The Fourier Transform

time

frequency
(Hz)
1
Jean Joseph Baptiste Fourier

March 21, 1768 to May 16, 1830

2
Review: Fourier Trignometric Series (for Periodic Waveforms)

∑(
f (t) = a0 + an cos(nω0t) + bn sin(nω0t)
)
n=1

2π and f 1
where ω = T0 0 =T
and
T
1 (DC term)
T∫
a0 = 0 f (t)dt
T
2 for n = 1, 2, 3,
T∫
an = 0 f (t) cos(nω0t)dt etc.
T
2 for n = 1, 2, 3,
T∫
bn = 0 f (t) sin(nω0t)dt etc.

3
Fourier Trigonometric Series in Amplitude-Phase
Format

∑
f (t) = A0 + ∞ An cos(nω0t + φn )
n=1
( )
a0 = A0

An 2n + bn2 and
a
φ=n= − tan ⎛⎜ n ⎟⎞
−1
b
⎝a n
Also known as⎠ Polar form of Fourier series.

4
Example: Periodic Square Wave as Sum of
Sinusoids Line
f0
Spectra

Even or
3f0 Odd?

5f0

7f0

EE 442 Fourier 5
Transform
Example: Periodic Square Wave (continued) Question:
What
This is an odd function
would
make this an
⎦
even
Fundamental
only
function?

Five
terms

Eleven
terms

Forty-nine
terms

http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx
6
 Fourier Series versus Fourier Transform

Continuous time Discrete time

Discrete
Fourier
Periodic Fourier
Series Transfor
m

Discrete
Aperiodic Fourier
Fourier
Transform Transfor
m

Fourier series for continuous-time periodic signals → discrete

spectra Fourier transform for continuous aperiodic signals →
continuous spectra

7
 Definition of Fourier Transform

The Fourier transform (i.e., spectrum) of f(t) is F

(ω): ∞

= f (t)e−
{ } ∫ ∞
F (ω) = F f (t) −∞
jωt
dt 1
−
f (t) = F 1 F 2π−∞ F (ω)e jωtdω
{ ∫
Therefore, f (t)⇔ F (ω) is a Fourier Transform pair
(ω) =
} Note: Remember ω = 2π f

1
1
Examples
 Example: Impulse Function δ(t)

− j ωt −
= e j0=
} ∫
F (ω) = F δ(t) = δ(t)e dt = e t
1
j ωt
{ =0

δ(t) ⇔ 1 −∞
1 ⇔ 2πδ(ω)

F δ
{

↔ (t) 1
}

ω
∞ if t = 0
δ(t) = 0 if t ≠ 0
Delta function has unity
area.
1
3
 Example: Fourier Transform of Single Rectangular Pulse

τ
τ
1 for − 2 ≤ t2≤
f (t) = rect(t ) = II(t /τ)
= τ
0 for all t> 2

f (t) = rect(t ) = II(t ∞ τ/ 2

/τ)
F (ω) = f (t)e− jωt dt = e−
Pulse 1 jωt
dt ∫ ∫
of τ/ 2 − jωτ/ 2
− jωt −∞ −τ/ 2 jωτ/ 2
width
= ⎛⎜ e ⎞⎟ e−e
τ =
⎝ − j ω⎠ − jω
−τ/ 2
τ time t ⎡ ⎤
τ
− 2 − j2sin(ωτ 2) sin(ωτ 2)
2 = =τ⋅ ⎢ ⎥
0 − jω ⎣ ⎦(ωτ 2 )
Remember ω =
2πf
1
4
Fourier Transform of Single Rectangular Pulse (continued)

⎡sin
⎢
(ωτ 2)⎤
F (ω) =τ⋅ ⎣ (ωτ 2) ⎦=τ⋅sinc(πfτ)
⎥
1
sinc
F(ω) function
τ

τ ττ time t
− 2 2
2
0
Note the
pulse is
− 3π −π π 2π 3π ω
time − 2π
τ τ τ τ τ
centered τ 0

ES 442 Fourier 1
Transform 5
 Properties of the Sinc
Function
Definition of the sinc
function:
sin(x)
sinc(x) = x
Sinc Properties:
1. sinc(x) is an even function of x.
2. sinc(x) = 0 at points where sin(x) = 0, that is,
sinc(x) = 0 when x = ±π, ±2π, ±3π, … .
3. Using L’Hôpital’s rule, it can be shown that sinc(0) = 1.
4. sinc(x) oscillates as sin(x) oscillates and
monotonically decreases as 1/ x decreases as |
x | increases.
5. sinc(x) is the Fourier transform of a single rectangular
pulse.

