1. 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理 ( 双语 ) 》 http://course.sdu.edu.cn/bdsp.html

2. 2 6.0 Introduction 6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations 6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations 6.3 Basic Structures for IIR Systems 6.4 Transposed Forms 6.5 Basic Network Structures for FIR Systems §6 structures for discrete-time system

3. 3 Structures for Discrete-Time Systems 6.0 Introduction

4. 4 Characterization of an LTI System:  Impulse Response  z-Transform: system function  Difference Equation  converted to a algorithm or structure that can be realized in the desired technology, when implemented with hardware.  Structure consists of an interconnection of basic operations of addition, multiplication by a constant and delay 6.0 Introduction → Frequency response

5. even if we only wanted to compute the output over a finite interval, it would not be efficient to do so by discrete convolution since the amount of computation required to compute y [n] would grow with n. 5 Example: find the output of the system Illustration for the IIR case by convolution IIR Impulse Response with input x[n]. Solution1:

6. 6 Example: find the output of the system computable recursively The algorithm suggested by the equation is not the only computational algorithm, there are unlimited variety of computational structures (shown later). with input x[n]. Solution2:

7. 7 Why Implement system Using Different Structures?  Equivalent structures with regard to their input-output characteristics for infinite-precision representation, may have vastly different behavior when numerical precision is limited.  Effects of finite-precision of coefficients and truncation or rounding of intermediate computations are considered in latter sections.  Computational structures(Modeling methods):  Block Diagram  Signal Flow Graph

8. 8 Structures for Discrete-Time Systems 6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations

9. 9 + x 1 [n] x 2 [n] x 1 [n] + x 2 [n] Adder x[n] a ax[n] Multiplier x[n] x[n-1] z1z1 Unit Delay (Memory, storage) Three basic elements: M sample Delay z -M x[n-M]

10. 10 Ex. 6.1 draw Block Diagram Representation of a Second-order Difference Equation x[n] + + b0b0 a1a1 z1z1 z1z1 a2a2 y[n] y[n-1] y[n-2] Solution:

11. 11 N th -Order Difference Equations Form changed to a[0] normalized to unity

12. 12 Block Diagram Representation (Direct Form I) v[n] + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N]

13. 13 + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N] Block Diagram Representation (Direct Form I) v[n]

14. 14 + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N] Block Diagram Representation (Direct Form I) v[n] Implementing zeros Implementing poles

15. 15 Block Diagram Representation (Direct Form I) v[n] How many Adders? How many multipliers? How many delays? How many Adders? How many multipliers? How many delays? + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N] N N N+M +M +M+1

16. 16 + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N] Block Diagram Representation (Direct Form I) v[n]

17. 17 Block Diagram Representation (Direct Form II) + z1z1 z1z1 + z1z1 + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N (or called Canonic direct Form)

18. 18 Block Diagram Representation (Direct Form II) + z1z1 z1z1 + z1z1 + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N

19. 19 + z1z1 z1z1 + z1z1 + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N Block Diagram Representation (Direct Form II)

20. 20 Block Diagram Representation (Direct Form II) + z1z1 z1z1 + z1z1 + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N Implementing zeros Implementing poles

21. 21 Block Diagram Representation (Direct Form II) How many Adders? How many multipliers? How many delays? How many Adders? How many multipliers? How many delays? + z1z1 z1z1 + z1z1 + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N N N N+M +M +M+1

22. 22 Block Diagram Representation (Canonic Direct Form or direct Form II) How many Adders? How many multipliers? How many delays? max(M, N) How many Adders? How many multipliers? How many delays? max(M, N) + + + b0b0 b1b1 bN1bN1 bNbN x[n] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] Assume M = N N N +M +M+1 N

23. 23 Ex. 6.2 draw Direct Form I and Direct Form II implementation of an LTI system x[n] + z1z1 z1z1 + 1.5  0.9 y[n] w[n-1] w[n-2] + 1 2 w[n] + z1z1 1 2 x[n] x[n-1] + z1z1 z1z1 + 1.5  0.9 y[n] y[n-1] y[n-2] v[n] Solution:

