pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products...
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Linear Systems
lecture 5
Fourier series
UNIVERSITY OF TWENTE.
academic year : 18-19lecture : 5build : August 28, 2018slides : 39
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UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
1 intro
LS
Today
1 State equations and stability2 Inner products3 Fourier series
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UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
2 1.1
LS
State equations and stability Section 2.6.5
TheoremSuppose an LTI system is described by an n-th order lineardifferential equation. Assume that the state equations invector form are
v′ = Av + x(t)b.y(t) = c · v + d x(t),
where A is an n × n-matrix, b and c are vectors, and d is aconstant. Let λ1, λ2, . . . , λn be the eigenvalues of A.
If all Reλk < 0 for all k = 1, . . . ,n, then the system isstable.If there exists an index k such that Reλk > 0, then thesystem is unstable.
If Reλk = 0 for some index k, then no conclusion canbe drawn.
![Page 4: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/4.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
2 1.1
LS
State equations and stability Section 2.6.5
TheoremSuppose an LTI system is described by an n-th order lineardifferential equation. Assume that the state equations invector form are
v′ = Av + x(t)b.y(t) = c · v + d x(t),
where A is an n × n-matrix, b and c are vectors, and d is aconstant. Let λ1, λ2, . . . , λn be the eigenvalues of A.
If all Reλk < 0 for all k = 1, . . . ,n, then the system isstable.If there exists an index k such that Reλk > 0, then thesystem is unstable.
If Reλk = 0 for some index k, then no conclusion canbe drawn.
![Page 5: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/5.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
3 1.2
LS
Trace and determinant
Let A =[
a bc d
]be a real 2× 2-matrix.
- The trace of A is tr(A) = a + d.- The determinant of A is det(A) = ad − bc.
Denote T = tr(A) and D = det(A). Thecharacteristic equation of A is
s2 − Ts + D = (s − λ1)(s − λ2) = 0. (∗)
Let λ1 and λ2 be the eigenvalues of A, thenT = λ1 + λ2 and D = λ1λ2.
The discriminant of (∗) is T 2 − 4D.T2 > 4D Equation (∗) has two real roots λ1 6= λ2.
T2 < 4D Equation (∗) has two non-real roots λ1 = λ2.
T2 = 4D Equation (∗) has one real roots λ1 = λ2.
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UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
3 1.2
LS
Trace and determinant
Let A =[
a bc d
]be a real 2× 2-matrix.
- The trace of A is tr(A) = a + d.- The determinant of A is det(A) = ad − bc.
Denote T = tr(A) and D = det(A). Thecharacteristic equation of A is
s2 − Ts + D = (s − λ1)(s − λ2) = 0. (∗)
Let λ1 and λ2 be the eigenvalues of A, thenT = λ1 + λ2 and D = λ1λ2.
The discriminant of (∗) is T 2 − 4D.T2 > 4D Equation (∗) has two real roots λ1 6= λ2.
T2 < 4D Equation (∗) has two non-real roots λ1 = λ2.
T2 = 4D Equation (∗) has one real roots λ1 = λ2.
![Page 7: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/7.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
3 1.2
LS
Trace and determinant
Let A =[
a bc d
]be a real 2× 2-matrix.
- The trace of A is tr(A) = a + d.- The determinant of A is det(A) = ad − bc.
Denote T = tr(A) and D = det(A). Thecharacteristic equation of A is
s2 − Ts + D = (s − λ1)(s − λ2) = 0. (∗)
Let λ1 and λ2 be the eigenvalues of A, thenT = λ1 + λ2 and D = λ1λ2.
The discriminant of (∗) is T 2 − 4D.T2 > 4D Equation (∗) has two real roots λ1 6= λ2.
T2 < 4D Equation (∗) has two non-real roots λ1 = λ2.
T2 = 4D Equation (∗) has one real roots λ1 = λ2.
![Page 8: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/8.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
3 1.2
LS
Trace and determinant
Let A =[
a bc d
]be a real 2× 2-matrix.
- The trace of A is tr(A) = a + d.- The determinant of A is det(A) = ad − bc.
Denote T = tr(A) and D = det(A). Thecharacteristic equation of A is
s2 − Ts + D = (s − λ1)(s − λ2) = 0. (∗)
Let λ1 and λ2 be the eigenvalues of A, thenT = λ1 + λ2 and D = λ1λ2.
The discriminant of (∗) is T 2 − 4D.T2 > 4D Equation (∗) has two real roots λ1 6= λ2.
T2 < 4D Equation (∗) has two non-real roots λ1 = λ2.
T2 = 4D Equation (∗) has one real roots λ1 = λ2.
![Page 9: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/9.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
4 1.3
LS
Second order LTI systems
TheoremSuppose an LTI system is described by a second-order lineardifferential equation. Assume that the state equations invector form are
v′ = Av + x(t)b.
y(t) = c · v + d x(t),
where A is a 2× 2-matrix.
If tr(A) < 0 and det(A) > 0, then the system is stable.
If tr(A) > 0 or det(A) < 0, then the system is unstable.
This theorem can be proved by using the results fromthe previous slide.
![Page 10: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/10.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
4 1.3
LS
Second order LTI systems
TheoremSuppose an LTI system is described by a second-order lineardifferential equation. Assume that the state equations invector form are
v′ = Av + x(t)b.
y(t) = c · v + d x(t),
where A is a 2× 2-matrix.
If tr(A) < 0 and det(A) > 0, then the system is stable.
If tr(A) > 0 or det(A) < 0, then the system is unstable.
This theorem can be proved by using the results fromthe previous slide.
![Page 11: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/11.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
5 1.4
LS
Example
Example Example 2.6.13
Investigate the stability of the LTI system whose statematrix is
A =[
2 −14 −3
].
T = tr(A) = 2 + (−3) = −1 < 0, so we can notconclude anything yet.D = det(A) = 2 · (−3)− 4 · (−1) = −2 < 0, hence thesystem is unstable.Alternatively: note that the discriminant isT 2 − 4D = 9 > 0, hence A has two distinct realeigenvalues.The eigenvalues are λ1 = −2 and λ2 = 1. SinceReλ2 = 1 > 0, the system is unstable.
