Fourier Transform in Signal and System of Telecom

1/39
EECE 301
Signals & Systems
Prof. Mark Fowler
Note Set #15
• C-T Signals: Fourier Transform Properties
• Reading Assignment: Section 3.6 of Kamen and Heck

2/39
Ch. 1 Intro
C-T Signal Model
Functions on Real Line
D-T Signal Model
Functions on Integers
System Properties
LTI
Causal
Etc
Ch. 2 Diff Eqs
C-T System Model
Differential Equations
D-T Signal Model
Difference Equations
Zero-State Response
Zero-Input Response
Characteristic Eq.
Ch. 2 Convolution
C-T System Model
Convolution Integral
D-T System Model
Convolution Sum
Ch. 3: CT Fourier
Signal Models
Fourier Series
Periodic Signals
Fourier Transform (CTFT)
Non-Periodic Signals
New System Model
New Signal
Models
Ch. 5: CT Fourier
System Models
Frequency Response
Based on Fourier Transform
New System Model
Ch. 4: DT Fourier
Signal Models
DTFT
(for “Hand” Analysis)
DFT & FFT
(for Computer Analysis)
New Signal
Model
Powerful
Analysis Tool
Ch. 6 & 8: Laplace
Models for CT
Signals & Systems
Transfer Function
New System Model
Ch. 7: Z Trans.
Models for DT
Signals & Systems
Transfer Function
New System
Model
Ch. 5: DT Fourier
System Models
Freq. Response for DT
Based on DTFT
New System Model
Course Flow Diagram
The arrows here show conceptual flow between ideas. Note the parallel structure between
the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).

3/39
Fourier Transform Properties
Note: There are a few of these we won’t cover….
see Table on Website or the inside front cover of the book for them.
As we have seen, finding the FT can be tedious (it can even be difficult)
But…there are certain properties that can often make things easier.
Also, these properties can sometimes be the key to understanding how the FT can
be used in a given application.
So… even though these results may at first seem like “just boring math” they are
important tools that let signal processing engineers understand how to build
things like cell phones, radars, mp3 processing, etc.
I prefer that you use the tables on the website… they are
better than the book’s

4/39
1. Linearity (Supremely Important)
If &
then
)
(
)
( ω
X
t
x ↔ )
(
)
( ω
Y
t
y ↔
[ ] [ ]
)
(
)
(
)
(
)
( ω
ω bY
aX
t
by
t
ax +
↔
+
To see why: { } [ ] dt
e
t
by
t
ax
t
by
t
ax t
j
∫
∞
∞
−
−
+
=
+ ω
)
(
)
(
)
(
)
(
F
dt
e
t
y
b
dt
e
t
x
a t
j
t
j
∫
∫
∞
∞
−
−
∞
∞
−
−
+
= ω
ω
)
(
)
(
By standard
Property of
Integral of sum
of functions
Use Defn
of FT
)
(ω
X
= )
(ω
Y
=
By Defn
of FT
By Defn
of FT
{ } { } { }
)
(
)
(
)
(
)
( t
y
b
t
x
a
t
by
t
ax F
F
F +
=
+
Another way to write this property:
Gets used virtually all the time!!

5/39
Example Application of “Linearity of FT”: Suppose we need to find the FT
of the following signal…
)
(t
x
1
2
2
− 2
t
1
Finding this using straight-forward application of the definition of FT is not
difficult but it is tedious:
{ } dt
e
dt
e
dt
e
t
x t
j
t
j
t
j
∫
∫
∫
−
−
−
−
−
−
+
+
=
2
1
1
1
1
2
2
)
( ω
ω
ω
F
− 1
So… we look for short-cuts:
• One way is to recognize that each of these integrals is basically the same
• Another way is to break x(t) down into a sum of signals on our table!!!

