Properties of the Fourier Transform
Properties of the Fourier Transform
Reference:
Sections 2.2 - 2.3 of
Professor Deepa Kundur
S. Haykin and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13
978-0-471-43222-7.
University of Toronto
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
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Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
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Properties of the Fourier Transform
The Fourier Transform (FT)
Z
Importance of FT Theorems and Properties
I
∞
G (f ) =
g (t)e
Z−∞
∞
g (t) =
−j2πft
dt
I
G (f )e +j2πft df
I
−∞
Notation:
g (t)
G (f )
G (f ) = F [g (t)]
g (t) = F−1 [G (f )]
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
I
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We live in the time-domain.
However, sometimes viewing information signals or system
operation as function of time does not easily provide insight.
The Fourier transform converts a signal or system representation
to the frequency-domain, which provides another way to
visualize a signal or system convenient for analysis and design.
The properties of the Fourier transform provided valuable insight
into how signal operations in the time-domain are described in
the frequency-domain.
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Properties of the Fourier Transform
Properties of the Fourier Transform
Importance of FT Theorems and Properties
FT Theorems and Properties
Property/Theorem
Notation:
LTI System
Linearity:
impulse response
Dilation:
Conjugation:
Duality:
Time Shifting:
Frequency Shifting:
Area Under G (f ):
Area Under g (t):
Time Differentiation:
Time Integration :
Modulation Theorem:
Convolution Theorem:
Correlation Theorem:
Rayleigh’s Energy Theorem:
LTI System
frequency response
I
Time Domain
g (t)
g1 (t)
g2 (t)
c1 g1 (t) + c2 g2 (t)
For systems that are linear time-invariant (LTI), the Fourier
transform provides a decoupled description of the system
operation on the input signal much like when we diagonalize a
matrix.
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
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Properties of the Fourier Transform
Frequency Domain
G (f )
G1 (f )
G2 (f )
c1 G1 (f ) +c2
G2 (f )
1
G fa
|a|
∗
G (−f )
g (at)
g ∗ (t)
G (t)
g (t − t0 )
e j2πfc t g (t)
R ∞g (0)
−∞ g (t)dt
d
g (t)
R t dt
−∞ g (τ )dτ
R ∞ g1 (t)g2 (t)
−∞ g1 (τ )g2 (t − τ )
R∞
∗
−∞Rg1 (t)g2 (t − τ )dt
∞
2 dt
|g
(t)|
∞
=
=
=
g (−f )
G (f )e −j2πft0
(f − fc )
RG
∞
−∞ G (f )df
G (0)
j2πfG (f )
1
G (f )
j2πf
R∞
G
(λ)G
2 (f − λ)dλ
−∞ 1
G1 (f )G2 (f )
G (f )G2∗ (f )
R ∞1
2
∞ |G (f )| df
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Properties of the Fourier Transform
Dilation Property
Dilation Property
g (at)
1
G
|a|
Let τ = at. Assume for now a > 0 and finite.
Three things must be changed:
1. integrand: substitute t = τ /a.
2. limits: for t = ∞, τ = ∞; for t = −∞, τ = −∞.
3. differential: dτ = adt or dt = dτ /a.
f
a
Proof: Let h(t) = g (at) and H(f ) = F[h(t)].
Z ∞
H(f ) =
h(t)e −j2πft dt
Z−∞
∞
=
g (at)e −j2πft dt
Z
Z
∞
dτ
g (τ )e −j2πf τ /a
a
−∞
−∞
Z ∞
f
1
1
f
=
g (t)e −j2π( a )t dt = G
a −∞
a
a
H(f ) =
−∞
Idea: Do a change of integrating variable to make it look more like
G (f ).
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
∞
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g (at)e
−j2πft
dt =
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Properties of the Fourier Transform
Properties of the Fourier Transform
Dilation Property
Dilation Property
For a < 0 and finite, all remains the same except the integration
limits:
1. integrand: substitute t = τ /a.
2. limits: for t = ∞, τ = −∞; for t = −∞, τ = +∞.
3. differential: dτ = adt or dt = dτ /a.
Therefore,
Z ∞
Z −∞
dτ
−j2πft
H(f ) =
g (at)e
dt =
g (τ )e −j2πf τ /a
a
−∞
+∞
Z ∞
f
1
1
f
= −
g (t)e −j2π( a )t dt = − G
a −∞
a
a
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
Therefore,
H(f ) =
 1
 +aG
f
a
− 1a G
f
a

a>0
a<0
1
=
G
|a|
f
a
for a 6= 0, and
1
G
H(f ) =
|a|
h(t) = g (at)
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f
a
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
Properties of the Fourier Transform
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Properties of the Fourier Transform
Inverse Relationship
Inverse Relationship
g (at)
1
G
|a|
f
a
AT sinc(fT)
AT
Arect(t/T)
T larger
A
I
I
A stretch in the time (or frequency) domain by a given factor a
leads to a compression in the frequency (or time) domain by
same same factor a.
There is also a corresponding amplitude change in the frequency
domain.
I
This is needed to keep the energies of the signals in both
domains equated (from Rayleigh’s Energy Theorem):
Z
∞
∞
|g (t)|2 dt =
Z
∞
T/2
-T/2
t
-4/T
2/T
-2/T
-3/T
-1/T
0
T/2
f
AT
A
-T/2
4/T
3/T
AT sinc(fT)
Arect(t/T)
T smaller
1/T
t
-2/T
2/T
-1/T
0
f
1/T
|G (f )|2 df
∞
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Properties of the Fourier Transform
Properties of the Fourier Transform
Time-Bandwidth Product
Time-Bandwidth Product
time-duration of a signal × frequency bandwidth = constant
time-duration of a signal × frequency bandwidth = constant
AT sinc(fT)
AT
Arect(t/T)
T larger
I
manifestation of the inverse relationship
I
the constant depends on the definitions of duration and
bandwidth and can change with the shape of signals being
considered
I
It can be shown that:
time-duration of a signal × frequency bandwidth ≥
with equality achieved for a Gaussian pulse.
