531: Digital Signal Processing Today IIR Filter Design Impulse
    Today ESE 531: Digital Signal Processing ! IIR Filter Design " Impulse Invariance " Bilinear Transformation Lec 18: March 30, 2017 ! Transformation of DT Filters IIR Filters and Adaptive Filters ! Adaptive Filters ! LMS Algorithm Penn ESE 531 Spring 2017 – Khanna Penn ESE 531 Spring 2017 - Khanna 2 IIR Filter Design Impulse Invariance ! Transform continuous-time filter into a discrete- ! Want to implement continuous-time system in time filter meeting specs discrete-time " Pick suitable transformation from s (Laplace variable) to z (or t to n) " Pick suitable analog Hc(s) allowing specs to be met, transform to H(z) ! We’ve seen this before… impulse invariance Penn ESE 531 Spring 2017 - Khanna 3 Penn ESE 531 Spring 2017 - Khanna 4 Impulse Invariance Impulse Invariance ! With Hc(jΩ) bandlimited, choose ! With Hc(jΩ) bandlimited, choose j ω j ω H(e ω ) = H ( j ), ω < π H(e ω ) = H ( j ), ω < π c T c T ! With the further requirement that T be chosen such ! With the further requirement that T be chosen such that that Hc ( jΩ) = 0, Ω ≥ π / T Hc ( jΩ) = 0, Ω ≥ π / T h[n] = Thc (nT ) Penn ESE 531 Spring 2017 - Khanna 5 Penn ESE 531 Spring 2017 - Khanna 6 1 IIR by Impulse Invariance Example jω ! If Hc(jω)≈0 for |ωd| > π/T, no aliasing and H(e ) = H(jω/T), ω<π jω ! To get a particular H(e ), find corresponding Hc and Td for which above is true (within specs) ! Note: Td is not for aliasing control, used for frequency scaling. Penn ESE 531 Spring 2017 - Khanna 7 Penn ESE 531 Spring 2017 - Khanna 8 Example Example 1 eat ←⎯L→ s − a Penn ESE 531 Spring
    [Show full text]
  • Chap 4 Sampling of Continuous-Time Signals Introduction
    Chap 4 Sampling of Continuous-Time Signals Introduction 4.1 Periodic Sampling 4.2 Frequency-Domain Representation of Sampling 4.3 Reconstruction of a Bandlimited Signal from its Samples 4.4 Discrete-Time Processing of Continuous-Time Signals 4.5 Continuous-Time Processing of Discrete-Time Signals 4.6 Changing the Sampling Rate Using Discrete-Time Processing 4.7 Multirate Signal Processing 4.8 Digital Processing of Analog Signals 4.9 Oversampling and Noise Shaping in A/D and D/A Conversion 2018/9/18 DSP 2 Periodic Sampling Sequence of samples x[n] is obtained from a continuous-time signal xc(t) : x[n] = xc(nT), - infinity < n < infinity T : sampling period fs = 1/T : sampling frequency, samples/second C/D Continuous-to-discrete-time Xc(t) converter X[n] = Xc(nT) T In a practical setting, the operation of sampling is often implemented by an analog-to-digital (A/D) converter which can be approximated to the ideal C/D converter. The sampling operation is generally not invertible. The inherent ambiguity in sampling is of primary concern in signal processing. 2018/9/18 DSP 3 Sampling with a Periodic Impulse Train s(t) C/D Converter Conversion from impulse train x[n] = x (nT) x (t) X C C xS(t)to discrete-time sequence xC(t) xC(t) xS(t) 2 Sampling Rates xS(t) -4T 0 2T -4T 0 2T x[n] 2 Output Sequences x[n] -4 0 2 -4 0 2 2018/9/18 DSP 4 Frequency-domain representation of sampling The conversion of xc(t) to xs(t) through modulating signal s(t) which is a periodic impulse train s(t) (t nT ) n xs (t) xc (t)s(t) xc (t) (t nT ) n by shifting property of the impulse xs (t) xc (nT)(t nT) n The Fourier transform of a periodic impulse train is a periodic impulse train.
