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Unformatted text preview: The Cooper Union
Department of Electrical Engineering
ECElll Signal Processing 8: Systems Analysis
Exam II April 28, 2009
Time: 2 hours. Closed book, closed notes. No calculators. 1. [3 pts.] Let a: (n) be a discretetime WSS process with correlation r (m) and PSD
S Write an explicit formula for r in terms of S (w) (i.e., a formula to compute
r (m) from S (0.2)). This result is called the _______ __ (2 words) Theorem. 2. [5 pts.] Let a: (t) have CTFT X (w). Starting from the Fourier transform integral,
express 7’ {x' (to — t)} in terms of X (on); work from the integral formula, showing
work, WITHOUT invoking general properties of the Fourier transform. 3. [5 pts.] Let X (w) be the Fourier transform of a discretetime signal a: (n), given by thefunction:
X(w)={ 1 —1r<w<0 1=‘f 05w<7r Do not attempt to compute a:(n) or any other timedomain signal in this
problem!! (3) Sketch X (w) over the range —21r S w 3 27r (yes, ~21r to 21r).
(b) True or false: :1: (n) could be real. No justiﬁcation needed.
(c) True or false: 2: (n) could be causal. No justiﬁcation needed. ((1) Let Y(w) = X (2w). Sketch Y(w) over the range —27r s w 5 21r. Is there
a discretetime signal y (n) whose Fourier transform is Y (w)? I am not asking
what 3/ (n) is, merely if such a signal exists. (e) Let V(w) = X (0.1/2). Sketch V(w) over the range —27r S w s 27r. Is there a
discretetime signal 2) (n) whom Fourier transform is V (w)? Again, do not try to
ﬁnd an expression for v (n), merely state whether the signal exists. (f) Either y (n) or v (n) exists but not the other. BRIEFLY indicate how you come to that conclusion, i.e., what property in the frequency domain are you looking
for? 4. [5 pts.] What do each of the following abbreviations stand for (do not give the
deﬁnition, merely write what they stand for):
(a) WSS
(b) 553
(c) ARMA
(d) PSD
(e) iid 5. [8 pts.] The WSS discretetime signal :3 (n) is modeled as:
v = — 0.51: (n — l) + 0.1x (n — 2) +0.21: (n — 3)
where v is white noise with 0,2, = 4. (a) Is :1: AR, MA or ARMA? (b) This ﬁlter, with a: as input and v as output, is called the _____ __ ﬁlter.
The inverse is called the ______ __ ﬁlter. The signal v(n) is called the
_______ __ signal of a: (c) Actually for this ﬁlter to truly be the ____ __ ﬁlter as you labeled it above, we would need to conﬁrm that a certain polynomial in z has all its roots inside the
unit circle. What is this polynomial? (Write it out explicitly). (d) If this polynomial has all its roots inside the unit circle, then this ﬁlter is ______ __
______ __ (two words connected by a hyphen), which is required for the
model. (e) Write an explicit formula (does not have to be simpliﬁed) for the PSD SI (f) True or false (no justiﬁcation needed): if a; (n) is also known to be a Gaussian
process, then the samples 11 (n) are iid. 6. [5 pts.] Write True or False, no justiﬁcation needed. (a) If a: (t) is SSS and y (t) m2 (t) then y (t) is SSS. (b) If x (t) is SSS and y (t) x2 (t) then y (t) is WSS. (c) If ac (t) is WSS and y (t) = 9:2 (t) then y (t) is SSS. (d) If a: (n) is Gaussian and y = a: (2n) then y (n) is Gaussian.
(e) If :1: is SSS and y = a: (2n) then y is SSS. 7. [3 pts.] For the real random variables X, Y, the following is known: E (X) = 3,
E(Y) = 2, E(Xz) = 10, E0”) = 5.
(a) For what value of E (X Y), if any, is X, Y guranteed to be orthogonal?