24. 24 Structures for Discrete-Time Systems 6.2 Signal Flow Graph( 信号流图 ) Representation of Linear Constant- Coefficient Difference Equations

25. 25 6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations Associated with each node is a variable or node value, being denoted w j [n].  A Signal Flow Graph is a network of directed branches ( 有向支路 )that connect at nodes( 节点 ). w j [n] w k [n] Node j Node k 梅森 (Mason) 信号流图

26. 26 Nodes And Branches w j [n] w k [n] a or z -1 Brach ( j, k ) Each branch has an input signal and an output signal. Input w j [n] Output: A linear transformation of input, such as constant gain and unit delay. We will only consider linear Signal Flow Graph An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node. Node j Node k if omitted, it indicates unity

27. 27 Source Nodes ( 源点 )  Nodes that have no entering branches x j [n]w k [n] Source node j Sink Nodes ( 汇点 )  Nodes that have only entering branches y k [n]w j [n] Sink node k

28. 28 Example : determine Linear Constant- Coefficient Difference Equations of SFG x[n] y[n]w 1 [n] w 2 [n] a b c d e Source Node Sink Node Solution:

29. 29 Block Diagram vs. Signal Flow Graph x[n] + a z1z1 + b1b1 b0b0 w[n]y[n] x[n] w 1 [n] w 2 [n] w 3 [n] a b1b1 b0b0 1 2 3 4 w 4 [n] y[n] Delay branch cannot be represented in time domain by a branch gain z1z1 Delay branch Canonic direct Form Source Node Sink Node =w 2 [n-1] by z-transform, a unit delay branch has a gain of z -l. By convention, variables is represented as sequences rather than as z-transforms branching point

30. Determine the difference equation (System Function) from the Flow Graph. 30 Block Diagram vs. Signal Flow Graph x[n] + a z1z1 + b1b1 b0b0 w[n] y[n] x[n] w 1 [n] w 2 [n]w 3 [n] a b1b1 b0b0 z1z1 1 2 3 4 w 4 [n] y[n] Solution:

31. Block Diagram vs. Signal Flow Graph 31 Determine difference equation difficult in time-domain

32. 32 Ex. 6.3 Determine the System Function from Flow Graph Solution:

33. 33 Ex. 6.3 Determine the System Function from Flow Graph for causal system :

34. 34 Ex. 6.3 compare two implementation -a-a a x[n]x[n] z -1 y[n]y[n] direct form I implementation requires only one multiplication and one delay (memory) element two multiplication and two delay

35. 35 Structures for Discrete-Time Systems 6.3 Basic Structure for IIR Systems

36. 36 6.3 Basic Structure for IIR Systems  Reduce the number of constant multipliers  Increase speed  Reduce the number of delays  Reduce the memory requirement  others: VLSI design;Modularity; multiprocessor implementations; effects of a finite register length and finite-precision arithmetic  for a rational system function, many equivalent difference equations or network structures exists. one criteria in the choice among these different structures is computational complexity:

37. 37 Basic Structures for IIR Systems  Direct Forms  Cascade Form  Parallel Form

38. 38 6.3.1 Direct Forms v[n] + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N]

39. 39 Direct Form I v[n] + z1z1 z1z1 + z1z1 + b0b0 b1b1 bM1bM1 bMbM x[n] x[n-1] x[n-2] x[n-M] + z1z1 z1z1 + z1z1 + a1a1 aN1aN1 aNaN y[n] y[n-1] y[n-2] y[n-N] b0b0 b1b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2b2 b N-1 bNbN x[n  M+1] a1a1 a2a2 a N-1 aNaN y[n-1] y[n-2] y[n-N] y[n  N+1] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 v[n] Block Diagram Signal Flow Graph

40. 40 Direct Form I Signal Flow Graph b0b0 b1b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2b2 b N-1 bNbN x[n  M+1] a1a1 a2a2 a N-1 aNaN y[n-1] y[n-2] y[n-N] y[n  N+1] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 v[n]

