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 2019/4/26 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
§6 structures for discrete-time system 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(转置) Forms 6.5 Basic Network Structures for FIR Systems
Structures for Discrete-Time Systems 6.0 Introduction
Characterization of an LTI System: 6.0 Introduction Characterization of an LTI System: Impulse Response z-Transform: system function Difference Equation → Frequency response 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
Example: find the output of the system with input x[n]. Solution1: IIR Impulse Response 长度无限 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 . Illustration for the IIR case by convolution
Example: find the output of the system with input x[n]. Solution2: LTI recursive computation of output initial-rest conditions(for n<0, if x[n]=0, then y[n]=0) The algorithm suggested by the equation is not the only computational algorithm, there are unlimited variety of computational structures (shown later).
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
Structures for Discrete-Time Systems 6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations
6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations Implementation of an LTI system by iteratively evaluating a recurrence formula needs three basic elements: 递推公式 system function x[n] x[n-1] z1 Unit Delay (Memory, storage) z-M x[n-M] M sample Delay x[n] b0 ax[n] Multiplier + x1[n] x2[n] x1[n] + x2[n] Adder
Ex. 6.1 draw Block Diagram Representation of a Second-order Difference Equation 1省略 Solution: x[n] + b0 a1 a2 y[n] z1 y[n-1] z1 y[n-2]
Nth-Order Difference Equations change Equation to a’[0] normalized to unity
Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] v[n] + z1 a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N]
Block Diagram Representation (Direct Form I) Implementing zeros Implementing poles v[n] + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N]
Block Diagram Representation (Direct Form I) Implementing zeros Implementing poles v[n] + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N]
Block Diagram Representation (Direct Form I) How many Adders? How many multipliers? How many delays? N +M N +M+1 N+M + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N] v[n]
Block Diagram Representation (Direct Form I) v[n] + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N]
Block Diagram Representation (Direct Form II) 规范直接型 (or called Canonic direct Form) + z1 b0 b1 bN1 bN x[n] a1 aN1 aN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N 保持不变 保持不变
Block Diagram Representation (Direct Form II) Implementing poles Implementing zeros + z1 b0 b1 bN1 bN x[n] a1 aN1 aN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N
Block Diagram Representation (Direct Form II) Implementing poles Implementing zeros + z1 b0 b1 bN1 bN x[n] a1 aN1 aN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N
Block Diagram Representation (Direct Form II) How many Adders? How many multipliers? How many delays? 与交换前相比 无变化 N +M+1 N+M + z1 b0 b1 bN1 bN x[n] a1 aN1 aN y[n] w[n-1] w[n-2] w[n-N] w[n] Assume M = N
Block Diagram Representation (Canonic Direct Form or direct Form II) How many Adders? How many multipliers? How many delays? max(M, N) N +M N +M+1 N + b0 b1 bN1 bN x[n] z1 a1 aN1 aN y[n] w[n] w[n-1] 规范直接型 Assume M = N w[n-2] 无 此 若M<N w[n-N]
Block Diagram Representation (Canonic Direct Form or direct Form II) w[n] + b0 b1 bN1 bN x[n] z1 a1 aN1 aN y[n] w[n-1] 规范直接型 w[n-2] Assume M = N w[n-N]
Ex. 6.2 draw Direct Form I(II) implementation of an LTI system Solution: + z1 1 2 x[n] x[n-1] v[n] + z1 1.5 0.9 y[n] y[n-1] y[n-2] Direct Form I
Ex. 6.2 draw Direct Form I（II ）implementation of an LTI system + z1 1 2 x[n] x[n-1] v[n] + z1 1.5 0.9 y[n] y[n-1] y[n-2] Solution: Direct Form I Direct Form II + z1 1 2 w[n-1] y[n] + z1 1.5 0.9 w[n] w[n-1] w[n-2] x[n] w[n-1] w[n-2]
Ex. 6.2 draw Direct Form I（II ）implementation of an LTI system x[n] + z1 1.5 0.9 y[n] w[n-1] w[n-2] 1 2 w[n] Solution: Direct Form II + z1 1 2 w[n-1] y[n] + z1 1.5 0.9 w[n] w[n-1] w[n-2] x[n]
Structures for Discrete-Time Systems 6.2 Signal Flow Graph(信号流图) Representation of Linear Constant-Coefficient Difference Equations
6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations A Signal Flow Graph is a network of directed branches (有向支路)that connect at nodes(节点). Associated with each node is a variable or node value, being denoted wj[n]. Node j Node k 梅森(Mason)信号流图 Sequence wj[n] wk[n] Signal Flow Graph (SFG)
Nodes And Branches Input wj[n] wj[n] a or z-1 wk[n] Node j We will only consider linear Signal Flow Graph Output: A linear transformation of input, such as constant gain and unit delay. Input wj[n] if omitted, it indicates unity unit delay a or z-1 Brach (j, k) Sequences wj[n] wk[n] Node j Node k Each branch has an input signal and an output signal. An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node.
