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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
antialiasing 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 ztransfonu 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 stepsize 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 ﬂoatingpoint
and _______ used in speech mmpression is that the SNR is (constant/variable) over
the dynamic range; this is accomplished with a. (logcﬁﬂzmtc/amtcngent/squoteroot)
nonlinearity, in which the stepsize 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 (steadystate) 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 Lmscaling 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 Lmnorrn 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'mscaling 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 complemt 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 statespace 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‘mscaled. (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 secondorder 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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