4: parseval’s theorem and convolution · convolution •parseval’s theorem (a.k.a....
TRANSCRIPT
![Page 1: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/1.jpg)
4: Parseval’s Theorem andConvolution
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 1 / 9
![Page 2: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/2.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
n=−∞ Vnei2πnFt
![Page 3: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/3.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
![Page 4: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/4.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
Now multiply u∗(t) and v(t) together and take the average over [0, T ].
![Page 5: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/5.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
Now multiply u∗(t) and v(t) together and take the average over [0, T ].
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
![Page 6: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/6.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
![Page 7: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/7.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
![Page 8: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/8.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
ei2π(m−n)Ft⟩
![Page 9: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/9.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
ei2π(m−n)Ft⟩
The quantity 〈· · · 〉 equals 1 if m = n and 0 otherwise, so the onlynon-zero element in the second sum is when m = n, so the second sumequals Vn.
![Page 10: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/10.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
ei2π(m−n)Ft⟩
The quantity 〈· · · 〉 equals 1 if m = n and 0 otherwise, so the onlynon-zero element in the second sum is when m = n, so the second sumequals Vn.
Hence Parseval’s theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
![Page 11: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/11.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
ei2π(m−n)Ft⟩
The quantity 〈· · · 〉 equals 1 if m = n and 0 otherwise, so the onlynon-zero element in the second sum is when m = n, so the second sumequals Vn.
Hence Parseval’s theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
If v(t) = u(t) we get:⟨
|u(t)|2⟩
=∑∞
n=−∞ U∗nUn
![Page 12: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/12.jpg)
Parseval’s Theorem (a.k.a. Plancherel’s Theorem)
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 2 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
⇒ u∗(t) =∑∞
n=−∞ U∗ne−i2πnFt [u(t) = u∗(t) if real]
v(t) =∑∞
n=−∞ Vnei2πnFt
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]
〈u∗(t)v(t)〉 =⟨∑∞
n=−∞ U∗ne−i2πnFt
∑∞m=−∞ Vmei2πmFt
⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
e−i2πnFtei2πmFt⟩
=∑∞
n=−∞ U∗n
∑∞m=−∞ Vm
⟨
ei2π(m−n)Ft⟩
The quantity 〈· · · 〉 equals 1 if m = n and 0 otherwise, so the onlynon-zero element in the second sum is when m = n, so the second sumequals Vn.
Hence Parseval’s theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
If v(t) = u(t) we get:⟨
|u(t)|2⟩
=∑∞
n=−∞ U∗nUn =
∑∞n=−∞ |Un|2
![Page 13: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/13.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
![Page 14: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/14.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
![Page 15: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/15.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
![Page 16: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/16.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
![Page 17: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/17.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
![Page 18: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/18.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)
![Page 19: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/19.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)-1 -0.5 0 0.5 1
0204060
u2 (t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)
![Page 20: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/20.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt⟨
|u(t)|2⟩
= 4 + 12
(
22 + 42 + (−2)2)
= 16
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)-1 -0.5 0 0.5 1
0204060
u2 (t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)
![Page 21: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/21.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt⟨
|u(t)|2⟩
= 4 + 12
(
22 + 42 + (−2)2)
= 16
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)-1 -0.5 0 0.5 1
0204060
u2 (t)
Pu=<u2>=16
U[0:3]=[2, 1-2j, 0, j]
Time (s)
![Page 22: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/22.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt⟨
|u(t)|2⟩
= 4 + 12
(
22 + 42 + (−2)2)
= 16
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)-1 -0.5 0 0.5 1
0204060
u2 (t)
Pu=<u2>=16
U[0:3]=[2, 1-2j, 0, j]
Time (s)
U0:3 = [2, 1− 2i, 0, i]
![Page 23: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/23.jpg)
Power Conservation
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9
The average power of a periodic signal is given by Pu ,
⟨
|u(t)|2⟩
.
This is the average electrical power that would be dissipated if thesignal represents the voltage across a 1Ω resistor.
Parseval’s Theorem: Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
= |U0|2 + 2∑∞
n=1 |Un|2 [assume u(t) real]
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
[U+n = an−ibn
2 ]
Parseval’s theorem ⇒ the average power in u(t) is equal to the sum of theaverage powers in each of its Fourier components.