Warning:
There are two definitions for the sinc(x) function. They are
sin(x) sin(πx)
sinc(x) = x and sinc(x) = πx

1
6
 Periodic Pulse Train Morphing Into a
Single Pulse
Frequency
resolution
Ck inversely
proportional to the
period.

Ck

Ck

https://www.quora.com/What-is-the-exact-difference-between-continuous-fourier-transform-discrete-Time-Fourier-Tran
sform-
DTFT-Discrete-Fourier-Transform-DFT-Fourier-series-and-Discrete-Fourier-Series-DFS-In-which-cases-is-which-one-
used
1
7
Sinc Function Tradeoff: Pulse Duration versus
Bandwidth G1 ( f
) T1
G2 ( f
) T2
1
− 1
T1
T1
f f
−1 1

❶ g1 (t) T2
g2 (t)
T2
❷

−T1 T1 t −T2 T2 t
2 2 G3 ( f 2 2
) T3

f
1
❸
1 Also called the
− g3 (t)
T1 > T2 > T3 T3 T3 Time Scaling
Property (Section 2)

−T3 T3 t
2
2
15
Properties of Fourier Transforms

1 Linearity (Superposition) Property

2. Time-Scaling Property
3. Time-Shifting Property
4. Frequency-Shifting Property
5. Time Differentiation Property
6. Frequency Differentiation Property
7. Time Integration Property
8. Time-Frequency Duality Property
9. Convolution Property

16
 1 Linearity (Superposition) Property

Given f(t) ↔ F(ω) and g(t) ↔ G(ω) ;

Then f(t) + g(t) ↔ F(ω) + G(ω) (additivity)

also kf(t) ↔ kF(ω) and mg(t) ↔ (homogeneity)

mG(ω) Note: k and m are
constants
Combining these we
have,
kf(t) + mg(t) ↔ kF (ω) + mG
(ω)
Hence, the Fourier Transform is a linear transformation.

18
 2 Time Scaling
Property

1 ⎛ω⎞
f (at) =a F⎜⎝ ⎟
{ } a
⎠
∞

f (at)e− jωt
} ∫
f (at) =
{
dt
−∞
λd ω
Let λ= at & dλ= adt, − j ωt ⎛ ⎞⎟
F a
} ∫ ⎝⎜ ⎠
f (at) =−∞
∞ f
{ 1
(λ)e
Hence, =
f ( - t) = F a(−ωa) = F
{* }
(ω)

2
1
 Time-Scaling Property (continued)

1 ⎛ω⎞
f (at) =a F⎝⎜ ⎟
{ } a
⎠
Time compression of a signal results in spectral
expansion and time expansion of a signal results in
spectral compression.

20
 3 Time Shifting Property

− jωt
f (t − t0 ) = e 0 F ω
{ } ( )
∞

f (t − t ) 0 ∫ f (t − t0 )e− jωt dt
{ } −∞

Let=λ= t − t0 , dλ= dt & t = λ+ t0

∞

f (t − 0t ) ∫ f (λ)e − jω(λ+t0 )
dλ=
{ } −∞
= ∞

− j ωt 9
=e f (λ)e− jωλdλ= e− jωt0 F ω
∫
−∞
( )

21
 Time-Shifting Property (continued)

− jωt
f (t − t0 ) = e 0 F ω
{ } ( )
Delaying a signal by t0 seconds does not change its
amplitude spectrum, but the phase spectrum is
changed by -2πft0.
⎡ ⎤ ⎡ ⎤
2 2
Note thatFthe phase ( )
spectrum
(ω) = ⎣ Re F (ω ⎦ ⎣shift
+ changes
Im ( ⎦
F ( )
linearly with
frequency f. ω
f (t) F
) (ω) )
A time shift
produces a t ω
Even Both
phase shift in function.
must be
its spectrum. identical.
ω
t
This time shifted pulse
is both even and odd.
22
 4 Frequency Shifting Property

