3.+Structures+for+Discrete-Time+Systems

# 3.+Structures+for+Discrete-Time+Systems - 3 Structures for...

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3. Structures for Discrete-Time Systems 3.1. Signal Flow Graphs (6.2) 3.2. Basic Structures for IIR Systems (6.3) 3.3. Basic Structures for FIR Systems (6.5) 3.4. Transposition Theorem (6.4)

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The input-output relation of a linear time-invariant discrete-time system can be characterized by an impulse response, a frequency response, a system function or a linear constant-coefficient difference equation. When the input-output relation is given, the system can be implemented in different structures. These structures are different in accuracy, speed, cost, and others. We discuss causal systems only. 3.1. Signal Flow Graphs The structure of a linear time-invariant discrete-time system can be represented by a signal flow graph. Basically, a signal flow graph is a network of nodes and directed branches (figure 3.1). A node carries out addition. Each output of a node equals the sum of all its inputs. Usually the number of the inputs of a node is limited to no more than 2. A directed branch carries out multiplication or delay. Its output is equal to its input multiplied by a constant (usually omitted if equal to
X 1 (z) X 2 (z) Y 1 (z) Y 2 (z) Figure 3.1. Elements of Signal Flow Graphs. (a) Y 1 (z)=Y 2 (z)=X 1 (z)+X 2 (z) X(z) Y(z) (b) Y(z)=aX(z) a z 1 X(z) Y(z) (c) Y(z)=z 1 X(z) 1) or delayed by 1.

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Example. Determine the system function of the signal flow graph given below. X(z) z 1 Y(z) 1 W 1 (z) W 2 (z) W 3 (z) W 4 (z) Figure 3.2. A Signal Flow Graph. 3.2. Basic Structures for IIR Systems The basic structures for IIR systems include the direct form I, the direct form II, the cascade form and the parallel form. These structures, as well as other structures for IIR systems, have feedback loops.
Consider an IIR system with system function , z a 1 z b ) z ( H N 1 k k k M 0 k k k (3.1) where a k and b k are assumed to be real numbers. Let us use different structures to implement this system. 3.2.1. Direct Form I From (3.1), we obtain , ) z ( X z b ) z ( Y z a ) z ( Y M 0 k k k N 1 k k k (3.2) where X(z) and Y(z) are the z-transforms of the input and the output, respectively. (3.2) can be used to construct the direct form I structure

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X(z) Y(z) z 1
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3.+Structures+for+Discrete-Time+Systems - 3 Structures for...

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