Example: u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt⟨
|u(t)|2⟩
= 4 + 12
(
22 + 42 + (−2)2)
= 16
-1 -0.5 0 0.5 1-4-202468
u(t)
U[0:3]=[2, 1-2j, 0, j]
Time (s)-1 -0.5 0 0.5 1
0204060
u2 (t)
Pu=<u2>=16
U[0:3]=[2, 1-2j, 0, j]
Time (s)
U0:3 = [2, 1− 2i, 0, i] ⇒ |U0|2 + 2∑∞
n=1 |Un|2 = 16
![Page 24: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/24.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
![Page 25: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/25.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
![Page 26: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/26.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
![Page 27: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/27.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
![Page 28: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/28.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
Fourier Coefficients: a0:3 = [4, 2, 0, 0]
![Page 29: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/29.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
Fourier Coefficients: a0:3 = [4, 2, 0, 0] b1:3 = [4, 0, −2]
![Page 30: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/30.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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]
![Page 31: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/31.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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
]
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Un|
![Page 32: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/32.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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
]
Power Spectrum:∣
∣U2−3:3
∣
∣ = [1, 0, 5, 4, 5, 0, 1]
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
5
Frequency (Hz)|U
n2 |
![Page 33: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/33.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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
]
Power Spectrum:∣
∣U2−3:3
∣
∣ = [1, 0, 5, 4, 5, 0, 1] [∑
=⟨
u2(t)⟩
]
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
5
Frequency (Hz)|U
n2 |
Σ=16
![Page 34: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/34.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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
]
Power Spectrum:∣
∣U2−3:3
∣
∣ = [1, 0, 5, 4, 5, 0, 1] [∑
=⟨
u2(t)⟩
]
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
5
Frequency (Hz)|U
n2 |
Σ=16
The magnitude and power spectra of a real periodic signal are symmetrical.
![Page 35: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/35.jpg)
Magnitude Spectrum and Power Spectrum
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 4 / 9
The spectrum of a periodic signal is the values of Un versus nF .
The magnitude spectrum is the values of |Un| =
12
√
a2|n| + b2|n|
.
The power spectrum is the values of
|Un|2
=
14
(
a2|n| + b2|n|
)
.
Example:u(t) = 2 + 2 cos 2πFt+ 4 sin 2πFt− 2 sin 6πFt
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
]
Power Spectrum:∣
∣U2−3:3
∣
∣ = [1, 0, 5, 4, 5, 0, 1] [∑
=⟨
u2(t)⟩
]
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
5
Frequency (Hz)|U
n2 |
Σ=16
The magnitude and power spectra of a real periodic signal are symmetrical.
A one-sided power power spectrum shows U0 and then 2 |Un|2 for n ≥ 1.
![Page 36: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/36.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
![Page 37: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/37.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm
![Page 38: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/38.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
![Page 39: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/39.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)
![Page 40: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/40.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
![Page 41: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/41.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
![Page 42: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/42.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
Now we change the summation variable to use r instead of n:r = m+ n
![Page 43: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/43.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
Now we change the summation variable to use r instead of n:r = m+ n ⇒ n = r −m
![Page 44: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/44.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
![Page 45: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/45.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
w(t) =∑∞
r=−∞
∑∞m=−∞ Ur−mVmei2πrFt
![Page 46: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/46.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
w(t) =∑∞
r=−∞
∑∞m=−∞ Ur−mVmei2πrFt=
∑∞r=−∞ Wre
i2πrFt
![Page 47: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/47.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
w(t) =∑∞
r=−∞
∑∞m=−∞ Ur−mVmei2πrFt=
∑∞r=−∞ Wre
i2πrFt
where Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr .
![Page 48: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/48.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
w(t) =∑∞
r=−∞
∑∞m=−∞ Ur−mVmei2πrFt=
∑∞r=−∞ Wre
i2πrFt
where Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr .
Wr is the sum of all products UnVm for which m+ n = r.
![Page 49: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/49.jpg)
Product of Signals
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 5 / 9
Suppose we have two signals with the same period, T = 1F
,
u(t) =∑∞
n=−∞ Unei2πnFt
v(t) =∑∞
m=−∞ Vnei2πmFt
If w(t) = u(t)v(t) then Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr
Proof:w(t) = u(t)v(t)=
∑∞n=−∞ Une
i2πnFt∑∞
m=−∞ Vmei2πmFt
=∑∞
n=−∞
∑∞m=−∞ UnVmei2π(m+n)Ft
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.
w(t) =∑∞
r=−∞
∑∞m=−∞ Ur−mVmei2πrFt=
∑∞r=−∞ Wre
i2πrFt
where Wr =∑∞
m=−∞ Ur−mVm , Ur ∗ Vr .