41. 41 Direct Form II x[n] y[n] w[n] b0b0 b1b1 b2b2 b N-1 bNbN a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1

42. 42 Direct Form II x[n] y[n] w[n] b0b0 b1b1 b2b2 b N-1 bNbN a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1

43. 43 Ex. 6.4 draw Direct Form I and Direct Form II structures of system x[n]y[n] z1z1 z1z1 z1z1 z1z1 0.75  0.125 2 x[n] y[n] z1z1 z1z1 0.75  0.125 2 Direct Form I Direct Form II Solution:

44. 44 6.3.2 Cascade Form( 串联形式 ) when all the coefficients are real 1 st -order factors represent real zeros at g k and real poles at c k, and the 2 nd -order factors represent complex conjugate pairs of zeros at h k and h * k and poles at d k, d * k

45. 45 Cascade Form 2nd Order System 2nd Order System 2nd Order System 2nd Order System 2nd Order System 2nd Order System A modular structure

46. 46 Cascade Form x[n]y[n] z1z1 z1z1 a 11 a 21 b 11 b 21 b 01 z1z1 z1z1 a 12 a 22 b 12 b 22 b02b02 z1z1 z1z1 a 13 a 23 b 13 b 23 b 03 123 For example, assume N s =3 It is used when implemented with fixed-point arithmetic, the structure can control the size of signals at various critical points because they make it possible to distribute the overall gain of the system.

47. 47 Ex. 6.5 draw the Cascade structures x[n] y[n] z1z1 z1z1 0.75  0.125 2 Direct Form II 1st-order Direct Form II 1st-order Direct Form I Solution:

48. 48 Another Cascade Form x[n] y[n] z1z1 z1z1 a 11 a 21 b 11 b 21 z1z1 z1z1 a 12 a 22 b 12 b 22 z1z1 z1z1 a 13 a 23 b 13 b 23 b0b0 ~ ~ ~ ~ ~ ~ implemented with fixed-point arithmetic when floating-point arithmetic is used and dynamic range is not a problem. used to decrease the amount of computation,

49. 49 6.3.3 Parallel Form

50. 50 Parallel Form Real Poles Complex Poles Poles at zero Group Real Poles in pairs

51. 51 Parallel Form z1z1 z1z1 a1ka1k a2ka2k e0ke0k e1ke1k x[n] y[n] C k z -k C0C0

52. 52 Ex. 6.6 draw parallel-form structures of system 8 x[n]y[n] z1z1 z1z1 0.75  0.125 8 77 Solution 1: If we use 2 nd –order sections,

53. 53 z1z1 0.5 18 8 x[n]y[n] z1z1 0.25  25 Solution 2: If we use 1 st –order sections, Ex. 6.6 draw parallel-form structures of system

54. 54 6.3.4 feedback in the IIR systems z1z1 z1z1 a x[n] y[n] -a2-a2 z1z1 a x[n] y[n] a x[n] y[n] systems with feedback may be FIR Noncomputable network z1z1 a x[n] y[n]

55. 55 Structures for Discrete-Time Systems 6.4 Transposed Forms

56. 56  There are many procedures for transforming signal flow graphs into different forms while leaving the overall system function between input and output unchanged. 6.4 Transposed Forms Flow Graph Reversal or Transposition x[n] y[n] z1z1 a x[n]y[n] z1z1 a  Changes the roles of input and output.  Reverse the directions of all arrows. Transposing doesn’t change the input-output relation

57. 57 Ex. 6.7 determine Transposed Forms for a first-order system z1z1 a x[n]y[n] z1z1 a x[n]y[n] z1z1 a x[n]y[n] Solution:

58. 58  Both have the same system function or difference equation Ex. 6.8 draw Transposed Forms for a basic second-order section Transpose Solution:

