Ece 310 Notes
description
Transcript of Ece 310 Notes
-
ECE 310
Discrete and Continuous Signals
and Systems
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Outline
Modulation
Linearity
Time and Frequency shifting
Time scaling
Time reversal
Conjugation
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Modulation
Vary a carrier signal (ie cos wave) to
contain information.
AM/FM Radios.
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{ } { }
( ) ( )
( ) ( )00
0
0
F2
1F
2
1
))((2
1
)cos()(
)()cos()(
00
++=
+=
=
=
+
dteetf
dtettf
tmttf
tjtj
tj
FF
-
Resulting the original signal to shift by 0
Allows communication to be more
efficient.
High Frequency
Shorter Antenna (Cellphones)
Low Frequency
Under water applications
fc =
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Demodulation
Multiply by the carrier signal again!
)2cos()(2
1)(
2
12
)2cos(1)(
)(cos)()cos()(
)cos()()(
0
0
02
0
ttftf
ttf
ttfttm
ttftm
+=
+=
=
=
The result is the original signal plus a modulated signal at twice the frequency.
We can use a low-pass filter to recover the original signal.
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Wireless Communication
Modulation
Modulation at transmitter
Multiply by cos (t) Demodulation at receiver
Multiply by cos (t) again LP filter (chap. 7)
Requires same frequency
Additional details of modulation will be covered in ECE311
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Additional Notes on
Demodulation
Demodulation requires the multiplication by cos (t)
If c Broadcast band
Tuning an AM radio is selecting the
cos (t) frequency!
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Linearity
)()()()(
)()()(
)()(
)()(
HbGaFtf
thbtgatf
Hth
Gtg
+=
+=
F
F
F
-
)()(
)()(
))()(()()}({
)()()(
HbGa
dtethbdtetga
dtethbtgaFtf
thbtgatf
tjtj
tj
+=
+=
+==
+=
F
-
Time Shift
)()}({
)]([2
1)(
2
1)(
)(2
1)(
0
00
0
)(0
Fettf
deFedeFttf
deFtf
tj
tjtjttj
tj
=
==
=
F
Inverse Fourier transform
Original Fourier Transform with a phase-shift
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Frequency Shift
Multiplication of a sinusoid will result a shift
in frequency spectrum.
As shown in modulation section
)()( 00 mFetf tj =
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Modulation & Frequency Shift
Example
)()(
)(})({
0)( 0
00
+==
=
+
Fdtetf
dteetfetf
tj
tjtjtjF
)2
3sinc(}){rect( 3
+= tjetF
Example }){rect( 3 tjet F
)()( 00 + Fetf tj)
2sinc()rect(
t
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Time Scaling
If a signal is expanded in time, the
frequency spectrum will contract.
= dteatfatf tj)()}({F at=
)(1
)(1
)()}({)(
aF
a
defa
adeff
a
j
aj
=
=
=
F
Consider the case where aaaa is negative
We may conclude
)(1
)(a
Fa
atfF
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Time Reversal
Consider the case of a = -1
)()(
)()1
(1
1)()(
=
=
Ftf
FFtfatf
F
F
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Conjugation
)()()()( ** FtfFtfFF
)(])([
)()}({
**
**
==
=
Fdtetf
dtetftf
tj
tjF
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Conjugation
)()()()( ** FtfFtfFF
Example
2)(
2
)(2
1)(
2
1
))((2
1)}({
)(2
)(2
00
0
0
0
0
)(00
00
0
0
tjtj
tjtj
tj
tj
tj
ed
e
dede
de
e
e
=+=
+=+=
+=
1-F
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Fourier Transform Properties
Linearity
Modulation & Frequency Shift
Time Shift
Time scale & Time Reversal
Conjugation
)()( 00 mFetf tj
)()()(
)()()(
HbGaFthbtgatf
+=+=
)()}({ 00 Fettf tj=F
( ) ( )000 F21
F2
1)cos()( ++ttf
)(1
)(a
Fa
atfF )()( Ftf
F
)()()()( ** FtfFtfFF
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Linearity + Time Shift + Scaling
Example
Show that the FT of rect(t-0.5)+rect(t+0.5) is the same as the FT of
rect(t/2)
))(2
sinc()2
sinc()2
sinc(
)}2
1rect()
2
1({
)2
rect()2
1rect()
2
1rect(
2222
jjjjeeee
ttrect
ttt
+=+=
++
=++
F
-
xxx
ee
eex
jj
jxjx
2sin2
1cossin
)sinc(2sin
2
2
)2
2sin(21
2
2
)2
cos()2
sin(2
)2
cos()2
sinc(2))(2
sinc(
2)cos(
22
=
==
==
=+
+=
-
)(1
)(a
Fa
atfF
)sinc(2)22
(sin
211
)}2
{rect(
2
1
)2
1rect()
2rect(
==
=
=
ct
a
tt
F
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Linearity + Scaling Example
})(
{dt
tdF
})(
{dt
tdF
dt
td )(
Find
)(t
-
25.0
25.0
)4
sinc(2
1
5.0
25.0rect
)4
sinc(2
1
5.0
25.0rect
)4
sinc(2
1
5.0rect
21
41
rect2
21
41
rect2})(
{
21
41
rect2
21
41
rect2)(
j
j
et
et
t
tt
dt
td
tt
dt
td
+
+
+=
+=
FF
)4
(sinc2
)4
sinc(4
2)
4sinc(
)4
sin(2)4
sinc()4
sinc(
)4
sinc()4
sinc(})(
{
2
44
44
jj
jee
eedt
td
jj
jj
==
=
=
=
F
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Differentiation
==
==
)}({)(2
)(2
1)(
2
1)(
)()(
1
FjdeFj
dedt
dFdeF
dt
d
dt
tdf
Fjdt
tdf
tj
tjtj
F
-
Example
)()( Fj
dt
tdf
dt
td )(
)4
(sinc2
1)( 2
t
Example: Find the FT of
)}4
(sinc2
1{
)( 2 jdt
td Which agrees with the result in p24
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Cumulative Integral in Time
)()0()(
)( F
j
Fdf
t+
)u()(11
)( tj
dt
+
Example
Cumulative Area moving from left to right
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Cross Correlation
)()()}({
)()()(
*
*
YXt
dtyxt
xy
xy
=
+=
F
2* )()()()(
)()(
when
XXXt
tytx
xx =
= Autocorrelation