Download 4: Parseval`s Theorem and Convolution

4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
4: Parseval’s Theorem and
Convolution
Parseval and Convolution: 4 – 1 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
v(t) =
∞
n=−∞
Un ei2πnF t
P∞
i2πnF t
V
e
n
n=−∞
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
• Summary
∗
hu (t)v(t)i =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
∗ −i2πnF t
e
U
n
n=−∞
P∞
i2πmF t
m=−∞
Vm e
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
∗
hu (t)v(t)i =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
∗ −i2πnF t
e
U
n
n=−∞
P∞
i2πmF t
m=−∞
Vm e
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
E1.10 Fourier Series and Transforms (2014-5543)
P∞
P
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
i2π(m−n)F t P∞
P∞
∗
= n=−∞ Un m=−∞ Vm e
E1.10 Fourier Series and Transforms (2014-5543)
P∞
P
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
P∞
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
i2π(m−n)F t P∞
P∞
∗
= n=−∞ Un m=−∞ Vm e
P
The quantity h· · · i equals 1 if m = n and 0 otherwise, so the only
non-zero element in the second sum is when m = n, so the second sum
equals Vn .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
P∞
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
i2π(m−n)F t P∞
P∞
∗
= n=−∞ Un m=−∞ Vm e
P
The quantity h· · · i equals 1 if m = n and 0 otherwise, so the only
non-zero element in the second sum is when m = n, so the second sum
equals Vn .
Hence Parseval’s theorem:
E1.10 Fourier Series and Transforms (2014-5543)
hu∗ (t)v(t)i =
P∞
n=−∞
Un∗ Vn
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
P∞
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
i2π(m−n)F t P∞
P∞
∗
= n=−∞ Un m=−∞ Vm e
P
The quantity h· · · i equals 1 if m = n and 0 otherwise, so the only
non-zero element in the second sum is when m = n, so the second sum
equals Vn .
P∞
hu∗ (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
|u(t)| = n=−∞ Un∗ Un
Hence Parseval’s theorem:
If v(t) = u(t) we get:
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
⇒
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
∗
u (t) = n=−∞ Un∗ e−i2πnF t
P∞
i2πnF t
V
e
n
n=−∞
[u(t) = u∗ (t) if real]
Now multiply u∗ (t) and v(t) together and take the average over [0, T ].
[Use different “dummy variables”, n and m, so they don’t get mixed up]
hu
∗
P∞
∞
∗ −i2πnF t
i2πmF t
(t)v(t)i =
e
U
V
e
m
n
n=−∞
m=−∞
−i2πnF t i2πmF t P∞
P
∞
∗
= n=−∞ Un m=−∞ Vm e
e
i2π(m−n)F t P∞
P∞
∗
= n=−∞ Un m=−∞ Vm e
P
The quantity h· · · i equals 1 if m = n and 0 otherwise, so the only
non-zero element in the second sum is when m = n, so the second sum
equals Vn .
P∞
hu∗ (t)v(t)i = n=−∞ Un∗ Vn
D
E P
P∞
2
∞
2
|u(t)| = n=−∞ Un∗ Un = n=−∞ |Un |
Hence Parseval’s theorem:
If v(t) = u(t) we get:
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 2 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Example:
u(t)
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
-1
E1.10 Fourier Series and Transforms (2014-5543)
-0.5
0
Time (s)
0.5
1
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Example:
U[0:3]=[2, 1-2j, 0, j]
u2(t)
u(t)
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
-1
E1.10 Fourier Series and Transforms (2014-5543)
-0.5
0
Time (s)
0.5
1
60
40
20
0
-1
-0.5
0
Time (s)
0.5
1
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
E
2
1
2
2
2
|u(t)| = 4 + 2 2 + 4 + (−2) = 16
u(t)
D
U[0:3]=[2, 1-2j, 0, j]
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
u2(t)
Example:
-1
E1.10 Fourier Series and Transforms (2014-5543)
-0.5
0
Time (s)
0.5
1
60
40
20
0
-1
-0.5
0
Time (s)
0.5
1
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
E
2
1
2
2
2
|u(t)| = 4 + 2 2 + 4 + (−2) = 16
u(t)
D
U[0:3]=[2, 1-2j, 0, j]
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
u2(t)
Example:
-1
E1.10 Fourier Series and Transforms (2014-5543)
-0.5
0
Time (s)
0.5
1
60
40
20
0
Pu=<u2>=16
-1
-0.5
0
Time (s)
0.5
1
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
E
2
1
2
2
2
|u(t)| = 4 + 2 2 + 4 + (−2) = 16
u(t)
D
U[0:3]=[2, 1-2j, 0, j]
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
u2(t)
Example:
-1
-0.5
0
Time (s)
0.5
1
60
40
20
0
Pu=<u2>=16
-1
-0.5
0
Time (s)
0.5
1
U0:3 = [2, 1 − 2i, 0, i]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 3 / 9
Power Conservation
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The average power of a periodic signal is given by Pu ,
D
2
|u(t)|
E
.
