7: Fourier Transforms: Convolution and Parseval`s Theorem

7: Fourier Transforms: Convolution and Parseval`s Theorem

7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Fourier Transform - Parseval and Convolution: 7 – 1 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
E1.10 Fourier Series and Transforms (2014-5559)
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
w(t) = u(t)v(t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
R +∞
= f =−∞ W (f )ei2πf t df
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
R +∞
= f =−∞ W (f )ei2πf t df
R +∞
where W (f ) = h=−∞ U (h)V (f − h)dh
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
R +∞
= f =−∞ W (f )ei2πf t df
R +∞
where W (f ) = h=−∞ U (h)V (f − h)dh , U (f ) ∗ V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
R +∞
= f =−∞ W (f )ei2πf t df
R +∞
where W (f ) = h=−∞ U (h)V (f − h)dh , U (f ) ∗ V (f )
This is the convolution of the two spectra U (f ) and V (f ).
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication of Signals
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Question: What is the Fourier transform of w(t) = u(t)v(t) ?
R +∞
i2πht
Let u(t) = h=−∞ U (h)e
dh
and
v(t) =
R +∞
i2πgt
V
(g)e
dg
g=−∞
[Note use of different dummy variables]
w(t) = u(t)v(t)
R +∞
R +∞
i2πht
dh g=−∞ V (g)ei2πgt dg
= h=−∞ U (h)e
R +∞
R +∞
= h=−∞ U (h) g=−∞ V (g)ei2π(h+g)t dg dh [merge e(··· ) ]
Now we make a change of variable in the second integral: g = f − h
R +∞
R +∞
= h=−∞ U (h) f =−∞ V (f − h)ei2πf t df dh
R∞
R +∞
R
i2πf t
dh df
= f =−∞ h=−∞ U (h)V (f − h)e
[swap ]
R +∞
= f =−∞ W (f )ei2πf t df
R +∞
where W (f ) = h=−∞ U (h)V (f − h)dh , U (f ) ∗ V (f )
This is the convolution of the two spectra U (f ) and V (f ).
w(t) = u(t)v(t)
E1.10 Fourier Series and Transforms (2014-5559)
⇔
W (f ) = U (f ) ∗ V (f )
Fourier Transform - Parseval and Convolution: 7 – 2 / 10
Multiplication Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
u(t) =
(
−at
e
0
E1.10 Fourier Series and Transforms (2014-5559)
t≥0
t<0
1
u(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
u(t) =
U (f ) =
(
−at
e
0
1
a+i2πf
E1.10 Fourier Series and Transforms (2014-5559)
1
t≥0
t<0
u(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
a=2
0.5
0
-5
0
Time (s)
5
[from before]
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
U (f ) =
−at
e
0
1
a+i2πf
E1.10 Fourier Series and Transforms (2014-5559)
1
t≥0
t<0
u(t)
u(t) =
(
a=2
0.5
0
[from before]
|U(f)|
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
e
0
1
a+i2πf
E1.10 Fourier Series and Transforms (2014-5559)
1
t≥0
t<0
u(t)
u(t) =
−at
a=2
0.5
0
[from before]
|U(f)|
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
e
0
1
t≥0
t<0
1
a+i2πf
v(t) = cos 2πF t
u(t)
u(t) =
−at
a=2
0.5
0
[from before]
|U(f)|
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
v(t)
1
0
-1
E1.10 Fourier Series and Transforms (2014-5559)
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
1
a+i2πf
t≥0
t<0
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
0
-1
E1.10 Fourier Series and Transforms (2014-5559)
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
1
a+i2πf
t≥0
t<0
u(t)
e
0
1
a=2
0.5
0
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
|U(f)|
u(t) =
−at
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
0
|V(f)|
-1
E1.10 Fourier Series and Transforms (2014-5559)
-5
0.5
0.4
0.3
0.2
0.1
-5
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.4 F=2