Wr is the sum of all products UnVm for which m+ n = r.
The spectrum Wr = Ur ∗ Vr is called the convolution of Ur and Vr.
![Page 50: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/50.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
Convolution behaves algebraically like multiplication:
![Page 51: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/51.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
![Page 52: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/52.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
Convolution behaves algebraically like multiplication:
1) Commutative: Ur ∗ Vr = Vr ∗ Ur
2) Associative: Ur ∗ Vr ∗Wr = (Ur ∗ Vr) ∗Wr = Ur ∗ (Vr ∗Wr)
![Page 53: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/53.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
![Page 54: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/54.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
![Page 55: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/55.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m∑
mUr−mVm
![Page 56: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/56.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm
![Page 57: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/57.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
![Page 58: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/58.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n∑
n((∑
mUn−mVm)Wr−n)
![Page 59: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/59.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)
![Page 60: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/60.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
![Page 61: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/61.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
=∑
k
∑
mUr−kVmWk−m
![Page 62: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/62.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
=∑
k
∑
mUr−kVmWk−m=
∑
k(Ur−k (
∑
mVmWk−m))
![Page 63: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/63.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
=∑
k
∑
mUr−kVmWk−m=
∑
k(Ur−k (
∑
mVmWk−m))
3)∑
mWr−m (Um + Vm)=
∑
mWr−mUm +
∑
mWr−mVm
![Page 64: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/64.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
=∑
k
∑
mUr−kVmWk−m=
∑
k(Ur−k (
∑
mVmWk−m))
3)∑
mWr−m (Um + Vm)=
∑
mWr−mUm +
∑
mWr−mVm
4) Ir−mUm = 0 unless m = r.
![Page 65: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/65.jpg)
Convolution Properties
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 6 / 9
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
0 r 6= 0, then Ir ∗ Ur = Ur
Proofs: (all sums are over ±∞)
1) Substitute for m: n = r −m⇔ m = r − n [1 ↔ 1 for any r]∑
mUr−mVm =
∑
nUnVr−n
2) Substitute for n: k = r +m− n⇔ n = r +m− k [1 ↔ 1]∑
n((∑
mUn−mVm)Wr−n)=
∑
k((∑
mUr−kVm)Wk−m)
=∑
k
∑
mUr−kVmWk−m=
∑
k(Ur−k (
∑
mVmWk−m))
3)∑
mWr−m (Um + Vm)=
∑
mWr−mUm +
∑
mWr−mVm
4) Ir−mUm = 0 unless m = r. Hence∑
mIr−mUm = Ur.
![Page 66: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/66.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt
-2 -1 0 1 20
10u(
t)
Time (s)
![Page 67: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/67.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πtU−1:1 = [4i, 10, −4i]
-2 -1 0 1 20
10u(
t)
Time (s)
![Page 68: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/68.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πtU−1:1 = [4i, 10, −4i]
-2 -1 0 1 20
10u(
t)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
![Page 69: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/69.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
![Page 70: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/70.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
![Page 71: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/71.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
![Page 72: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/72.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
w(t) = u(t)v(t)
![Page 73: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/73.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
w(t) = u(t)v(t)= (10 + 8 sin 2πt) 4 cos 6πt
![Page 74: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/74.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
w(t) = u(t)v(t)= (10 + 8 sin 2πt) 4 cos 6πt
= 40 cos 6πt+ 32 sin 2πt cos 6πt
![Page 75: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/75.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
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
![Page 76: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/76.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
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]
![Page 77: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/77.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
-4 -3 -2 -1 0 1 2 3 40
10
20
Frequency (Hz)
|Wn|
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]
![Page 78: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/78.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
-4 -3 -2 -1 0 1 2 3 40
10
20
Frequency (Hz)
|Wn|
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:
![Page 79: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/79.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
-4 -3 -2 -1 0 1 2 3 40
10
20
Frequency (Hz)
|Wn|
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 anyoverlapping harmonics.
