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 Equivalent LTI DT system in z-transform can be expressed as

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

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.,

Implementation Methods for FIR Systems 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

Direct-Form Structure z-1 +

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., 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

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

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

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) + +

Linear-Phase FIR Systems