59. 59 Ex. 6.8 Transposed Forms for a basic second-order section Transpose b0b0 b1b1 x[n] y[n] b2b2 a1a1 a2a2 z1z1 z1z1 v 1 [n] z1z1 z1z1 b0b0 b1b1 x[n] y[n] b2b2 a1a1 a2a2 z1z1 z1z1 v 1 [n] x[n] y[n] b0b0 b1b1 b2b2 z1z1 z1z1 v 2 [n] a1a1 a2a2 z1z1 z1z1

60. 60 Transposed Direct Form I b0b0 b1b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2b2 b N-1 bNbN x[n  M+1] a1a1 a2a2 a N-1 aNaN y[n-1] y[n-2] y[n-N] y[n  N+1] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 v[n] b0b0 b1b1 x[n] y[n] b2b2 b N-1 bNbN a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 v'[n]

61. 61 Transposed Direct Form II x[n] y[n] w[n] b0b0 b1b1 b2b2 b N-1 bNbN a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1 y[n] x[n] w ' [n] b0b0 b1b1 b2b2 b N-1 bNbN a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1

62. 62 Structures for Discrete-Time Systems 6.5 Basic Structure for FIR Systems

63. 63 6.5 Basic Structure for FIR Systems  For causal FIR systems, the system function has only zeros(except for poles at z = 0). 6.5. 1 Direct Form

64. 64 Direct Form I x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M1]h[M1] h[M]h[M] b0b0 b1b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2b2 b M-1 bMbM x[n  M+1] a1a1 a2a2 a N-1 aNaN y[n-1] y[n-2] y[n-N] y[n  N+1] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 v[n] y[n] x[n-1] x[n-2] x[n  M+1] x[n-M]

65. 65 Direct Form II x[n] y[n] z1z1 z1z1 z1z1 x[n] y[n] w[n] b0b0 b1b1 b2b2 b M-1 bMbM a1a1 a2a2 a N-1 aNaN z1z1 z1z1 z1z1 y[n] h[0]h[1]h[2] h[M1]h[M1] h[M]h[M]

66. 66 x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M1]h[M1] h[M]h[M] Traspostion of Direct Form x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M1]h[M1] h[M]h[M] x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M  1] h[M] tapped delay line structure or transversal filter structure. 抽头延迟线结构 or 横向滤波器结构.

67. 67 6.5.2 Cascade Form x[n] y[n] z1z1 z1z1 b 01 b 11 b 21 z1z1 z1z1 b 02 b 12 b 22 z1z1 z1z1 b 1Ms b 2Ms b 0Ms

68. 68 Cascade Form x[n] y[n] z1z1 z1z1 b 01 b 11 b 21 z1z1 z1z1 b 02 b 12 b 22 z1z1 z1z1 b 1Ms b 2Ms b 0Ms x[n]y[n] z1z1 z1z1 a 11 a 21 b 11 b 21 b 01 z1z1 z1z1 a 12 a 22 b 12 b 22 b 01 z1z1 z1z1 a 13 a 23 b 13 b 23 b 03

69. 69 M is evenM is odd h[M-n]= h[n] h[M-n]=  h[n] 6.5.3 Structures for Linear Phase Systems  A causal FIR system has generalized linear phase if h[n] satisfies: h[M-n]= h[n] for n = 0,1,…,M h[M-n]=  h[n] for n = 0,1,…,M or Type I Type III Type II Type VI x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M  1] h[M] M+1 multiplications

70. 70  For even M and type I or type III systems: 0 M/2 M 0 M  Symmetry means we can half the number of multiplications

71. 71 Type I and III x[n-1] x[n-2]x[n-M/2+1] x[n-M/2] x[n-M] x[n] x[n-M+1] x[n-M+2]x[n-M/2-1] x[n] y[n] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 h[M/2] h[M/2  1] h[0]h[1]h[2] 0 M/2 M 0 M Type III =0 -- - - Type I x[n] y[n] z1z1 z1z1 z1z1 h[0] h[1]h[2] h[M  1] h[M] x[n-1]x[n-2] x[n-M] x[n] x[n-M+1]

72. 72 Type II or Type IV FIR Systems  For odd M and type II or type IV systems: 0 M/2M 0 M