![Page 12: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/12.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
5 1.4
LS
Example
Example Example 2.6.13
Investigate the stability of the LTI system whose statematrix is
A =[
2 −14 −3
].
T = tr(A) = 2 + (−3) = −1 < 0, so we can notconclude anything yet.
D = det(A) = 2 · (−3)− 4 · (−1) = −2 < 0, hence thesystem is unstable.Alternatively: note that the discriminant isT 2 − 4D = 9 > 0, hence A has two distinct realeigenvalues.The eigenvalues are λ1 = −2 and λ2 = 1. SinceReλ2 = 1 > 0, the system is unstable.
![Page 13: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/13.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
5 1.4
LS
Example
Example Example 2.6.13
Investigate the stability of the LTI system whose statematrix is
A =[
2 −14 −3
].
T = tr(A) = 2 + (−3) = −1 < 0, so we can notconclude anything yet.D = det(A) = 2 · (−3)− 4 · (−1) = −2 < 0, hence thesystem is unstable.
Alternatively: note that the discriminant isT 2 − 4D = 9 > 0, hence A has two distinct realeigenvalues.The eigenvalues are λ1 = −2 and λ2 = 1. SinceReλ2 = 1 > 0, the system is unstable.
![Page 14: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/14.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
5 1.4
LS
Example
Example Example 2.6.13
Investigate the stability of the LTI system whose statematrix is
A =[
2 −14 −3
].
T = tr(A) = 2 + (−3) = −1 < 0, so we can notconclude anything yet.D = det(A) = 2 · (−3)− 4 · (−1) = −2 < 0, hence thesystem is unstable.Alternatively: note that the discriminant isT 2 − 4D = 9 > 0, hence A has two distinct realeigenvalues.
The eigenvalues are λ1 = −2 and λ2 = 1. SinceReλ2 = 1 > 0, the system is unstable.
![Page 15: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/15.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
5 1.4
LS
Example
Example Example 2.6.13
Investigate the stability of the LTI system whose statematrix is
A =[
2 −14 −3
].
T = tr(A) = 2 + (−3) = −1 < 0, so we can notconclude anything yet.D = det(A) = 2 · (−3)− 4 · (−1) = −2 < 0, hence thesystem is unstable.Alternatively: note that the discriminant isT 2 − 4D = 9 > 0, hence A has two distinct realeigenvalues.The eigenvalues are λ1 = −2 and λ2 = 1. SinceReλ2 = 1 > 0, the system is unstable.
![Page 16: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/16.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
6 1.5
LS
Example
Example
Investigate the stability of the LTI system whose statematrix is
A =[−3 −1
5 1
].
T = tr(A) = 2 + (−3) = −2 < 0 andD = det(A) = (−3) · 1− 5 · (−1) = 2 > 0, hence thesystem is stable.Alternatively: note that the discriminant isT 2 − 4D = −4 < 0, hence A has two conjugatenon-real eigenvalues.The eigenvalues are λ1 = −1 + i and λ2 = −1− i.Since Reλ1 = Reλ2 = −1 < 0, the system is stable.
![Page 17: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/17.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
6 1.5
LS
Example
Example
Investigate the stability of the LTI system whose statematrix is
A =[−3 −1
5 1
].
T = tr(A) = 2 + (−3) = −2 < 0 andD = det(A) = (−3) · 1− 5 · (−1) = 2 > 0, hence thesystem is stable.
Alternatively: note that the discriminant isT 2 − 4D = −4 < 0, hence A has two conjugatenon-real eigenvalues.The eigenvalues are λ1 = −1 + i and λ2 = −1− i.Since Reλ1 = Reλ2 = −1 < 0, the system is stable.
![Page 18: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/18.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
6 1.5
LS
Example
Example
Investigate the stability of the LTI system whose statematrix is
A =[−3 −1
5 1
].
T = tr(A) = 2 + (−3) = −2 < 0 andD = det(A) = (−3) · 1− 5 · (−1) = 2 > 0, hence thesystem is stable.Alternatively: note that the discriminant isT 2 − 4D = −4 < 0, hence A has two conjugatenon-real eigenvalues.
The eigenvalues are λ1 = −1 + i and λ2 = −1− i.Since Reλ1 = Reλ2 = −1 < 0, the system is stable.
![Page 19: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/19.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
6 1.5
LS
Example
Example
Investigate the stability of the LTI system whose statematrix is
A =[−3 −1
5 1
].
T = tr(A) = 2 + (−3) = −2 < 0 andD = det(A) = (−3) · 1− 5 · (−1) = 2 > 0, hence thesystem is stable.Alternatively: note that the discriminant isT 2 − 4D = −4 < 0, hence A has two conjugatenon-real eigenvalues.The eigenvalues are λ1 = −1 + i and λ2 = −1− i.Since Reλ1 = Reλ2 = −1 < 0, the system is stable.
![Page 20: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/20.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
7 1.6
LS
Intermediate test demarcation line
This slide marks the end of the first part of the course.
The material required for the intermediate test consistsof everything up to this point.
![Page 21: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/21.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
8 2.1
LS
Inner products
DefinitionLet ϕ and ψ be two complex-valued functions defined on theinterval [a, b]. The inner product of ϕ and ψ is
〈ϕ,ψ〉 =∫ b
aϕ(t)ψ(t) dt.
The inner product of ϕ and ψ is a complex number.
Properties
(1) 〈ϕ,ψ〉 = 〈ψ,ϕ〉,
(2) 〈αϕ1 + βϕ2, ψ〉 = α〈ϕ1, ψ〉+ β〈ϕ2, ψ〉;〈ϕ, αψ1 + βψ2〉 = α〈ϕ,ψ1〉+ β〈ϕ,ψ2〉,
(3) 〈ϕ,ϕ〉 ∈ R and 〈ϕ,ϕ〉 ≥ 0,
(4) 〈ϕ,ϕ〉 = 0 if and only if ϕ = 0.