    [Show full text]
  • IIR Filters (II)
    Lecture 8 - IIR Filters (II) James Barnes ([email protected]) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423 – 1 / 27 Lecture 8 Outline ● Introduction ● Digital Filter Design by Analog → Digital Conversion ● (Probably next lecture) ”All Digital” Design Algorithms ● (Next lecture) Conversion of Filter Types by Frequency Transformation Colorado State University Dept of Electrical and Computer Engineering ECE423 – 2 / 27 ❖ Lecture 8 Outline Introduction ❖ IIR Filter Design Overview Method: Impulse Invariance for IIR FIlters Approximation of Derivatives Bilinear Transform Matched Z-Transform Introduction Colorado State University Dept of Electrical and Computer Engineering ECE423 – 3 / 27 IIR Filter Design Overview ● Methods which start from analog design ✦ Impulse Invariance ✦ Approximation of Derivatives ✦ Bilinear Transform ✦ Matched Z-transform All are different methods of mapping the s-plane onto the z-plane ● Methods which are ”all digital” ✦ Least-squares ✦ McClellan-Parks Colorado State University Dept of Electrical and Computer Engineering ECE423 – 4 / 27 ❖ Lecture 8 Outline Introduction Method: Impulse Invariance for IIR FIlters ❖ Impulse Invariance ❖ Impulse Invariance (2) ❖ Impulse Invariance (3) ❖ Impulse Invariance (5) ❖ Impulse Invariance Procedure ❖ Impulse Invariance Example Method: Impulse Invariance for IIR FIlters ❖ Impulse Invariance Example (2) Approximation of Derivatives Bilinear Transform Matched Z-Transform Colorado State University Dept of Electrical and Computer Engineering ECE423 – 5 / 27 Impulse Invariance We start by sampling the impulse response of the analog filter: ha(t) h[n]= ha(nt0) t0 Sampling Theorem gives relation between Fourier Transform of sampled and continuous ”signals”: ∞ 1 ω 2πk H(z)|z=ejω = Ha(j − j ), (1) t0 t0 t0 k=X−∞ where ω =Ωt0 = 2πf/fs and f is the analog frequency in Hz.
    [Show full text]
  • IIR Filters (II)
    Lecture 8 - IIR Filters (II) James Barnes ([email protected]) Spring 2014 Colorado State University Dept of Electrical and Computer Engineering ECE423 – 1 / 29 Lecture 8 Outline ● Introduction ● Digital Filter Design by Analog → Digital Conversion ● (Probably next lecture) ”All Digital” Design Algorithms ● (Next lecture) Conversion of Filter Types by Frequency Transformation Colorado State University Dept of Electrical and Computer Engineering ECE423 – 2 / 29 ❖ Lecture 8 Outline Introduction ❖ IIR Filter Design Overview Method: Impulse Invariance for IIR FIlters Approximation of Derivatives Bilinear Transform Matched Z-Transform Introduction Colorado State University Dept of Electrical and Computer Engineering ECE423 – 3 / 29 IIR Filter Design Overview ● Methods which start from analog design ✦ Impulse Invariance ✦ Approximation of Derivatives ✦ Bilinear Transform ✦ Matched Z-transform All are different methods of mapping the s-plane onto the z-plane ● Methods which are ”all digital” ✦ Least-squares ✦ McClellan-Parks Colorado State University Dept of Electrical and Computer Engineering ECE423 – 4 / 29 ❖ Lecture 8 Outline Introduction Method: Impulse Invariance for IIR FIlters ❖ Impulse Invariance ❖ Impulse Invariance (2) ❖ Impulse Invariance (3) ❖ Impulse Invariance (4) ❖ Impulse Invariance Procedure ❖ Impulse Invariance Example Method: Impulse Invariance for IIR FIlters ❖ Impulse Invariance Example (2) Approximation of Derivatives Bilinear Transform Matched Z-Transform Colorado State University Dept of Electrical and Computer Engineering ECE423 – 5 / 29 Impulse Invariance We start by sampling the impulse response of the analog filter: ha(t) h[n]= ha(nt0) t0 Sampling Theorem gives relation between Fourier Transform of sampled and continuous ”signals”: ∞ 1 ω 2πk H(z)|z=ejω = Ha(j − j ), (1) t0 t0 t0 k=X−∞ where ω =Ωt0 = 2πf/fs and f is the analog frequency in Hz.
    [Show full text]
  • Dsp Course File
    DSP COURSE FILE Contents Course file Contents: 1. Cover Page 2. Syllabus copy 3. Vision of the Department 4. Mission of the Department 5. PEOs and POs 6. Course objectives and outcomes 7. Instructional Learning Outcomes 8. Prerequisites, if any 9. Brief note on the importance of the course and how it fits into the curriculum 10. Course mapping with PEOs and POs 11. Class Time Table 12. Individual Time Table 13. Micro Plan with dates and closure report 14. Detailed notes 15. Additional topics 16. University Question papers of previous years 17. Question Bank 18. Assignment topics 19. Unit wise Quiz Questions 20. Tutorial problems 21. Known gaps ,if any 22. References, Journals, websites and E-links 23. Quality Control Sheets 24. Student List 25. Quality measurement sheets a) Course end survey b) Teaching evaluation 26. Group-Wise students list for discussion topics 1. COVER PAGE GEETHANJALI COLLEGE OF ENGINEERING AND TECHNOLOGY DEPARTMENT OF Electronics and Communication Engineering (Name of the Subject / Lab Course) : Digital Signal Processing JNTU CODE -56027 Programme : UG Branch: ECE Version No : 02 Year: III Year ECE ( C ) Document Number: GCET/ECE/DSP/02 Semester: II No. of pages : Classification status (Unrestricted / Restricted ) Distribution List : Prepared by : 1) Name : 1) Name : M.UMARANI 2) Sign : 2) Sign : 3) Design : Asst.Prof 3) Design :Asst.Prof 4) Date : 11/11/13 4) Date :18/11/15 Verified by : 1) Name : * For Q.C Only. 2) Sign : 1) Name : 3) Design : 2) Sign : 4) Date : 3) Design : 4) Date : Approved by : (HOD ) 1) Name : Dr.P.SRIHARI 2) Sign : 3) Date : 2.