Source Nodes (源点 ) Sink Nodes (汇点, 阱点) Nodes that have no entering branches xj[n] wk[n] Source node j yk[n] wj[n] Sink node k inputs outputs Sink Nodes (汇点, 阱点) Nodes that have only entering branches
Example : determine Linear Constant-Coefficient Difference Equations of SFG x[n] y[n] w1[n] w2[n] a b c d e Source Node Sink Node Solution:
Block Diagram vs. Signal Flow Graph x[n] w[n] y[n] + a z1 b1 b0 branching point Canonic direct Form Source Node w1[n] Sink Node a b1 b0 1 2 3 4 w2[n] x[n] y[n] w3[n] By convention, variables is represented as sequences rather than as z-transforms Delay branch z1 w4[n] =w2[n-1] Delay branch cannot be represented in time domain by a branch gain by z-transform, a unit delay branch has a gain of z-l.
Block Diagram vs. Signal Flow Graph Determine the difference equation (System Function) from the Flow Graph. Solution: x[n] + a z1 b1 b0 w[n] y[n] a b1 b0 z1 1 2 3 4 w1[n] x[n] y[n] w2[n] w3[n] w4[n]
Block Diagram vs. Signal Flow Graph Determine difference equation difficult to find final solution in time-domain
Ex. 6.3 Determine the System Function from Flow Graph causal system Solution:
Ex. 6.3 Determine the System Function from Flow Graph for causal system :
Ex. 6.3 compare two implementation requires only one multiplication and one delay (memory) element direct form I implementation -a a x[n] z-1 y[n] two multiplication and two delay
Structures for Discrete-Time Systems 6.3 Basic Structure for IIR Systems
6.3 Basic Structure for IIR Systems for a rational system function, there are many equivalent difference equations or network structures. A criteria in the choice among these different structures is computational complexity: Reduce the number of constant multipliers Increase speed Reduce the number of delays Reduce the memory requirement
Basic Structures for IIR Systems Direct Forms Cascade Form Parallel Form
6.3.1 Direct Forms v[n] x[n] y[n] z1 b0 b1 bM1 bM x[n-1] x[n-2] 2019年4月26日1时45分 6.3.1 Direct Forms v[n] + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] + z1 a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N]
v[n] x[n] y[n] Direct Form I x[n] v[n] y[n] Block Diagram + z1 b0 b1 bM1 bM x[n] x[n-1] x[n-2] x[n-M] a1 aN1 aN y[n] y[n-1] y[n-2] y[n-N] Direct Form I Block Diagram b0 b1 x[n] x[n-1] x[n-2] x[n-M] b2 bN-1 bN x[nM+1] z1 v[n] y[n] a1 a2 aN-1 aN y[n-1] y[n-2] y[n-N] y[nN+1] z1 Signal Flow Graph
Direct Form I Signal Flow Graph Draw SFG Directly v[n] b0 b1 x[n] x[n-1] x[n-2] x[n-M] b2 bN-1 bN x[nM+1] z1 v[n] y[n] a1 a2 aN-1 aN y[n-1] y[n-2] y[n-N] y[nN+1] z1
Direct Form II
Direct Form II
Direct Form II w[n] x[n] y[n] b0 z1 a1 b1 a2 b2 aN-1 bN-1 aN bN w[n-N-1] w[n-N]
Direct Form II w[n] x[n] y[n] Draw SFG Directly b0 z1 a1 b1 a2 b2 bN-1 bN a1 a2 aN-1 aN z1
Ex. 6.4 draw Direct Form I and Direct Form II structures of system Solution: x[n] y[n] z1 1 1 2 0.75 Direct Form I 1 0.125 x[n] y[n] z1 2 Direct Form II 0.75 0.125
6.3.2 Cascade Form(串联形式) when all the coefficients are real 1st-order factors represent real zeros at gk and real poles at ck , and the 2nd-order factors represent complex conjugate pairs of zeros at hk and h*k and poles at dk ,d*k
Cascade Form A modular structure 2nd Order System
Cascade Form For example, assume Ns=3 x[n] 1 2 3 y[n] z1 a11 a21 b11 b21 b01 z1 a12 a22 b12 b22 b02 z1 a13 a23 b13 b23 b03 It is used(see 6.9)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.