6/39
)
(t
x
1
2
2
− 2
t
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
=
π
ω
π
ω
ω sinc
2
2
sinc
4
)
(
X
From FT Table we have a known result for the FT of a pulse, so…
Break a complicated signal down into simple signals before finding FT:
)
(
4 t
p
1
2
− 2
t
)
(
2 t
p
1
1
− 1
t
Add
to get
)
(
)
(
)
( 2
4 ω
ω
ω P
P
X +
=
)
(
)
(
)
( 2
4 t
p
t
p
t
x +
=
Mathematically we write:

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2. Time Shift (Really Important!)
ω
ω
ω jc
e
X
c
t
x
X
t
x −
↔
−
↔ )
(
)
(
then
)
(
)
(
If
Shift of Time Signal ⇔ “Linear” Phase Shift of Frequency Components
Used often to understand practical
issues that arise in audio,
communications, radar, etc.
Note: If c > 0 then x(t – c) is a delay of x(t)
)
(
)
( ω
ω ω
X
e
X c
j
=
−
So… what does this mean??
First… it does nothing to the magnitude of the FT:
That means that a shift doesn’t change “how much” we need of each of
the sinusoids we build with
ω
ω
ω
ω ω
ω
c
X
e
X
e
X jc
jc
+
∠
=
∠
+
∠
=
∠ −
−
)
(
)
(
}
)
(
{
This gets added
to original phase
Line of slope –c
Second… it does change the phase of the FT:
Phase shift increases linearly
as the frequency increases

8/39
Example Application of Time Shift Property: Room acoustics.
Practical Questions: Why do some rooms sound bad? Why can you fix this by
using a “graphic equalizer” to “boost” some frequencies and “cut” others?
0
>
c
1
0 <
< α
Delayed signal
Attenuated signal
[ ]
c
j
e
X
Y ω
α
ω
ω −
+
= 1
)
(
)
( This is the FT of what you hear…
It gives an equation that shows how the
reflection affects what you hear!!!!
{ } { } { }
c
j
e
X
X
c
t
x
t
x
c
t
x
t
x
Y
ω
ω
α
ω
α
α
ω
−
+
=
−
+
=
−
+
=
)
(
)
(
)
(
)
(
)
(
)
(
)
( F
F
F
Use linearity and time shift to get the FT at your ear:
So… You hear: )
(
)
(
)
( c
t
x
t
x
t
y −
+
= α instead of just x(t)
speaker
ear
)
(t
x
)
( c
t
x −
α
Reflecting Surface
Very simple case of a single reflection:

9/39
c
j
e
X
Y ω
α
ω
ω −
+
= 1
)
(
)
(
)
(ω
H
≡
changes
shape of
)
(ω
H
)
(ω
X
The big
picture!
The room changes how much of each frequency you hear…
)
sin(
)
cos(
1
1
)
( ω
α
ω
α
α
ω ω
c
j
c
e
H jc
−
+
=
+
= −
Let’s look closer at |H(ω)| to see what it does… Using Euler’s formula
gives Rectangular Form
)
(
sin
)
(
cos
)
cos(
2
1
)
(
sin
))
cos(
1
( 2
2
2
2
2
2
2
c
c
c
c
c ω
α
ω
α
ω
α
ω
α
ω
α +
+
+
=
+
+
=
( ) ( )2
2
Im
Re +
=
mag
Expand 1st squared term
2
α
=
Use Trig ID
)
cos(
2
)
1
(
|
)
(
| 2
c
H ω
α
α
ω +
+
=

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)
cos(
2
)
1
(
)
(
)
( 2
c
X
Y ω
α
α
ω
ω +
+
=
The big picture… revisited:
Effect of the room… what does it look like as
a function of frequency?? The cosine term
makes it wiggle up and down… and the value
of c controls how fast it wiggles up and down
Speed of sound in air ≈ 340 m/s
Typical difference in distance ≈ 0.167m
sec
5
.
0
m/s
340
m
167
.
0
m
c =
=
What is a typical value for delay c???
Î Spacing = 2 kHz
“Dip-to-Dip”
“Peak-to-Peak”
Spacing = 1/c Hz c controls spacing between dips/peaks
α controls depth/height of dips/peaks
The next 3 slides explore these effects

