What is discretization in signals and systems

Properties and application of time-discrete systems

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1 Faculty of Computer Science Institute for Applied Computer Science, Professorship for Technical Information Systems Properties and Application of Discrete-Time Systems Dresden, 3.8.2

2 Outline Preliminary remarks Properties of time-discrete systems Application of time-discrete systems Summary and outlook TU Dresden, Time-discrete systems Slide 2 of 65

3 Outline Preliminary remarks Properties of time-discrete systems Application of time-discrete systems Summary and outlook TU Dresden, Time-discrete systems Slide 3 of 65

4 Preliminary remarks If you have any questions or problems Author and contact person: Dipl.-Inf. Denis Stein Website: TU Dresden, Discrete Time Systems Slide 4 of 65

5 Outline Preliminary remarks Properties of time-discrete systems Repetition Description and discretization Signals System properties Significance of the weight sequence Basic system types Nomenclature Convolution Image area [] TU Dresden, Time-Discrete Systems Slide 5 of 65

6 Repetition Definition of a time-discrete system In time-discrete systems, only time-discrete signals occur. Their value is only known for integer multiples k of the sampling period T (duration between two sampling times). Description by difference equations Example: Processing of measured value sequences in the computer x (kt) system y (kt) TU Dresden, Discrete Time Systems Slide 6 of 65

7 Repetition Classification of signals x (t) x (t) time-continuous, value-continuous t time-continuous, value-discrete t x (t) x (t) time-discrete, value-continuous t time-discrete, value-discrete TU Dresden, time-discrete systems Slide 7 of 65 t

8 Repetition of definition of LTI system A system is an LTI system if applicable. it has the following properties: linear, time-invariant and causal. LTI is the English abbreviation for Linear, Time-Invariant TU Dresden, Zeitdiskrete Systeme Slide 8 of 65

9 Repetition Description of any continuous-time LTI systems linear differential equation with constant coefficients: n n d d d an y t an y t a y t a y t n n dt dt dt m m d d d b x t b x t bm x t b m m x t m dt dt dt a i n; i, n i b j m; j, m j x (t) LTI system y (t) TU Dresden, Discrete Time Systems Slide 9 of 65

10 Description and discretization Description of any time-discrete LTI systems x (kt) LTI system linear difference equation with constant coefficients: m an ykn T an ykn T ayk T ay kt bx kt bxk T bxkm T bxkm T m Prerequisite: equidistant sampling y (kt ) TU Dresden, Discrete Time Systems Slide from 65

11 Description and discretization Description of any time-discrete LTI systems II linear difference equation with constant coefficients: n m a y k n T a y k n T n a y k T a y kt b x kt b x k T b x k m T b x k m T m a i n; i, n i b j m; j, m j k TU Dresden, Discrete Time Systems Slide from 65

12 Description and discretization Discretization of any time-continuous LTI systems Discretization is the determination of difference equations from differential equations (each linear and with constant coefficients). in the form of three simple replacement rules that can be applied to any differential equation. TU Dresden, Discrete Time Systems Slide 2 of 65

13 Description and discretization Discretization of any time-continuous LTI systems II Procedure Backward rectangular rule: Replace all occurrences of in the differential equation. dt through T, 2. t through kt and 3. d through. The following applies: T is the sampling period, k is its integer multiple and applied to any time-discrete signal x (kt): x kt x kt x k T (backward difference). TU Dresden, Discrete Time Systems Slide 3 of 65

14 Description and discretization Discretization of any time-continuous LTI systems III Example: Approximation of a trapezoidal signal x (t) (time-continuous signal; orange solid line) by backwards rectangles (dark blue dotted line) results in a time-discrete signal x (kt) (the values ​​are marked by black triangles ) x (t), x (kt) t (in s) TU Dresden, Discrete Time Systems Slide 4 of 65

15 Signals repetition: Definition of test signals Selected signals for identifying and testing systems in some cases not physically realizable, but mathematically easy to handle Examples: Impulse jump ramp TU Dresden, Discrete-Time Systems Slide 5 of 65

16 Signals Definition of time-discrete (unit) impulse Impulse: x (kt) kkx (kt) Unit impulse: x kt k (kt) k Response to time-discrete unit impulse: Weight sequence (time-discrete unit impulse response) g (kt) k TU Dresden, Time-Discrete Systems Slide 6 from 65