A
T /2
-T/2
t
-4/T
2/T
-2/T
-3/T
duration
-1/T
0
1/T
4/T
3/T
f
null-to-null
bandwidth
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Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
Properties of the Fourier Transform
Time Shifting Property
Let τ = t − t0 where t0 ∈ R. Three things must be changed:
1. integrand: substitute t = τ + t0 .
2. limits: for t = ∞, τ = ∞; for t = −∞, τ = −∞.
3. differential: dτ = dt.
G (f )e −j2πft0
Proof: Let h(t) = g (t − t0 ) and H(f ) = F[h(t)].
Z ∞
Z ∞
−j2πft
g (t − t0 )e −j2πft dt
H(f ) =
h(t)e
dt =
−∞
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Properties of the Fourier Transform
Time Shifting Property
g (t − t0 )
1
4π
Z
∞
g (t − t0 )e
H(f ) =
Z
∞
dt =
g (τ )e −j2πf (τ +t0 ) dτ
−∞
Z−∞
∞
−∞
g (τ )e −j2πf τ · e −j2πf t0 dτ
−∞
Z ∞
−j2πf t0
= e
g (τ )e −j2πf τ dτ = e −j2πf t0 G (f )
=
Idea: Do a change of integrating variable to make it look more like
G (f ).
−∞
∴
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
−j2πft
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g (t − t0 )
G (f )e −j2πft0
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Properties of the Fourier Transform
Properties of the Fourier Transform
Time Shifting Property
I
Time Shifting Property
G (f )e −j2πft0 results in a change of phase only of G (f ).
Magnitude:
−j2πft0
|G (f )e
−j2πft0
| =
| = |G (f )| |e
| {z }
=1
I
Recall, that the phase of the FT determines how the complex
sinusoid e j2πft lines up in the synthesis of g (t).
I
A delayed signal g (t − t0 ), requires all the corresponding
sinusoidal components {e j2πft } for −∞ < f < ∞ to be delayed
by t0 thus changing their individual absolute phases.
|G (f )|
| {z }
no mag change
Phase:
∠[G (f )e −j2πft0 ]
=
∠[|G (f )|e j∠G (f ) e −j2πft0 ]
=
∠|G (f )| +∠G (f ) + ∠e −j2πft0
| {z }
=
∠G (f ) − 2πft0
|
{z
}
phase change!
=0
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Properties of the Fourier Transform
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Properties of the Fourier Transform
Convolution Theorem
Convolution Theorem
∞
Z
g1 (τ )g2 (t − τ )
G1 (f )G2 (f )
−∞
Z
G1 (f )G2 (f )e j2πft df
Z ∞
Z−∞
∞
0 −j2πft 0
0
G1 (f )
g2 (t )e
dt e j2πft df
=
−∞
Z−∞
∞ Z ∞
0
=
G1 (f )g2 (t 0 )e j2πf (t−t ) dt 0 df
Proof: Let H(f ) = G1 (f )G2 (f ).
From the synthesis equation:
Z
∞
h(t) =
H(f )e
∞
h(t) =
j2πft
Z
∞
df =
G1 (f )G2 (f )e j2πft df
−∞
−∞
−∞
From the analysis equation, substitute in:
Z
Idea: Substitute for the integrating variable t 0 .
∞
G2 (f ) =
−∞
0
g2 (t 0 )e −j2πft dt 0
−∞
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Properties of the Fourier Transform
Properties of the Fourier Transform
Convolution Theorem
LTI Systems and Filtering
0
Let τ = t − t .
Z
∞
Z
LTI System
−∞
G1 (f )g2 (t − τ )e j2πf τ (−dτ )df
h(t) =
∞
Z−∞
∞ Z ∞
impulse response
j2πf τ
G1 (f )g2 (t − τ )e
dτ df
Z ∞
j2πf τ
=
g2 (t − τ )
G1 (f )e
df dτ
−∞
−∞
Z ∞
Z ∞
g2 (t − τ )g1 (τ )dτ =
=
g1 (τ )g2 (t − τ )dτ
=
Z−∞
∞
−∞
−∞
LTI System
frequency response
I
−∞
I
Z
∞
g1 (τ )g2 (t − τ )
∴
The convolution theorem provides a filtering perspective to how
a linear time-invariant system operates on an input signal.
The LTI system scales the sinusoidal component corresponding
to frequency f by H(f ) providing frequency selectivity.
G1 (f )G2 (f )
−∞
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Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
Properties of the Fourier Transform
Properties of the Fourier Transform
Conjugation Property and Conjugate Symmetry
g ∗ (t)
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Conjugate Symmetry
G ∗ (−f )
G (f ) = G ∗ (−f )
|G (f )| = |G ∗ (−f )| = |G (−f )|
∠G (f ) = ∠G ∗ (−f ) = −∠G (−f )
If g (t) is real (i.e., not complex), then we can say:
Therefore,
g (t) = g ∗ (t)
F[g (t)] = F[g ∗ (t)]
G (f ) = G ∗ (−f )
|G (f )| = |G (−f )|
∠G (f ) = −∠G (−f )
That is, G (f ) obeys conjugate symmetry.
Professor Deepa Kundur (University of Toronto) Properties of the Fourier Transform
mag is EVEN
phase is ODD
for real time-domain signals.
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