rec3.pdfrec3solRecitation 3 -EECS 451, Winter 2010 Jan 27, 2010 OUTLINE - Review of important concepts (Lecture 3-4) - Practice problems Concepts: Discrete Time Systems 1. Classification of DT systems: For y(n)= T (x(n)), - static (memoryless): y(n) depends only on the present input - causal: y(n) depends only on the present and past inputs - time-invariant: T(x(n − k)) = y(n − k), where k is an integer - linear: T (a1x1(n)+ a2x2(n)) = a1 T (x1(n)) + a2 T (x2(n)) - stable (BIBO): every bounded input provides a bounded output 2. Linear Time Invariant (LTI) System: y(n)= T (x(n)) - completely characterized by the impulse response h(n)= T (δ(n)) - output given by simple convolution operation, y(n) := h(n)x(n)=kx(n-k)h(k) 3. Properties of convolution - Commutative: x(n)  h(n)= h(n)  x(n) - Associative: (x(n)  h1(n))  h2(n)= x(n)  (h1(n)  h2(n)) - Distributive: x(n)  (h1(n)+ h2(n)) = x(n)  h1(n)+ x(n)  h2(n) 4. Classification of LTI systems by the impulse response - Causal: The given LTI system is causal if and only if h(n)=0 n< 0.- Stable (BIBO): The given LTI system is stable if and only if n|h(n)| < ∞ 5. Two classes of LTI systems characterized by the impulse response - Finite Impulse Response (FIR) system: number of non-zero h(n)’s are finite. - Infinite Impulse Response (IIR) system: number of non-zero h(n)’s are infinite. 6. The system defined by linear constant coefficient difference equation 10() () ()NMkkkkyn ayn k bxn k    - is LTI and causal - can be implemented by direct form I or direct form II (more efficient). - is recursive and the impulse response is IIR if N ≥ 1. - is non-recursive and the impulse response is FIR if N = 0. h(n)= {b0,b1, ··· ,bM }. Problems 1. Determine which of the following systems is static, linear, time-invariant, causal, stable (a) y(n)= x2(n + 1) (b) (), 0()( ), 0xn nynxn n (c) y(n)= kx(n − k)p(k), where p(n)= {−10, ··· , −1, 0, 1, ··· , 10} 2. Compute the output of the following LTI systems (a) x(n)= {0, 0, 1, 1, 1, 1},h(n)= {1, −2, 3} (b) x(n)= {1, 1, 2},h(n)= u(n)3. Determine the impulse response of a discrete-time system realized by the structure shown in Fig. 1. Fig.