Ex. 6.5 draw the Cascade structures Solution: x[n] y[n] z1 0.75 0.125 2 Direct Form II 1st-order Direct Form I 1st-order Direct Form II
Another Cascade Form x[n] y[n] implemented with fixed-point arithmetic used to decrease the amount of computation, when floating-point arithmetic is used and dynamic range is not a problem. b0 z1 a11 a21 b11 b21 z1 a12 a22 b12 b22 z1 a13 a23 b13 b23 x[n] y[n] ~ ~ ~
all coefficients are real 6.3.3 Parallel Form all coefficients are real
( Real ) Poles are Grouped in pairs Parallel Form Complex Poles Poles at zero Real Poles ( Real ) Poles are Grouped in pairs
Parallel Form x[n] y[n] z1 a1k a2k e0k e1k Ckz-k C0 a11 a21 e01 e11 2019年4月26日1时45分 Parallel Form x[n] y[n] Ckz-k C0 z1 a1k a2k e0k e1k a11 a21 e01 e11 a1k a2k e0k e1k
Ex. 6.6 draw parallel-form structures of system Solution 1: If we use 2nd –order sections (二阶节), 8 x[n] y[n] z1 8 7 0.75 0.125
Ex. 6.6 draw parallel-form structures of system Solution 2: If we use 1st –order sections(一阶节), z1 8 x[n] y[n] 0.5 18 0.25 25
6.3.4 feedback in the IIR systems z1 a x[n] y[n] systems with feedback may be FIR z1 a x[n] y[n] -a2 z1 a x[n] y[n] Noncomputable network delay-free loop, not Implementable a x[n] y[n] equation solvable n时刻输出(未知)=n时刻的输出(未知, 可用n-1时刻输出(已知))×a+x[n]
Structures for Discrete-Time Systems 6.4 Transposed Forms
Flow Graph Reversal or Transposition 6.4 Transposed Forms There are many procedures for transforming signal flow graphs into different forms while leaving the overall system function between input and output unchanged. Flow Graph Reversal or Transposition Changes the roles of input and output. Reverse the directions of all arrows. Transposing doesn’t change the input-output relation z1 a z1 a x[n] y[n] y[n] x[n]
Ex. 6.7 determine Transposed Forms for a first-order system x[n] y[n] z1 a Transposing Solution: y[n] x[n] x[n] y[n] z1 a z1 a by convention left input, right output
Ex. 6.8 draw Transposed Forms for a basic second-order section Solution: Both have the same system function or difference equation proof or it is proved
Ex. 6.8 Transposed Forms for a basic second-order section x[n] y[n] b2 a1 a2 z1 v[n] x[n] y[n] b0 b1 b2 z1 w[n] a1 a2 b0 b1 x[n] y[n] b2 a1 a2 z1 v[n]
Transposed Direct Form I b0 b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2 bN-1 bN x[nM+1] a1 a2 aN-1 aN y[n-1] y[n-2] y[n-N] y[nN+1] z1 v[n] b0 b1 x[n] y[n] b2 bN-1 bN a1 a2 aN-1 aN z1 v'[n] Transposing
