1 / 37

CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain

CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain. YANG Jian jianyang@ynu.edu.cn School of Information Science and Technology Yunnan University. Outline. Discrete-Time Signals Typical Sequences and Sequence Representation The Sampling Process Discrete-Time Systems

qabil
Télécharger la présentation

CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain YANG Jian jianyang@ynu.edu.cn School of Information Science and Technology Yunnan University

  2. Outline • Discrete-Time Signals • Typical Sequences and Sequence Representation • The Sampling Process • Discrete-Time Systems • Time-Domain Characterization of LTI Discrete-Time Systems • Finite-Dimensional LTI Discrete-Time Systems • Correlation of Signals • Summary 云南大学滇池学院课程：现代信号处理数字信号处理

  3. Discrete-Time Signals • Basic signals • Unit sample or unit impulse sequence • Unit step sequence • Exponential sequence • Signal classification • Continuous-time / discrete-time signals • Deterministic / random signals • Energy signals • signals with finite energy • Power signals • signals with finite power • Energy signals have zero power, and power signals have infinite energy 云南大学滇池学院课程：现代信号处理数字信号处理

  4. Discrete-Time Signals • Time-Domain Representation • Sequence of numbers: • — sequence • — samples • — sample value or nth samples, a real or complex value • Figure of sequence: • is defined only for integer value of 云南大学滇池学院课程：现代信号处理数字信号处理

  5. Discrete-Time Signals • Operation on sequences • Basic operation • Adder / Subtraction: • Scalar multiplication ( gain / attenuation ): • Delay / Advance: • Combination of Basic Operations • Multiplier: • Linear combination: 云南大学滇池学院课程：现代信号处理数字信号处理

  6. Discrete-Time Signals • Operation on sequences • Sampling Rate Alteration ( special operations of for discrete-time signals ) • Up-sampling: • Down-sampling: 云南大学滇池学院课程：现代信号处理数字信号处理

  7. Discrete-Time Signals • Classification of Sequences • The number of sequences: finite / infinite • Finite-length sequences: • Symmetry • conjugate-symmetric ( even ): • conjugate-antisymmetric ( odd ): • Periodity: periodic / aperiodic • Periodic sequence: 云南大学滇池学院课程：现代信号处理数字信号处理

  8. Discrete-Time Signals • Classification of Sequences • Energy and Power Signals 云南大学滇池学院课程：现代信号处理数字信号处理

  9. Discrete-Time Signals • Classification of Sequences • Other types of Classification • Bounded: • Absolutely summable: • Square-summable: 云南大学滇池学院课程：现代信号处理数字信号处理

  10. Typical Sequences and Sequence Representation • Some Basic Sequences • Unite sample sequence: • An arbitrary sequence can be represented by unite sample sequence in time-domain • Unite step sequence: 云南大学滇池学院课程：现代信号处理数字信号处理

  11. Typical Sequences and Sequence Representation • Sinusoidal and Exponential Sequences • The real sinusoidal sequence: • The exponential sequence: • The sinusoidal sequence are periodic of period N as long as is an integer multiple of . The smallest possible N is the fundamental period of the sequence. 云南大学滇池学院课程：现代信号处理数字信号处理

  12. Typical Sequences and Sequence Representation • Some Typical Sequences • Regular window sequence: • Real exponential sequence: • Representation of an Arbitrary Sequence • An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version. 云南大学滇池学院课程：现代信号处理数字信号处理

  13. The Sampling Process • Uniform sampling: • Often the discrete-time sequence is developed by uniformly sampling a continuous-time signal : • the sampling frequency • the sampling angular frequency 云南大学滇池学院课程：现代信号处理数字信号处理

  14. The Sampling Process • Aliasing: • When , a continuous-time sinusoidal signal of higher frequency would acquire the identity of a sinusoidal sequence of lower frequency after sampling. e.g. 云南大学滇池学院课程：现代信号处理数字信号处理

  15. H [ ] Output y(n) Input x(n) Discrete-Time System • Discrete-time system • Simple Discrete-Time Systems • The accumulator • The M-point moving-average filter • The factor-of-L interpolator 云南大学滇池学院课程：现代信号处理数字信号处理

  16. Discrete-Time System • Classification of Discrete-Time System • Linear system: • Shift-Invariant System: • LTI System: The linear time-invariable discrete-time system satisfies both the linear and the time-invariable properties. 云南大学滇池学院课程：现代信号处理数字信号处理

  17. Discrete-Time System • Classification of Discrete-Time System • Causal System: In a causal discrete-time system, the th output sample depends only on input samples for and does notdepend on input samples for . 云南大学滇池学院课程：现代信号处理数字信号处理

  18. Discrete-Time System • Classification of Discrete-Time System • Stable System: Definition of bounded-input, bounded-output ( BIBO ) stable. • Passive and Lossless Systems • The passivity: • The losslessness: the same energy 云南大学滇池学院课程：现代信号处理数字信号处理

