1 DIGITAL SIGNAL PROCESSING Digital Filter Structures HW on State-Space Realization Department of Electrical and Computer Engineering University of Miami Question Points Q1 25 Q2 25 Q3 25 Q4 25 TOTAL 100 I. QUESTION 1 Problems 2.26 and 2.27 of [1]: Consider the state-space model fA;B;C;Dg described by x(k + 1) = 240 1 0 335x(k) + 241135u(k); y(k) = h??2 1i x(k): 1) Find the TF Y (z)=U(z). 2) Find a similarity transformation that yields the following ‘diagonal’ state-space model f ^ A; ^B; ^ C; ^Dg for this same system: ^x(k + 1) = 24d1 0 0 d235 ^x(k) + ^Bu(k); y(k) = ^ C ^x(k): 3) Find ^Band ^ C.
2 II. QUESTION 2 Problem 3.44 of of [2]: Consider the state-space model fA;B;C;Dg of a digital filter where A = 240:5 ??0:1 0:1 0:5 35; B = 240135; C = h1 2i ; D = h0:2i : 1) Find the TF of this filter. 2) What are its poles and zeros? 3) Find the zero state solution when the input is a unit step. 4) Find the zero input solution when the initial conditions are x(0) = h1 ??1iT . III. QUESTION 3 The direct form II implementation of a digital filter is given in Figure 1. z”1 z”1 z”1 ++++… … a1 a2 aN”1 aN ++++… b1 b2 bN”1 bN u[n] w[n] b0 y[n] Fig. 1. Direct form II implementation of an order N digital filter. 1) Construct a state-space model corresponding to this realization. 2) What is its TF?
3 IV. QUESTION 4 Problem 8.18 of [3]: Let (z) = 2666664 1z… zn??13777775: Suppose a(z) = zn + an??1zn??1 + + a0; b(z) = bnzn + bn??1zn??1 + + b0: 1) Find the unique matrix A and vector B that satisfies (zI ?? A)(z) = a(z)B () (zI ?? A)??1B = 1 a(z) (z): 2) Determine C and D such thatC (zI ?? A)??1B + D = b(z) a(z) : 3) If a() = 0, then is an eigenvalue of A. What is the corresponding eigenvector? REFERENCES [1] C. L. Phillips and H. T. Nagle, Digital Control System Analysis and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. [2] T. Bose, Digital Signal and Image Processing. Hoboken, NJ: John Wiley and Sons, 2004. [3] R. A. Roberts and C. T. Mullis, Digital Signal…