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ECE114_2005FALL_EXAM3__[0] - The Cooper Union Department of...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering EOE114 Digital Signal Processing Exam III _ December 21, 2005 Time: 90 min. Closed book, closed notes. No calculators. 1. [8 pts.] The multirate system shown in Figure 1a is equivalent to either Figure lb ‘61- Figure 1c, with appropriate choice for multirate factor M and LTI system G (2). Specify whether Figure lb or 1c is the correct choice, and specify M and G (3). Show all work! 2. [8 pts.] A dig'tal spectrum X (w) is shown in Figure 2. In this problem, assume no anti-aliasing or mti—imag‘ing filtering is performed. (a) First, the signal is decimated by 2. Then it is interpolated by 3. Show the spectrum of the final signal (only) with frequency in the range -11' to 1r. (b) First, the signal is interpolated by 3. Then it is decimated by 2. Show the spectrum of the final signal (only) with frequency in the range —rr to 1r. 3. [8 pics.) Consider the HR filter: 2' H(z)= z — o: where |oz| < 1. Let £0,131 be the polyphase components of H with respect to a. decimation factor of 2. (3) Find h(n) and, from that, ea {11.) and 61 [n]. (b) Find Eu (2) and E1(z) from 80(11): 8,01). (0) Verify the z-transfonu relationship between H and E0, E1. 4. [6 pta] An experiment with the LMS algorithms results in the performance curve shown in Figure 4, where J (n) is the value of the cost function at iteration n. The theoretical minimum cost, associated with an “optimal" filter under stationary condi- tions, is Jmin. (a) Suppose the step-size a is increased in such a way that the algorithm mains stable. In general, would you expect the misadjustment to increase or decrease? Would you expect the rate of convergence to increase or decrease? No explanation is necessary. ' "[13] Copy the sketch in your exam book, and superimpose a. sleetch of what you would expect the performance curve to look like if the step size is increased. Your sketch should clearly reflect your predictions above, and clearly labei “before” and “after.” ((3) Does the rate of convergence describe the behavior of the algorithm under non— stationary conditions? If not, what term is normally used in such cases? 5. [3 pts.] Associate each of the following with LMS or RIB: (a) Implements a recursive update for the inverse correlation matrix. (b) Employs a deterministic (not a stochastic) cost function. (c) Generally requires significantly fewer iterations to converge, when the algorithm is working properly. 6. [8 pts.] In a differential coding scheme, the input to the quantizer should be the input signal minus the (pick one): (a) output of the prediction filter. (b) input to the prediction filter. (c) output of the quantizer. (cl) previous input sample value. 7. [7 pts.] There are 7 parts to this question: just write your answers in order, don’t copy over the whole paragraph. For a uniform quantizer, the SNR is {constant/variable) over the dynamic range. A rule of thumb for such quantizers is that every additional bit can be used to increase SNR by ___dB for the same dynamic range, or increase the dynamic range by ___dB for the same SNR. A common feature of nonuniform quantizers such as floating-point and _______ used in speech mmpression is that the SNR is (constant/variable) over the dynamic range; this is accomplished with a. (logcfiflzmtc/amtcngent/squote-root) nonlinearity, in which the step-size is (proportional to / inversely proportional to / the same regardless of ) the input signal amplitude. 8. [5 pts.] Consider FIR or HR filters implemented with lattice structures. Although such structures often require more multipliers than transversal or direct form struc- tures, they do not necessarily'require a higher computational complexity, because the multplier coefficients in lattice filters can often be ...... finish this train of thought. 2 9. [9 pts.] This problem refers to scaling to control the variance of state variables in a digital filter. (a) The scheme is called _____ -scaling. (b) The goal of 'this scaling rule is to make the (steady-state) wvarimce matrix of the stain: vector satisfy the followhig: 1. it should be identity; 2. it should have all elements equal to 1; 3. it should have diagonal elements equal to 1. 0:) True or false: this generally requires changing the structure of the filter. 10. [8 pts.] The Lm-scaling rule has been applied to a cascade H1 H2 - . . HL (where H; is the first stage and HL is the last stage]. (a) What is the Lm-norrn of a digital filter? (I am not asking what is the scaling rule. I’m asking you to define the norm.) (h) After scaling has been applied, each stage H; does not have infinity norm equal to 1. Explain briefly. (c) Does applying I'm-scaling force the sequence of stages to be changed? That is, do the stages have to be moved around? 11. [2 pts.] Filters with sharper magnitude response characteristics tend to have pole; with flemsrflenmllw') magnitudes. 12. [8 pts.] Draw a sketch showing the general relationship between SNR and input amplitude for a uniform quantizer; both axes should be in decibels. Label the following on the curve: (a) The region where rounded error dominates. (b) The region where overflow occurs. (c) How the dynamic range corresponding to a target SNR can be determined. 13. [4 pts.] Find the value of the following two's complement number: 11111111011010 14. [8 pts.] The following two’s complem-t number is to be stored in (4) . (2) form: 001010.01 1 Specify the code (not the value) for each of the following cases: (a) saturation overflow and roundofl' by rounding. (b) saturation overflow and roundoff by truncation. . (c) two’s complement overflow and roundofi by rounding. (d) two’s complement overflow and roundoif by truncation. 15. [12 pts.] True or false: (:1) If the state-space realization {A, B. C. D} of a digital filter satisfies the condition that “A" < I (here |H| denotes spectral norm of a matrix), then limit. cycles caused by quantization cannot occur. (b) Overflow cannot occur is a system that has been ”-scleed. (0) Limit cycles camiot occur in 8. system that. has been I‘m-scaled. (d) LMS works in situations where RLS does not because limit cycles cannot occur in the LMS algorithm. (e) A digital filter implemented as a cascade of second-order stations, where every section is designed to be limit cycle free, cannot exhibit limit cycles. (f) Difl‘erentiel coding schemes avoid limit cycles by placing feedback around the quantizer. 16. [4 ptn.] Briefly define DPCM gain. ...
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