VE451Su2010C9

# VE451Su2010C9 - VE451 Lecture Notes Dianguang Ma Summer...

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VE451 Lecture Notes Dianguang Ma Summer 2010

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Chapter 9 Implementation of Discrete- Time Systems
Structures for the Realization of Discrete- Time Systems 1 0 0 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 = = - = - = = - - + - = +

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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.

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Structures for FIR Systems 1 0 1 0 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, , 0 1 ( ) 0, otherwise k k b b k M h k - =

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1 0 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.

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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 -

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1 1 2 0 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

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0 0 10 20 0 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.

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* * 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 * 1 1 1 * 0 1 2 3 4 0 1 2 1 0 Consequently, we gain some simplification by forming fourth-order sections of the FIR system as follows ( ) (1 )(1 )(1 / )(1 / ) Thus, by combin k k k k k

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