73. 73 Type II and IV  Structure for odd M x[n-(M-1)/2] x[n-(M+1)/2] x[n-1] x[n-2] x[n-M] x[n-M+1] x[n-M+2] x[n] 0 M/2M 0 M - -- - Type II Type IV x[n] y[n] z1z1 z1z1 z1z1 h[0]h[1]h[2] h[M  1] h[M]

74. 74 Type I, and II x[n-1] x[n-2]x[n-M/2+1] x[n-M/2] x [ n-M] x[n] x [ n-M+1] x[n-M+2]x[n-M/2-1] x[n] y[n] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 h[M/2] h[M/2  1] h[0]h[1]h[2] x[n-(M-1)/2] x[n-(M+1)/2] x[n-1] x[n-2] x[n-M] x[n-M+1] x[n-M+2] x[n] Type I Type II

75. 75 6.6 OVERVIEW OF FINITE-PRECISION NUMERICAL EFFECTS  6.6.1 Number Representations A real number can be represented with infinite precision in two's-complement form as where X m is an arbitrary scale factor and the b i s are either 0 or 1. The quantity b 0 is referred to as the sign bit. If b 0 = 0, then 0 ≤ x < X m, and if b 0 = 1, then -X m ≤ x < 0.

76. 76  For a finite number of bits (B +1), the equation above must be modified to so the smallest difference between numbers is 6.6.1 Number Representations the quantized numbers are in the range : -X m ≤ < X m.

77. 77  quantizing a number to (B +1) bits can be implemented by rounding or by truncation, which is a nonlinear memoryless operation. define the quantization error as 6.6.1 Number Representations  The fractional part of can be represented with the positional notation

78. 78  For the case of two's-complement rounding, - Δ/2 < e <Δ/2, and for two's-complement truncation, - Δ< e <0 6.6.1 Number Representations truncation rounding For B =2

79. 79 6.6.2Quantization in Implementing Systems  Consider the following system  A more realistic model would be

80. 80 6.6.2 Quantization in Implementing Systems  In order to analyze it we would prefer

81. 81 6.7.1 Effects of Coefficient Quantization in IIR Systems  When the parameters of a rational system are quantized, The poles and zeros of the system function move.  If the system structure of the system is sensitive to perturbation of coefficients,  The resulting system may no longer meet the original specifications,  and may no longer be stable.

82. 82 6.7 Effects of Coefficient Quantization in IIR Systems  Detailed sensitivity analysis for general case is complicated. Using simulation tools, in specific cases,  Quantize the coefficients and analyze frequency response  Compare frequency response to original response  We would like to have a general sense of the effect of quantization

83. 6.7.1 Effects of Coefficient Quantization in IIR Systems 83  Each root is affected by quantization errors in ALL coefficient  Tightly clustered roots are significantly effected  Narrow-bandwidth lowpass or bandpass filters can be very sensitive to quantization noise Quantization

84. 84 Effects on Roots(poles and zeros)  The larger the number of roots in a cluster the more sensitive it becomes  So second order cascade structures are less sensitive to quantization error than higher order system  Each second order system is independent from each other Quantization

85. 6.7.2 Example of Coefficient Quantization in an Elliptic Filter 85  An IIR bandpass elliptic filter was designed to meet the following specifications:

86. 6.7.2 Example of Coefficient Quantization in an bandpass Elliptic Filter 86  Poles and zeros of H(z) for unquantized Coefficients and 16-bit quantization of the direct form unquantized 16-bit quantization the direct form system cannot be implemented with 16-bit coefficients because it would be unstable

87. 6.7.2 Example of Coefficient Quantization in an bandpass Elliptic Filter 87 16-bit quantizatio n  the cascade form is much less sensitive to coefficient quantization  Magnitude in passband for 16-bit quantization of the cascade form

88. 6.7.2 Example of Coefficient Quantization in an Elliptic Filter 88

89. 89 6.7.3 Poles of Quantized 2nd-Order Sections  Consider a 2nd order system with complex-conjugate pole pair

90. 90 6.7.3 Poles of Quantized 2nd-Order Sections  3-bits  The pole locations after quantization will be on the grid point

91. 91 6.7.3 Poles of Quantized 2nd-Order Sections  7-bits  The pole locations after quantization will be on the grid point

92. 92 Coupled-Form Implementation of Complex-Conjugate Pair  Equivalent implementation of the 2nd order system Twice as many constant multipliers are required to achieve more uniform density.

