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Discrete-Time Structure Hafiz Malik Realization of Discrete-Time Systems • Let us consider the important of LTI DT system characterized by the general linear constant-coefficient difference equation p 1 y (n) ak y (n k ) k 1 q 1 b x(n l ) l l0 • Equivalent LTI DT system in z-transform can be expressed as q 1 H z bl z l l0 p 1 k 0 , where a 0 1 ak z k Structures for FIR Systems • In general, an FIR system is described as, q 1 y (n) b x(n l ) l (2) l0 • Or equivalently, system function H z q 1 bl z l l0 3 Structures for FIR Systems • In general, an FIR system is described as, q 1 y (n) b x(n l ) l l0 • Or equivalently, system function H z q 1 bl z l l0 • The unit sample response of FIR system is identical to the coefficients {bl}, i.e., bn , h n 0 0 n q 1 otherwise 4 Implementation Methods for FIR Systems 1. 2. 3. 4. Direct-Form Structure Cascade-Form Structure Frequency-Sampling Structure Lattice Structure Direct-Form Realization • The direct-form realization follows immediately from the non-recursive difference equation (2) or equivalently by the following convolution summation q 1 y (n) h (l )x ( n l ) l0 Direct-Form Structure x(n) z-1 z-1 z-1 h (1) h (0) z-1 h (3) h (2) h ( q 2) h ( q 1) y (n) + + + + + Complexity of Direct-Form Structure • Requires q – 1 memory locations for storing q – 1 previous inputs, • Complexity of q – 1 multiplications and q – 1 addition per output point • As output consists of a weighted linear combination of q – 1 past inputs and the current input, which resembles a tapped-delay line or a transversal system. • The direct-form realization is called a transversal or tapped-delay-line filter. Linear-Phase FIR System • When the FIR system is linear phase, the unit sample response of the system satisfies either the symmetry or asymemtry condition, i.e., h(n) h(q 1 n) • For such system the number of multiplicaitons is reduced from M to 1. M/2 for M is even 2. (M – 1)/2 for M is odd Direct-Form Realization of Linear-Phase FIR System x(n) z-1 z-1 + + z-1 y (n) z-1 h (2) + + + z-1 h (1) + z-1 + z-1 h (0) + z-1 z-1 z-1 h (3) + q3 h 2 + q 1 h 2 Cascade-Form Structures • Cascade realization follows naturally from the LTI DT system given by equation (3). Simply factorize H(z) into second-order FIR systems, i.e., K H (z) H k (z) l 1 • where, H k ( z ) bk 0 bk 1 z 1 bk 2 z 2 , k 1, 2 , , K • Here K is integer part of (q – 1)/2 Cascade-Form Realization of FIR System y1(n) = x2(n) X(n) = x1(n) H1(z) y3(n) = x4(n) y2(n) = x3(n) H2(z) H3(z) x(n) z-1 z-1 bk1 bk0 bk2 yk(n) = xk+1(n) + + yK-1(n) = xK(n) HK(z) yK(n) = y(n) Linear-Phase FIR Systems