ECE 563 Exam 1 with Solutions

B 11 points now suppose that we set the dt systems

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Unformatted text preview: sider the system shown below: xC ( t ) C/D$ TS H(F)$ h[n]% D/C$ TS yr (t ) Assume that xc ( t ) is bandlimited to 6 kHz. 1 a. (11 points) Suppose that we set = 12 kHz. Find H ( F ) and h [ n ] if we want the Ts € system to be equivalent to an ideal CT high- pass filter (HPF) having frequency response € € H ( f ) = 1, 3 kHz ≤ f ≤ 6 kHz €c Hc ( f ) = 0, f < 3 kHz. b. (11 points) Now suppose that we set the DT system’s frequency response to be 1 H ( F ) = 1, F ≤ € 4 1 1 H ( F ) = 0, < F ≤ . 4 2 1 What is the smallest sample rate (that is, the smallest value of ) that we can Ts use and not see any effects of aliasing in the system output y r ( t ) ? (Hint: Note that € we will not see effects of aliasing in y r ( t ) if the effects are not seen in the DT signal that is input to the interpolator.) € € € 5. (21 points) Say we have a real- valued 64- point signal x [ n ], n = 0, ..., 63. We calculate a 64- point DFT X [ k ], k = 0, ..., 63. When we examine the positive frequency part of the DFT, we find that X [16] = − j and X [ k ] = 0 for k = 0, ..., 15 and k = 17, ..., 32....
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