11/39
Longer delay causes closer spacing… so more dips/peaks over audio range!
Attenuation: α = 0.2 Delay: c = 0.5 ms (Spacing = 1/0.5e-3 = 2 kHz)
FT magnitude at
the speaker
(a made-up
spectrum… but
kind of like audio)
|H(ω)|… the effect
of the room
FT magnitude at
your ear… room
gives slight boosts
and cuts at closely
spaced locations

12/39
Stronger reflection causes bigger boosts/cuts!!
FT magnitude at
the speaker
of the room
FT magnitude at
your ear… room
gives large boosts
and cuts at closely
spaced locations

13/39
Shorter delay causes wider spacing… so fewer dips/peaks over audio range!
FT magnitude at
the speaker
of the room
FT magnitude at
your ear… room
gives small boosts
and cuts at widely
spaced locations

14/39
Matlab Code to create
the previous plots

15/39
3. Time Scaling (Important)
Q: If , then for
)
(
)
( ω
X
t
x ↔ ???
)
( ↔
at
x 0
≠
a
⎟
⎠
⎞
⎜
⎝
⎛
↔
a
X
a
at
x
ω
1
)
(
A:
If the time signal is
Time Scaled by a
Then… The FT is
Freq. Scaled by 1/a
An interesting “duality”!!!

16/39
To explore this FT property…first, what does x(at) look like?
|a| > 1 makes it “wiggle” faster ⇒ need more high frequencies
|a| < 1 makes it “wiggle” slower ⇒ need less high frequencies
)
2
( t
x
t
1
3.5
|a| > 1 “squishes” horizontally
)
(t
x
t
1 2 3 4 5 6 7
Original
Signal
Time-Scaled
w/ a = 2
⎟
⎠
⎞
⎜
⎝
⎛
t
x
2
1
t
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|a| < 1 “stretches” horizontally
Time-Scaled
w/ a = 1/2

17/39
When |a| > 1 ⇒ |1/a| < 1 ⎟
⎠
⎞
⎜
⎝
⎛
↔
a
X
a
at
x
ω
1
)
(
Time Signal is Squished FT is Stretched Horizontally
and Reduced Vertically
)
(t
x
)
2
( t
x
t
t
⎟
⎠
⎞
⎜
⎝
⎛
2
2
1 ω
X
.5A
ω
( )
ω
X
A
ω
Original Signal & Its FT
Squished Signal & Its FT

18/39
When |a| < 1 ⇒ |1/a| > 1 ⎟
⎠
⎞
⎜
⎝
⎛
↔
a
X
a
at
x
ω
1
)
(
)
(t
x
t
⎟
⎠
⎞
⎜
⎝
⎛
t
x
2
1
t
( )
ω
2
2X
2A
ω
( )
ω
X
A
ω
Time Signal is Stretched
FT is Squished Horizontally
and Increased Vertically
Original Signal & Its FT
Stretched Signal & Its FT
Rough Rule of Thumb we can extract from this property:
↑ Duration ⇒ ↓ Bandwidth
↓ Duration ⇒ ↑ Bandwidth
Very Short Signals tend to take up Wide Bandwidth

19/39
4. Time Reversal (Special case of time scaling: a = –1)
)
(
)
( ω
−
↔
− X
t
x
∫
∞
∞
−
−
−
=
− dt
e
t
x
X t
j )
(
)
(
)
( ω
ω
Note: ∫
∞
∞
−
+
= dt
e
t
x t
jω
)
(
double conjugate
= “No Change”
∫
∞
∞
−
+
= dt
e
t
x t
jω
)
(
)
(
)
( ω
ω
X
dt
e
t
x t
j
=
= ∫
∞
∞
−
−
)
(
)
( ω
ω X
X =
)
(
)
( ω
ω X
X −∠
=
∠
Recall: conjugation
doesn’t change abs.
value but negates the
angle
= x(t) if x(t) is real
Conjugate changes to –j
So if x(t) is real, then we get the special case:
)
(
)
( ω
X
t
x ↔
−