![Page 22: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/22.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
8 2.1
LS
Inner products
DefinitionLet ϕ and ψ be two complex-valued functions defined on theinterval [a, b]. The inner product of ϕ and ψ is
〈ϕ,ψ〉 =∫ b
aϕ(t)ψ(t) dt.
The inner product of ϕ and ψ is a complex number.
Properties
(1) 〈ϕ,ψ〉 = 〈ψ,ϕ〉,
(2) 〈αϕ1 + βϕ2, ψ〉 = α〈ϕ1, ψ〉+ β〈ϕ2, ψ〉;〈ϕ, αψ1 + βψ2〉 = α〈ϕ,ψ1〉+ β〈ϕ,ψ2〉,
(3) 〈ϕ,ϕ〉 ∈ R and 〈ϕ,ϕ〉 ≥ 0,
(4) 〈ϕ,ϕ〉 = 0 if and only if ϕ = 0.
![Page 23: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/23.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
9 2.2
LS
Norm and distance
DefinitionThe norm of a function ϕ : [a, b]→ C is defined as
‖ϕ‖ =√〈ϕ,ϕ〉.
The norm of a function is sometimes called the lengthof a function.In some books, the norm is denoted with single bars:
‖ϕ‖ = |ϕ|.
DefinitionLet ϕ,ψ : [a, b]→ C be two complex-valued functions. Thedistance beween ϕ and ψ is defined as
dist(ϕ,ψ) = ‖ϕ− ψ‖ .
![Page 24: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/24.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
9 2.2
LS
Norm and distance
DefinitionThe norm of a function ϕ : [a, b]→ C is defined as
‖ϕ‖ =√〈ϕ,ϕ〉.
The norm of a function is sometimes called the lengthof a function.
In some books, the norm is denoted with single bars:‖ϕ‖ = |ϕ|.
DefinitionLet ϕ,ψ : [a, b]→ C be two complex-valued functions. Thedistance beween ϕ and ψ is defined as
dist(ϕ,ψ) = ‖ϕ− ψ‖ .
![Page 25: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/25.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
9 2.2
LS
Norm and distance
DefinitionThe norm of a function ϕ : [a, b]→ C is defined as
‖ϕ‖ =√〈ϕ,ϕ〉.
The norm of a function is sometimes called the lengthof a function.In some books, the norm is denoted with single bars:
‖ϕ‖ = |ϕ|.
DefinitionLet ϕ,ψ : [a, b]→ C be two complex-valued functions. Thedistance beween ϕ and ψ is defined as
dist(ϕ,ψ) = ‖ϕ− ψ‖ .
![Page 26: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/26.jpg)
UNIVERSITYOF TWENTE.
State equationsand stability
Inner products
Fourier series
Fourier series ofreal signals
Linear Systems
LS.18-19[5]28-8-2018
9 2.2
LS
Norm and distance
DefinitionThe norm of a function ϕ : [a, b]→ C is defined as
‖ϕ‖ =√〈ϕ,ϕ〉.
The norm of a function is sometimes called the lengthof a function.In some books, the norm is denoted with single bars:
‖ϕ‖ = |ϕ|.
DefinitionLet ϕ,ψ : [a, b]→ C be two complex-valued functions. Thedistance beween ϕ and ψ is defined as
dist(ϕ,ψ) = ‖ϕ− ψ‖ .
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Orthogonal and orthonormal sets
Definition
Two functions ϕ,ψ : [a, b]→ C are said to beorthogonal if 〈ϕ,ψ〉 = 0.A set of non-zero functions {ϕ1, ϕ2, . . .} is anorthogonal set if 〈ϕi , ϕj〉 = 0 for all i 6= j.
Note that if i = j then 〈ϕi , ϕj〉 = 〈ϕi , ϕi〉 > 0.
DefinitionThe Kronecker delta function is defined as
δij ={
1 if i = j,0 if i 6= j.
A set of functions {ϕ1, ϕ2, . . .} is an orthonormal setif 〈ϕi , ϕj〉 = δij .
In an orthonormal set, every function has length 1.
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Orthogonal and orthonormal sets
Definition
Two functions ϕ,ψ : [a, b]→ C are said to beorthogonal if 〈ϕ,ψ〉 = 0.A set of non-zero functions {ϕ1, ϕ2, . . .} is anorthogonal set if 〈ϕi , ϕj〉 = 0 for all i 6= j.
Note that if i = j then 〈ϕi , ϕj〉 = 〈ϕi , ϕi〉 > 0.
DefinitionThe Kronecker delta function is defined as
δij ={
1 if i = j,0 if i 6= j.
A set of functions {ϕ1, ϕ2, . . .} is an orthonormal setif 〈ϕi , ϕj〉 = δij .
In an orthonormal set, every function has length 1.
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Periodic signals
LemmaLet x(t) be a periodic signal with period T > 0, then∫ T
0x(t) dt =
∫ a+T
ax(t) dt for all a ∈ R.
0 a T a+T
x(t)
NotationThe integral over the interval [a, a + T ] is denoted as∫
〈T〉x(t) dt.
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Periodic signals
LemmaLet x(t) be a periodic signal with period T > 0, then∫ T
0x(t) dt =
∫ a+T
ax(t) dt for all a ∈ R.
0 a T a+T
x(t)
NotationThe integral over the interval [a, a + T ] is denoted as∫
〈T〉x(t) dt.
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Proof of the lemma Self-tuition
We prove the case 0 < a < T .∫ a+T
ax(t) dt
=∫ T
ax(t) dt +
∫ a+T
Tx(t) dt
τ = t − T
=∫ T
ax(t) dt +
∫ a
0x(τ + T ) dτ
x(t) isperiodic
=∫ T
ax(t) dt +
∫ a
0x(t) dt
swapintegrals
=∫ a
0x(t) dt +
∫ T
ax(t) dt
=∫ T
0x(t) dt.
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LS
Time-harmonic signals Recap
DefinitionA time-harmonic signal is a continuous-time signalf : R→ C of the form f (t) = ceiωt with constants c ∈ Cand ω ∈ R.
The constant ω is called the (angular) frequency ofthe signal.
If c = Aeiϕ0 with A, ϕ0 ∈ R then f (t) = Aei(ωt+ϕ0).The constant A is called the amplitude of the signal,and ϕ0 is called the phase.
