2 k uk 2 k uk e 02 0819 hk lesson 31 continuous

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Unformatted text preview: sson 31 Continuous to Discrete Given H(s)=s/(s2+400s+2x105) Using published mapping rules: • • • • • Impulse invariant: H(z)=kz(z-0.9135)/(z2-1.508z+0.6703) Step invariant: H(z)=k(z-1)/(z2-1.509z+0.6708) Finite difference: H(z)=kz(z-1)/(z2-1.5z+0.625) Matched z-transform: H(z)=kz(z-1)/(z2-1.509z+0.6708) Bilinear z-transform: H(z)=k(z2-1)/(z2-1.52z+0.68) Lesson 31 Continuous to Discrete Lesson 31 Continuous to Discrete Phase response Lesson 31 Continuous to Discrete Given H (s) = 2 s + 2 L-1 h(t ) = 2e −2t u (t ) h(t) Sample fs=10 Sa/s Impulse invariant h[k ] = 2e −2 kTs u[k ] = 2 e −0.2 k u[k ] = 2 α k u[k ] α = e −0.2 = 0.819 h[k] Lesson 31 Continuous to Discrete h[k ] = 2α k u[k ] α = e −0.2 = 0.819 Z H ( z ) = Ts 2 0.2 0.2 z = = 1 − 0.819 z −1 1 − 0.819 z −1 z − 0.819 Plausible? Analog DC gain Discrete DC gain H(0) = 2/2=1 H(1) = 0.2/(1-0.819) = 1.105 ~ 1 Lesson 31 Continuous to Discrete 1 2 H (s) = s+2 0.9 0.8 0.7 0.6 0.5 » na=[2];da=[1 2]; » [Ha,wa]=freqs(na,da); 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 π 3.5 1.4 H ( z) = 0 .2 z z − 0.819 1.2 1 0.8 0.6 0.4 » nd=[.2 0];dd=[1 -0.819]; » [Hd,fd]=freqz(nd,dd); Lesson 31 0.2 0 0 0.5 1 1.5 2 2.5 3 Continuous to Discrete Given H (s) = 2 s + 2 L-1 h(t ) = 2e −2t u (t ) h(t) fs=100 Sa/s Same problem using a much higher sample rate. h[k ] = 2e −2 kTs u[k ] = 2 e −0.02 k u[k ] = 2 α k u[k ] α = e −0.02 = 0.9802 h[k] Lesson 31 Continuous to Discrete h[k ] = α k u[k ] α = e −0.02 = 0.9802 Z H ( z) = 2Ts 0.02 0.02 z = = 1 − 0.9802 z −1 1 − 0.9802 z −1 z − 0.9802 Plausible? Analog DC gain Discrete DC gain H(0) = 2/2=1 H(1) = 0.02/(1-0.9802) = 1.01 Lesson 31 (Much closer fit) Suggested problems: 5.3-5 (a): 4 y [k ] + 4 y [k − 1] + y [k − 2] = x[k − 1] y [−1] = 0; y [−2] = 1; x[k ] = u[k ]. Show 5 1 13 y [k ] = − (−0.5) k − k (−0.5) k u[k ] 12 9 36 Lesson 31 Mini Quiz Given a linear continuous time system: H(s) = 2/(s2+3s+2) The system is sampled at a fs = 10 Hz rate. What is H(z) (approximate)? Lesson 31 Challenge: 5.3-5 (b): What is the (steady-state) value of y[k]? Lesson 31...
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