0.2
0
-5
-5
0
Time (s)
0.5
0
Frequency (Hz)
5
0.5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
0
-1
w(t)
|V(f)|
w(t) = u(t)v(t)
E1.10 Fourier Series and Transforms (2014-5559)
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
E1.10 Fourier Series and Transforms (2014-5559)
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
0
|V(f)|
-1
w(t)
w(t) = u(t)v(t)
W (f ) = U (f ) ∗ V (f )
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
E1.10 Fourier Series and Transforms (2014-5559)
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
0
|V(f)|
-1
0.5
a+i2π(f −F )
w(t)
w(t) = u(t)v(t)
W (f ) = U (f ) ∗ V (f )
0.5
= a+i2π(f
+F ) +
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
0
|V(f)|
-1
0.5
a+i2π(f −F )
|W(f)|
E1.10 Fourier Series and Transforms (2014-5559)
-5
1
w(t)
w(t) = u(t)v(t)
W (f ) = U (f ) ∗ V (f )
0.5
= a+i2π(f
+F ) +
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0.2
0.1
0
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
0
Frequency (Hz)
5
5
5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
0
|V(f)|
-1
0.5
a+i2π(f −F )
|W(f)|
<W(f) (rad/pi)
E1.10 Fourier Series and Transforms (2014-5559)
-5
1
w(t)
w(t) = u(t)v(t)
W (f ) = U (f ) ∗ V (f )
0.5
= a+i2π(f
+F ) +
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0.2
0.1
0
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
5
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Multiplication Example
U (f ) =
t≥0
t<0
1
a+i2πf
u(t)
e
0
1
[from before]
v(t) = cos 2πF t
V (f ) = 0.5 (δ(f + F ) + δ(f − F ))
0.5
0.4
0.3
0.2
0.1
-5
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
0
|V(f)|
-1
0.5
a+i2π(f −F )
|W(f)|
<W(f) (rad/pi)
If V (f ) consists entirely of Dirac impulses
then U (f ) ∗ V (f ) iust replaces each
impulse with a complete copy of U (f )
scaled by the area of the impulse and shifted
so that 0 Hz lies on the impulse. Then add
the overlapping complex spectra.
-5
1
w(t)
w(t) = u(t)v(t)
W (f ) = U (f ) ∗ V (f )
0.5
= a+i2π(f
+F ) +
a=2
0.5
0
|U(f)|
u(t) =
−at
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
<U(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
-5
0.4 F=2
0.2
0
-5
1
a=2, F=2
0.5
0
-0.5
-5
0.2
0.1
0
-5
0
Time (s)
0.5
5
0.5
0
Frequency (Hz)
0
Time (s)
5
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
Fourier Transform - Parseval and Convolution: 7 – 3 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
E1.10 Fourier Series and Transforms (2014-5559)
[duality]
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
y(t) = V (t) ⇔ Y (f ) = v(−f )
E1.10 Fourier Series and Transforms (2014-5559)
[duality]
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
y(t) = V (t) ⇔ Y (f ) = v(−f )
z(t) = W (t) ⇔ Z(f ) = w(−f )
E1.10 Fourier Series and Transforms (2014-5559)
[duality]
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
y(t) = V (t) ⇔ Y (f ) = v(−f )
z(t) = W (t) ⇔ Z(f ) = w(−f )
[duality]
Now the convolution property becomes:
w(−f ) = u(−f )v(−f ) ⇔ W (t) = U (t) ∗ V (t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
y(t) = V (t) ⇔ Y (f ) = v(−f )
z(t) = W (t) ⇔ Z(f ) = w(−f )
[duality]
Now the convolution property becomes:
w(−f ) = u(−f )v(−f ) ⇔ W (t) = U (t) ∗ V (t) [sub t ↔ ±f ]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution Theorem:
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
Convolution in the time domain is equivalent to multiplication in the
frequency domain and vice versa.