20/39
5. Multiply signal by tn
n
n
n
n
d
X
d
j
t
x
t
ω
ω)
(
)
(
)
( ↔ n = positive integer
Example Find X(ω) for this x(t)
)
(t
x
t
1
-1
1
-1
Notice that: )
(
)
( 2 t
tp
t
x =
This property is mostly useful for finding the FT of typical signals.
t
t
1
-1
1
-1
)
(
2 t
p
t
1
1
-1

21/39
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
=
π
ω
ω
ω
ω
ω sinc
2
)
(
)
( 2
d
d
j
P
d
d
j
X
So… we can use this property as follows:
⎥
⎦
⎤
⎢
⎣
⎡ −
= 2
)
sin(
)
cos(
2
ω
ω
ω
ω
j
From entry #8 in
Table 3.2 with τ = 2.
From entry on FT
Table with τ = 2.
Now… how do you get the
derivative of the sinc???
Use the definition of sinc and then use the rule for the
derivative of a quotient you learned in Calc I:
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
x
g
dx
x
dg
x
f
dx
x
df
x
g
x
g
x
f
dx
d
−
=
⎥
⎦
⎤
⎢
⎣
⎡

22/39
6. Modulation Property Super important!!! Essential for understanding
practical issues that arise in
communications, radar, etc.
There are two forms of the modulation property…
1. Complex Exponential Modulation … simpler mathematics, doesn’t
directly describe real-world cases
2. Real Sinusoid Modulation… mathematics a bit more complicated,
directly describes real-world cases
Euler’s formula connects the two… so you often can use the Complex
Exponential form to analyze real-world cases

23/39
Complex Exponential Modulation Property
)
(
)
( 0
0
ω
ω
ω
−
↔ X
e
t
x t
j
Multiply signal by a
complex sinusoid
Shift the FT
in frequency
( )
t
x
t
( )
ω
X
A
ω
( )
o
t
j
X
e
t
x o
ω
ω
ω
−
=
}
)
(
{
F
A
ω
o
ω

24/39
Real Sinusoid Modulation
Based on Euler, Linearity property, & the Complex Exp. Modulation Property
{ } [ ]
[ ]
{ } [ ]
{ }
[ ]
[ ]
)
(
)
(
2
1
)
(
)
(
2
1
)
(
)
(
2
1
)
cos(
)
(
0
0
0
0
0
o
o
t
j
t
j
t
j
t
j
X
X
e
t
x
e
t
x
e
t
x
e
t
x
t
t
x
ω
ω
ω
ω
ω
ω
ω
ω
ω
+
+
−
=
+
=
⎭
⎬
⎫
⎩
⎨
⎧
+
=
−
−
F
F
F
F
Linearity of FT
Euler’s Formula
Comp. Exp. Mod.
[ ]
)
(
)
(
2
1
)
cos(
)
( 0
0
0 ω
ω
ω
ω
ω −
+
+
↔ X
X
t
t
x
Shift Down Shift Up
The Result:
Related Result: [ ]
)
(
)
(
2
)
sin(
)
( 0
0
0 ω
ω
ω
ω
ω −
−
+
↔ X
X
j
t
t
x
Exercise: ??
)
cos(
)
( 0
0 ↔
+φ
ω t
t
x

25/39
)
(ω
X
ω
{ }
)
cos(
)
( 0t
t
x ω
F
ω
0
ω
0
ω
−
[ ]
)
(
)
(
2
1
)
cos(
)
( 0
0
0 ω
ω
ω
ω
ω +
+
−
↔ X
X
t
t
x
Shift up Shift down
Shift Up
Shift Down
Interesting… This tells us how to move a signal’s spectrum up to higher
frequencies without changing the shape of the spectrum!!!
What is that good for??? Well… only high frequencies will radiate from an
antenna and propagate as electromagnetic waves and then induce a signal in a
receiving antenna….
Visualizing the Result

26/39
Application of Modulation Property to Radio Communication
FT theory tells us what we need to do to make a simple radio system… then
electronics can be built to perform the operations that the FT theory calls for:
{ }
)
cos(
)
( 0t
t
x ω
F
ω
0
ω
0
ω
−
amp multiply
oscillator
amp
Sound and
microphone
antenna
)
(t
x
)
cos( 0t
ω
Transmitter
Modulator
)
(ω
X
ω
FT of Message Signal
Choose f0 > 10 kHz to enable efficient radiation (with ω0 = 2πf0 )
AM Radio: around 1 MHz FM Radio: around 100 MHz
Cell Phones: around 900 MHz, around 1.8 GHz, around 1.9 GHz etc.