(b) For what value of E (X Y), if any, is X, Y guaranteed to be uncorrelated? (0) Suppose it is also known that aX + bY is Gaussian for all coefﬁcients a, b. For
what value of E (X Y), if any, is X, Y guaranteed to be independent? 8. [3 pts.] A block of 100 samples of complex random data is collected: a: (n) for
1 S n g 100. Assume that a: is WSS with correlation function 'r Write an
explicit formula that can be used to estimate 7' (5); the formula can only use data inside
the block (i.e., only indices in the range 1 to 100), but as many terms as possible should be included. The underlying property that is being assumed here is ______ __
(one word, does not begin with ’s’). 9. [6 pts.] Use the method of partial fractions to ﬁnd h (t) corresponding to: 232+s+'1 H“): (2s+5)(s+4) 10. [6 pts.] Use the method of partial fractions to ﬁnd h corresponding to: 2z2+z+1 H“): (5z+2)(4z1) 11. [8 pts.] Given the following analog transfer function: (3 + 10) (s + 0.4) H(s)=8(s+2)(s+40)2 (3.) Compute the gain at DC in dB (the numbers should be such that you can do this
by hand). (b) Use the graph paper provided to draw straight line asymptotes and the correct curve for the magnitude response of the ﬁlter. Use ldB/ div as your vertical
scale. Neatness counts! 12. [2 pts.] Given the following: * s (3 +10)
H“) _ (s + 2)2 (s + 100) Specify the phase at DC and at w —» oo, in degrees. 13. [7 pts.] An analog system is characterized by the statespace representation {A, B, C', D},
and transfer flmction matrix H (s). The input vector is 21' (t), the output vector is 37 (t)
and the state vector is a? (a) Under the condition that _______ __ (a short phrase), we have m (t) =
(D (t) a: (0), where <I> (t) is called the _____ __ matrix. It is expressed in terms
of {A,B,C’, D} as !I>(t) = _____ __. Its Laplace transform is (in terms of
{A, B, C, D}) _______ __. (b) The general formula for H (s) in terms of {A, B, C, D} is ________ __. (c) The common denominator (not counting pole/ zero cancellations) in the entries in
H (s) is called the __________________ __ (two words) because it is
the ___________ __ (repeat one of the words used before) of the matrix
_______ __ (write the matrix in terms of {A, B,C', D}).. ((1) Consider the entry Hi3 It is the transfer fimction from the ___"‘ input to
the _ _‘h output (write 1' or j in the appropriate location). 14. [7 pts.] Continue the above analog system with {A, B, C, D} state—space represen
tation and transfer function matrix H (3). Given that the H23( entry in H (3) has
zeros at 2 :l: j (simple) and poles at —3 :l: 23' (simple) and at —4' triple). (a) Can the system be externally stable? (b) There is enough information to determine minimium values of the number of
inputs, outputs and state variables. Specify these values. (c) Assuming these minimum values are in fact the correct values, specify the sizes
of the A, B, C, D matrices. ((1) True or false: Assuming these minimum values, the system is guaranteed to be
internally stable Justify brieﬂy. (e) Write the most general time domain expression for h23 (t), written explicitly as a
real function (i.e., no ejis stuff). For each mode that appears, specify the time
constant in seconds and, if there is oscillation, the frequency in Hertz. Also
identify the dominant mode. 15. [4 pts.] Consider a discretetime system with statespace representation {A, B, C, D}
and transfer function matrix H Let if be the state vector. (a) Under the condition that _________ __, we have :1: = <I> at (0) where
1P (n) = _____ (express in terms of A, B, C, D). (b) Suppose the dominant mode in H (z) is associated with the poles 0.5 exp (:1: j7r / 4).
Assume the system is internally stable, and consider the dominant mode of (I) Of course, it is possible that the dominant mode of <I> dies out just as rapidly as that of H We consider when that does not happen: Remember that
your answer must be consistent with internal stability. 1. Suppose the dominant mode of <1) dies out faster than that of H (2).
Either this is not possible (say so), or it is possible only if there is an eigenvalue
of A that... give the most general condition on that eigenvalue that causes this
to occur, while not violating intemal stability. I’m looking for a mathematical
constraint on the value of this eigenvalue. 2. Suppose the dominant mode of (I) (n) dies out slower than that of H Ei
ther that is not possible (say so), or it is possible only if there is an eigenvalue
of A that... give the most general condition on that eigenvalue that causes
this to occur, while not violating intemal stability. 16. [6 pts.] A stable second order system has system determinant: 32 +83 + K
where K > O.
(a) Express the damping factor and natural frequency (in rad/sec) in terms of K. (b) Specify the values or ranges of K for which the system is overdamped, critically
damped and underdamped. (c) In the case where K is such that the poles are complex (not real), express the fre—
quency of oscillation of the transient response in Hertz in terms of K. Describe what happens to K as this frequency approaches 0, and as it approaches 00 (i.e.,
what are the limiting values of K in each case). 17. [4 pts.] Let a: be a complex WSS process with correlation function r Let R
be the correlation matrix for the following vector: 29(71)
:1:(n—1)
$(n—2) (a) Find R (express every entry in the form 'r (m) for some m). Show work. (b) Check if the following is possible: r (1) = r (2) = %r 18. 19. 20. 21. [5 pts.] For a digital transfer function H (z), the paraconjugate is deﬁned as:
H(z) = H*(1/z*) In this problem, you should show enough work that I am convinced you actually
derived results, and did not just guess or perhaps recall something you may have read
somewhere. But try to be concise. (a) Consider the basic relation between the transform and frequency domains to de
termine how H (w) relates to H (w). (b) Starting with the formula for the ztransform that takes h to H (2), take the
paraconjugate of the formula and study the result to express h (n) (the impulse
response of the paraconjugate) to h Do not assume that h (n) is real. (c) Consider:
(3222 + blz + b0
(1.2.22 + alz + a0 H(z)= where ai, b. are possibly complex coefﬁcients. Write an explicit formula for H (2)
as a ratio of polynomials in z (there should be no 2‘1 or 2"). [2 pts.] Let a: have DTFT 6 (w — 1r/ 3) + 26' (w + 7r/2). Use the IDFT formula
and the properties (actually, deﬁnitions) of 6, 6' to compute z You are not allowed
to use integration by parts, properties of Fourier transform, etc. Use 6, 6' properties. [2 pts.] The unit step response of a ﬁrst order stable discretetime system oscillates
around its steady state value. What must be true of the system pole? [4 pts.] Write an explicit formula (not simpliﬁed) for the transfer function of the block
diagram shown in Figure 1. ...
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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.
 Spring '08
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