Test1Solution - Solution of ECE 316 Test 1 Su08 1....

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Solution of ECE 316 Test 1 Su08 1. Determine whether or not the systems with these frequency responses are causal and explain how you know. (a) H j ± (29 = 1 ² e ² j h t = t ² t ² 1 h t = 0 , t < 0 ± Causal (b) H f = rect f /20 e ± j 40 ² f h t = 20sinc 20 t ± 20 h t ± 0 , t < ² Non-Causal (c) H e j ± = e j ± e j ± ² 1.3 (I should have used a number less than one in magnitude instead of 1.3 since, strictly speaking, the DTFT will not converge for numbers greater than one in magnitude. But I counted a solution correct if the student did not notice that since it is my error.) H e j ± = 1 1 ² 1.3 e ² j ± h n ± ² ³ ´ = 1.3 n u n ± ² ³ ´ h n ± ² ³ ´ = 0 , n < µ Causal (d) H e j ± = e j 2 ± e j ± ² 1.3 H e j ± = e j ± 1 ² 1.3 e ² j ± h n ± ² ³ ´ = 1.3 n + 1 u n + 1 ± ² ³ ´ h n ± ² ³ ´ µ 0 , n < Non-Causal
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2. With reference to this circuit schematic diagram, L RC v ( t ) in v ( t ) out + - + - (a) Find a general expression for the frequency response H j ± (29 = V out j V in j in terms of R , L and C of the form H j = A j 2 + j a + b where A , a b are constants involving only R , L and C (not ). H j = R / j C R + 1/ j C j L + R / j C R + j C = R / j C j LR + L / C + R / j C = R j 2 LRC + j L + R H j = 1 LC 1 j 2 + j / RC + LC A = 1/ LC , a = 1/ RC , b = 1/ LC (b) Is this circuit a practical lowpass, highpass, bandpass or bandstop filter? Lowpass (c) What is the slope in dB/decade of the magnitude Bode diagram for the frequency response of this filter at very low frequencies? lim ² 0 H j = 1 Slope = 0 dB/decade (d) What is the slope in dB/decade of the magnitude Bode diagram for the frequency response of this filter at very high frequencies? lim ²³ H j = 1 2 LC Slope = -40 dB/decade
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3. In the single-sideband modulation system below let x t (29 = 4sin 10 ± t , f c = 1000 , f m = 10 and let the magnitude of H f in its passband be 5.
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Test1Solution - Solution of ECE 316 Test 1 Su08 1....

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