Discrete-Time Structure

1. Discrete-Time Structure Hafiz Malik

2. Let us consider the important of LTI DT system characterized by the general linear constant-coefficient difference equation Equivalent LTI DT system in z-transform can be expressed as Realization of Discrete-Time Systems

3. Structures for FIR Systems • In general, an FIR system is described as, • Or equivalently, system function

4. Structures for FIR Systems • In general, an FIR system is described as, • Or equivalently, system function • The unit sample response of FIR system is identical to the coefficients {bl}, i.e.,

5. Implementation Methods for FIR Systems • Direct-Form Structure • Cascade-Form Structure • Frequency-Sampling Structure • Lattice Structure

6. Direct-Form Realization • The direct-form realization follows immediately from the non-recursive difference equation (2) or equivalently by the following convolution summation

7. Direct-Form Structure z-1 z-1 z-1 z-1 + + + + +

8. 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.

9. 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., • For such system the number of multiplicaitons is reduced from M to • M/2 for M is even • (M – 1)/2 for M is odd

10. Direct-Form Realization of Linear-Phase FIR System z-1 z-1 z-1 z-1 + + + + + z-1 z-1 z-1 z-1 z-1 z-1 + + + + +

11. 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., • where, • Here K is integer part of (q – 1)/2

12. Cascade-Form Realization of FIR System yK-1(n) = xK(n) X(n) = x1(n) y1(n) = x2(n) y3(n) = x4(n) yK(n) = y(n) y2(n) = x3(n) H1(z) H2(z) H3(z) HK(z) z-1 z-1 bk1 bk2 bk0 yk(n) = xk+1(n) + +

13. Linear-Phase FIR Systems