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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 finitewordlength 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 directform 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 directform realization is often called a or tappeddelayline 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 secondorder 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 secondorder FIR systems. It is always desirable to pair complexconjugate zeros such that the coefficients are real valued. On the other hand, realvalued zeros can be paired in any arb ki b itrary manner. * * In the case of linearphase 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.
 Summer '10
 DianguangMa

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