Let T = 2π/ |ω |, then f (t) = ceiωt is periodic withperiod T :
f (t) = f (t + T ) for all t ∈ R.
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Time-harmonic signals Recap
DefinitionA time-harmonic signal is a continuous-time signalf : R→ C of the form f (t) = ceiωt with constants c ∈ Cand ω ∈ R.
The constant ω is called the (angular) frequency ofthe signal.
If c = Aeiϕ0 with A, ϕ0 ∈ R then f (t) = Aei(ωt+ϕ0).The constant A is called the amplitude of the signal,and ϕ0 is called the phase.
Let T = 2π/ |ω |, then f (t) = ceiωt is periodic withperiod T :
f (t) = f (t + T ) for all t ∈ R.
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Time-harmonic signals Recap
DefinitionA time-harmonic signal is a continuous-time signalf : R→ C of the form f (t) = ceiωt with constants c ∈ Cand ω ∈ R.
The constant ω is called the (angular) frequency ofthe signal.
If c = Aeiϕ0 with A, ϕ0 ∈ R then f (t) = Aei(ωt+ϕ0).The constant A is called the amplitude of the signal,and ϕ0 is called the phase.
Let T = 2π/ |ω |, then f (t) = ceiωt is periodic withperiod T :
f (t) = f (t + T ) for all t ∈ R.
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LS
Time-harmonic signals
TheoremLet ω0 > 0. The set {einω0t |n ∈ Z} is an orthogonal setconsisting of periodic time-harmonic signals withperiod T = 2π/ω0.
Note that ϕn(t) = einω0t is periodic with periodpn = 2π/(|n |ω0). Observe that T = |n | pn , thereforeT is also a period of ϕn .
For the inner product we integrate over the domain[0,T ] for all signals ϕn .
〈ϕm , ϕn〉 =∫ T
0eimω0t einω0t dt
=∫ T
0eimω0te−inω0t dt
=∫ T
0ei(m−n)ω0t dt.
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Time-harmonic signals
TheoremLet ω0 > 0. The set {einω0t |n ∈ Z} is an orthogonal setconsisting of periodic time-harmonic signals withperiod T = 2π/ω0.
Note that ϕn(t) = einω0t is periodic with periodpn = 2π/(|n |ω0). Observe that T = |n | pn , thereforeT is also a period of ϕn .For the inner product we integrate over the domain[0,T ] for all signals ϕn .
〈ϕm , ϕn〉 =∫ T
0eimω0t einω0t dt
=∫ T
0eimω0te−inω0t dt
=∫ T
0ei(m−n)ω0t dt.
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Time-harmonic signals
TheoremLet ω0 > 0. The set {einω0t |n ∈ Z} is an orthogonal setconsisting of periodic time-harmonic signals withperiod T = 2π/ω0.
Note that ϕn(t) = einω0t is periodic with periodpn = 2π/(|n |ω0). Observe that T = |n | pn , thereforeT is also a period of ϕn .For the inner product we integrate over the domain[0,T ] for all signals ϕn .
〈ϕm , ϕn〉 =∫ T
0eimω0t einω0t dt
=∫ T
0eimω0te−inω0t dt
=∫ T
0ei(m−n)ω0t dt.
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Time-harmonic signals
If m 6= n then
〈ϕm , ϕn〉 =∫ T
0ei(m−n)ω0t dt
= 1(m − n)ω0i ei(m−n)ω0t
∣∣∣T0
= 1(m − n)ω0i
[ei(m−n)ω0T − 1
]= 1
(m − n)ω0i[e(m−n)2πi − 1
]= 0,
This shows that {eint |n ∈ Z} is an orthogonal set.
If m = n then
‖ϕm ‖2 = 〈ϕm , ϕm〉 =∫ T
0ei(m−m)t dt =
∫ T
01 dt = T .
Corollary
The set{
1√T einω0t
∣∣∣ n ∈ Z}is an orthonormal set.
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Time-harmonic signals
If m 6= n then
〈ϕm , ϕn〉 =∫ T
0ei(m−n)ω0t dt
= 1(m − n)ω0i ei(m−n)ω0t
∣∣∣T0
= 1(m − n)ω0i
[ei(m−n)ω0T − 1
]= 1
(m − n)ω0i[e(m−n)2πi − 1
]= 0,
This shows that {eint |n ∈ Z} is an orthogonal set.If m = n then
‖ϕm ‖2 = 〈ϕm , ϕm〉 =∫ T
0ei(m−m)t dt =
∫ T
01 dt = T .
Corollary
The set{
1√T einω0t
∣∣∣ n ∈ Z}is an orthonormal set.
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Time-harmonic signals
If m 6= n then
〈ϕm , ϕn〉 =∫ T
0ei(m−n)ω0t dt
= 1(m − n)ω0i ei(m−n)ω0t
∣∣∣T0
= 1(m − n)ω0i
[ei(m−n)ω0T − 1
]= 1
(m − n)ω0i[e(m−n)2πi − 1
]= 0,
This shows that {eint |n ∈ Z} is an orthogonal set.If m = n then
‖ϕm ‖2 = 〈ϕm , ϕm〉 =∫ T
0ei(m−m)t dt =
∫ T
01 dt = T .
Corollary
The set{
1√T einω0t
∣∣∣ n ∈ Z}is an orthonormal set.
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LS
Decomposition of periodic signals
Holy grail
Let x(t) be a periodic signal with period T > 0. Withω0 = 2π/T , find coefficients cn ∈ C (with n ∈ Z) such that
x(t) =∞∑
n=−∞cn einω0t .
Questions
For which t ∈ R does the series converge?
For which t ∈ R does the equation hold?
Are the coefficients cn unique?
How can you find the coefficients cn?
