Structures for Discrete-Time Systems 主講人：虞台文

Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed Forms Basic Structure for FIR Systems Lattice Structures

Structures for Discrete-Time Systems Introduction

Characterize an LTI System Impulse Response z-Transform Difference Equation

Example Noncomputable Computable

Basic Operations Addition Multiplication Delay In fact, there are unlimited variety of computational structures. Computable

Why Implement Using Different Structures? Finite-precision number representation of a digital computer. Truncation or rounding error. Modeling methods: Block Diagram Signal Flow Graph

Block Diagram Representation + x1(n) x2(n) x1(n) + x2(n) Adder x(n) a ax(n) Multiplier x(n) x(n1) z1 Unit Delay

Example + b a1 z1 a2 x(n) y(n) y(n1) y(n2)

Higher-Order Difference Equations

Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x(n) x(n1) x(n2) x(nM) a1 aN1 aN y(n) y(n1) y(n2) y(nM) v(n)

Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x(n) x(n1) x(n2) x(nM) a1 aN1 aN y(n) y(n1) y(n2) y(nM) v(n)

Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x(n) x(n1) x(n2) x(nM) a1 aN1 aN y(n) y(n1) y(n2) y(nM) v(n)

Block Diagram Representation (Direct Form I) Implementing zeros Implementing poles Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x(n) x(n1) x(n2) x(nM) a1 aN1 aN y(n) y(n1) y(n2) y(nM) v(n)

Block Diagram Representation (Direct Form I) How many Adders? How many multipliers? How many delays? Block Diagram Representation (Direct Form I) + z1 b0 b1 bM1 bM x(n) x(n1) x(n2) x(nM) a1 aN1 aN y(n) y(n1) y(n2) y(nM) v(n)

Block Diagram Representation (Direct Form II) + z1 b0 b1 bN1 bN x(n) a1 aN1 aN y(n) w(n1) w(n2) w(nN) w(n) Assume M = N

Block Diagram Representation (Direct Form II) + z1 b0 b1 bN1 bN x(n) a1 aN1 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 Block Diagram Representation (Direct Form II) + z1 b0 b1 bN1 bN x(n) a1 aN1 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? Block Diagram Representation (Direct Form II) + z1 b0 b1 bN1 bN x(n) a1 aN1 aN y(n) w(n1) w(n2) w(nN) w(n) Assume M = N

Block Diagram Representation (Canonic Direct Form) + b0 b1 bN1 bN x(n) z1 a1 aN1 aN y(n) Assume M = N

Block Diagram Representation (Canonic Direct Form) How many Adders? How many multipliers? How many delays? max(M, N) Block Diagram Representation (Canonic Direct Form) + b0 b1 bN1 bN x(n) z1 a1 aN1 aN y(n) Assume M = N

Structures for Discrete-Time Systems Signal Flow Graph

Nodes And Branches wj(n) wk(n) Associated with each node is a variable or node value. wj(n) wk(n)

Nodes And Branches Input wj(n) wj(n) wk(n) Brach (j, k) Output: A linear transformation of input, such as constant gain and unit delay. wj(n) wk(n) Brach (j, k) Each branch has an input signal and an output signal.

More on Nodes wj(n) wk(n) An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node. wj(n) wk(n)

Source Nodes Nodes without entering branches xj(n) wk(n) Source node j

Sink Nodes yk(n) wj(n) Nodes that have only entering branches Sink node k

Example x(n) y(n) w1(n) w2(n) a b c d e Source Node Sink Node

Block Diagram vs. Signal Flow Graph x(n) w(n) y(n) + a z1 b1 b0 a b1 b0 z1 1 2 3 4 w1(n) x(n) y(n) w2(n) w3(n) w4(n)

Block Diagram vs. Signal Flow Graph x(n) + a z1 b1 b0 w(n) y(n) w1(n) w2(n) w3(n) 1 2 3 4 w4(n)

Block Diagram vs. Signal Flow Graph

Structures for Discrete-Time Systems Basic Structure for IIR Systems

Criteria Reduce the number of constant multipliers Increase speed Reduce the number of delays Reduce the memory requirement Modularity: VLSI design The effects of finite register length and finite-precision arithmetic.

Basic Structures Direct Forms Cascade Form Parallel Form

Direct Forms

Direct Form I x(n) v(n) y(n) b0 b1 x(n1) x(n2) x(nN) b2 bN-1 bN a1 a2 aN-1 aN y(n1) y(n2) y(nN) y(nN+1) z1 v(n)

Direct Form I x(n) v(n) y(n) b0 b1 x(n1) x(n2) x(nN) b2 bN-1 bN a1 a2 aN-1 aN y(n1) y(n2) y(nN) y(nN+1) z1 v(n)

Direct Form II x(n) y(n) w(n) b0 b1 b2 bN-1 bN a1 a2 aN-1 aN z1

Direct Form II x(n) y(n) w(n) b0 b1 b2 bN-1 bN a1 a2 aN-1 aN z1

Example x(n) y(n) x(n) y(n) Direct Form I Direct Form II z1 2 0.75 0.125 x(n) y(n) z1 2 Direct Form II 0.75 0.125

Cascade Form

Cascade Form

Cascade Form 2nd Order System

Cascade Form x(n) y(n) z1 a11 a21 b11 b21 b01 z1 a12 a22 b12 b22 b01

Another Cascade Form

Parallel Form

Parallel Form Group Real Poles Complex Poles Poles at zero Real Poles

Parallel Form z1 a1k a2k e0k e1k

Parallel Form x(n) y(n)

Example 8 x(n) y(n) z1 0.75 0.125 7

Example z1 0.5 18 8 x(n) y(n) 0.25 25

Structures for Discrete-Time Systems Transposed Forms

Signal Flow Graph Transformation To transform signal graphs into different forms while leaving the overall system function between input and output unchanged.