This is the average electrical power that would be dissipated if the
signal represents the voltage across a 1 Ω resistor.
D
E P
2
∞
2
Parseval’s Theorem: Pu = |u(t)|
= n=−∞ |Un |
P∞
2
= |U0 | + 2 n=1 |Un |2
[assume u(t) real]
P∞
n
]
= 14 a20 + 21 n=1 a2n + b2n [U+n = an −ib
2
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of the
average powers in each of its Fourier components.
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
E
2
1
2
2
2
|u(t)| = 4 + 2 2 + 4 + (−2) = 16
u(t)
D
U[0:3]=[2, 1-2j, 0, j]
U[0:3]=[2, 1-2j, 0, j]
8
6
4
2
0
-2
-4
u2(t)
Example:
-1
-0.5
0
Time (s)
0.5
U0:3 = [2, 1 − 2i, 0, i]
E1.10 Fourier Series and Transforms (2014-5543)
1
⇒
60
40
20
0
Pu=<u2>=16
-1
|U0 |2 + 2
-0.5
0
Time (s)
P∞
0.5
1
2
|U
|
= 16
n
n=1
Parseval and Convolution: 4 – 3 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
The spectrum of a periodic signal is the values of {Un } versus nF .
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
√
√
Magnitude Spectrum: |U−3:3 | = 1, 0, 5, 2, 5, 0, 1
|Un|
2
1
0
-3
-2
E1.10 Fourier Series and Transforms (2014-5543)
-1
0
1
Frequency (Hz)
2
3
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
√
√
Magnitude Spectrum: |U−3:3 | = 1, 0, 5, 2, 5, 0, 1
2 Power Spectrum: U−3:3 = [1, 0, 5, 4, 5, 0, 1]
5
|U2n|
|Un|
2
1
0
-3
-2
E1.10 Fourier Series and Transforms (2014-5543)
-1
0
1
Frequency (Hz)
2
3
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
√
√
Magnitude Spectrum: |U−3:3 | = 1, 0, 5, 2, 5, 0, 1
2 P 2 Power Spectrum: U−3:3 = [1, 0, 5, 4, 5, 0, 1] [
= u (t) ]
5
1
0
Σ=16
|U2n|
|Un|
2
-3
-2
E1.10 Fourier Series and Transforms (2014-5543)
-1
0
1
Frequency (Hz)
2
3
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
√
√
Magnitude Spectrum: |U−3:3 | = 1, 0, 5, 2, 5, 0, 1
2 P 2 Power Spectrum: U−3:3 = [1, 0, 5, 4, 5, 0, 1] [
= u (t) ]
5
1
0
Σ=16
|U2n|
|Un|
2
-3
-2
-1
0
1
Frequency (Hz)
2
3
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
The magnitude and power spectra of a real periodic signal are symmetrical.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
The spectrum of a periodic signal is the values of {Un } versus nF .
n q
o
The magnitude spectrum is the values of {|Un |} = 21 a2|n| + b2|n| .
o
n
o n 2
1
2
2
The power spectrum is the values of |Un |
= 4 a|n| + b|n| .