Proof of second line:
Given u(t), v(t) and w(t) satisfying
w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
define dual waveforms x(t), y(t) and z(t) as follows:
x(t) = U (t) ⇔ X(f ) = u(−f )
y(t) = V (t) ⇔ Y (f ) = v(−f )
z(t) = W (t) ⇔ Z(f ) = w(−f )
[duality]
Now the convolution property becomes:
w(−f ) = u(−f )v(−f ) ⇔ W (t) = U (t) ∗ V (t) [sub t ↔ ±f ]
Z(f ) = X(f )Y (f ) ⇔ z(t) = x(t) ∗ y(t)
[duality]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 4 / 10
Convolution Example
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
-2
0
2
Time t (s)
4
6
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
-2
0
2
Time t (s)
4
6
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
1
u(t)
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
w(t) = u(t) ∗ v(t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
1
u(t)
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
1
u(t)
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
1
u(t)
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
min(t,1)
= [(2 − τ ) eτ −t ]τ =0
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
u(t)
1
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
min(t,1)
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
u(t)
1
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
w(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.3
0.2
0.1
0
min(t,1)
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
u(t)
1
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
w(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.3
0.2
0.1
0
1
∫ = 0.307
0.5
0
-2
u(τ)
0
t=0.7
v(0.7-τ)
2
Time τ (s)
4
6
min(t,1)
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
min(t,1)
u(t)
1
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
0.5
0
0.3
0.2
0.1
0
1
∫ = 0.307
0.5
0
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
E1.10 Fourier Series and Transforms (2014-5559)
0.5
0
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
w(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
1
-2
∫ = 0.16
0.5
0
-2
u(τ)
0
u(τ)
0
t=0.7
v(0.7-τ)
2
Time τ (s)
4
t=1.5
v(1.5-τ)
2
Time τ (s)
6
4
6
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
min(t,1)
u(t)
1
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
0.5
0
0.3
0.2
0.1
0
1
∫ = 0.307
0.5
0
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
E1.10 Fourier Series and Transforms (2014-5559)
0.5
0
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
w(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
1
-2
∫ = 0.16
0.5
0
1
-2
∫ = 0.059
0.5
0
-2
u(τ)
0
u(τ)
0
2
Time τ (s)
4
2
Time τ (s)
4
v(2.5-τ)
2
Time τ (s)
6
t=1.5
v(1.5-τ)
u(τ)
0
t=0.7
v(0.7-τ)
4
6
t=2.5
6
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Example
w(t) = u(t) ∗ v(t)
R∞
= −∞ u(τ )v(t − τ )dτ
R min(t,1)
(1 − τ )eτ −t dτ
= 0
min(t,1)
u(t)
1
0.5
0
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
-2
0
2
Time t (s)
4
6
1
v(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
(
1−t 0≤t<1
u(t) =
0
otherwise
(
e−t t ≥ 0
v(t) =
0
t<0
0.5
0
w(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.3
0.2
0.1
0
1
∫ = 0.307
0.5
0
= [(2 − τ ) eτ −t ]τ =0


t<0
0
= 2 − t − 2e−t 0 ≤ t < 1


(e − 2) e−t
t≥1
1
-2
∫ = 0.16
0.5
0
1
-2
∫ = 0.059
0.5
0
-2
u(τ)
0
u(τ)
0
2
Time τ (s)
4
2
Time τ (s)
4
v(2.5-τ)
2
Time τ (s)
6
t=1.5
v(1.5-τ)
u(τ)
0
t=0.7
v(0.7-τ)
6
t=2.5
4
6
Note how v(t − τ ) is time-reversed (because of the −τ ) and time-shifted
to put the time origin at τ = t.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 5 / 10
Convolution Properties
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
R∞
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t)
5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t))
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t)
5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t))
Proof: In the frequency domain, convolution is multiplication.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t)
5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t))
Proof: In the frequency domain, convolution is multiplication.
Also, if u(t) ∗ v(t) = w(t), then
6) Time Shifting: u(t + a) ∗ v(t + b) = w(t + a + b)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Convolution Properties
R∞
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
Convloution: w(t) = u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t)
2) Associative:
u(t) ∗ v(t) ∗ w(t) = (u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t))
3) Distributive over addition:
w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t)
4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t)
5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t))
Proof: In the frequency domain, convolution is multiplication.