![Page 80: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/80.jpg)
Convolution Example
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 7 / 9
u(t) = 10 + 8 sin 2πt v(t) = 4 cos 6πtU−1:1 = [4i, 10, −4i] V−3:3 = [2, 0, 0, 0, 0, 0, 2] [0 = V0]
-2 -1 0 1 20
10u(
t)
Time (s)-2 -1 0 1 2
-4-2024
v(t)
Time (s)-2 -1 0 1 2
-50
0
50
w(t
)
Time (s)
-1 0 10
5
10
Frequency (Hz)
|Un|
-3 -2 -1 0 1 2 30
1
2
Frequency (Hz)
|Vn|
-4 -3 -2 -1 0 1 2 3 40
10
20
Frequency (Hz)
|Wn|
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 anyoverlapping harmonics.
![Page 81: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/81.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
![Page 82: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/82.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
![Page 83: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/83.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5
![Page 84: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/84.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4
![Page 85: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/85.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
![Page 86: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/86.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2
![Page 87: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/87.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x
![Page 88: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/88.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
![Page 89: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/89.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r.
![Page 90: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/90.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2
![Page 91: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/91.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
![Page 92: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/92.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
If all the missing coefficients are assumed to be zero, we can write
Wr =∑∞
m=−∞ Ur−mVm
![Page 93: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/93.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
If all the missing coefficients are assumed to be zero, we can write
Wr =∑∞
m=−∞ Ur−mVm, Ur ∗ Vr
![Page 94: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/94.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
If all the missing coefficients are assumed to be zero, we can write
Wr =∑∞
m=−∞ Ur−mVm, Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
![Page 95: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/95.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
If all the missing coefficients are assumed to be zero, we can write
Wr =∑∞
m=−∞ Ur−mVm, Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
Actually, the complex Fourier Series is iust a polynomial:
u(t) =∑∞
n=−∞ Unei2πnFt
![Page 96: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/96.jpg)
Convolution and Polynomial Multiplication
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 8 / 9
Two polynomials: u(x) = U3x3 + U2x
2 + U1x+ U0
v(x) = V2x2 + V1x+ V0
Now multiply the two polynomials together:w(x) = u(x)v(x)
= U3V2x5 +(U3V1 + U2V2) x
4 +(U3V0 + U2V1 + U1V2)x3
+(U2V0 + U1V1 + U0V2)x2 +(U1V0 + U0V1)x+U0V0
The coefficient of xr consists of all the coefficient pair from U and V wherethe subscripts add up to r. For example, for r = 3:
W3 = U3V0 + U2V1 + U1V2 =∑2
m=0 U3−mVm
If all the missing coefficients are assumed to be zero, we can write
Wr =∑∞
m=−∞ Ur−mVm, Ur ∗ Vr
So, to multiply two polynomials, you convolve their coefficient sequences.
Actually, the complex Fourier Series is iust a polynomial:
u(t) =∑∞
n=−∞ Unei2πnFt =
∑∞n=−∞ Un
(
ei2πFt)n
![Page 97: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/97.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
![Page 98: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/98.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
![Page 99: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/99.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
![Page 100: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/100.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• Linearity: w(t) = au(t) + bv(t) ⇔ Wn = aUn + bVn
![Page 101: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/101.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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
![Page 102: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/102.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
![Page 103: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/103.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• Convolution acts like multiplication: Commutative: U ∗ V = V ∗ U
![Page 104: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/104.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• Convolution acts like multiplication: Commutative: U ∗ V = V ∗ U Associative: U ∗ V ∗W is unambiguous
![Page 105: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/105.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• 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
![Page 106: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/106.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• 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
![Page 107: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/107.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• 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
![Page 108: 4: Parseval’s Theorem and Convolution · Convolution •Parseval’s Theorem (a.k.a. Plancherel’s Theorem) •Power Conservation •Magnitude Spectrum and Power Spectrum •Product](https://reader034.vdocument.in/reader034/viewer/2022043003/5f80c558d24a7652ad29b143/html5/thumbnails/108.jpg)
Summary
4: Parseval’s Theorem andConvolution• Parseval’s Theorem (a.k.a.Plancherel’s Theorem)
• Power Conservation• Magnitude Spectrum andPower Spectrum
• Product of Signals
• Convolution Properties
• Convolution Example
• Convolution andPolynomial Multiplication
• Summary
E1.10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 9 / 9
• Parseval’s Theorem: 〈u∗(t)v(t)〉 =∑∞
n=−∞ U∗nVn
Power Conservation:⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
or in terms of an and bn:⟨
|u(t)|2⟩
= 14a
20 +
12
∑∞n=1
(
a2n+ b2
n
)
• 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 ,
∑∞m=−∞ Un−mVm
• 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.