93. 93 Coupled-Form Implementation of Complex-Conjugate Pair  3-bits  7-bits  Twice as many constant multipliers are required to achieve this more uniform density of quantization grid

94. 94 6.7.4 Effects of Coefficient Quantization in FIR Systems  No poles to worry about only zeros  Direct form is commonly used for FIR systems  Suppose the coefficients are quantized

95. 95 6.7.4 Effects of Coefficient Quantization in FIR Systems  No poles to worry about only zeros  Direct form is commonly used for FIR systems

96. 96 6.7.4 Effects of Coefficient Quantization in FIR Systems  Quantized system is linearly related to the quantization error  Again quantization noise is higher for clustered zeros  However, most FIR filters have spread zeros

97. 97 6.8 EFFECTS OF ROUND-OFF NOISE IN DIGITAL FILTERS  Difference equations implemented with finite-precision arithmetic are non-linear systems.  Second order direct form I system

98. 98 6.8 EFFECTS OF ROUND-OFF NOISE IN DIGITAL FILTERS  Model with quantization effect  Density function error terms for rounding

99. 99 6.8.1 Analysis of the Direct Form IIR Structures  Combine all error terms to single location to get

100. 100 6.8.1 Analysis of the Direct Form IIR Structures  The variance of e[n] in the general case is  The contribution of e[n] to the output is

101. 101 6.8.1 Analysis of the Direct Form IIR Structures  The variance of the output error term f[n] is

102. 102 Example 6.9 Round-Off Noise in a First- Order System  Suppose we want to implement the following stable system  The quantization error noise variance is

103. 103 Example 6.9 Round-Off Noise in a First- Order System  Noise variance increases as |a| gets closer to the unit circle  As |a| gets closer to 1 we have to use more bits to compensate for the increasing error

104. 104 6.9 Zero-Input Limit Cycles in Fixed- Point Realization of IIR Filters  For stable IIR systems the output will decay to zero when the input becomes zero  A finite-precision implementation, however, may continue to oscillate indefinitely  Nonlinear behaviour is very difficult to analyze, so we will study by example

105. 105 6.9 Zero-Input Limit Cycles in Fixed- Point Realization of IIR Filters  Example: Limite Cycle Behavior in First-Order Systems  Assume x[n] and y[n-1] are implemented by 4 bit registers

106. 106 Example Cont’d  Assume that a=1/2=0.100b and the input is ny[n]Q(y[n]) 07/8=0.111b 17/16=0.011100b1/2=0.100b 21/4=0.010000b1/4=0.010b 31/8=0.001000b1/8=0.001b 41/16=0.00010b1/8=0.001b  If we calculate the output for values of n  A finite input caused an oscillation with period 1

107. 107 Example: Limite Cycles due to Overflow  Consider a second-order system realized by  Where Q() represents two’s complement rounding  Word length is chosen to be 4 bits  Assume a 1 =3/4=0.110b and a 2 =-3/4=1.010b  Also assume  The output at sample n=0 is

108. 108 Example: Limite Cycles due to Overflow  Binary carry overflows into the sign bit changing the sign  When repeated for n=1  The output at sample n=0 is  After rounding up we get

109. 109 Avoiding Limit-Cycles  Desirable to get zero output for zero input: Avoid limit-cycles  Generally adding more bits would avoid overflow  Using double-length accumulators at addition points would decrease likelihood of limit cycles

110. 110 Avoiding Limit-Cycles  Trade-off between limit-cycle avoidance and complexity  FIR systems cannot support zero- input limit cycles

111. 111 2015-10-15 111 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 6 HW  6.5, 6.6, 6.19  6.1, 6.3, 6.20 上一页下一页 返 回