    [Show full text]
  • Sampling and Quantization for Optimal Reconstruction by Shay Maymon Submitted to the Department of Electrical Engineering and Computer Science
    Sampling and Quantization for Optimal Reconstruction by Shay Maymon Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of MASSACHUSETTS INSTITUTE OF TECHNOLOGY Doctor of Philosophy in Electrical Engineering JUN 17 2011 at the LIBRARIES MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARCHiVES June 2011 @ Massachusetts Institute of Technology 2011. All rights reserved. Author ..... .......... De 'artment of'EYectrical Engineerin /d Computer Science May 17,2011 Certified by.. Alan V. Oppenheim Ford Professor of Engineering Thesis Supervisor Accepted by............ Profeksor Lelif". Kolodziejski Chairman, Department Committee on Graduate Theses 2 Sampling and Quantization for Optimal Reconstruction by Shay Maymon Submitted to the Department of Electrical Engineering and Computer Science on May 17, 2011, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Abstract This thesis develops several approaches for signal sampling and reconstruction given differ- ent assumptions about the signal, the type of errors that occur, and the information available about the signal. The thesis first considers the effects of quantization in the environment of interleaved, oversampled multi-channel measurements with the potential of different quan- tization step size in each channel and varied timing offsets between channels. Considering sampling together with quantization in the digital representation of the continuous-time signal is shown to be advantageous. With uniform quantization and equal quantizer step size in each channel, the effective overall signal-to-noise ratio in the reconstructed output is shown to be maximized when the timing offsets between channels are identical, result- ing in uniform sampling when the channels are interleaved.
    [Show full text]
  • Understanding Digital Signal Processing Third Edition This Page Intentionally Left Blank Understanding Digital Signal Processing Third Edition
    Understanding Digital Signal Processing Third Edition This page intentionally left blank Understanding Digital Signal Processing Third Edition Richard G. Lyons Upper Saddle River, NJ • Boston • Indianapolis • San Francisco New York • Toronto • Montreal • London • Munich • Paris • Madrid Capetown • Sydney • Tokyo • Singapore • Mexico City Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trade- marks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed with initial capital letters or in all capitals. The author and publisher have taken care in the preparation of this book, but make no expressed or implied war- ranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or con- sequential damages in connection with or arising out of the use of the information or programs contained herein. The publisher offers excellent discounts on this book when ordered in quantity for bulk purchases or special sales, which may include electronic versions and/or custom covers and content particular to your business, training goals, marketing focus, and branding interests. For more information, please contact: U.S. Corporate and Government Sales (800) 382-3419 [email protected] For sales outside the United States please contact: International Sales [email protected] Visit us on the Web: informit.com Library of Congress Cataloging-in-Publication Data Lyons, Richard G., 1948- Understanding digital signal processing / Richard G. Lyons.—3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-13-702741-9 (hardcover : alk. paper) 1.
    [Show full text]
  • Discrete Time Systems and Signal Processing
    EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING M.I.E.T. ENGINEERING COLLEGE (Approved by AICTE and Affiliated to Anna University Chennai) TRICHY – PUDUKKOTTAI ROAD, TIRUCHIRAPPALLI – 620 007 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING COURSE MATERIAL EE6403 & DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING II YEAR- IV Semester M.I.E.T 3 DEPT OF ECE EE6403 DISCRETE TIME SYSTEMS & SIGNAL PROCESSING M.I.E.T. ENGINEERING COLLEGE (Approved by AICTE and Affiliated to Anna University Chennai) Sub.code:EE6403 TRICHY – PUDUKKOTTAI ROAD, TIRUCHIRAPPALLIBranch/Year/Sem – 620 :EEE/II/IV 007 Sub Name: DISCRETE TIME SYSTEM AND SIGNAL PROCESSING Staff Name: Ms.P.Delphine Mary _____________________________________________________________________________ UNIT I INTRODUCTION 9 Classification of systems: Continuous, discrete, linear, causal, stable, dynamic, recursive, time variance; classification of signals: continuous and discrete, energy and power; mathematical representation of signals; spectral density; sampling techniques, quantization, quantization error, Nyquist rate, aliasing effect. UNIT II DISCRETE TIME SYSTEM ANALYSIS 9 Z-transform and its properties, inverse z-transforms; difference equation – Solution by z-transform, application to discrete systems - Stability analysis, frequency response – Convolution – Discrete Time Fourier transform , magnitude and phase representation. UNIT III DISCRETE FOURIER TRANSFORM & COMPUTATION 9 Discrete Fourier Transform- properties, magnitude and phase representation - Computation of DFT using FFT algorithm
    [Show full text]