Transposed Direct Form I b0 b1 x[n] y[n] b2 bN-1 bN a1 a2 aN-1 aN z1 v'[n] left input, right output by convention b0 b1 x[n] y[n] b2 bN-1 bN a1 a2 aN-1 aN z1 v'[n]
Transposed Direct Form II x[n] y[n] w[n] b0 b1 b2 bN-1 bN a1 a2 aN-1 aN z1 y[n] x[n] w' [n] b0 b1 b2 bN-1 bN a1 a2 aN-1 aN z1 Transposing
Transposed Direct Form II y[n] x[n] w' [n] b0 a1 a2 aN-1 aN b1 b2 bN-1 bN z1 left input, right output by convention y[n] x[n] w' [n] b0 b1 b2 bN-1 bN a1 a2 aN-1 aN z1
Structures for Discrete-Time Systems 6.5 Basic Structure for FIR Systems
6.5 Basic Structure for FIR Systems 6.5. 1 Direct Form the system function For causal FIR systems, the system function has only zeros(except for poles at z = 0).
Direct Form I v[n] x[n] y[n] x[n] y[n] 对折轴 按绿线对折 z1 …… …… z1 y[n] b0 b1 x[n] x[n-1] x[n-2] x[n-M] y[n] b2 bM-1 bM x[nM+1] a1 a2 aN-1 aN y[n-1] y[n-2] y[n-N] y[nN+1] z1 v[n] y[n] 按绿线对折 x[n] y[n] z1 x[n-1] x[n-2] x[nM+1] x[n-M] …… h[0] h[1] h[2] h[M1] h[M] ……
Direct Form II w[n] x[n] y[n] x[n] y[n] 转动轴 b0 b1 b2 bM-1 bM a1 a2 aN-1 aN z1 y[n] 绕绿线转1800 x[n] y[n] z1 …… same with direct form I h[0] h[1] h[2] h[M1] h[M] ……
Traspostion of Direct Form x[n] y[n] z1 h[0] h[1] h[2] h[M1] h[M] …… …… The structure is called: tapped delay line structure or transversal filter structure. 抽头延迟线结构 or 横向滤波器结构. x[n] y[n] z1 h[0] h[1] h[2] h[M1] h[M] …… …… Trasposed form usually x[n] y[n] z1 h[0] h[1] h[2] h[M1] h[M] …… left input …… right output
6.5.2 Cascade Form y[n] x[n] z1 b01 b11 b21 b02 b12 b22 b1Ms b2Ms 参考IIR Cascade Form
6.5.3 Structures for Linear Phase Systems Causal FIR system has generalized linear phase if h[n] satisfies: h[M-n]=± h[n], for n = 0,1,…,M M is even M is odd h[M-n]= h[n] h[M-n]= h[n] Type I Type II Type III Type VI x[n] y[n] z1 h[0] h[1] h[2] h[M1] h[M] …… …… multiplications: M+1 左右对折后共用系数(h[M-n]= ±h[n])可减少一半乘法次数
Type I, and II x[n] y[n] x[n-(M-1)/2] x[n-(M+1)/2] x[n-1] x[n-2] Type II h[0] h[1] h[2] h[(M-3)/2] h[(M-1)/2] 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] z1 h[M/2] h[M/21] h[0] h[1] h[2] Type I
Type III, and IV x[n] y[n] x[n-(M-1)/2] x[n-(M+1)/2] x[n-1] x[n-2] Type IV - 比II 型多负号 h[0] h[1] h[2] h[(M-3)/2] h[(M-1)/2] 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] z1 h[M/2]=0 h[M/21] h[0] h[1] h[2] Type III - 比I 型多负号 且无中项
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2019年4月26日1时45分 Chapter 6 HW 6.6, 6.19, 6.20 6.5, 6.18, 66.9,.10 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 85 2019/4/26 返 回 上一页 下一页