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Fourier coefficients
Theorem
Let x(t) be a periodic signal with period T > 0, and letω0 = 2π/T . Suppose that
x(t) =∞∑
n=−∞cn einω0t ,
then for all n ∈ Z:
cn = 1T
∫〈T〉
x(t) e−inω0t dt Eq. 3.3.4
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Proof of the theorem
Let ϕn(t) = einω0t , then
〈x, ϕn〉 =∫〈T〉
x(t) einω0t dt
=∫〈T〉
x(t) e−inω0t dt. (1)
Now use that {ϕn |n ∈ Z} is an orthogonal set:
〈x, ϕn〉 =⟨ ∞∑
k=−∞ck eikω0t , ϕn
⟩
=⟨ ∞∑
k=−∞ck ϕk , ϕn
⟩=
∞∑k=−∞
ck 〈ϕk , ϕn〉
= · · ·+ 0 + cn 〈ϕn , ϕn〉+ 0 + · · ·= cnT . (2)
Combining (1) and (2) proves the theorem.
![Page 44: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/44.jpg)
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Proof of the theorem
Let ϕn(t) = einω0t , then
〈x, ϕn〉 =∫〈T〉
x(t) einω0t dt
=∫〈T〉
x(t) e−inω0t dt. (1)
Now use that {ϕn |n ∈ Z} is an orthogonal set:
〈x, ϕn〉 =⟨ ∞∑
k=−∞ck eikω0t , ϕn
⟩
=⟨ ∞∑
k=−∞ck ϕk , ϕn
⟩=
∞∑k=−∞
ck 〈ϕk , ϕn〉
= · · ·+ 0 + cn 〈ϕn , ϕn〉+ 0 + · · ·= cnT . (2)
Combining (1) and (2) proves the theorem.
![Page 45: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/45.jpg)
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Proof of the theorem
Let ϕn(t) = einω0t , then
〈x, ϕn〉 =∫〈T〉
x(t) einω0t dt
=∫〈T〉
x(t) e−inω0t dt. (1)
Now use that {ϕn |n ∈ Z} is an orthogonal set:
〈x, ϕn〉 =⟨ ∞∑
k=−∞ck eikω0t , ϕn
⟩
=⟨ ∞∑
k=−∞ck ϕk , ϕn
⟩=
∞∑k=−∞
ck 〈ϕk , ϕn〉
= · · ·+ 0 + cn 〈ϕn , ϕn〉+ 0 + · · ·= cnT . (2)
Combining (1) and (2) proves the theorem.
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Fourier coefficients
DefinitionLet x(t) be a piecewise continuous periodic signal withperiod T , and let ω0 = 2π/T . The Fourier coefficientsof x(t) are defined as
cn = 1T
∫〈T〉
x(t) e−inω0t dt
If cn are the Fourier coefficients of x(t), then this isdenoted as x(t)↔ cn .If x(t) is not continuous at t0, the value of x at t0 doesnot affect the Fourier coefficients.
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Fourier coefficients
Example Example 3.3.1
Find the Fourier coefficients ofthe periodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0
with period 2.
t
K
−Kx(t)
0 1−1
Note that T = 2 and ω0 = π.
cn = 1T
∫ T/2
−T/2x(t) e−inω0t dt
= 12
[∫ 0
−1−K e−inπt dt +
∫ 1
0K e−inπt dt
]= 1
2K[−∫ 0
−1e−inπt dt +
∫ 1
0e−inπt dt
].
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Fourier coefficients
Example Example 3.3.1
Find the Fourier coefficients ofthe periodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0
with period 2.
t
K
−Kx(t)
0 1−1
Note that T = 2 and ω0 = π.
cn = 1T
∫ T/2
−T/2x(t) e−inω0t dt
= 12
[∫ 0
−1−K e−inπt dt +
∫ 1
0K e−inπt dt
]= 1
2K[−∫ 0
−1e−inπt dt +
∫ 1
0e−inπt dt
].
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Fourier coefficients
Example Example 3.3.1
Find the Fourier coefficients ofthe periodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0
with period 2.
t
K
−Kx(t)
0 1−1
Note that T = 2 and ω0 = π.
cn = 1T
∫ T/2
−T/2x(t) e−inω0t dt
= 12
[∫ 0
−1−K e−inπt dt +
∫ 1
0K e−inπt dt
]= 1
2K[−∫ 0
−1e−inπt dt +
∫ 1
0e−inπt dt
].
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LS
Example (continued)
If n 6= 0:
cn = 12K
[−∫ 0
−1e−inπt dt +
∫ 1
0e−inπt dt
]= 1
2K[ 1
inπ e−inπt∣∣∣0−1− 1
inπ e−inπt∣∣∣10
]= K
2inπ[(
1− einπ)− (e−inπ − 1)]
= K2inπ
(2− 2(−1)n
)=
2Kinπ if n is odd,
0 if n is even.
If n = 0:c0 = −1
2K∫ 0
−1e−i 0πt dt + 1
2K∫ 1
0e−i 0πt dt
= −12K
∫ 0
−11 dt + 1
2K∫ 1
01 dt
= −12K · 1 + 1
2K · 1 = 0.
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Example (continued)
If n 6= 0:
cn = 12K
[−∫ 0
−1e−inπt dt +
∫ 1
0e−inπt dt
]= 1
2K[ 1
inπ e−inπt∣∣∣0−1− 1
inπ e−inπt∣∣∣10
]= K
2inπ[(
1− einπ)− (e−inπ − 1)]
= K2inπ
(2− 2(−1)n
)=
2Kinπ if n is odd,
0 if n is even.
If n = 0:c0 = −1
2K∫ 0
−1e−i 0πt dt + 1
2K∫ 1
0e−i 0πt dt
= −12K
∫ 0
−11 dt + 1
2K∫ 1
01 dt
= −12K · 1 + 1
2K · 1 = 0.
![Page 52: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/52.jpg)
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LS
Piecewise continuity and smoothness
Definition
A function f is called piecewise continuous on [a, b]if f is continuous in every point of [a, b], except possiblyin a finite number points t1, t2, . . . , tn ∈ [a, b].Moreover, f (a+), f (b−) and f (t+
i ), f (t−i ) should existfor i = 1, 2, . . . ,n.A function f is called piecewise continuous on R if fis piecewise continuous on every interval [a, b].
If f is continuous in t then f (t−) = f (t+) = f (t).
