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Department of Electrical Engineering University of Arkansas EE 2000 SIGNALS AND SYSTEMS ELEG 3124 Signals & Systems Ch Discrete-Time System Dr Jingxian Wu wuj@uark.edu (These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.) SIGNAL • Discrete-time signal – The time takes discrete values n x(n) = cos 4 n x ( n) = exp 4 SIGNAL: CLASSIFICATION • Energy signal v.s Power signal – Energy: E = lim N → N x ( n) n=− N – Power: N P = lim x ( n) N → N + n=− N – Energy signal: E – Power signal: P SIGNAL: CLASSIFICATION • Periodic signal v.s aperiodic signal – Periodic signal x(n) = x(n + N ) • The smallest value of N that satisfies this relation is the fundamental periods – Is cos(n) periodic? cos(n) is periodic if – Example: cos(3n) cos(n) cos( n) k is integer for integer k 5 SIGNAL: ELEMENTARY SIGNAL • Unit impulse function 1, n = 0, ( n) = 0, n • Unit step function 0, n 0, u ( n) = 1, n • Relation between unit impulse function and unit step function (n) = u(n) − u(n − 1) u ( n) = n (k ) k = − SIGNAL: ELEMENTARY SIGNAL • Exponential function x(n) = exp(n) • Complex exponential function x(n) = exp( j0n) = cos(0n) + j sin( 0n) OUTLINE • Discrete-time signals • Discrete-time systems • Z-transform SYSTEM: IMPULSE RESPONSE • Impulse response of LTI system – The response of the system when the input is (n) x(n) = (n) y(n) = h(n) System LTI system • System response for arbitrary input – Any signal can be decomposed as the sum of time-shifted impulses x ( n) = – Time invariant (n − k ) + x(k ) (n − k ) k = − h( n − k ) System LTI system – Linear + + x(k ) (n − k ) k = − x(k )h(n − k ) System k = − LTI system SYSTEM: CONVOLUTION SUM • Convolution sum – The convolution sum of two signals x(n) and h(n) is x ( n) h( n) = + x(k )h(n − k ) k = − • Response of LTI system – The output of a LTI system is the convolution sum of the input and the impulse response of the system x(n) h(n) x(n) h(n) LTI system 10 SYSTEM: CONVOLUTION SUM • Example – x(n) (n − m) – x(n) = nu(n), x(n) h(n) = h(n) = nu(n) 11 SYSTEM: CONVOLUTION SUM • Example: – Let x(n) = [1,3,−1,−2] sequences, find h(n) = [1,2,0,−1,1], be two x(n) h(n) 12 STSTEM: COMBINATION OF SYSTEMS • Combination of systems ➔ Two systems in series + ➔ Two systems in parallel 13 SYSTEM: DIFFERENCE EQUATION REPRESENTATION • Difference equation representation of system N a k =0 M k y (n − k ) = bk x(n − k ) k =0 14 OUTLINE • Discrete-time signals • Discrete-time systems • Z-transform 15 Z-TRANSFORM • Bilateral Z-transform X ( z) = + −n x ( n ) z n = − • Unilateral Z-transform + X ( z ) = x(n)z − n n =0 • Z-transform: – Ease of analysis – Doesn’t have any physical meaning (the frequency domain representation of discrete-time signal can be obtained through discrete-time Fourier transform) – Counterpart for continuous-time systems: Laplace transform 16 Z-TRANSFORM • Example: find Z-transforms – x(n) = (n) n – x(n) = u (n) 2 17 Z-TRANSFORM • Example – n 1 x(n) = − u (− n − 1) 2 • Region of convergence (ROC) Region of convergence 18 Z-TRANSFORM: CONVERGENCE • Convergence of causal signal x(n) = nu(n) • Convergence of anti-causal signal x(n) = nu(−n −1) 19 Z-TRANSFORM: TIME SHIFTING PROPERTY • Time Shifting – Let x(n ) be a causal sequence with the Z-transform X (z ) – Then Z x(n + n0 ) = z X ( z ) − z n0 Z x(n − n0 ) = z − n0 X ( z) + z n0 − n0 n0 −1 x ( m) z −m m =0 −1 x(m) z m = − n0 −m 20 Z-TRANSFORM: LTI SYSTEM • LTI System – Difference equation representation N a k =0 M k y (n − k ) = bk x(n − k ) k =0 – Z-domain representation N M −k −k a z Y ( z ) = b z k k X ( z ) k =0 k =0 – Transfer function M −k b z k Y ( z ) k =0 H ( z) = = N X ( z) −k a z k k =0 21 Z-TRANSFORM: LTI SYSTEM • Example – Find the transfer function of the system described by the following difference equation y (n) − y (n − 1) + y (n − 2) = x(n) + x(n − 1) 22 Z-TRANSFORM: STABILITY • Stability z H ( z) = z−a h( n) = a n u ( n) – A LTI system is BIBO stable is all the poles are within the unit circle (|a| < 1) – A LTI system is unstable is at least one pole is on or outside of the unit circult ( | a | ) ... Discrete-time signals • Discrete-time systems • Z-transform SYSTEM: IMPULSE RESPONSE • Impulse response of LTI system – The response of the system when the input is (n) x(n) = (n) y(n) = h(n) System. .. System LTI system – Linear + + x(k ) (n − k ) k = − x(k )h(n − k ) System k = − LTI system SYSTEM: CONVOLUTION SUM • Convolution sum – The convolution sum of two signals x(n) and h(n)... = − • Response of LTI system – The output of a LTI system is the convolution sum of the input and the impulse response of the system x(n) h(n) x(n) h(n) LTI system 10 SYSTEM: CONVOLUTION SUM
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