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Midterm Exam Solutions | Communication and Information Theory | ECE 460, Exams of Theories of Communication

Material Type: Exam; Professor: Paris; Class: Comm and Information Thry; Subject: Electrical & Computer Enginrg; University: George Mason University; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 02/10/2009

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ECE 460: Communication & Information Theory
Midterm Exam Solution โ€“ Problem 2
Willy W Huaracha
April 1, 2002
AM Modulation
Throughout this problem assume that the message signal ()mt is given as
() sin(2 )
m
mt f t
ฯ€
=. The carrier frequency c
f
is much higher than m
f
.
(a) Assume ()mt is modulated using conventional DSB AM. Sketch the resulting
time domain signal.
[
]
() ().cos(2 )
c
x
tAmt ft
ฯ€
=+
E
nvelo
p
e Carrier
Figure 1
[
]
() cos(2 ).cos(2 )
mc
x
tA ft ft
ฯ€ฯ€
=+
Then we simplify it by using the identity
[]
1
cos( ) cos( ) cos( ) cos( )
2
x
yxyxy=โˆ’++
( ) .cos(2 ) cos(2 ).cos(2 )
cmc
x
t A ft f t ft
ฯ€ฯ€ฯ€
=+
[]
1
( ) .cos(2 ) cos(2 ( ) ) cos(2 ( ) )
2
ccmcm
tA ft fft fft
ฯ€ฯ€ ฯ€
=+โˆ’++
(b) Compute and sketch the Fourier transform of the modulated signal
Taking the Fourier transform of ()
x
t results in the following equation:
pf3
pf4
pf5

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ECE 460: Communication & Information Theory

Midterm Exam Solution รฑ Problem 2

Willy W Huaracha

April 1, 2002

AM Modulation

Throughout this problem assume that the message signal m t ( ) is given as

m t ( ) = sin(2 ฯ€ f tm ). The carrier frequency fc is much higher than fm.

(a) Assume m t ( ) is modulated using conventional DSB AM. Sketch the resulting

time domain signal.

x t ( ) = [ A + m t ( ) .cos(2 ] ฯ€ f tc )

Envelope Carrier

Figure 1

x t ( ) = [ A +cos(2 ฯ€ f tm ) .cos(2] ฯ€ f tc )

Then we simplify it by using the identity [ ]

1 cos( )cos( ) cos( ) cos( ) 2

x y = x โˆ’ y + x + y

x t ( ) = A .cos(2 ฯ€ f tc ) +cos(2ฯ€ f tm ).cos(2 ฯ€ f tc )

[ ]

( ) .cos(2 ) cos(2 ( ) ) cos(2 ( ) ) 2

x t = A ฯ€ f t c + ฯ€ fc โˆ’ fm t + ฯ€ f (^) c + fm t

(b) Compute and sketch the Fourier transform of the modulated signal

Taking the Fourier transform of x ( ) t results in the following equation:

[ ] [ ]

[ ]

c c c m c m

c m c m

A

X f f f f f f f f f f f

f f f f f f

Below is the Fourier transform plot of the modulated signal.

X ( f )

2

A 2

A

1 4

1 4

1 4

1 4

โˆ’ +( fc f (^) m ) โˆ’ f (^) c โˆ’ โˆ’( fc f (^) m ) ( fc โˆ’ fm ) fc ( fc + fm )

Figure 2

(c) Repeat parts (a) and (b) if DSB-SC AM is used.

Taking the same steps as for part (a) we determine the modulated signal x ( ) t

using a DSB-SC AM.

x t ( ) = m t ( ).cos(2 ฯ€ f tc ) =cos(2 ฯ€ f tm ).cos(2ฯ€ f tc )

[ ]

( ) cos(2 ( ) ) cos(2 ( ) ) 2

x t = ฯ€ fc โˆ’ f (^) m t + ฯ€ fc + f (^) mt

Envelope Carrier

Figure 3

First, we simplify (^) x 1 (^) ( ) t by using the identity (^) [ ]

cos( )cos( ) cos( ) cos( ) 2

x y = x โˆ’ y + x + y , and

take its Fourier transform as follows.

x 1 ( ) t = m t ( ).cos(2 ฯ€ f tm ) = cos(2ฯ€ f tm ) cos(2 โ‹… ฯ€ f tm )

2 1

( ) cos (2 ) (1 cos(4 ) 2

x t = ฯ€ f t m = + ฯ€ f tm

[ ]

X f = ฮด f โˆ’ f (^) m + ฮด f + fm

X ( f )

1 4

1 4

โˆ’ 2 fm โˆ’ f^ m f (^) m 2 fm

Figure 6

(e) Compute the outputs of the lowpass filters, y 1 (^) ( ) t and y 2 (^) ( ) t , as well as their

Fourier transforms Y 1 (^) ( f )and Y 2 (^) ( f ).

When the signals x 1 (^) ( ) t and x 2 (^) ( ) t are passed through the lowpass filters, the filter

cuts off all-higher frequencies signals and the following signals result:

1

Y f = ฮด f

1

y t =

Y 2 (^) ( f ) = 0

y 2 (^) ( ) t = 0

(f) Compute the output signal s t ( ) and its Fourier transform S ( f ).

First we determine from the block diagram that s 2 (^) ( ) t = z 1 (^) ( ) t โˆ’ z 2 ( ) t , which results

as follows.

1 1

( ) ( ) cos(2 ( ) ) cos(2 ( ) ) 2

z t = y t โ‹… ฯ€ f c โˆ’ f (^) m t = โ‹… ฯ€ fc โˆ’ f (^) mt

z 2 ( ) t = y 2 ( ) sin(2 t โ‹… ฯ€ ( f c โˆ’ fm ) ) t = 0

( ) cos(2 ( ) ) 2

s t = โ‹… ฯ€ f c โˆ’ f m t

Ultimately we determine we the Fourier transform of s ( t )

[ ]

S f = ฮด f โˆ’ f c โˆ’ f m + ฮด f + fc โˆ’ fm

(g) How would you describe what this system does?

This system is a LSB-SSB AM Modulator.