Definition
A function f is called piecewise smooth on [a, b]if f ′ is piecewise continuous on [a, b].A function f is called piecewise smooth on R if f ispiecewise smooth on every interval [a, b].
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Piecewise continuity and smoothness
Definition
A function f is called piecewise continuous on [a, b]if f is continuous in every point of [a, b], except possiblyin a finite number points t1, t2, . . . , tn ∈ [a, b].Moreover, f (a+), f (b−) and f (t+
i ), f (t−i ) should existfor i = 1, 2, . . . ,n.A function f is called piecewise continuous on R if fis piecewise continuous on every interval [a, b].
If f is continuous in t then f (t−) = f (t+) = f (t).
Definition
A function f is called piecewise smooth on [a, b]if f ′ is piecewise continuous on [a, b].A function f is called piecewise smooth on R if f ispiecewise smooth on every interval [a, b].
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LS
The Dirichlet conditions Sec. 3.4
TheoremLet f be piecewise smooth, then for every a < b1. the function f is absolutely integrable over [a, b], i.e.∫ b
a|f (t)| dt <∞.
2. the function f has a finite number of extrema in [a, b],3. the function f has a finite number of discontinuities in
[a, b], none of the discontinuities are infinite,4. the function f is bounded.
The above four conditions are called the Dirichletconditions.
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LS
The Fundamental Theorem of Fourier Series
TheoremLet x(t) be a periodic signal with Fourier coefficients cn ,that satisfies the Dirichlet conditions. Then for every t ∈ Rwe have
∞∑n=−∞
cn einω0t = x(t+) + x(t−)2 .
The series∞∑
n=−∞cn einω0t is called the Fourier series
of x(t). It converges for every t ∈ R.
The expression x(t+) + x(t−)2 is denoted as x(t).
If x is continuous in t, then x(t) = x(t).
![Page 56: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/56.jpg)
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The Fundamental Theorem of Fourier Series
TheoremLet x(t) be a periodic signal with Fourier coefficients cn ,that satisfies the Dirichlet conditions. Then for every t ∈ Rwe have
∞∑n=−∞
cn einω0t = x(t+) + x(t−)2 .
The series∞∑
n=−∞cn einω0t is called the Fourier series
of x(t). It converges for every t ∈ R.
The expression x(t+) + x(t−)2 is denoted as x(t).
If x is continuous in t, then x(t) = x(t).
![Page 57: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/57.jpg)
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The Fundamental Theorem of Fourier Series
TheoremLet x(t) be a periodic signal with Fourier coefficients cn ,that satisfies the Dirichlet conditions. Then for every t ∈ Rwe have
∞∑n=−∞
cn einω0t = x(t+) + x(t−)2 .
The series∞∑
n=−∞cn einω0t is called the Fourier series
of x(t). It converges for every t ∈ R.
The expression x(t+) + x(t−)2 is denoted as x(t).
If x is continuous in t, then x(t) = x(t).
![Page 58: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/58.jpg)
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LS
The Fundamental Theorem of Fourier Series
TheoremLet x(t) be a periodic signal with Fourier coefficients cn ,that satisfies the Dirichlet conditions. Then for every t ∈ Rwe have
∞∑n=−∞
cn einω0t = x(t+) + x(t−)2 .
The series∞∑
n=−∞cn einω0t is called the Fourier series
of x(t). It converges for every t ∈ R.
The expression x(t+) + x(t−)2 is denoted as x(t).
If x is continuous in t, then x(t) = x(t).
![Page 59: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/59.jpg)
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The Fundamental Theorem of Fourier Series
Example Example 3.3.1
Find the Fourier series of theperiodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
The Fourier coefficients are cn =
2Kinπ if n is odd,
0 if n is even.The Fourier series of x(t) is:
2Kiπ
∞∑n=−∞
n odd
1n einπt = x(t) = x(t+)+x(t−)
2 =
−K −1<t<0,K 0<t<1,0 n∈Z.
1−1
![Page 60: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/60.jpg)
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The Fundamental Theorem of Fourier Series
Example Example 3.3.1
Find the Fourier series of theperiodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
The Fourier coefficients are cn =
2Kinπ if n is odd,
0 if n is even.
The Fourier series of x(t) is:
2Kiπ
∞∑n=−∞
n odd
1n einπt = x(t) = x(t+)+x(t−)
2 =
−K −1<t<0,K 0<t<1,0 n∈Z.
1−1
![Page 61: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/61.jpg)
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The Fundamental Theorem of Fourier Series
Example Example 3.3.1
Find the Fourier series of theperiodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
The Fourier coefficients are cn =
2Kinπ if n is odd,
0 if n is even.The Fourier series of x(t) is:
2Kiπ
∞∑n=−∞
n odd
1n einπt = x(t) = x(t+)+x(t−)
2 =
−K −1<t<0,K 0<t<1,0 n∈Z.
1−1
![Page 62: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/62.jpg)
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Spectra
DefinitionLet cn be the Fourier coefficients of x(t).
The sequence |cn | is called the amplitude spectrum ormagnitude spectrum of x(t).The sequence Arg cn is called the phase spectrumof x(t).Amplitude spectrum and phase spectrum are linespectra of x(t).
If cn = 0 then the argument of cn is defined as 0.
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Spectra
DefinitionLet cn be the Fourier coefficients of x(t).
The sequence |cn | is called the amplitude spectrum ormagnitude spectrum of x(t).The sequence Arg cn is called the phase spectrumof x(t).Amplitude spectrum and phase spectrum are linespectra of x(t).
If cn = 0 then the argument of cn is defined as 0.
