EE538 Digital Signal Processing I Session 13 Exam 1 Live: Wed., Sept. 18, Cover Sheet

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1 EE538 Digital Signal Processing I Session 3 Exam Live: Wed., Sept. 8, 00 Cover Sheet Test Duration: 50 minutes. Coverage: Sessions -0. Open Book but Closed Notes. Calculators not allowed. This test contains three problems. All work should be done in the blue books provided. You must show all work for each problem to receive full credit. Do not return this test sheet, just return the blue books. Prob. No. Topic(s) Points. DT Correlation 30. LTI Systems (Causal): 30 Properties, Frequency Response Interconnection, Pole-Zero Diagram 3. Digital Upsampling 0

2 Problem. [30 points] Consider a very simplistic CDMA system with only two users assigned the following length orthogonal complex-value codes, respectively, which are two rows of a -point DFT matrix: User s code: c [n] ={, j,,j} User s code: c [n] ={,j,, j} Consider transmitting a block of two QAM symbols for each of the two users, User s two info. symbols: b [n] ={b [0],b []} User s two info. symbols: b [n] ={b [0],b [], } where b k [n] can take on one of four different complex values (for each value of k and n):. +j representing the bit pair {,} ;. -+j representing the bit pair {0,} ; 3. --j representing the bit pair {0,0} ;. -j representing the bit pair {,0} ; The transmitted code-division multiplexed block may be mathematically expressed as x[n] = b k [m]c k [n m], n =0,,..., 7 k= m=0 Given that the received block has the following numerical values x[n] ={ j, j, j, j, j, j, j, j} }{{} where the first entry above is the value of x[0], determine the numerical values of b [n],n= 0, andb [n],n=0,. Your answer should consist of numerical values all together. Show all work in arriving at your answer. Note that code is orthogonal to code and that the two users are synchronized.

3 EE538 Digital Signal Processing I Exam Fall 000 Problem. [30 points] Consider the causal, second-order difference equation below where it is noted that.905 =.95. y[n] = 0.95 y[n ] y[n ] + x[n]+ x[n ] + x[n ] Consider implementing this second-order difference equation as two first-order systems in parallel (one pole each) as shown in the diagram. x[n] + y[n] (a) Determine and write the first-order difference equation for each of the two first-order systems in parallel. The upper first-order system has impulse response h [n] andis described by the difference equation y [n] =a () y [n ] + b () 0 x[n]+b () x[n ] The lower first-order system has impulse response h [n] and is described by the difference equation y [n] =a () y [n ] + b () 0 x[n]+b () x[n ] Determine the numerical values of a (i), b (i) 0,andb (i), i =, six values total. (b) For EACH of the two first-order systems, i =,, do the following: (i) Plot the pole-zero diagram. (ii) State and plot the region of convergence for H i (z). (iii) Determine the DTFT of h i [n] and plot the magnitude H i (ω) over the interval <ω<showing as much detail as possible. In particular, explicitly point out if there are any values of ω for which H i (ω) is exactly zero. 3

4 EE538 Digital Signal Processing I Exam Fall 00 Problem 3. [0 points] X a (F) /W H (ω) W W F 3 3 ω x a (t) Ideal A/D F = W s x [n] w [n] Lowpass Filter ω = ω = 3 p s gain = Figure. The analog signal x a (t) with CTFT X a (F ) shown above is input to the system above, where x[n] =x a (n/f s )withf s =W,and h [n] = sin( n) cos( n) n, <n<, n such that H (ω) =for ω, H (ω) =0for 3 ω, andh (ω) has a cosine roll-off from at ω p = to 0 at ω s = 3. Finally, the zero inserts may be mathematically described as { x( n), n even w[n] = 0, n odd (a) Plot the magnitude of the DTFT of the output y[n], Y (ω), over <ω<. (b) Given that x[n] = sin( n) <n<, n provide an analytical expression for y[n] for <n< (similar to the expression for either x[n] orh [n] above, for example.) THIS PROBLEM IS CONTINUED ON THE NEXT PAGE.

5 (c) The up-sampling by a factor of in Figure can be efficiently done via the block diagram in Figure below. (i) Provide an analytical expression for h 0 [n] =h [n] for <n<. Simplify as much as possible. (ii) Plot the magnitude of the DTFT of h 0 [n], H 0 (ω), over <ω<. (iii) Provide an analytical expression for the output y 0 [n] for <n<. (iv) Plot the magnitude of the DTFT of h [n], H (ω), over <ω<. (v) Plot the phase of the DTFT of h [n], H (ω), over <ω<. (vi) Provide an analytical expression for the output y [n] for <n<. 0 = =y[n] 0 y[n] x [n] interleaver = h [n+] Figure. =y[n+] 5

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