86 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 Forms 6.5 Basic Network structures for Fir Systems
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
Structures for discrete-time Systems 6.0 Introduction
3 Structures for Discrete-Time Systems 6.0 Introduction
6.0 Introduction haracterization of an LTI System: ◆ Impulse Response→ Frequency 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
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
xample: find the output of the system h()=+h zab with input x[nI Solution 1 I -az IIR Impulse n=bauIntba-uln Response {m]=xp]*小]=∑x[l]h[n-1]=∑小]x[n- k=0 even if we only wanted to compute the output over a finite interval. it would not be efficient h(n-k k 5 Illustration for the iir case by convolution
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 1 0 1 1 ( ) , | | | |, 1 b b z H z z a az − − + = − 1 0 1 1 − = + − n n h n b a u n b a u n 0 n k k y n x n h n x k h n k h k x n k =− = = = − = − Illustration for the IIR case by convolution IIR Impulse Response with input x[n]. Solution1:
Example: find the output of the system bo+bE Y() H(E)1-a2(),|=plal, with input x[n] Solution2: yn n-avln =bx{小]+bx1[n-1] y[n]=ay[n-1]+box[n]+bjx[n-1] computable recursively The algorithm suggested by the equation is not the only computational algorithm, there are unlimited variety of computational structures(shown later)
6 Example: find the output of the system 1 0 1 1 ( ) , | | | |, 1 ( ) ( ) b b z H z z a az Y z X z − − = + = − y n ay n b x n b x n − − = + − 1 1 0 1 y n ay n b x n b x n = − + + − 1 1 0 1 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:
Why Implement system Using DIfferent structures K 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 l Computational structures(Modeling methods ◆ Block Diagram ◆ Signal Flow Graph
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
Structures for discrete-time Systems 6.1 Block Diagram Representation of Linear Constant-Coefficient difference Equations
8 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 yIn=ayln-+bor n+xiN Three basic elements: Unit Delay Memory, storage) In/12 xin M sample Delay z-Mm17 a Multiplier axin Adder xin+x2n
9 6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations + x1 [n] x2 [n] x1 [n] + x2 Adder [n] x[n] a Multiplier ax[n] x[n] x[n-1] z Unit Delay −1 (Memory, storage) y n ay n b x n b x n = − + + − 1 1 0 1 Three basic elements: M sample Delay z -M x[n-M]
EX. 6.1 draw Block Diagram Representation of a Second-order Difference Equation y[n]=ay[n-1]+a2y[n-2]+box[n y[n]-aiy[n-1]-a2y[n-2]=bo-in Solution b Y(=) X(=)1-a1 n-1] =H(z)
10 Ex. 6.1 draw Block Diagram Representation of a Second-order Difference Equation 1 2 0 y n y n y n b x n [ ] [ 1] [ 2] [ ] = − + − + a a x[n] + + b0 a1 z −1 z −1 a2 y[n] y[n-1] y[n-2] 0 2 1 2 1 ( ) ( ) 1 Y z b X z z a a z − − − − = Solution: = H z( ) 1 2 0 y n y n y n b x n [ ] [ 1] [ 2] [ ] − − a a − − =
Nth-Order Difference Equations N Form ayn-k=∑对m一k1 changed k=0 k=0 to N M 川-∑an-6]=∑bn-k alo] k=1 k=0 normalized M to unity yinI=∑4m-+ kn k=1 k=0 ∑b-6 H()=-k ∑a=k k=1
11 Nth-Order Difference Equations 1 0 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = − − = − = − = − − = N k k k M k k k a z b z H z 1 1 1 ( ) 0 0 [ ] [ ] = = − = − N M k k k k a y n k b x n k 1 0 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = = − + − Form changed to a[0] normalized to unity