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Spectra
Example Example 3.3.1
Find the line spectra of the pe-riodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
n cn |cn | Arg cn
odd 2Kinπ
2K|n |π sgn(n)π/2
even 0 0 0
n
|cn |1 2 3 4−4 −3 −2 −1
2K|x |π
n
Arg cn
π2
−π2
1 2 3 4
−4 −3 −2 −1
![Page 65: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/65.jpg)
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Spectra
Example Example 3.3.1
Find the line spectra of the pe-riodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
n cn |cn | Arg cn
odd 2Kinπ
2K|n |π sgn(n)π/2
even 0 0 0
n
|cn |1 2 3 4−4 −3 −2 −1
2K|x |π
n
Arg cn
π2
−π2
1 2 3 4
−4 −3 −2 −1
![Page 66: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/66.jpg)
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Spectra
Example Example 3.3.1
Find the line spectra of the pe-riodic signal
x(t) ={
K if 0 < t < 1,−K if −1 < t < 0.
t
K
−Kx(t)
0 1−1
n cn |cn | Arg cn
odd 2Kinπ
2K|n |π sgn(n)π/2
even 0 0 0
n
|cn |1 2 3 4−4 −3 −2 −1
2K|x |π
n
Arg cn
π2
−π2
1 2 3 4
−4 −3 −2 −1
![Page 67: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/67.jpg)
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LS
Real signals
TheoremIf x(t)↔ cn , then x(t)↔ c−n .
Corollary
The signal x(t) is real if and only if cn = c−n .
cn
c−nC
We may assume that if x(t) is real, then x(t) is real:
Corollary (as applied in practice)
The signal x(t) is real if and only if cn = c−n .
![Page 68: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/68.jpg)
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Real signals
TheoremIf x(t)↔ cn , then x(t)↔ c−n .
Corollary
The signal x(t) is real if and only if cn = c−n .
cn
c−nC
We may assume that if x(t) is real, then x(t) is real:
Corollary (as applied in practice)
The signal x(t) is real if and only if cn = c−n .
![Page 69: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/69.jpg)
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Real signals
TheoremIf x(t)↔ cn , then x(t)↔ c−n .
Corollary
The signal x(t) is real if and only if cn = c−n .
cn
c−nC
We may assume that if x(t) is real, then x(t) is real:
Corollary (as applied in practice)
The signal x(t) is real if and only if cn = c−n .
![Page 70: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/70.jpg)
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Real signals
TheoremIf x(t)↔ cn , then x(t)↔ c−n .
Corollary
The signal x(t) is real if and only if cn = c−n .
cn
c−nC
We may assume that if x(t) is real, then x(t) is real:
Corollary (as applied in practice)
The signal x(t) is real if and only if cn = c−n .
![Page 71: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/71.jpg)
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LS
Proofs Self tuition
Proof of the theorem:Let x(t) be periodic with period T , and let ω0 = 2π/T .The Fourier coefficient of x(t) is1T
∫〈T〉
x(t) e−inω0t dt = 1T
∫〈T〉
x(t) e−i(−n)ω0t dt = c−n .
Proof of the corollary:⇒ This follows directly from the theorem.⇐ If cn = c−n , then
12[x(t+) + x(t−)
]=
∞∑n=−∞
cn einω0t
=∞∑
n=−∞cn e−inω0t =
−∞∑−n=∞
c−n ei(−n)ω0t
=−∞∑
m=∞cm eimω0t = 1
2[x(t+) + x(t−)
].
![Page 72: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/72.jpg)
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LS
The trigonometric Fourier series
Rewrite the Fourier series with sines and cosines:∞∑
n=−∞cn einω0t = c0 +
∞∑n=1
cn einω0t +−1∑
n=−∞cn einω0t
= c0 +∞∑
n=1cn einω0t +
∞∑n=1
c−n einω0t
= c0 +∞∑
n=1cn(cos nω0t + i sin nω0t)
+ c−n(cos nω0t − i sin nω0t)
= c0 +∞∑
n=1(cn + c−n) cos nω0t + i(cn − c−n) sin nω0t.
Define
an ={
c0 if n = 0,cn + c−n if n ≥ 1, , bn = i(cn − c−n), n ≥ 1,
then∞∑
n=−∞cn einω0t =
∞∑n=0
an cos nω0t +∞∑
n=1bn sin nω0t.
![Page 73: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/73.jpg)
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The trigonometric Fourier series
Rewrite the Fourier series with sines and cosines:∞∑
n=−∞cn einω0t = c0 +
∞∑n=1
cn einω0t +−1∑
n=−∞cn einω0t
= c0 +∞∑
n=1cn einω0t +
∞∑n=1
c−n einω0t
= c0 +∞∑
n=1cn(cos nω0t + i sin nω0t)
+ c−n(cos nω0t − i sin nω0t)
= c0 +∞∑
n=1(cn + c−n) cos nω0t + i(cn − c−n) sin nω0t.
Define
an ={
c0 if n = 0,cn + c−n if n ≥ 1, , bn = i(cn − c−n), n ≥ 1,
then∞∑
n=−∞cn einω0t =
∞∑n=0
an cos nω0t +∞∑
n=1bn sin nω0t.
![Page 74: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/74.jpg)
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Trigonometric Fourier coefficients
From cn to an and bn :
an ={
c0 if n = 0,cn + c−n if n ≥ 1
bn = i(cn − c−n), n ≥ 1
From an and bn to cn :
cn =
an − i bn
2 if n ≥ 1,a0 if n = 0,a−n + i b−n
2 if n < 0
an and bn are real if and only if cn = c−n .
![Page 75: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/75.jpg)
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Trigonometric Fourier coefficients
From cn to an and bn :
an ={
c0 if n = 0,cn + c−n if n ≥ 1
bn = i(cn − c−n), n ≥ 1
From an and bn to cn :
cn =
an − i bn
2 if n ≥ 1,a0 if n = 0,a−n + i b−n
2 if n < 0
an and bn are real if and only if cn = c−n .
![Page 76: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/76.jpg)
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Trigonometric Fourier coefficients
From cn to an and bn :
an ={
c0 if n = 0,cn + c−n if n ≥ 1
bn = i(cn − c−n), n ≥ 1
From an and bn to cn :
cn =
an − i bn
2 if n ≥ 1,a0 if n = 0,a−n + i b−n
2 if n < 0
an and bn are real if and only if cn = c−n .
![Page 77: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/77.jpg)
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Trigonometric Fourier coefficients of real signals
TheoremLet x(t) be a periodic signal.
The signal x(t) is real if and only if all trigonometricFourier coefficients an and bn are real.If x(t) is real then
an ={
c0 if n = 0,2 Re cn if n ≥ 1
andbn = −2 Im cn , n ≥ 1.