Example:
u(t) = 2 + 2 cos 2πF t + 4 sin 2πF t − 2 sin 6πF t
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
b1:3 = [4, 0, −2]
Spectrum: U−3:3 = [−i, 0, 1 + 2i, 2, 1 − 2i, 0, i]
√
√
Magnitude Spectrum: |U−3:3 | = 1, 0, 5, 2, 5, 0, 1
2 P 2 Power Spectrum: U−3:3 = [1, 0, 5, 4, 5, 0, 1] [
= u (t) ]
5
1
0
Σ=16
|U2n|
|Un|
2
-3
-2
-1
0
1
Frequency (Hz)
2
3
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
The magnitude and power spectra of a real periodic signal are symmetrical.
2
A one-sided power power spectrum shows U0 and then 2 |Un | for n ≥ 1.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 4 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Suppose we have two signals with the same period, T = F1 ,
P
u(t) =
v(t) =
∞
i2πnF t
n=−∞ Un e
P∞
i2πmF t
m=−∞ Vn e
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
E1.10 Fourier Series and Transforms (2014-5543)
, Ur ∗ Vr
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
w(t) = u(t)v(t)
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
w(t) = u(t)v(t) =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
i2πnF t
n=−∞ Un e
P∞
, Ur ∗ Vr
m=−∞
Vm ei2πmF t
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
P∞
P∞
w(t) = r=−∞ m=−∞ Ur−m Vm ei2πrF t
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
P∞
P∞
P∞
i2πrF t
w(t) = r=−∞ m=−∞ Ur−m Vm e
= r=−∞ Wr ei2πrF t
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
P∞
P∞
P∞
i2πrF t
w(t) = r=−∞ m=−∞ Ur−m Vm e
= r=−∞ Wr ei2πrF t
P∞
where Wr =
m=−∞ Ur−m Vm , Ur ∗ Vr .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
P∞
P∞
P∞
i2πrF t
w(t) = r=−∞ m=−∞ Ur−m Vm e
= r=−∞ Wr ei2πrF t
P∞
where Wr =
m=−∞ Ur−m Vm , Ur ∗ Vr .
Wr is the sum of all products Un Vm for which m + n = r .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Product of Signals
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Suppose we have two signals with the same period, T = F1 ,
P
∞
i2πnF t
n=−∞ Un e
P∞
v(t) = m=−∞ Vn ei2πmF t
P
u(t)v(t) then Wr = ∞
m=−∞ Ur−m Vm
u(t) =
If w(t) =
Proof:
, Ur ∗ Vr
P∞
P∞
w(t) = u(t)v(t) = n=−∞ Un ei2πnF t m=−∞ Vm ei2πmF t
P∞
P∞
= n=−∞ m=−∞ Un Vm ei2π(m+n)F t
Now we change the summation variable to use r instead of n:
r =m+n⇒n=r−m
This is a one-to-one mapping: every pair (m, n) in the range ±∞
corresponds to exactly one pair (m, r) in the same range.
P∞
P∞
P∞
i2πrF t
w(t) = r=−∞ m=−∞ Ur−m Vm e
= r=−∞ Wr ei2πrF t
P∞
where Wr =
m=−∞ Ur−m Vm , Ur ∗ Vr .
Wr is the sum of all products Un Vm for which m + n = r .
