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ECE114_2006FALL_EXAM1__[0]

# ECE114_2006FALL_EXAM1__[0] - The Cooper Uniori Department...

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Unformatted text preview: The Cooper Uniori Department of Electrical Engineering ECE114 Digital Signal Processing Exam I September 28, 2006 Time: 2 hours. Closed book, closed notes. No calculators. 1. [20 pts.] This question references a senior project undertaken last year by Alin Cosmanescu, Michael Dooley and Chris Shaver. A magnetic ﬁeld is measured by a planar microcoil array. Each sensor detects a localized magnetic ﬁeld (actually, the component of the magnetic ﬁeld perpendicular to the array), producing a continuous-time output signal. For simplicity, assume a large, planar array with unit distance between sensors. Thus, the array output can be expressed as a (ii, i) where it = (711,112) is a discrete spatial coordinate vector, with n1, n2 integers ranging from —00 to co, and t is continuousmtime, ranging from —oo to 00. In general, a can be complex (it is actually a baseband signal). Let the spectrum be denoted by U (law) where E = (Jul, 192) is the wavenumber vector and w is the temporal radian frequency. (a) Specify I; and w (i.e., continuous, discrete, inﬁnite, periodic}. (b) Write the Fourier transform and inverse transform formulas. (c) Write Parseval’s theorem. (d) PROVE Parseval’s theorem. (e) Write the formula for convolution in space-time: a at o. l l (f Write the formula for convolution in frequencyiwavenumber: U * V. (g If h (ﬁl, t) is the impulse response of LTI processing performed on the array Output, write an explicit condition on it that holds if the processing is BIBO stable. T j ’s. 2. [8 pts.] Given the following transfer function: (22 — 3)2 (z + 5) (2 +4) Hm: (z—3)(4z+3)(22+1) (a) Specify all poles and zeros1 including any at inﬁnity, with multiplicity. (b) Specify the R00, if any, associated with: 1. a causal system; 2. a stable system; 3. a system with a well-deﬁned frequency response; 4. an anti-causal system. (c) The system z‘lH (2) (can / can not) be causal and (can / can not) be anti-causal. (No justiﬁcation needed) (d) The system ZR (2) (can / can not) be causal and (can / can not) be anti—causal. (No justiﬁcation needed) 3. [8 pts.) Given the following discrete-time signal: M )_ A(—2/3)" + (Bn+ C) (1/4)” +D6(n) n 2 U n _ E (4/5)“ + F2” 11 < o where A, B , C, D, E, F are undetermined constants. (a) Without attempting to actually compute the z-transform of 11(71), specify any information (including multiplicity) regarding the poles and/or zeros of H (:5) that can be discerned from the above. (b) Specify the ROC of H corresponding to h (n) as given above. IF there is not enough information to determine the R00, say so. No justiﬁcation needed. 4. [8 pts.) Given the following transfer function: 1 2 __ 2 M: (32+ ) (z s) (z + 4) (32 + 2) Find all-pass A (z) and minimum—phase Hmin (2:) such that Hmin = H - A. 5. [10 pts.] The power spectral density of a discretetime WSS random process is: 5 — 4cosw 8(w) = ———~—2 (13 + 5cosw) (a) Find its innovation and whitening ﬁlters, expressed in the z-domain (clearly iden~ tify which is which). (Assume these are deﬁned with reference to an innovations process with unit variance). (b) Name the method you used. —_:——\$ 6. [8 pts.] Given a 5 X 5 2—D FIR ﬁlter h(n1, n2). (3.} If the ﬁlter is separable, how many parameters need to be determined to Specify r the ﬁlter completely? In other words, how many independent variables are there which can be chosen in the ﬁlter design process? (b) Suppose a 16 X 15 block of pixels is to be ﬁltered with a separable 5 x 5 FIR. ﬁlter. First, what is the size of the output pixel block, and, second, how many multiply operations are needed in total assuming the operation is implemented as a separable structure. (You do not need to simplify your answer; writing something like 7 x 8 x 1347 or whatever is ﬁne). 7. [5 pits] An analog ﬁlter has denominator polynomial 834 + 253 +452 + 53 + 7. Specify the numerator polynomial terms of s, not w) such that the ﬁlter is all-pass. Hint: If ,8 is a complex number, then what is |ﬁ*/ﬂ| ? 8. [12 13135.] A digital spectrum analyzer samples a real signal at IOOiMHz, collects a block of 1000 samples, and computes a DFT. Suppose the DFT exhibits a peak at index k m 50. (a) Specify the bin spacing in Hertz. (b) Specify the input frequency in Hertz, assuming no aliasing occurred. (c) There should be another peak among the DFT coefﬁcients, if they are indexed in the “usual” fashion. What is the other index value it? 9. [6 pts.] Let :5 = {3,1,4,—2} be a. block of 4 samples. An 8—point DFT is to be computed, where the “usual” convention is employed. Compute X (3), the DFT coefﬁcient corresponding to k = 3. Your ﬁnal answer must be simpliﬁed to the form a —i— jb where e,b are real; there should be no e3“, sin or cos stuff left. 10. [6 pts.] Let a: = {3,1,4,—2} and h = {2,3, —3,5}. Let y4 denote the result of 4-point circular convolution and ya (n) the result of 8-point circular convolution, and ’91,; (n) the result of ordinary (linear) convolution, where the “usual” conventions are employed. (a) Compute 314(2) only.’ Show work (i.e., write out the terms you are multiplying and adding, not just a ﬁnal numerical result). (13) Given ya, for all in (you do not know the original values of nigh}, is there enough information to compute ya (2) or in, (2)? If so, write the expressions. No justiﬁcation needed. (c) Given yg for all it (you do not know the original values of r, h], is there enough information to compute yr (2] and/ or '91,; (2)? If so, write the expressions. No justiﬁcation needed. ((1) Given pl, (71) for all in. (you do not know the original values of x, it), is there enough information to compute yr (2} and/or pg (2)? If so, write the expressions. No justiﬁcation needed. I’Wiwll d3 -:1[i 10 11. [9 pts.] Figure 11 shows the magnitude spectrum of a window function that is em- ployed by a digital spectrum analyzer prior to computing the DFT of the input data. (a) (b) Specify the null-to—null mainlobe width and the peak sidelobe level. Roughly sketch the spectrum in your exam books and indicate how you read off these values from the curve; don’t reproduce a detailed sketch the point is simply to show how you got these numbers. The ability to distinguish two input frequencies that are close together refers to the spectral _____ __ of the analyzer. Which of the two parameters (mainlobe width or sidelobe level) is associated with this feature? The appearance of “false” peaks at frequencies far from the actual input frequency refers to the spectral ____ __ of the analyzer. Which of the two parameters (mainlobe width or sidelobe level) is associated with this feature? ...
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