17 Signals Definition of time-discrete (unit) jump Jump: x (kt) kkx (kt) Unit jump: x kt k (kt) k Response to time-discrete unit jump: Transition sequence (time-discrete unit jump response) h (kt) k The following applies (without proof): h (kt) g (jt) kj TU Dresden, Discrete Time Systems Slide 7 of 65

18 Signals Definition of time-discrete (unitary) ramp Ramp: x (kt) k m k k x (kt) Unit ramp: x kt k k k k Response to time-discrete unit ramp: Time-discrete unit ramp response TU Dresden, Time-discrete systems Slide 8 of 65

19 System properties Question Which known system properties can be transferred from the time-continuous to the time-discrete? Idea: Replace the continuous time parameter t with the discrete kt in the respective definitions if necessary TU Dresden, Time Discrete Systems Slide 9 of 65

20 System properties Answer all, including: dynamics (static or dynamic system) memory q (kt) linearity (linear or nonlinear system) amplification and superposition principle (superposition principle) causality (causal or acausal system) causality principle time variance (time invariant or time variant system) displacement principle stability ((BIBO-) stable or (BIBO-) unstable system) Limitation of the amplitude of the output signal TU Dresden, Discrete Time Systems Slide 2 of 65

21 Significance of the weight sequence Observation If the weight sequence g (kt) is known, the behavior of a time-discrete LTI system and thus its reaction to any input signal x (kt) is clearly described. x (kt) g (kt) y (kt) Question: Is it possible to prove system properties again by simply examining g (kt)? Answer: yes. TU Dresden, Discrete Time Systems Slide 2 of 65

22 Significance of the weight sequence Reconsidered system properties Dynamics: static system: g (kt) = dynamic system: g (kt) in other words: The amplitude of weight sequences of static systems is only different for k = from. TU Dresden, Discrete Time Systems Slide 22 of 65

23 Significance of the weight sequence System properties considered again II Type of time parameter: time-discrete system: g (kt) time-discrete time-continuous system: g (t) time-continuous in other words: The signal property (time parameter of g (t) or g (kt)) corresponds to System property. TU Dresden, Discrete Time Systems Slide 23 of 65

24 Meaning of the Weight Sequence System Properties Considered Again III Causality: causal system: g (kt <) = acausal system: g (kt <) in other words: the amplitude of weight sequences of causal systems is different only for k. TU Dresden, Discrete Time Systems Slide 24 of 65

25 Significance of the weight sequence System properties considered again IV Stability: (BIBO-) stable system: T kg kt (BIBO-) unstable system: in other words: the sum of the absolute values ​​of the weight sequence values ​​is finite in (BIBO-) stable systems (T is per se finite) .. Consequence: Stability verification is possible directly at g (kt) 2. Consequence: Instead of an infinite number of limited input signals, only one signal has to be examined for the weight sequence. TU Dresden, Time Discrete Systems Slide 25 of 65 T k g kt

26 Significance of the weight sequence Examples g (kt) 2 k Properties of the system: dynamic g (kt) time-discrete g (kt) time-discrete acausal g (kt <) (BIBO-) unstable T k g kt TU Dresden, Time-Discrete Systems Slide 26 of 65

27 Meaning of the weight sequence Examples II g (kt) 2 k Properties of the system: static g (kt) = time-discrete g (kt) time-discrete causal g (kt <) = (BIBO-) stable T kg kt TU Dresden, time-discrete systems Slide 27 from 65

28 Significance of the weight sequence Examples III g (kt) 2 k Properties of the system: dynamic g (kt) time-discrete g (kt) time-discrete causal g (kt <) = (BIBO-) stable T kg kt TU Dresden, time-discrete systems Slide 28 from 65

29 Basic system types Definition of proportional system (P system) (differential) equation: Difference equation: P yt K xty kt K x kt characteristic parameter: proportional coefficient KPP x (kt) y (kt) x (kt) P y (kt) kk TU Dresden , Discrete Time Systems Slide 29 of 65

30 basic system types Definition of integral system (I system) (differential) equation: Difference equation: I yt K xd characteristic parameter: Integrating factor KI t I y kt yk TKT x kt x (kt) y (kt) x (kt) I y (kt ) kk TU Dresden, Discrete Time Systems Slide 3 of 65

