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The Cooper lJnion
Department of Electrical & Computer Engineering
ECE111 Signal Processing & Systems Analysis
Exam I
February 24, 2005
Time: 90 min. Closed book, closed notes. No calculators,
Note: cos
Acos B
:
,l
cos
(,4 + A) +cos(AA).
1.
f10
pts.] The impedance
of a circuit at 1Mffz is Z
=
4 j3O.
In this problem, given
the appropdate units with all answers.
(a) Determino
th" r"sistance,
reactance,
admittance,
conductance,
susceptance.
Also
provide
the standard
letter (e.g., Z for irnpedance)
used
for all of these quantities.
(b) Is the impedance
equivalent
to a resistor
in seies with an inductor or capacitor?
(c) Find the value of the inductance
or capacitance
leave
the answer as an explicit
numerical expression
(i.e.,
ready
to be plugged
into a calculator, no symbols).
2.
16
pts.] Express
the following
in the form Acos (1011
d) bywingphasorc, specillcally
complex 0, thmetic not trigonomet c identities, and not phasor diagrams:
3 cos
(10?rl
+ ?r/4) +
\cos(t,rtr12)
'JZ
3.
f9
pts.l A chi.rp
signal
is given by:
t(t):
Acos
(2r
J.t
+ dt+
Bt')
(a) Find the instantaneou,s
frequency
j"", (l).
(b) The input voltage
to a VCO rl,hich generates
this chirp signal must be:
1. a step
function;
2. a ramp function (grows
linearly with time);
3. a quadratic
function (grows

i2).
(c) Assume
o,
B
and the tine span we axe examining are small enough
that the banci
widthof r(t)
is srnall
compared
with
l"
(o<<
l",P<<f.,and
j""1(t) l"l
<<
/.
for the timespan of interest). Suppose
e (l) is mixed (multiplied) with another
chirp given by:
b (t)
:
B cos
(2r
f.t

ci

A*)
and the result
is iowpass
fllteredto reject
terms near
/"
and above. That is:
y (t)
:
[z
(t)
.u
(t)]*:"".
."". .,  ,_*,
i.
Show
that y(l) has
the lorllr 1lcos(P(l)),
i.e.,
its em,elope
is constant.
Find K
and the instantaneous
frequency
of g (t).
s
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I
4.
[10
pts.] A signal c (t) is transmitted tbtou'gh a multipath
Jad'r'ng
channel, testittng
in a received
signal g (t) given byr
Y
(t) =
aP
lt

rr) + a2r
lt
't2)
where or,
(1,2
axe constant attenuatiorN
(positive
real values) and 7r, r,
axe
f.xeddelays'
For example,
the first term may be a lineofsight
(LOS) path' which is a direct path
from transmitter to receivet, and the second
term may repreient a reflected path
lf
the input signal has a baseband
equivalent
rse (l) and ca.rrier
frequency
f.,
find the
baseba.nd
representation
of g (i), yes (t). Specifically,
show
that:
ltee
ft)
:
AxsB
(t

11) + A2t
BB
(t

12)
where.4t are complex valued but do not depend on t
(they may depend on 7i, oi,
f",
but not i).
Thus, the baseband model
is similar to the bandpass model,except
that
the attenuation
factors become
complex
valued.
5.
l8
pts.]
A real periodic signal with period 5sec
is expanded
in a complex exponential
Fourierseries,
resulting
in coefficients
{c.}.
It is known that .{
:
2, cr:5+
j and
c24
2j.
(a)
(b)
(")
Find the ftindamental
and second
harmonic
frequencies,
in ,42
Find the DC power.
You have enough
information
to determine
certain other c," coeficients,
but not all
of them. Write the values
for a.llother coeficients which are uniquely detexmined
fionl
the above
information.
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 Spring '08
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 Signal Processing, pts

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