1. COMMUNICATIONS: (a) AWGN noise passing through linear filter system, calculate and draw output autocorrelation function. Given t0, calculate PDF of w(t0), where w(t) is the output. (b) X(t) = A(t)cos(wt + B) where A(t) is a stationary random process and B is a uniformly distributed random variable in [0, 2pi]. Calculate the autocorrelation function and PSD of X(t). If A(t) = A and region of B is changed to [0, pi/2], calculate autocorrelation function again and the average power of X(t).
COMMENTS: All the question is from the first 4 chapters in EE4102 - Digital Communications (Melissa Tao's part), which is actually the basic math knowledge of communication and is, to some extent, irrelevant to the major comm contents (modulation, optimum receiver, etc.)
2. SIGNAL PROCESSING: (a) h(n) = (a^-n)u(n), where u(n) is unit step function. a>1. Find DFT in terms of H(e^jw). Calculate |H(e^jw)|^2 and explain what will happen when 'a' becomes large. Let f(n) = [h(n)]^2, Find DFT. Discuss the PSD of f(n) when 'a' is large. (b) (i) Butterworth low-pass filter design given delta1(dB), delta2(dB), -3dB freq and stop-band freq. (ii) Transfer to digital filter H(z) using the derivative method.
COMMENTS: Part (a) is from EE2009 - Signals, textbook chapter 9. Part (b) is from EE3101 - DSP, lecture notes (Samir's part). Questions are simple, though the large amount takes a while for the writing.
3. COMPUTER SCIENCE: (a) (i) Given a hash function h(k) = m[kA mod 1], where m = 1000; A = (5^(1/2) - 1)/2; "mod 1" gives the fraction of kA, i.e. if kA = 119.3475, then "kA mod 1" = 0.3475; [x] is the integer floor function. Find h(k)'s range. (ii) Define the term "perfect hashing". (iii) h(k) = k mod m. Find a m<=12 such that h(k) is perfect given the input keys {0, 6, 9, 12, 22, 31}. Prove that this is the only solution. (b) can't remember but it's about universal hashing and some proof questions.
COMMENTS: CS3230's notes and tutorials are useful. The MIT book's level is too high for QE. Thought sorting or trees may come out but I was disappointed. Didn't do well on this question.
4. CIRCUIT DESIGN: (a) Wilson current mirror circuit (configuration with resistor load). DC analysis. (b) Output resistance calculation of the configuration in (a). (c) Explain the drawback of this configuration and draw an improved schematic.
COMMENTS: Can be found in EE5507 - Analog Circuit Design lecture notes. Can't remember whether it's in EE3408's notes, but the knowledge still belongs to undergraduate level, though the feedback invovled in calculation in part (b). Didn't find them in the reference book though.
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