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Homework ME 10, Spring 2012
Problem 1
sin(C1t)
sin(C1t)
x1(t)
Filter
x1(t)
Demodulation
Modulation
y(t)
x2(t)
Filter
Modulation
sin(C2t)
x2(t)
Demodulation
(Receiver)
sin(C2t)
(Transmitter)
You are interested in transmitting x1 (t ) = 10 cos 2 10t and x2 (t ) = 20 cos 2 30t using
the carrier signals V1 (t ) = 2 sin 2 500t and V2 (t ) = 2 sin 2 530t respectively via amplitude
modulation.
a) What is the sum of the two modulated signals: y (t ) ? This is the signal that will be
transmitted.
b) Sketch the amplitude spectrum of the signal in part a.
c) What is the problem? How can it be avoided?
d) Repeat parts a and b with V2 = 2 sin 2 3030t . Is the problem fixed for now?
Before demodulation, the two modulated signals must be separated by passing y (t )
through two separate filters.
e) Using two ideal band-pass filters with unity gain calculate and sketch the
amplitude spectra of the two signals just before they go through demodulation.
Call these signals y1 (t ) and y2 (t ) .
f) Repeat part e with a low pass and a high pass filter. Both filters are first order
with unity gain with 600 Hz and 2600 Hz corner (i.e. cut-off) frequencies.
Now you need to perform the demodulation by multiplication by carrier signal
method (Recall that after the multiplication, you need to use low-pass filtering. This
is a part of the demodulation process and is separate from the filters you have already
used just before demodulation)
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g) Using your result from part e, obtain x1 (t ) and x2 (t ) . Assume that your low pass
filter is ideal.
h) Perform the same with your answer from part f.
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Problem 2
Below are the power spectral density of a white noise signal (on the left), and the
frequency response magnitude of the system G ( j ) it is passed through (on the right).
30 2/Hz
1
frequency (mV) [Hz]
100
300 400
frequency [Hz]
a) Sketch the power spectral density of the noise after is goes through system G.
b) Calculate the power of this modified noise nm (t ) .
This modified noise is then added to a measurement s (t ) where
s (t ) = 5.0 + 0.2sin(1t ) + 0.5sin(2t ) Volts
c) Calculate the power of s (t ) .
d) Calculate the signal-to-noise ratio of s (t ) + nm (t )
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Problem 3.
Consider the following system:
1st order filter
s (t ) + n1 (t ) + n2 (t )
s f (t ) + n f 1 (t ) + n f 2 (t )
Where:
s (t ) : desired signal,
n1 (t ) : noise component 1,
n2 (t ) : noise component 2
x f (t ) : desired signal after filter,
n f 1 (t ) : noise component 1 after filter
n f 2 (t ) : noise component 2 after filter
Let:
s (t ) = 5 + 2sin(2 10t ) + 1sin(2 20t + / 6)
n1 (t ) = 0.1sin(2 120t + / 3) ,
n2 (t ) : Johnson Noise, with T=600K, device resistance of 1Mohm
a) What is the power of the uncorrupted, unfiltered signal, s 2 (t ) ?
b) What is the power of the 1st unfiltered noise component, n12 (t ) ?
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c) What is the power of the 2nd unfiltered noise component, n2 (t ) ?
d) What is the signal-to-noise ratio of the unfiltered signal?
s (t ) + n1 (t ) + n2 (t ) ?
Now design a unity-gain low pass 1st-order filter which gives you as much SNR as
possible for s f (t ) + n f 1 (t ) + n f 2 (t ) while making sure that the desired signals power
doesnt drop by more than 2%.
e) Write down the 1st-order filter designed to meet the above criteria.
s 2 (t )
Show your calculation that verifies that f
is greater than 0.98.
s 2 (t )
f) Calculate the new signal-to-noise ratio of s f (t ) + n f 1 (t ) + n f 2 (t ) .
g) Describe what your filter is doing to reject noise. Can you remove all of
the unwanted noise with a 1st order filter? Why or why not?
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