27/39
Amp &
Filter
multiply
oscillator
Amp &
Filter
Speaker
Receiver
)
cos( 0t
ω
ω
0
ω
0
ω
−
Signals from Other
Transmitters
Signals from Other
Transmitters
Signals from Other
Transmitters
Signals from Other
Transmitters
De-Modulator
The next several slides show how these ideas are used to make a receiver:
ω
0
ω
0
ω
−
The “Filter” removes the Other signals
(We’ll learn about filters later)

28/39
Amp &
Filter
multiply
oscillator
Speaker
Receiver
)
cos( 0t
ω
ω
0
ω
0
ω
−
De-Modulator
ω
0
ω
0
ω
− 0
2ω
0
2ω
−
ω
0
ω
0
ω
− 0
2ω
0
2ω
−
By the Real-Sinusoid Modulation Property… the De-Modulator shifts up & down:
Shifted Up
Shifted Down
ω
0
ω
0
ω
− 0
2ω
0
2ω
−
Add… gives double

29/39
Amp &
Filter
multiply
oscillator
Amp &
Filter
Speaker
Receiver
)
cos( 0t
ω
De-Modulator
ω
0
ω
0
ω
− 0
2ω
0
2ω
−
Extra Stuff we don’t want
ω
0
ω
0
ω
− 0
2ω
0
2ω
−
The “Filter” removes the Extra Stuff
The “Filter” removes the Extra Stuff
Speaker is driven by desired message signal!!!

30/39
1. Key Operation at Transmitter is up-shifting the message spectrum:
a) FT Modulation Property tells the theory then we can build…
b) “modulator” = oscillator and a multiplier circuit
2. Key Operation at Transmitter is down-shifting the received spectrum
a) FT Modulation Property tells the theory then we can build…
b) “de-modulator” = oscillator and a multiplier circuit
c) But… the FT modulation property theory also shows that we need
filters to get rid of “extra spectrum” stuff
i. So… one thing we still need to figure out is how to deal with
these filters…
ii. Filters are a specific “system” and we still have a lot to learn
about Systems…
iii. That is the subject of much of the rest of this course!!!
So… what have we seen in this example:
Using the Modulation property of the FT we saw…

31/39
7. Convolution Property (The Most Important FT Property!!!)
The ramifications of this property are the subject of the
entire Ch. 5 and continues into all the other chapters!!!
It is this property that makes us study the FT!!
)
(
)
(
)
(
)
( ω
ω H
X
t
h
t
x ↔
∗
Mathematically we state this property like this:
{ } )
(
)
(
)
(
)
( ω
ω H
X
t
h
t
x =
∗
F
Another way of stating this is:

32/39
h(t)
)
(t
x )
(
)
(
)
( t
h
t
x
t
y ∗
=
System’s H(ω) changes the
shape of the input’s X(ω)
via multiplication to create
output’s Y(ω)
Now… what does this mean and why is it so important??!!
Recall that convolution is used to described what comes out of an LTI system:
Now we can take the FT of the input and the output to see how we can
view the system behavior “in the frequency domain”:
)
(ω
X )
(
)
(
)
( ω
ω
ω H
X
Y =
h(t)
)
(t
x )
(
)
(
)
( t
h
t
x
t
y ∗
=
FT FT
Use the Conv. Property!!
It is easier to think about and analyze the operation of a system
using this “frequency domain” view because visualizing
multiplication is easier than visualizing convolution

33/39
speaker
ear
Let’s revisit our “Room Acoustics” example:
amp
)
(t
x
)
(
*
)
(
)
( t
h
t
x
t
y room
=
c
j
e
X
Y ω
α
ω
ω −
+
= 1
)
(
)
(
Recall:
Hroom(ω)
Plot of |Hroom(ω)|
What we hear
is not right!!!