31 Basic system types Definition of differential system (D system) d (differential) equation: yt KD xt K dt Difference equation: y kt D x kt xk TT Characteristic parameter: Difference coefficient KD x (kt) y (kt) x (kt) D y ( kt) kk TU Dresden, Discrete Time Systems Slide 3 of 65

32 basic system types Definition of dead time system (T t system) (differential) equation: ytxt T t Tt difference equation: y kt xkn T, n T characteristic parameters: dead time T tx (kt) y (kt) x (kt) T ty (kt ) kk TU Dresden, Discrete Time Systems Slide 32 of 65

33 Basic system types Definition of the delay system. Order (T system) d (differential) equation: T ytytxt dt Difference equation: y kt yk T x kt, T characteristic parameter: delay time TTT x (kt) y (kt) x (kt) T y (kt) kk TU Dresden, Discrete Time Systems Slide 33 of 65

34 Basic System Types Repetition: Observations Properties of the five basic system types (P, I, D, T t, T): All five are LTI systems (linear, time-invariant, causal). All five can optionally be time-continuous or time-discrete. The P-system is static; the other four systems are dynamic. An interconnection of LTI systems in the form of a series, parallel or circular structure results in an LTI system. TU Dresden, Discrete Time Systems Slide 34 of 65

35 Nomenclature Designation of the five basic system types P: proportional system I: integral system D: differential system T t: dead time system T: delay system. Regulations TU Dresden, Discrete Time Systems Slide 35 of 65

36 Nomenclature Series structure Characterization of the structure: Hyphen x (kt) x (kt) = x (kt) System x 2 (kt) = System 2 y (kt) x 3 (kt) = y 2 (kt) System 3 y 3 ( kt) y (kt) = y 3 (kt) Nomenclature for this action plan (signal flow graph): System System 2 System 3 (System i is the nomenclature for the i-th subsystem) TU Dresden, Time Discrete Systems Slide 36 of 65

37 Nomenclature Parallel structure Characterization of the structure: space x (kt) = x (kt) system y (kt) x (kt) x 2 (kt) = x (kt) system 2 y 2 (kt) y (kt) = y ( kt) + y 2 (kt) + y 3 (kt) x 3 (kt) = x (kt) System 3 y 3 (kt) Nomenclature for this action plan (signal flow graph): System System 2 System 3 TU Dresden, Discrete Time Systems Slide 37 of 65

38 Nomenclature circular structure (feedback circuit) Characterization of the structure: none defined (regardless of whether positive (+) or negative (-)) x (kt) x (kt) = x (kt) ± y 2 (kt) system y (kt) y (kt) = y (kt) y 2 (kt) System 2 x 2 (kt) = y (kt) Nomenclature for this action plan (signal flow graph): none defined TU Dresden, Discrete Time Systems Slide 38 of 65

39 Convolution Definition of convolution x (kt) LTI system (g (kt)) y (kt) Determination of y (kt) from g (kt) and x (kt) using the convolution sum: y kt g kt x kt g (kt) is weight sequence (time-discrete unit impulse response) Note: j: gkj T x jt * is a convolution operator stands for multiplication TU Dresden, Time-Discrete Systems Slide 39 of 65

40 convolution observations so far have also been convolved by using the solution of the convolution sum, e.g. y (kt) = KP x (kt) for a P system knowledge of the weight sequence (time-discrete unit impulse response) g (kt) is necessary. Consequence: If g (kt) is known, the behavior of a time-discrete LTI system and thus its reaction to any input signal x (kt) is clearly described. x (kt) LTI system (g (kt)) y (kt) Problem: Convolution sum is difficult to solve! TU Dresden, Discrete Time Systems Slide 4 of 65

41 Convolution Observations II The following applies: g kt kt g kt in other words: Convolution with a time-discrete unit impulse kt delivers weight sequence (time-discrete unit impulse response) g (kt) g kt kt h kt, in other words: Convolution with a time-discrete unit jump kt delivers a transition sequence (time-discrete unit jump response ) h (kt) g kt x kt x kt g kt in other words: Convolution is commutative TU Dresden, Discrete Time Systems Slide 4 of 65

42 Image area, overview Transformation of time-discrete functions (sequences) f (kt) here: input and output signals as well as weight sequence in the time domain in functions F (z) in the image area: time area: integer variable k lowercase letters for formula symbols Image area: complex variable Ts e uppercase letters for formula symbols Here and below, z image area stands for image area of ​​the z-transformation TU Dresden, Discrete-Time Systems Slide 42 of 65