{} f (t)e jω0t = F ( ) 0
ω−ω
∞

f (t)e jω0t f (t)e jω0te−

} ∫
=
{dt
jωt ∞

∫ f (t)e − j (ω−∞
−ω0 )t
= dλ= F (ω−ω
)0
−∞

Special application:
Apply to cos (ω 0t) 1
(e 0
jωt − jω0t
2
+e );
=
{f (t)cos (ω t)} (F (ω−ω) + F ( ω+ω) )
0
1
2 0 0
=

23
 An Important Formula to Remember
Euler's formula

exp ± jθ = cos(θ) ± j
[ ]
sin(θ)
1
sin(θ) = 2 exp jθ − exp −
j ( [ ] [
θ θ) = 21 exp jθ + exp −
jcos(
]) ( [ ] [
jθ ⎧1 for n even
and
± j π/2
e ( )=±j ]) ± jnπ
e = ⎩⎨
−1 for n odd
aa++jbjb==re jθ where r ⎛b⎞
2
a + b2 , θ= tan−1 ⎝⎜ ⎟
reiθ = a

⎠
24
Frequency Shifting Property is Very Useful in
Communications
Multiplication of a signal g(t) by the factor
[cos(2πfCt)]
places G(f) centered at f = ± fC.
Carrier frequency is fc G( f
& ) 2
g(t) is the message A
g(t)
signal g(t) ⇔ G( f
)
t −B B f
2B

g(t) g(t) cos(2πfC t)

USB A LS USB
LSB B

t − fC fC f
2B
g(t)

ES 442 Fourier 25
Transform
 Frequency-Shifting Property
(continued)
g(t)
(a) Message G( G(
f) f) (b)
signal
θg ( f )

g(t) cos(2πfC t) c
( (d)
)
-fC fC

g(t) sin(2πfC t) e Note phase

( shift (f
) π2 )
−π 2

Multiplication of a signal g(t) by the cos(2πfCt)

factor places G(f) centered
at f = ±EEfC442
. Fourier 26
Transform
Modulation Comes From Frequency Shifting
Property
Given FT pair: f (t) ⇔ F ω
( )
then, f (t) e jω t ⇔ F ω−ω 0
0

Amplitude Modulation
( )
Example:
Audio tone:
~ sin(ωt)

Sinusoidal carrier signal:

Amplitude
Modulate
d Signal

EE 442 Fourier 27
Transform
 Fourier Transform of AM Tone Modulated
Signal

f(t)

Carrier
signal F(ω
fC = 500 Hz
)
Modulated AM
sidebands

Only positive frequencies

shown;
Must include negative
frequencies.
ES 442 Fourier 28
Transform
Modulation of Baseband and Carrier
Signals

fC = f 0

https://slideplayer.com/slide/1093995
1/ ES 442 Fourier 29
Transform
 Transform Duality
g(t) ⇔ G( f ),
Property
Given
then g(t) ⇔ G( f
) and
G(t) ⇔ g(− f )

Note the minus sign!

Because of the minus sign they are not

perfectly symmetrical – See the illustration
on next slide.

EE 442 Fourier 3
Transform 2
 Illustration of Fourier Transform
Duality G1 ( f
g1
(t)
) τT
−1 1
1
3
τ τ
τ
−τ τ t − 2 f
2 2 2τ τ

g2 (t) G2 ( f
)
−11
τ 2π What
− 1
τT does
τ this
−τ
1
t τ imply? f
4π 4π
−τ τ ω
EE 442 Fourier 2 2 3
Transform 3
Fourier Transform of Complex
Exponentials
F−1 [δ ( f] − fc ) = ∫
∞ δ( f − fC )e
− j2πf t

df −∞

Evaluate for f = fc

F−1 [δ ( f]− f c ) = ∫ e
− j2πfct
df = e
− j2πfct

f = fc

∴ δ( f − fc ) ⇔ e − j2πf tc an
∞ d
F −1
[δ ( f] + fc ) = ∫ δ( f + f C )e− j2πf t df
−∞