Also, if u(t) ∗ v(t) = w(t), then
6) Time Shifting: u(t + a) ∗ v(t + b) = w(t + a + b)
1
7) Time Scaling: u(at) ∗ v(at) = |a|
w(at)
How to recognise a convolution integral:
the arguments of u(· · · ) and v(· · · ) sum to a constant.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 6 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) =
E1.10 Fourier Series and Transforms (2014-5559)
R
δ(f − g)ei2πf t df
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) =
E1.10 Fourier Series and Transforms (2014-5559)
R
δ(f − g)ei2πf t df = ei2πgt
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
⇒ X(f ) = e
e
dt
E1.10 Fourier Series and Transforms (2014-5559)
R
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt
E1.10 Fourier Series and Transforms (2014-5559)
R
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
E1.10 Fourier Series and Transforms (2014-5559)
R
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
E1.10 Fourier Series and Transforms (2014-5559)
∗
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
E1.10 Fourier Series and Transforms (2014-5559)
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
R∞
∗
u
(t)v(t)dt
t=−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
[Note use of different dummy variables]
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
R∞
∗
u
(t)v(t)dt
t=−∞
R +∞
R +∞
R∞
∗
−i2πf t
df g=−∞ V (g)ei2πgt dgdt
= t=−∞ f =−∞ U (f )e
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
[Note use of different dummy variables]
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
R∞
∗
u
(t)v(t)dt
t=−∞
R +∞
R +∞
R∞
∗
−i2πf t
df g=−∞ V (g)ei2πgt dgdt
= t=−∞ f =−∞ U (f )e
R∞
R +∞
R +∞
∗
= f =−∞ U (f ) g=−∞ V (g) t=−∞ ei2π(g−f )t dtdgdf
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
[Note use of different dummy variables]
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
R∞
∗
u
(t)v(t)dt
t=−∞
R +∞
R +∞
R∞
∗
−i2πf t
df g=−∞ V (g)ei2πgt dgdt
= t=−∞ f =−∞ U (f )e
R∞
R +∞
R +∞
∗
= f =−∞ U (f ) g=−∞ V (g) t=−∞ ei2π(g−f )t dtdgdf
R +∞
R +∞
∗
= f =−∞ U (f ) g=−∞ V (g)δ(g − f )dgdf
[lemma]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Parseval’s Theorem
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Lemma:
X(f ) = δ(f − g) ⇒ x(t) = δ(f − g)ei2πf t df = ei2πgt
R i2πgt −i2πf t
R i2π(g−f )t
⇒ X(f ) = e
e
dt = e
dt = δ(g − f )
R
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
∗
Proof:
R +∞
Let u(t) = f =−∞ U (f )ei2πf t df
and
v(t) =
R +∞
g=−∞
V (g)ei2πgt dg
[Note use of different dummy variables]
Now multiply u∗ (t) = u(t) and v(t) together and integrate over time:
R∞
∗
u
(t)v(t)dt
t=−∞
R +∞
R +∞
R∞
∗
−i2πf t
df g=−∞ V (g)ei2πgt dgdt
= t=−∞ f =−∞ U (f )e
R∞
R +∞
R +∞
∗
= f =−∞ U (f ) g=−∞ V (g) t=−∞ ei2π(g−f )t dtdgdf
R +∞
R +∞
∗
= f =−∞ U (f ) g=−∞ V (g)δ(g − f )dgdf
[lemma]
R +∞
= f =−∞ U ∗ (f )V (f )df
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 7 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
R∞
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
∗
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R +∞
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
E1.10 Fourier Series and Transforms (2014-5559)
t=−∞
u (t)u(t)dt =
f =−∞
U ∗ (f )U (f )df
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
E1.10 Fourier Series and Transforms (2014-5559)
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
e−at t ≥ 0
0
t<0
u(t)
1
0
E1.10 Fourier Series and Transforms (2014-5559)
a=2
0.5
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
u(t)
1
0
E1.10 Fourier Series and Transforms (2014-5559)
a=2
0.5
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
u(t)
1
a=2
0.5
0
E1.10 Fourier Series and Transforms (2014-5559)
1
2a
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
U (f ) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
1
a+i2πf
[from before]