43 Image area Definition of z-transformation Forward transformation (transformation from time to image area): F z Z f kt Prerequisite: f (kt <) = (always fulfilled here) Notation: F zf kt: Designation as one-sided z-transformation, since lower Summation limit is kf kt zk TU Dresden, Discrete Time Systems Slide 43 of 65

44 Image area definition z-transformation II Here, too, use of so-called correspondences in order to avoid solving the sum or the integral for the forward and backward transformation. Identifiers used: a, a 2 are any real numbers n is any natural number f (kt), f (kt), f 2 (kt) are any time-discrete signals TU Dresden, Time-Discrete Systems Slide 44 of 65

45 Image area definition z-transformation III Linearity: in other words: amplification by constant factors (a, a 2) and addition remain unaffected by (back and forth) transformation Convolution in the time domain: af kt af kt a Z f kt a Z f kt f kt f kt Z f kt Z f kt 2 2 in other words: the convolution operation (convolution sum) in the time domain becomes multiplication in the image domain solving the convolution sum in the time domain is simplified to solve products in the image domain TU Dresden, Time Discrete Systems Slide 45 from 65

46 Image area definition z-transformation IV kz summation in the time domain: f jt Z f kt jz in other words: z the sum becomes factor zz discrete-time unit jump: kt z in other words: z discrete-time unit jump is simplified to z convolution with discrete-time unit jump in the time domain to multiply by z in the image area z transition sequence is the sum of the weight sequence TU Dresden, Discrete Time Systems Slide 46 of 65

47 Image area definition z-transformation V time-discrete unit pulse: kt in other words: time-discrete unit pulse becomes a number Convolution with time-discrete unit pulse in the time domain simplifies to multiplication with in the image area right shift in the time domain: fk nt Z nf kt z in other words: right shift factor z -n the solving of difference equations in the time domain is simplified for solving algebraic equations (mostly polynomials in z) in the picture area TU Dresden, Zeitdiskrete Systeme Slide 47 of 65

48 Image area Definition of z-transformation VI Definition of the inverse transformation, further correspondence and comments see, for example, the collection of formulas for the course Methods of Quality Control in Technical Processes (on request, sent by). More on transformations in the lecture Methods of Quality Control in Technical Processes in the summer semester 22nd TU Dresden, Time Discrete Systems Slide 48 of 65

49 Image area Solution of the convolution sum using the z-transformation. Transformation of the input signal x (kt) and the weight sequence g (kt): X z Z x kt Z g kt G z 2nd solution of the convolution in the image area where this is simplified for multiplication Transformed Y (z) of the output signal y (kt) : G z X z Y z 3. Reverse transformation of the output signal: y kt ZY z Compare an earlier handout (Fig. 6.24) on convolution using the Laplace transformation. TU Dresden, Discrete Time Systems Slide 49 of 65

50 Transfer function screen area If the transfer function G (z) is known, the behavior of a time-discrete LTI system and thus its reaction to any input signal x (kt) is clearly described. x (kt) X (z) g (kt) G (z) The following applies: Z g kt G z Y z X z Note: The transfer function G (z) in the image area should not be confused with the transition sequence (time-discrete standard step response) h (kt) in the time domain. y (kt) Y (z) TU Dresden, Discrete Time Systems Slide 5 of 65

51 Image area transfer function II The well-known transformation rules for effect plans (signal flow graphs) also apply to discrete-time LTI systems (see also an earlier handout (Fig. 6.28)). The statements on transfer functions of complex, static as well as dynamic LTI systems continue to apply (replace s with z in the respective equations): Series structure: Product of the individual transfer functions Parallel structure: Sum of the individual transfer functions Circular structure: Quotient with both transfer functions TU Dresden, time-discrete systems Slide 5 of 65

52 Image area transfer function III The statements for the proportional value of complex static LTI systems still apply: Series structure: Product of the individual proportional coefficients Parallel structure: Sum of the individual proportional coefficients Circular structure: Quotient with both proportional coefficients TU Dresden, Discrete Time Systems Slide 52 of 65

53 Outline Preliminary remarks Properties of time-discrete systems Application of time-discrete systems Time-discrete filter Realization Summary and outlook TU Dresden, time-discrete systems Slide 53 of 65