Evaluate for f =− fc

F−1 [δ ( f]+ f c ) = ∫ e
j2πfct
df = e
f =− fc j2πf tc

∴ δ( f + fc ) ⇔ e j2πf tc

EE 442 Fourier 32
Transform
 Fourier Transform of Sinusoidal
Functions − j2πf t j2πf t n!
Taking δ( f − fc ) ⇔ e c
and δ( f + fc ) ⇔ e
c r! n−r
We use these results to find FT of cos(2πft) and sin(2πft) (
!
Using the identities for cos(2πft) and sin(2πft), )
* cos(2πft) = 12⎡ej2πfct + e− j2πfct ⎤ & co (2πft) = 1 ⎡ej2πfct − e− j2πfct ⎤
2j ⎦ ⎦sin
Therefore, ⎣ ⎣ s

cos(2πft) ⇔ δ21 [ ( f + f c) +δ ( f − f c) ], and

sin(2πft) ⇔ 1 [δ( f + f )c −δ ( f − f c) ]
2j

R I
cos(2πfct e sin(2πfct m
) )
-fc fc f -fc fc f

EE 442 Fourier 33
Transform
Summary
of Several
Fourier
Transform
Pairs

http://media.cheggcd
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34
 Spectrum Analyzer Shows Frequency
Domain
A spectrum analyzer measures
the magnitude of an input signal
versus frequency within the full
frequency range of the
instrument. It measures
frequency, power, harmonics,
distortion, noise, spurious
signals and bandwidth.
It is an electronic receiver
Measure magnitude of signals
Does not measure phase of
signals
Complements time domain

Courtesy: Keysight
Technologies

Bluetooth Spectrum 39
 Fourier Transform of Cosine
Signal
A
A cos(2πf ct) = c 2 δ( f + f ) +δ( f − f )c
A
[ ]
2
A cos(2πf t)
Re

-fc
3D A
2
View

fc

Blue arrows indicate

positive phase
directions
EE 442 Fourier 40
Transform
Fourier Transform of Cosine Signal (as shown in
textbooks)

Real
cos(t) δ(− f0 ) axis

F
−T0 T0 t
T
f
− f0 f0

f0 =
1 T0

ES 442 Fourier 3
Transform 9
 Fourier Transform of Sine
Signal
B
B sin(2πf ct) = cj 2 δ( f + f ) −δ( f − f )c
[ ]
B sin(2πf t)
Re
B -fc
2

B
2
fc

We must subtract
90° from cos(x) to get
sin(x)
EE 442 Fourier 42
Transform
 Fourier Transform of Sine Signal (as usually shown in
textbooks)

Imaginary
axis
B⋅sin(ω0t δ (− f0 ) B
) B j2

t F − f0 f0 f
T
B
−j 2

EE 442 Fourier 43
Transform
 Visualizing Fourier Spectrum of Sinusoidal
Signals

Multiplying
by j is a
phase shift

ES 442 Fourier 44
Transform
Fourier Transform of a Phase Shifted
Sinusoidal(with phase information
Signal
shown)

jφ j2πft − jφ − j2πf t
R ⋅e e + R ⋅e e
A
2 R
− j2πft
R ⋅e - e
φ -fc
B
j2πft
2 R ⋅e
A
2 φ B
2

fc

A
⎛⎜ ⎞2 ⎛ ⎟⎞2
B ⎛ A
R + and φ= tan − −1 ⎟⎞
= 2 2
⎝ ⎠⎝ ⎠ ⎝ ⎜ ⎜ B
⎠
⎟
http://www.ece.iit.edu/~biitcomm/research/references/Other/Tutorials%20in%20Communications
%20E
ngineering/Tutorial%207%20-%20Hilbert%20Transform%20and%20the%20Complex%20Envelo
pe.pdf
EE 442 Fourier 45
Transform