EE 518 — Midterm Exam
Your Name :
Instructions :
1. Open book and notes.
2. Do not open this exam until 6:00 pm.
3. Put all youranswersin the
appropriate space. If necessary, you can use extra worksheets, but please try
to put solutions in the correct spot on this exam.
4. Turn in your work and put your name
at the top of all loose worksheets. This work will be looked at for possible
partial credit.
5.Justify all of your answers. A “correct” answer with no justification will not receive fullcredit. An incorrect
answer with even partially correct partial derivation and/or justification will
get partial credit.
6. The exam can be turned in at the
front of the room before 8:50 pm. You must stop your work at 8:50 pm. Anyone
working past this time will be penalized.
7. You can take bathroom breaks any time
you wish and without special permission, during the exam, but cannot converse
with anyone during these breaks.
8. This exam has total of 7 pages
(including this page).
9. The total weight for the exam is 100
points.
Problem
1
Suppose we have a cascade systems shown
in the following figure where1
1
( )
1
1
2
j
j
H e
e
w
w
=

and2
1
( )
1
1
3
j
j
H e
e
w
w
=

.
a) Find []hnsuch that the input
[]xnand output []ynsatisfy the relationship [ ] [ ] [ ]y nx
nhn=*.(10 points)
b) Findg[n] such thatx[n]
=
g[n]*y[n].(10
points)
Problem
2
Consider the system shown in the
following figure,
C/D L M D/C
The linear timeinvariant filter ()jHewhas DTFT as shown in the following figure where we know .cLwp<Assume that the Fourier transform of the input,{( )}( )c cF x t=X jW, is bandlimited,
i.e.
( ) 0cX jW =for0W³ W.
a) We would like to choose a sampling
periodTsuch that there is no aliasing by the C/D converter or, if
there is any aliasing, the frequency component contaminated by aliasing is
rejected by the filter ()jHew. Given0W,cwandL, determine the most general condition on sampling
periodT.(10 points)
b) Let ( )jLX ew, ()jXew, ()jYew, ()j MY ewbe the discretetime Fourier transforms
of []Lx n, []x n, []y n, and []Myn, respectively.
Express ( )jX ew, ( )jY ew, and ( )j MY ewrespectively in terms of ( )j
LX ewand, if needed, also ( )jH ew.(10 points)
d) Again, ifLM=, what is the overall
equivalent continuoustime frequency responseeff(
(( ) ) )c cY j X H j j= W W W?(5
points)
Problem
3
A causal LTI system has system functionH(z)
with the polezero plot shown in the figure below. You are also told thatH(1)
2.
a) Is ()Hzstable? Justify your
answer. (Hint: Consider that01( ) 2.jz eHz) (5 points)
b) What is the region of convergence for
()Hz? Motivate your answer. (10 points)
c) Ish[n] real? Justify
your answer. (Hint: you don’t need to solve forh[n] to answer
this question.) (10 points)
d) What is the polezero plot and region
of convergence for theztransform of []hn? (5 points)
e) What is the region of convergence for
the ztransform of2[ ]n jh n? (5 points)
Problem
4
Letx[n] be a causal
stable sequence withztransformX(z) . The complex
cesptrum , which we might see used later for an estimator for undoing the
effect of convolution (“deconvolution,”) is defined as the inverse transform of
the logarithm of ()Xz; i.e.{ }1ˆ ( ) ln ( ) ˆ[ ]ZX z X z x n = «
where ln{ }log{ }e× = ×is the natural logarithm. The region of convergence of thisXˆ
(z) includes the unit circle. (Strictly speaking, taking the logarithm
of a complex number often requires some carefulconsiderations. Moreover, the logarithm of a
validztransform may not be a validztransform. For now, just
assume that this logarithm of a complex number operation is valid.) Given the
sequencex[n]=d[n]+ad[nN], whereNis an integer anda<1.
a) Find ()Xz, theztransform
of this []xn. (Easy) (10 points)
b)Nis an arbitrary positive
integer. Determine the complex cesptrum ˆ[]xnof the above sequence []xn
. Your answer need not be in closed form. (It can be an infinite sum.) (Hint:
feel free to use the infinite Mercator series definition:( ) ( )1 11 ln 1k k kx x k¥ + = + =å) (5 points)