ece 313 hw 4 - ECE 313 Fall 2010 Homework 4 –...

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Unformatted text preview: ECE 313 Fall 2010 Homework 4 – Due Thursday, October 7 For each of the CT system input ­output equations given below, determine if the system is memoryless or has memory; is causal or non ­causal; is linear or nonlinear; is time invariant or time varying; and is stable or unstable: 1. () (a) y ( t ) = x t 2 (b) y ( t ) = ( t + 1) x ( t ) € (c) y ( t ) = € 2. 1 x ( t + 1) For each of the CT system input ­output equations given below, determine if the system is causal or non ­causal; linear or nonlinear; time invariant or time € varying; and stable or unstable: (a) y ( t ) = t /3 ∫ x (r) dr −∞ (b) y ( t ) = t € t ∫ x (r) dr −∞ t € (c) y(t) = ∫ r x ( r) dr for t ≥ 0; −∞ y ( t ) = 0 for t < 0. 3. € The RC circuit described in class has impulse response h( t ) = 1 − t / RC e u( t ) RC (that is: h ( t ) is the output signal when the input is δ ( t ) ). € 1 (a) Plot h ( t ) for −1 ≤ t ≤ 4 for the case RC = . 2 € € € € € 1 (b) Again with RC = , find the system output y ( t ) if the input is 2 x ( t ) = 2 δ ( t ) − 3δ ( t − 1) , and plot y ( t ) for −1 ≤ t ≤ 4 . 4. € Suppose that a CT LTI system has step response € € € s( t ) = 4 1 − e −( t −1) u( t − 1) € ( ) (that is, s( t ) is the output signal when the input is u( t ) ). € (a) Plot s( t ) for −2 ≤ t ≤ 5 . 5. € € ȹ t ȹ (b) Find the system output y ( t ) if the input is x ( t ) = rectȹ ȹ , and plot y ( t ) for ȹ 2 Ⱥ € € −2 ≤ t ≤ 5 . For each of the DT system input ­output equations given below, determine if the € € € is causal or non ­causal; is linear or system is memoryless or has memory; nonlinear; is time invariant or time varying; and is stable or unstable: € x [n ] u[ n ] n +1 (a) y[ n ] = Ⱥ n Ⱥ y[ n ] = x Ⱥ Ⱥ for n even; Ⱥ 2 Ⱥ (b) y[ n ] = 0 for n odd. € € 6. (c) y[ n ] = n ∑ x[ k ] . k =−n Consider the DT system implementation shown below: € (where the D block represents a 1 ­sample delay). (a) Find the system’s input ­output difference equation. € 7. (b) Assuming that y[ −1] = 0 , find y[ n ] for 0 ≤ n ≤ 5 when the input is x[ n ] = u[ n ] . (Use the computational (recursive) approach discussed in class.) Based on your results, do you think this system is stable or unstable? € € ȹ 1 ȹ n Suppose we have a DT system whose impulse response is h[ n ] = ȹ − ȹ u[ n ] . ȹ 2 Ⱥ (a) Plot h[ n ] for −2 ≤ n ≤ 7 . € (b) Find the system output y[ n ] if the system input is x[ n ] = δ [ n + 1] − 2δ [ n ] + δ [ n − 1], and plot y[ n ] for −2 ≤ n ≤ 7 . € € € € € € ...
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