  19. Discrete-Time System • Impulse and Step Responses • Input sequence → output sequence • Impulse response : • Step response : 云南大学滇池学院课程：现代信号处理数字信号处理

  20. Time-Domain Characterization of LTI Discrete-Time Systems • Input-Output Relationship • The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum: • The operation • Step 1, time-reverse: • Step 2, shift n sampling period: • Step 3, product: • Step 4, summing all samples: 云南大学滇池学院课程：现代信号处理数字信号处理

  21. Time-Domain Characterization of LTI Discrete-Time Systems • Some useful properties of the convolution operation • Commutative: • Associative for stable and single-sided sequences: • Distributive: 云南大学滇池学院课程：现代信号处理数字信号处理

  22. Time-Domain Characterization of LTI Discrete-Time Systems • Simple Interconnection Schemes • Cascade Connection: • Parallel Connection: • Inverse System: 云南大学滇池学院课程：现代信号处理数字信号处理

  23. Time-Domain Characterization of LTI Discrete-Time Systems • Stability Condition in Terms of the Impulse Response • An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.: • Causality Condition in Terms of the Impulse Response • An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition: 云南大学滇池学院课程：现代信号处理数字信号处理

  24. Finite-Dimensional LTI Discrete-Time Systems • The difference equation: • An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form: • The order of the system is given by max( N, M ) 云南大学滇池学院课程：现代信号处理数字信号处理

  25. Finite-Dimensional LTI Discrete-Time Systems • Total Solution Calculation • The complementary solution • The homogeneous difference equation: • The characteristic equations: 云南大学滇池学院课程：现代信号处理数字信号处理

  26. Finite-Dimensional LTI Discrete-Time Systems • Total Solution Calculation • The particular solution is of the same form as specified input . • The total solution: 云南大学滇池学院课程：现代信号处理数字信号处理

  27. Finite-Dimensional LTI Discrete-Time Systems • Zero-Input Response and Zero-State Response • zero-input response = complementary solution with initials; • zero-state response = the convolution sum of x(n) and h(n). 云南大学滇池学院课程：现代信号处理数字信号处理

  28. Finite-Dimensional LTI Discrete-Time Systems • Impulse Response Calculation 云南大学滇池学院课程：现代信号处理数字信号处理

  29. Finite-Dimensional LTI Discrete-Time Systems • Impulse Response Calculation • The solutions 云南大学滇池学院课程：现代信号处理数字信号处理

  30. Finite-Dimensional LTI Discrete-Time Systems • Location of Roots of Characteristic Equation for BIBO Stability • A casual LTI system characteristic of a linear constant coefficient difference equation is BIBO stable, if the magnitude of each of the roots its characteristic equation is less than 1. • The necessary and sufficient condition: 云南大学滇池学院课程：现代信号处理数字信号处理

  31. Finite-Dimensional LTI Discrete-Time Systems • Classification of LTI System • Based on impulse response length • Finite impulse response ( FIR ): • Infinite impulse response ( IIR ): 云南大学滇池学院课程：现代信号处理数字信号处理

  32. Finite-Dimensional LTI Discrete-Time Systems • Classification of LTI System • Based on the output calculation process • Non-recursive system: If the output sample can be calculated sequentially, knowing only the present and pass input samples. • Recursive system: If the computation of the output involves past output samples. • Remarks: • FIR — Non-recursive • IIR— Recursive 云南大学滇池学院课程：现代信号处理数字信号处理

  33. Correlation of Signals • Definitions • A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence defined by: • The autocorrelation sequence of x(n) is given by: 云南大学滇池学院课程：现代信号处理数字信号处理

  34. Correlation of Signals • Properties of Autocorrelation and Cross-correlation Sequences • Set and as energies of the sequences x(n) and y(n) , then we can get or equivalently • If y(n) = x(n), then • The sample value of the autocorrelation sequence has its max value at zero lag ( l = 0 ). 云南大学滇池学院课程：现代信号处理数字信号处理

  35. Correlation of Signals • Properties of Autocorrelation and Cross-correlation Sequences • If , where N is integer and b>0 is an arbitrary number. In this case , so 云南大学滇池学院课程：现代信号处理数字信号处理

  36. Correlation of Signals • Normalized Forms of Correlation: • Correlation Computation for Power and Periodic Signals • Power signals: • Periodic signals: 云南大学滇池学院课程：现代信号处理数字信号处理

  37. Summary • The LTI system has numerous applications in practice. • The LTI system can be described by an input-output relation composed of a linear constant coefficient difference equation. • The LTI discrete-time system is usually classified in terms of the length of its impulse response. 云南大学滇池学院课程：现代信号处理数字信号处理

More Related