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20085ee102_1_F08-102-HW-1

20085ee102_1_F08-102-HW-1 - Is S TI or TV C or NC 3 Compute...

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FALL 2008: Put First Letter of LAST Name in the corner →→ ( Otherwise Your HW May Be LOST! ) PRINT: (LAST , Middle, First):——————————————————– EE102: SYSTEMS & SIGNALS HW: # 1 A LATE HW CANNOT BE A HW! Posted: October 1 Hand In To Me 1 : October 8 Attach This Sheet To Your HW (Otherwise It May Be Lost!) 0. Given the complex numbers: z = 1 + i and w = 3 - 4 i . (i) Compute the Real (Re.) Part and the Imaginary (Im.) Part of: ( z - w ) 2 , z w , and z w. (ii) Let s = a + ib . Write down: Re. i s and Im. i s , where s is the complex conjugate of s . 1. Consider the di ff erential equation — together with the initial conditions: dy ( t ) dt + y ( t ) = dx ( t ) dt - x ( t ) , t > 0 , y (0) = 0 , x (0) = 0 . (called “Initial-Valued Problem”). (i) Solve for y ( · ) in terms of x ( · ). (ii) Calculate y ( · ) given that x ( t ) = te - t , t 0 . 2. A system S is described by the following IPOP description: y ( · ) = T [ x ( · )] , y ( t ) = -∞ ( t τ ) U ( τ - t ) x ( τ ) d τ , t > -∞ , 1 during the break of Wednesday Lecture 1
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where x ( · ) is input and
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Unformatted text preview: Is S : TI or TV? C or NC? 3. Compute: I 1 ( t ) = ± t-∞ ( t-τ ) e-2( t-τ ) U ( τ-2) d τ , t >-∞ , I 2 = ± 4-1 e-2 t U ( t-2) dt. 4. Given the following IPOP description: y ( t ) = T [ x ( t )] , y ( t ) = x ( t )-± t-∞ e-t e τ x ( τ ) d τ , t >-∞ , (i) Rewrite y ( t ) as y ( t ) = ± ∞-∞ [ ? ] d σ where [ ? ] is TBD (To Be Determined – by You). (ii) State the following properties of the system: L, NL, TI, TV, C, NC, if applicable. 5. Given dy ( t ) dt + 1 t + 2 y ( t ) = x ( t ) , t > , y (0) = . Solve for y ( · ) in terms of x ( · ) . 6. A system admits the IPOP description: y ( · ) = T [ x ( · )] , y ( t ) = ± ∞-∞ e-( t-σ ) U ( t-σ ) x ( σ ) d σ . Find y ( · ) given that x ( t ) = e-t , t ≥ , = e t , t < . 2...
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