34/39
So, we fix it by putting in an “equalizer” (a system that fixes things)
amp
)
(t
heq
)
(t
x
Equalizer
)
(
)
(
)
(
2 ω
ω
ω X
H
X eq
=
(by convolution property)
c
j
eq e
H
X
Y ω
α
ω
ω
ω −
+
= 1
)
(
)
(
)
(
Then:
Recall: Peaks and dips
)
(
*
)
(
)
(
2 t
h
t
x
t
x eq
=
[ ] )
(
*
)
(
*
)
(
)
(
*
)
(
)
( 2
t
h
t
h
t
x
t
h
t
x
t
y
room
eq
room
=
=
(by convolution property,
applied twice!)
)
(
)
(
)
(
)
(
)
(
)
( 2
ω
ω
ω
ω
ω
ω
X
H
H
X
H
Y
eq
room
room
=
=
Want this whole thing to be = 1 so )
(
)
( ω
ω X
Y =

35/39
Equalizer’s |Heq(ω)| should peak at frequencies
where the room’s |Hroom(ω)| dips and vice versa
Room
&
Equalizer
Room
Equalizer

36/39
8. Multiplication of Signals
∫
∞
∞
−
−
=
∗
↔ λ
λ
ω
λ
π
ω
ω
π
d
Y
X
Y
X
t
y
t
x )
(
)
(
2
1
)
(
)
(
2
1
)
(
)
(
This is the “dual” of the convolution property!!!
“Convolution in the
Time-Domain”
gives “Multiplication in the
Frequency-Domain”
“Multiplication in
the Time-Domain”
gives “Convolution in the
Frequency-Domain”

37/39
9. Parseval’s Theorem (Recall Parseval’s Theorem for FS!)
∫ ∫
∞
∞
−
∞
∞
−
= ω
ω
π
d
X
dt
t
x
2
2
)
(
2
1
)
(
Energy computed in time domain Energy computed in frequency domain
Generalized Parseval’s Theorem:
ω
ω
ω
π
d
Y
X
dt
t
y
t
x )
(
)
(
2
1
)
(
)
( ∫
∫
∞
∞
−
∞
∞
−
=
= energy at time t
dt
t
x
2
)
(
= energy at freq. ω
π
ω
ω
2
)
(
2 d
X

38/39
10. Duality:
)
(t
x )
(ω
X
∫
∞
∞
−
−
= dt
e
t
x
X t
jω
ω )
(
)
(
∫
∞
∞
−
= ω
ω
π
ω
d
e
X
t
x t
j
)
(
2
1
)
(
Both FT & IFT are pretty much the “same machine”: λ
λ λξ
d
e
f
c j
∫
∞
∞
−
±
)
(
So if there is a “time-to-frequency” property we would expect a
virtually similar “frequency-to-time” property
)
(
)
( 0
0
ω
ω
ω
−
↔ X
e
t
x t
j
Modulation Property:
Other Dual Properties: (Multiply by tn) vs. (Diff. in time domain)
(Convolution) vs. (Mult. of signals)
c
j
e
X
c
t
x ω
ω −
↔
− )
(
)
(
Delay Property:
Illustration:

39/39
Pair B
)
(t
pτ ⎟
⎠
⎞
⎜
⎝
⎛
π
ωτ
τ
2
sinc
Here is an example… We found the FT pair for the pulse signal:
Pair A
Also, this duality structure gives FT pairs that show duality.
Suppose we have a FT table that a FT Pair A… we can get the dual
Pair B using the general Duality Property:
⎟
⎠
⎞
⎜
⎝
⎛
π
τ
τ
2
sinc
t
Step 1
)
(
2 ω
π τ
p
Step 2
Here we have used the
fact that pτ(-ω) = pτ(ω)
1. Take the FT side of (known) Pair A and replace ω by t and move it
to the time-domain side of the table of the (unknown) Pair B.
2. Take the time-domain side of the (known) Pair A and replace t by
–ω, multiply by 2π, and then move it to the FT side of the table of
the (unknown) Pair B.

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