102_1_hw02 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #2 Due: Tuesday, April 20, 2010 at 5 PM. 1. State whether the following systems are linear or nonlinear; time invariant or time variant; and why. (a) y ( t ) = x ( t ) sin( ω t + φ ) (b) y ( t ) = x ( t ) x ( t - 1) (c) y ( t ) = 1 + x ( t ) (d) y ( t ) = cos( ω t + x ( t )) (e) y ( t ) = t -∞ x ( τ ) d τ (f) y ( t ) = t/ 2 -∞ x ( τ ) d τ 2. A periodic signal x ( t ) , with a period T , is applied to a linear, time-invariant system H . Show that the output y ( t ) y ( t ) = H ( x ( t )) is also periodic, with period T . 3. Sample and hold system. A sample and hold (S/H) system, with sample time h , is described by y ( t ) = x ( h t/h ) , where a denotes the largest integer that is less than or equal to a . Sketch an input and corresponding output signal for a S/H, to illustrate that you under- stand what it does. Is a S/H system linear? 4. Consider a system that takes a signal x ( t ) and returns the even part of x ( t ) as it’s output x e ( t ) = H ( x ( t )) where x e ( t ) is the even part of x ( t ) . Is this system linear? Is it time invariant?
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2 5. Write the following signals as a combination (sums or products) of scaled and shifted unit rectangles rect ( t ) and unit triangles Δ ( t ) . 2 -1 1 0 -2 1 2 -2 -1 x(t) a) 2 -1 1 0 -2 1 2 -2 -1 t x(t) b) 6. System Equations from Block Diagrams Find the differential equation that correspond to this block diagram. Hint: Label all of the intermediate signals, and write equations for each.
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