The spectrum Wr = Ur ∗ Vr is called the convolution of Ur and Vr .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 5 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
Convolution behaves algebraically like multiplication:
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗ (Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
E1.10 Fourier Series and Transforms (2014-5543)
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m
P
m
Ur−m Vm
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm
E1.10 Fourier Series and Transforms (2014-5543)
[1 ↔ 1 for any r ]
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
E1.10 Fourier Series and Transforms (2014-5543)
P
n
Un Vr−n
[1 ↔ 1 for any r ]
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n
P
P
n ((
m Un−m Vm ) Wr−n )
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
P
P
n ((
m Un−m Vm ) Wr−n )
E1.10 Fourier Series and Transforms (2014-5543)
[1 ↔ 1]
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
P
[1 ↔ 1]
P
P P
n ((
m Un−m Vm ) Wr−n ) =
k ((
m Ur−k Vm ) Wk−m )
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
[1 ↔ 1]
P P
P P
Un−m Vm ) Wr−n ) = k (( m Ur−k Vm ) Wk−m )
n ((
P Pm
= k m Ur−k Vm Wk−m
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
[1 ↔ 1]
P P
P P
Un−m Vm ) Wr−n ) = k (( m Ur−k Vm ) Wk−m )
n ((
P Pm
P
P
= k m Ur−k Vm Wk−m = k (Ur−k ( m Vm Wk−m ))
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
[1 ↔ 1]
P P
P P
Un−m Vm ) Wr−n ) = k (( m Ur−k Vm ) Wk−m )
n ((
P Pm
P
P
= k m Ur−k Vm Wk−m = k (Ur−k ( m Vm Wk−m ))
P
P
P
3)
m Wr−m (Um + Vm ) =
m Wr−m Um +
m Wr−m Vm
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
[1 ↔ 1]
P P
P P
Un−m Vm ) Wr−n ) = k (( m Ur−k Vm ) Wk−m )
n ((
P Pm
P
P
= k m Ur−k Vm Wk−m = k (Ur−k ( m Vm Wk−m ))
P
P
P
3)
m Wr−m (Um + Vm ) =
m Wr−m Um +
m Wr−m Vm
4) Ir−m Um = 0 unless m = r .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Properties
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗ Wr = (Ur ∗ Vr ) ∗ Wr = Ur ∗ (Vr ∗ Wr )
3) Distributive over addition: Wr ∗(
(Ur + Vr ) = Wr ∗ Ur + Wr ∗ Vr
4) Identity Element or “1”: If Ir =
1 r=0
, then Ir ∗ Ur = Ur
0 r=
6 0
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r − m ⇔ m = r − n
P
m
Ur−m Vm =
P
n
Un Vr−n
[1 ↔ 1 for any r ]
2) Substitute for n: k = r + m − n ⇔ n = r + m − k
[1 ↔ 1]
P P
P P
Un−m Vm ) Wr−n ) = k (( m Ur−k Vm ) Wk−m )
n ((
P Pm
P
P
= k m Ur−k Vm Wk−m = k (Ur−k ( m Vm Wk−m ))
P
P
P
3)
m Wr−m (Um + Vm ) =
m Wr−m Um +
m Wr−m Vm
P
4) Ir−m Um = 0 unless m = r . Hence
m Ir−m Um = Ur .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 6 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
u(t) = 10 + 8 sin 2πt
• Power Conservation
• Magnitude Spectrum and
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
10
0
-2
-1
0
Time (s)
1
2
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt
U−1:1 = [4i, 10, −4i]
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
10
0
-2
-1
0
Time (s)
1
2
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt
U−1:1 = [4i, 10, −4i]
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
0
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
5
0
-1
0
1
Frequency (Hz)
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt
U−1:1 = [4i, 10, −4i]
v(t) = 4 cos 6πt
10
0
-2
-1
0
Time (s)
1
2
4
2
0
-2
-4
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
v(t)
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
5
0
-1
0
1
Frequency (Hz)
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
10
0
-2
-1
0
Time (s)
1
2
4
2
0
-2
-4
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
v(t)
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
[ 0 = V0 ]
5
0
-1
0
1
Frequency (Hz)
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
[ 0 = V0 ]
-1
0
1
Frequency (Hz)
E1.10 Fourier Series and Transforms (2014-5543)
1
0
-3
3
Parseval and Convolution: 4 – 7 / 9
Convolution Example
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