54 Use of time-discrete systems Time-discrete filter Repetition: Description of any time-discrete LTI systems x (kt) LTI system y (kt) linear difference equation with constant coefficients: a y k n T a y k n T n bm x k ​​m T bm x k ​​m T a i n; i, n b j m; j, m k n a y k T a y kt b x kt b x k T i j TU Dresden, Time Discrete Systems Slide 54 of 65

55 Application of time-discrete systems Time-discrete filter Overview A distinction is made between two cases of the linear difference equation with constant coefficients: FIR filter IIR filter Use: Interference suppression (e.g. in audio and video technology) TU Dresden, Time-Discrete Systems Slide 55 of 65

56 Use of time-discrete systems Time-discrete filter FIR filter FIR is the abbreviation for Finite Impulse Duration Response (finite duration of the impulse response). FIR filters are also referred to as mean value filters or non-recursive filters. Difference equation: y kt b x kt b x k T m m a, ai i n; i, n b j m; j, m j k b x k m T b x k m T TU Dresden, Discrete Time Systems Slide 56 of 65

57 Application of time-discrete systems Time-discrete filter FIR filter II Examples of type FIR filters: P, D and T t system Further example: y kt x kt xk T xk 2T 4 2 x (kt) bbb 2 y (kt) x (kt) FIR y (kt) observations: k impulse response finite (equal from k = 3) ky (t) =, 25 = b, y (t) = = b, y (2t) =, 5 = b 2 TU Dresden, Discrete Time Systems Slide 57 of 65

58 Use of Discrete-Time Systems Discrete-Time Filter IIR-Filter IIR is the abbreviation for Infinite Impulse Duration Response (infinite duration of the impulse response) IIR filters are also known as recursive filters. Difference equation: y kt a, ai i n; i, n b j m; j, m j k m n k T x k m T bm x k ​​mt a y k n T a y k n T n a y b x kt b x k T b TU Dresden, Time Discrete Systems Slide 58 of 65

59 Use of time-discrete systems Time-discrete filter IIR filter II Examples of type IIR filters: I- and T -system Example T -system (time-discrete low-pass filter): y kt yk T x kt ,, x (kt) y (kt) x ( kt) IIR y (kt) kk Observations: Impulse response infinite: Disturbance almost undamped: Disturbance heavily damped, but also useful signal TU Dresden, Discrete Time Systems Slide 59 of 65

60 Use of Discrete-Time Systems Discrete-Time Filters Outlook More on the design of filters, especially the consideration in the frequency range in the 7th slide set on SOI from summer semester 29 and in the script (p. 59ff.). TU Dresden, Time Discrete Systems Slide 6 of 65

61 Application of time-discrete systems Realization of the signal processor Linear difference equations with constant coefficients can be processed by (digital) signal processors. They specialize in the fast (vectorial) calculation of addition and multiplication. More on this in the script (p. 65f.). TU Dresden, Time Discrete Systems Slide 6 of 65

62 Outline Preliminary remarks Properties of time-discrete systems Application of time-discrete systems Summary and outlook TU Dresden, Time-discrete systems Slide 62 of 65

63 Summary and Outlook Summary Many findings from the world of continuous-time systems are similar to those of discrete-time systems (mostly except for changing the time parameter and replacing integrals and differentials with sums). These include system properties (e.g. LTI, stability), the importance of the weight function / sequence, convolution and the two image areas. This penultimate co-lecture served not only to introduce new knowledge about time-discrete systems, but also to repeat the knowledge about time-continuous systems and thus as a review of almost all previous co-lecture contents. As the first application of time-discrete systems, time-discrete filters (FIR, IIR) were presented. TU Dresden, Discrete Time Systems Slide 63 of 65

64 Summary and Outlook Outlook Exercise 5 deepens the handling of time-discrete systems (including convolution and discretization). The first application example is the time-discrete filter. Exercise 6 looks at the time-discrete controller as a further application example of time-discrete systems. The next co-lecture and exercise 6 deal with: Control: How can I specifically influence the behavior of one system through another system? Studies: Which courses are recommended after successfully completing process control? TU Dresden, Discrete Time Systems Slide 64 of 65

65 Knowledge closes gaps TU Dresden, Discrete-Time Systems Slide 65 of 65