VE451Su2010C9 - VE451 Lecture Notes Dianguang Ma Summer...

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Unformatted text preview: VE451 Lecture Notes Dianguang Ma Summer 2010 Chapter 9 Implementation of Discrete- Time Systems Structures for the Realization of Discrete- Time Systems 1 1 Let us consider an LTI system characterized by the difference equation ( ) ( ) ( ) or, equivalently, by the system function ( ) 1 N M k k k k M k k k N k k k y n a y n k b x n k b z H z a z = =- =- = = -- +- = + ∑ ∑ ∑ ∑ The system can be described a block diagram consiting of an interconnection of adders, multipliers, and dealy elements. We refer to such a block diagram as a or of the syste realization implementation m or, equivalently, as a for realizing the system. structure We will show that an LTI system can be realized in a variety of ways. The major factors that influence our choice of a specific realization are computational complexity, memory requirements, and finite-word-length effects in the computations. Structures for FIR Systems 1 1 In general, an FIR system is described by the difference equation ( ) ( ) or, equivalently, by the system function ( ) M k k M k k k y n b x n k H z b z- =-- = =- = ∑ ∑ { } Furthermore, the unit sample response of the FIR system is identical to the coefficients , that is, , 1 ( ) 0, otherwise k k b b k M h k ≤ ≤- = 1 The simplest structure, called the direct-form realization, or , follows directly from the difference equation or, equivalently, by the convolution summation ( ) ( ) ( ) It is ill M k direct form y n h k x n k- = =- ∑ ustrated in Figure 9.2.1. We observe that this structure requires 1 memory locations for storing the 1 previous inputs, and has a complexity of multiplications and 1 additions per output point. The structure in Figure M M M M--- 9.2.1 resembles a tapped delay line or a transversal system. Consequenctly, the direct-form realization is often called a or tapped-delay-line filter. transversal When the FIR system has a linear phase, the unit sample response satisfies either the symmetry or asymmetry condition ( ) ( 1 ) For such a system, the number of multiplications is reduced from t h n h M n M = ±- - o / 2 for even and to ( 1) / 2 for odd. M M M M- 1 1 2 1 2 The realization follows naturally from the system function. It is a simple matter to factor ( ) into second-order FIR systems so that ( ) ( ) where ( ) , 1,2, , and K k k k k k k cascade H z H z H z H z b b z b z k K =-- = = + + = ∏ K is the integer part of / 2. K M 10 20 The filter coefficient may be equally distributed among the filter sections, such that or it may be assigned to a single filter section. The zeros of ( ) are grouped in pairs to pr K b K b b b b H z = L { } oduce the second-order FIR systems. It is always desirable to pair complex-conjugate zeros such that the coefficients are real valued. On the other hand, real-valued zeros can be paired in any arb ki b itrary manner. * * In the case of linear-phase FIR filters, the symmetry in ( ) implies that the zeros of ( ) also exhibit a form of symmetry. In particular, if and are a pair of zeros then 1/ and 1/ are a k k k k h n H z z z z z lso a pair of zeros. 1...
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This note was uploaded on 10/14/2011 for the course EE 451 taught by Professor Dianguangma during the Summer '10 term at Shanghai Jiao Tong University.

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VE451Su2010C9 - VE451 Lecture Notes Dianguang Ma Summer...

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