ECE114_2006FALL_EXAM2__[0]

ECE114_2006FALL_EXAM2__[0] - The Cooper Union Department of...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering ECE114 Digital Signal Processing Exam II - October 20, 2006 Time: 2 hours. Closed book, closed notes. No calculators. 1. [6 pts.] Figure 1 shows the spectrum X (to) of a discrete—time signal. No anti-aliasing or anti~imaging filtering is performed unless explicitly stated. (a) Let u : (T 2) a: and y = {13)o. Sketch the spectra of v and y. (b) Let u = (L 3) a: and y = (T 2) a. Sketch the spectra of u and y. (c) Suppose in part a the signal 12 is filtered by H before it is decimated to yield y. Specify H, if possible, so that no aliasing or imaging distortion occurs, while retaining the full spectral information in 9: (i.e., no part of the spectrum of :I: should be cut off; the spectrum may be "stretched" or "squeezed" but no information is lost). You are not allowed to use other filters in other positions: just one filter between the two multirate operations. I do not require sketches from you — just give the answer. (d) Suppose in part b the signal a is filtered by H before it is interpolated to yield 3:. Specify H, if possible, so that the conditions described in part c can be achieved. 2. [3 pts.] In problem 1 above, suppose the original sampling rate of :r was 104M H z. Specify the appropriate sampling rates for n, u, y. 3. [6 pts.] Figure 3a shows a multirate filter structure. It simplifies to a structure of the form shown in Figure 3b. (a) Specify the multirate factors M and N and express H in terms of F and G. (b) Let H0 (z) , H1 be the polyphase components of H (with respect to a multirate factor of 2). Specify Hg (z), H1 in terms ofF and G. 4. [8 pts.] Let u (n) be a discrete-time WSS process where fM (n) denotes the output of the MM order FPEF at time n, and bM (n) denotes the output of the M1th order BPEF at time a. Complete the foliowing train of thought (additional justification not needed): (a) fm (n) l span {what range of n’s?} (b) 33;, (n — l) E span {what range of n’s?} (c) Therefore, for fixed m 3 1, fm (n) .L 1);, (n —- 1) for ? g is S? 5. [5 pts.] A FFT algorithm applied to a 16~point DFT, in which each successive step is either a decimation—in—time 0r decnnation-imfrequency operation, results in the fol- lowing time—domain input sequence: 0,4, 2,6, 8, 12, 10,14, 1,5, 3,7, 9,13, 11,15 Specify the sequence of decimation-inatime / decimationeinefrequency steps. 1 6. [8 pts.l For a uniform quantizer1 graph the general relationship between SNR and input amplitude, both axes in decibels. On the graph, clearly indicate: (i)how the dynamic range for a prescribed minimal SNR can be determined; (ii)the region where roundoff noise dominates; (iii)the region where overflow dominates. 7. [4 pts.] Consider an A/ D converter employing a uniform quantizer. If the quantized word length is extended by 2 bits in order to increase the dynamic range by QdB, then does the minimum SNR that is sustained over the dynamic range increase1 decrease or stay the same? If it changes, by how much? 8. [4 pts.] Toll quality speech can be represented with 8 bits per sample by employing a(n) wwwwww __, which is a nonuniform quantizer based on a (log / tan / quadratic) nonlinearity. This ensures that the step-size is ( proportional to / inversely proportional to / constant regardless of ) the input amplitude and that, in turn, ensures the SNR is (proportional to / inversely preparations! to / constant regardless of) the input ampli- tude over the dynamic range of the quantizer. 9. [12 13135.] The follOWing questions refer to mitigation of quantization effects in digital filters. Just provide the short answer- justification not needed. (a) The —scaling rule controls the size of the state—variable variance at steady— (b) TRUE or FALSE: The scaling rule described in part a generally requires changing the particular filter structure (e.g., we may need to "abandon" a direct form II structure in favor of another), as a consequence of a transformation of state 33’ = Tr. (c) TRUE or FALSE: If we apply the Lm-scaling rule to a cascade of filter blocks, then the scaling performed on each block will change if the ordering of the blocks in the cascade changes. (d) In the context of Lm-scaling, the L°°~norm for a digital filter is defined as _ u _i i. (e) TRUE or FALSE: Given a stable System, it is always possible to apply a trans— formation of state, if necessary, so that limit cycles associated with quantiza- tion effects {or other noniinearities), say so —> f cannot occur as long as S for all :13. (f) The condition that |f g for all a (see part e) applies to which of the fol~ 10wing? (some, all or none— specify which) saturation overflow, two’s complement overflow, roundoff by rounding, truncation towards zero. 10. [2 pts.] Briefly define the term limit cycle (in general, not just in reference to quan— tization effects). One sentence only! 11. [3 pts.] An adaptive algorithm is tested with a simulated stationary input signal. The learning curve, which shows the cost function as it evolves with successive iterations, is shown in Figure 11. Jmin denotes the theoretical minimum value achievable with the optimal (Wiener) filter, and J (00) denotes the actual (mean) steady/estate cost. (a) The fact that J (00) exists (is finite) refers to the _____ ___ of the algorithm. (b) The ratio J (00) mein is called the _____ fi_ of the algorithm. (c) TRUE or FALSE: This experiment provides useful information regarding the trucking ability of the algorithm. 12. [8 pts.] The modified Welch periodogram is to be computed for a block of data. The key parameters are: 5127point DFT computed on 100 frames with 256 point overlap. (a) What is the total length of the data block that is required? You do not have to calculate the actual value, just write an explicit formula that can be plugged into a hand—held calculator. (b) What is the appropriate length of the window function that should be employed? (c) If it is desired to impmve the spectral resolution, changing which of the following will have the most direct effect? The size of the DFT (512), the number of frames (100) or the amount of overlap (256)? (d) If it is desired to reduce the variance (the “fuzz”) of the periodogram, changing which of the following will have the most direct effect? The size of the DFT (512), the number of frames (100) or the amount of overlap (256)? 13. [8 pts.) A (causal, real) linear-phase FIR filter of length 7 is partially specified by: M0) : 2,h(1): 3, M2) : 5 (a) Are the other coefficients completely determined? If not, how many choices are there? Give all possible solutions. If one or more coefficients are completely free (no constraints at all), say so. (b) Which, if any, of the solutions could perhaps be used for a digital realization of a Hilbert transformer? (c) Pick one of the possible solutions. Draw a filter structure that not only has a minimum number of delays but also has a minimum number of multipliers. (d) Does your realization require more, the same or fewer adders as a transversal filter? 14. [10 pts.] A digital bandpass filter is to be designed using the bilinear transform method. There are four specificaiton frequencies f1 < f2 < f3 < )2; in Hertz, plus the sampling rate fwmp; specifically: o passband variation rp (dB) in the range f2 g f g f3 0 stopband attenuation r3 (dB) in the range f 5 f1 and f 2 f4. o sampling rate fsnmp You have a filter design CAD routine which can determine the order of a lowpass analog filter with passband edge at lmd/ sec, and then design this lowpass filter. The only help I will give you is what you might see if you pick up a random book (yes, Jared, pun intended) on signals & systems: (3.) Describe how you would determine the stopband frequency you need to specify to this filter design routine. Your explanation should be brief and complete! In particular, write the formulas for the specification frequencies (passband and stopband edges) at all stages, and draw appropriate sketches of the specifications, and also briefly indicate how you would pick the stopband specification (note the bandpass filter has two stopband edges, but your lowpass filter design routine only takes one stopband value). Your explanation should be professional quality: clear and to the point. "Derivations" of these results are not needed; explain only enough so that someone who knows MATLAB but absolutely nothing about EE stuff can write a program to do this. (b) Suppose the CAD routine computes a “raw” filter order of n = 3.15. What value of it should actually be used (for the lowpass analog prototype)? (c) What is the order (defined as minimum number of state variables in a realization) for the final digital bandpass filter? 15. [4 pts.] Characterize the passband and stopband of each of the following as either monotonic or equiripple: (a) Butterworth (b) Chebyshevl (c) Chebyshev H (d) elliptic ...
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This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.

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ECE114_2006FALL_EXAM2__[0] - The Cooper Union Department of...

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