[ 0 = V0 ]
50
0
-50
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
-2
-1
0
Time (s)
1
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-1
0
1
Frequency (Hz)
1
0
-3
3
w(t) = u(t)v(t)
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
[ 0 = V0 ]
50
0
-50
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
-2
-1
0
Time (s)
1
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-1
0
1
Frequency (Hz)
1
0
-3
3
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
[ 0 = V0 ]
50
0
-50
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
-2
-1
0
Time (s)
1
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-1
0
1
Frequency (Hz)
1
0
-3
3
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
[ 0 = V0 ]
50
0
-50
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
-2
-1
0
Time (s)
1
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-1
0
1
Frequency (Hz)
1
0
-3
3
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
[ 0 = V0 ]
50
0
-50
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
-2
-1
0
Time (s)
1
2
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-1
0
1
Frequency (Hz)
1
0
-3
3
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
W−4:4 = [8i, 20, −8i, 0, 0, 0, 8i, 20, −8i]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
-1
0
1
Frequency (Hz)
0
-50
-2
-1
0
Time (s)
1
2
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
20
1
0
[ 0 = V0 ]
50
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
|Wn|
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-3
-2
-1
0
1
Frequency (Hz)
2
3
10
0
-4
-3
3
4
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
W−4:4 = [8i, 20, −8i, 0, 0, 0, 8i, 20, −8i]
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
-1
0
1
Frequency (Hz)
0
-50
-2
-1
0
Time (s)
1
2
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
20
1
0
[ 0 = V0 ]
50
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
|Wn|
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-3
-2
-1
0
1
Frequency (Hz)
2
3
10
0
-4
-3
3
4
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
W−4:4 = [8i, 20, −8i, 0, 0, 0, 8i, 20, −8i]
To convolve Un and Vn :
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
-1
0
1
Frequency (Hz)
0
-50
-2
-1
0
Time (s)
1
2
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
20
1
0
[ 0 = V0 ]
50
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
|Wn|
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-3
-2
-1
0
1
Frequency (Hz)
2
3
10
0
-4
-3
3
4
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
W−4:4 = [8i, 20, −8i, 0, 0, 0, 8i, 20, −8i]
To convolve Un and Vn :
Replace each harmonic in Vn by a scaled copy of the entire {Un }
and sum the complex-valued coefficients of any
overlapping harmonics.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution Example
0
v(t)
-2
-1
0
Time (s)
1
2
10
|Un|
Polynomial Multiplication
• Summary
10
4
2
0
-2
-4
5
0
-1
0
1
Frequency (Hz)
0
-50
-2
-1
0
Time (s)
1
2
-2
-1
0
Time (s)
1
-2
-1
0
1
Frequency (Hz)
2
2
20
1
0
[ 0 = V0 ]
50
2
|Vn|
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
u(t)
Power Spectrum
w(t)
• Power Conservation
• Magnitude Spectrum and
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πt
U−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2]
|Wn|
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
-3
-2
-1
0
1
Frequency (Hz)
2
3
10
0
-4
-3
3
4
w(t) = u(t)v(t) = (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt + 32 sin 2πt cos 6πt
= 40 cos 6πt + 16 sin 8πt − 16 sin 4πt
W−4:4 = [8i, 20, −8i, 0, 0, 0, 8i, 20, −8i]
To convolve Un and Vn :
Replace each harmonic in Vn by a scaled copy of the entire {Un }
(or vice versa) and sum the complex-valued coefficients of any
overlapping harmonics.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 7 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
• Power Conservation
• Magnitude Spectrum and
v(x) =
V2 x2 + V1 x + V0
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r .