Transposition of Signal Flow Graph Reverse the directions of all arrows. Changes the roles of input and output. z1 a z1 a x(n) y(n) y(n) x(n)

Transposition of Signal Flow Graph Are there any relations between the two systems? x(n) y(n) z1 a

Example: x(n) y(n) z1 a y(n) x(n) x(n) y(n) z1 a z1 a

Transposition of Signal Flow Graph Reverse the directions of all arrows. Changes the roles of input and output. x(n) y(n) z1 a Detail proof see reference

Structures for Discrete-Time Systems Basic Structure for FIR Systems

FIR For causal FIR systems, the system function has only zeros.

Direct Form x(n) y(n) z1 h(0) h(1) h(2) h(M1) h(M)

Direct Form x(n) y(n) y(n) x(n) z1 h(0) h(1) h(2) h(M1) h(M) z1

Direct Form x(n) y(n) y(n) x(n) z1 h(0) h(1) h(2) h(M1) h(M) z1

Cascade Form

Cascade Form x(n) y(n) z1 b01 b11 b21 b02 b12 b22 b1Ms b2Ms b0Ms

Structures for Linear Phase Systems A generalized linear phase system satisfies: h(Mn) = h(n) for n = 0,1,…,M or 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

Type I

Type I x(n) y(n) z1 h(M/2) h(M/21) h(0) h(1) h(2)

Type II, III and VI Construct them in a similar manner by yourselves.

Structures for Discrete-Time Systems Lattice Structures

Consider x(n)=(n), one will see FIR Lattice

Consider x(n)=(n), one will see FIR Lattice

Consider x(n)=(n), one will see FIR Lattice

Define Consider x(n)=(n), one will see FIR Lattice

Define FIR Lattice Show that

FIR Lattice FIR Lattice i=1: Show that

FIR Lattice FIR Lattice i = n: Assumed true i = n+1 also true. Prove i = n+1 also true. Show that

FIR Lattice FIR Lattice =

FIR Lattice FIR Lattice

FIR Lattice FIR Lattice Given the lattice, to find A(z). m=0 k1 k2 k3

FIR Lattice FIR Lattice Given A(z), to find the lattice. m=0 m=1 m=2

FIR Lattice FIR Lattice Given A(z), to find the lattice. m=0 m=1 m=2

Example 1 m=0 m=1 m=2 m=3 0.6728 0.7952 0.9 0.1820 0.64 0.576

Example 0.576 0.1820  0.6728 1 m=0 m=1 m=2 m=3 0.6728 0.7952 0.1820 0.9 0.64 0.576

Inverse Filter

All-Pole Filter

All-Pole Filter

All-Pole Filter

All-Pole Filter

Example 0.576 0.1820  0.6728 0.6728 0.6728 0.1820 0.1820 0.576  0.576

Example 0.9  0.64 0.576 0.6728 0.6728 0.1820 0.1820 0.576  0.576

Stability of All-Pole Filter All zeros of A(z) have to lie within the unit circle. Necessary and sufficient conditions: All of k-parameters ki’s satisfy |ki| < 1.

Normalized Lattice

Normalized Lattice

Normalized Lattice Section i

Normalized Lattice Section i Section N N1 1

Normalized Lattice Section i Three-Multiplier Form

Normalized Lattice Four-Multiplier, Kelly-Lochbaum Form Three-Multiplier Form Four-Multiplier, Normalized Form

Normalized Lattice Three-Multiplier Form Section N N1 1

Normalized Lattice Four-Multiplier, Normalized Form Section N N1 1

Normalized Lattice Four-Multiplier, Kelly-Lochbaum Form Section N N1

Lattice Systems with Poles and Zeros Section N1 1 N c0 c1 cN2 cN1 cN

Lattice Systems with Poles and Zeros Section N1 1 N c0 c1 cN2 cN1 cN

Lattice Systems with Poles and Zeros

Example 0.6728 0.6728 0.1820 0.1820 0.576  0.576 c3 c2 c1 c0

Example 1 c3 c2 c1 c0 m=0 m=1 m=2 m=3 0.6728 0.7952 0.1820 0.9 0.64 0.6728 0.1820 0.1820 0.576  0.576 c3 c2 c1 c0 Example 1 m=0 m=1 m=2 m=3 0.6728 0.7952 0.1820 0.9 0.64 0.576

Example 0.6728 0.6728 0.1820 0.1820 0.576  0.576 1 3.9 5.4612 4.5404