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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 antialiasing
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 discretetime 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 decnnationimfrequency 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 decimationinatime / 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 stepsize 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 Lmscaling 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 Lmscaling, 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) linearphase 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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