1
u(t)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
a=2
0.5
|U(f)|
0
E1.10 Fourier Series and Transforms (2014-5559)
1
2a
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
1
U (f ) = a+i2πf
R
R
2
⇒
|U (f )| df =
E1.10 Fourier Series and Transforms (2014-5559)
[from before]
1
u(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
1
2a
a=2
0.5
0
df
2
a +4π 2 f 2
|U(f)|
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
1
U (f ) = a+i2πf
[from before]
R
R
2
df
⇒
|U (f )| df = a2 +4π
2f 2
−1 2πf ∞
tan ( a )
=
2πa
1
u(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
1
2a
a=2
0.5
0
|U(f)|
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
1
U (f ) = a+i2πf
[from before]
R
R
2
df
⇒
|U (f )| df = a2 +4π
2f 2
−1 2πf ∞
tan ( a )
π
=
=
2πa
2πa
1
u(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
1
2a
a=2
0.5
0
|U(f)|
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Conservation
∗
R∞
∗
R +∞
Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
For the special case v(t) = u(t), Parseval’s theorem becomes:
R +∞
u (t)u(t)dt = f =−∞ U ∗ (f )U (f )df
R +∞
R∞
2
2
⇒ Eu = t=−∞ |u(t)| dt = f =−∞ |U (f )| df
t=−∞
Energy Conservation: The energy in u(t) equals the energy in U (f ).
Example: (
u(t) =
h −2at i∞
R
e−at t ≥ 0
2
=
⇒ Eu = |u(t)| dt = −e2a
0
0
t<0
1
U (f ) = a+i2πf
[from before]
R
R
2
df
⇒
|U (f )| df = a2 +4π
2f 2
−1 2πf ∞
tan ( a )
1
π
=
=
=
2πa
2πa
2a
1
u(t)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
R∞
1
2a
a=2
0.5
0
|U(f)|
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 8 / 10
Energy Spectrum
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
E1.10 Fourier Series and Transforms (2014-5559)
w(t)
Energy Spectrum
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
5
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
W (f ) =
0.5
a+i2π(f +F )
E1.10 Fourier Series and Transforms (2014-5559)
+
w(t)
Energy Spectrum
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
5
0.5
a+i2π(f −F )
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
W (f ) =
=
0.5
a+i2π(f +F )
+
w(t)
Energy Spectrum
1
a=2, F=2
0.5
0
-0.5
-5
0
Time (s)
5
0.5
a+i2π(f −F )
a+i2πf
a2 +i4πaf −4π 2 (f 2 −F 2 )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
E1.10 Fourier Series and Transforms (2014-5559)
|W(f)|
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
<W(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
E1.10 Fourier Series and Transforms (2014-5559)
|W(f)|
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
<W(f) (rad/pi)
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
a2 +4π 2 f 2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
E1.10 Fourier Series and Transforms (2014-5559)
2
2
a +4π f
|W(f)|
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
<W(f) (rad/pi)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
|W(f)| 2
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0
Frequency (Hz)
5
0.5
0
-0.5
-5
0.06
0.04
0.02
0
-5
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
2
2
a +4π f
|W(f)|
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
<W(f) (rad/pi)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
|W(f)| 2
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0
Frequency (Hz)
5
0.5
0
-0.5
-5
0.06
0.04
0.02
0
-5
2
• The units
of |W (f )| are “energy per Hz” so that its integral,
R∞
Ew = −∞ |W (f )|2 df , has units of energy.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
2
2
a +4π f
|W(f)|
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
<W(f) (rad/pi)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
|W(f)| 2
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0
Frequency (Hz)
5
0.5
0
-0.5
-5
0.06
0.04
0.02
0
-5
Energy Spectrum
2
• The units
of |W (f )| are “energy per Hz” so that its integral,
R∞
Ew = −∞ |W (f )|2 df , has units of energy.