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
E1.10 Fourier Series and Transforms (2014-5543)
P2
m=0
U3−m Vm
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
P2
m=0
U3−m Vm
If all the missing coefficients are assumed to be zero, we can write
Wr =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
m=−∞
Ur−m Vm
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
P2
m=0
U3−m Vm
If all the missing coefficients are assumed to be zero, we can write
Wr =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
m=−∞
Ur−m Vm , Ur ∗ Vr
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
P2
m=0
U3−m Vm
If all the missing coefficients are assumed to be zero, we can write
Wr =
P∞
m=−∞
Ur−m Vm , Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
v(x) =
V2 x2 + V1 x + V0
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
P2
m=0
U3−m Vm
If all the missing coefficients are assumed to be zero, we can write
Wr =
P∞
m=−∞
Ur−m Vm , Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
Actually, the complex Fourier Series is iust a polynomial:
u(t) =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
n=−∞
Un ei2πnF t
Parseval and Convolution: 4 – 8 / 9
Convolution and Polynomial Multiplication
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
Two polynomials: u(x) = U3 x3 + U2 x2 + U1 x + U0
V2 x2 + V1 x + V0
v(x) =
Now multiply the two polynomials together:
w(x) = u(x)v(x)
= U3 V2 x5 + (U3 V1 + U2 V2 ) x4 + (U3 V0 + U2 V1 + U1 V2 ) x3
+ (U2 V0 + U1 V1 + U0 V2 ) x2 + (U1 V0 + U0 V1 ) x +U0 V0
The coefficient of xr consists of all the coefficient pair from U and V where
the subscripts add up to r . For example, for r = 3:
W3 = U3 V0 + U2 V1 + U1 V2 =
P2
m=0
U3−m Vm
If all the missing coefficients are assumed to be zero, we can write
Wr =
P∞
m=−∞
Ur−m Vm , Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
Actually, the complex Fourier Series is iust a polynomial:
u(t) =
E1.10 Fourier Series and Transforms (2014-5543)
P∞
n=−∞
i2πnF t
Un e
=
P∞
n=−∞
i2πF t n
Un e
Parseval and Convolution: 4 – 8 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
∗
• Parseval’s Theorem: hu (t)v(t)i =
• Power Conservation
• Magnitude Spectrum and
P∞
∗
Vn
U
n
n=−∞
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
∗
P∞
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
Polynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
E1.10 Fourier Series and Transforms (2014-5543)
1
2
P∞
2
a
n
n=1
+
b2n
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
• Product of signals ⇔ Convolution of complex Fourier coefficients:
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
◦ Associative: U ∗ V ∗ W is unambiguous
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
◦ Associative: U ∗ V ∗ W is unambiguous
◦ Distributes over addition: U ∗ (V + W ) = U ∗ V + U ∗ W
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
◦ Associative: U ∗ V ∗ W is unambiguous
◦ Distributes over addition: U ∗ (V + W ) = U ∗ V + U ∗ W
◦ Has an identity: Ir = 1 if r = 0 and = 0 otherwise
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
◦ Associative: U ∗ V ∗ W is unambiguous
◦ Distributes over addition: U ∗ (V + W ) = U ∗ V + U ∗ W
◦ Has an identity: Ir = 1 if r = 0 and = 0 otherwise
• Polynomial multiplication ⇔ convolution of coefficients
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9
Summary
4: Parseval’s Theorem and
Convolution
• Parseval’s Theorem (a.k.a.
Plancherel’s Theorem)
• Power Conservation
• Magnitude Spectrum and
Power Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution and
Polynomial Multiplication
• Summary
P∞
∗
• Parseval’s Theorem: hu (t)v(t)i = n=−∞ Un∗ Vn
D
E P
2
∞
2
◦ Power Conservation: |u(t)| = n=−∞ |Un |
◦ or in terms of an and bn :
D
E
2
|u(t)| = 14 a20 +
1
2
P∞
2
a
n
n=1
+
b2n
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
Fourier coefficients:
• Product of signals ⇔ Convolution of complex
P∞
w(t) = u(t)v(t) ⇔ Wn = Un ∗ Vn , m=−∞ Un−m Vm
• Convolution acts like multiplication:
◦ Commutative: U ∗ V = V ∗ U
◦ Associative: U ∗ V ∗ W is unambiguous
◦ Distributes over addition: U ∗ (V + W ) = U ∗ V + U ∗ W
◦ Has an identity: Ir = 1 if r = 0 and = 0 otherwise
• Polynomial multiplication ⇔ convolution of coefficients
For further details see RHB Chapter 12.8.
E1.10 Fourier Series and Transforms (2014-5543)
Parseval and Convolution: 4 – 9 / 9