2
• The quantity |W (f )| is called the energy spectral density of w(t) at
frequency f and its graph is the energy spectrum of w(t).
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
2
2
a +4π f
|W(f)|
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
<W(f) (rad/pi)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
|W(f)| 2
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0
Frequency (Hz)
5
0.5
0
-0.5
-5
0.06
0.04
0.02
0
-5
Energy Spectrum
2
• The units
of |W (f )| are “energy per Hz” so that its integral,
R∞
Ew = −∞ |W (f )|2 df , has units of energy.
2
• The quantity |W (f )| is called the energy spectral density of w(t) at
frequency f and its graph is the energy spectrum of w(t). It shows
how the energy of w(t) is distributed over frequencies.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
W (f ) =
=
0.5
a+i2π(f +F )
+
0.5
a+i2π(f −F )
a+i2πf
2
a +i4πaf −4π 2 (f 2 −F 2 )
2
|W (f )| =
2
2
a +4π f
|W(f)|
(
e−at cos 2πF t t ≥ 0
w(t) =
0
t<0
<W(f) (rad/pi)
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
Example from before:
|W(f)| 2
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
w(t)
Energy Spectrum
2
(a2 −4π 2 (f 2 −F 2 ))2 +16π 2 a2 f 2
1
a=2, F=2
0.5
0
-0.5
-5
0.25
0.2
0.15
0.1
0.05
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0
Frequency (Hz)
5
0.5
0
-0.5
-5
0.06
0.04
0.02
0
-5
Energy Spectrum
2
• The units
of |W (f )| are “energy per Hz” so that its integral,
R∞
Ew = −∞ |W (f )|2 df , has units of energy.
2
• The quantity |W (f )| is called the energy spectral density of w(t) at
frequency f and its graph is the energy spectrum of w(t). It shows
how the energy of w(t) is distributed over frequencies.
2
• If you divide |W (f )| by the total energy, Ew , the result is non-negative
and integrates to unity like a probability distribution.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 9 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
R +∞
R∞
∗
• Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
R +∞
R∞
∗
• Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
• Energy Spectrum:
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
R +∞
R∞
∗
• Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
• Energy Spectrum:
2
◦ Energy spectral density: |U (f )| (energy/Hz)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
R +∞
R∞
∗
• Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
• Energy Spectrum:
2
◦ Energy spectral density:
|U
(f
)|
(energy/Hz)
R
R
◦ Parseval: Eu = |u(t)|2 dt = |U (f )|2 df
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10
Summary
7: Fourier Transforms:
Convolution and Parseval’s
Theorem
• Multiplication of Signals
• Multiplication Example
• Convolution Theorem
• Convolution Example
• Convolution Properties
• Parseval’s Theorem
• Energy Conservation
• Energy Spectrum
• Summary
• Convolution:
R∞
◦ u(t) ∗ v(t) , −∞ u(τ )v(t − τ )dτ
⊲ Arguments of u(· · · ) and v(· · · ) sum to t
◦ Acts like multiplication + time scaling/shifting formulae
• Convolution Theorem: multiplication ↔ convolution
◦ w(t) = u(t)v(t) ⇔ W (f ) = U (f ) ∗ V (f )
◦ w(t) = u(t) ∗ v(t) ⇔ W (f ) = U (f )V (f )
R +∞
R∞
∗
• Parseval’s Theorem: t=−∞ u (t)v(t)dt = f =−∞ U ∗ (f )V (f )df
• Energy Spectrum:
2
◦ Energy spectral density:
|U
(f
)|
(energy/Hz)
R
R
◦ Parseval: Eu = |u(t)|2 dt = |U (f )|2 df
For further details see RHB Chapter 13.1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform - Parseval and Convolution: 7 – 10 / 10