If x(t) is real then its Fourier series
a0 +∞∑
n=1
(an cos nω0t + bn sin nω0t
),
is called the real Fourier series of x(t).
![Page 78: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/78.jpg)
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Trigonometric Fourier coefficients
The Fourier coefficients an and bn can be found directly byintegration:
an =
1T
∫〈T〉
x(t) dt, n = 0,
2T
∫〈T〉
x(t) cos(2nπt
T
)dt, n ≥ 1
eq. 3.3.9a
eq. 3.3.9b
bn = 2T
∫〈T〉
x(t) sin(2nπt
T
)dt, n ≥ 1 eq. 3.3.9c
![Page 79: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/79.jpg)
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Example: the block wave
DefinitionLet T > 0 and 0 ≤ a ≤ T . The even block wave withpulse width a is the periodic signal ba,T with period Tdefined by
ba,T (t) =
1 if |t| ≤ a/2,
0 if a/2 < t ≤ T/2.
t−3T/2 −T −T/2 T/2 T 3T/2−a/2 a/2
1
0
ba,T
a
T
![Page 80: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/80.jpg)
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The block wave
Compute the trigonometric Fourier coefficients an .If n = 0 then
a0 = 1T
∫〈T〉
ba,T (t) dt = 1T
∫ a/2
−a/21 dt = a
T .
If n ≥ 1 then
an = 2T
∫〈T〉
ba,T (t) cos(nω0t) dt
= 2T
∫ a/2
−a/2cos
(2nπtT
)dt
= 2T ·
T2nπ
[sin(2nπt
T
) ]a/2
−a/2
= 1nπ
[sin(nπa
T
)− sin
(−nπa
T
)]= 2
nπ sin(nπa
T
)
![Page 81: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/81.jpg)
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The block wave
Compute the trigonometric Fourier coefficients an .If n = 0 then
a0 = 1T
∫〈T〉
ba,T (t) dt = 1T
∫ a/2
−a/21 dt = a
T .
If n ≥ 1 then
an = 2T
∫〈T〉
ba,T (t) cos(nω0t) dt
= 2T
∫ a/2
−a/2cos
(2nπtT
)dt
= 2T ·
T2nπ
[sin(2nπt
T
) ]a/2
−a/2
= 1nπ
[sin(nπa
T
)− sin
(−nπa
T
)]= 2
nπ sin(nπa
T
)
![Page 82: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/82.jpg)
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The block wave
Compute the trigonometric Fourier coefficients bn .
For n ≥ 1 we have
bn = 2T
∫〈T〉
ba,T (t) sin(nω0t) dt
= 2T
∫ a/2
−a/2sin(2nπt
T
)dt
= − 2T ·
T2nπ
[cos
(2nπtT
) ]a/2
−a/2
= − 1nπ
[cos
(nπaT
)− cos
(−nπa
T
)]= 0.
![Page 83: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/83.jpg)
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The block wave
For all n ∈ Z we have
cn =
an − i bn
2 if n > 0,a0 if n = 0,a−n + i b−n
2 if n < 0
=
a0 if n = 0,
12a|n | if n 6= 0
=
aT if n = 0,
1nπ sin
(nπaT
)if n 6= 0.
![Page 84: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/84.jpg)
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The block wave
Note that if n 6= 0 then
cn = 1nπ sin
(nπaT
)= 1
nπ ·nπaT Sa
(nπaT
)= a
T sinc(n a
T
).
Also, for n = 0
c0 = aT = a
T sinc(
0 aT
),
by definition of sinc(0).
Conclusion:
cn = aT sinc
(n a
T
)for all n ∈ Z
![Page 85: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/85.jpg)
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The block wave
Note that if n 6= 0 then
cn = 1nπ sin
(nπaT
)= 1
nπ ·nπaT Sa
(nπaT
)= a
T sinc(n a
T
).
Also, for n = 0
c0 = aT = a
T sinc(
0 aT
),
by definition of sinc(0).
Conclusion:
cn = aT sinc
(n a
T
)for all n ∈ Z
![Page 86: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/86.jpg)
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The block wave
Note that if n 6= 0 then
cn = 1nπ sin
(nπaT
)= 1
nπ ·nπaT Sa
(nπaT
)= a
T sinc(n a
T
).
Also, for n = 0
c0 = aT = a
T sinc(
0 aT
),
by definition of sinc(0).
Conclusion:
cn = aT sinc
(n a
T
)for all n ∈ Z
![Page 87: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/87.jpg)
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The block wave
Definition
Define the duty cycle of the block wave ba,T as ρ = aT .
TheoremFor the Fourier coefficients of the block wave ba,T we have
cn = ρ sinc(nρ)
−3ρ
−2ρ
−ρ 0 ρ
2ρ
3ρ
−7 −6 −5 −4 −3 −2 1 2 3 4 5 6 7
ρ
ρ sinc
![Page 88: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/88.jpg)
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The block wave
Definition
Define the duty cycle of the block wave ba,T as ρ = aT .
TheoremFor the Fourier coefficients of the block wave ba,T we have
cn = ρ sinc(nρ)
−3ρ
−2ρ
−ρ 0 ρ
2ρ
3ρ
−7 −6 −5 −4 −3 −2 1 2 3 4 5 6 7
ρ
ρ sinc
![Page 89: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/89.jpg)
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The block wave
Definition
Define the duty cycle of the block wave ba,T as ρ = aT .
TheoremFor the Fourier coefficients of the block wave ba,T we have
cn = ρ sinc(nρ)
−3ρ
−2ρ
−ρ 0 ρ
2ρ
3ρ
−7 −6 −5 −4 −3 −2 1 2 3 4 5 6 7
ρ
ρ sinc
![Page 90: pdfs.semanticscholar.org · UNIVERSITY OFTWENTE. State equations and stability Inner products Fourier series Fourier series of real signals Linear Systems LS.18-19[5] 28-8-2018 132.6](https://reader035.fdocuments.in/reader035/viewer/2022062610/6111d4c60681646275387aa3/html5/thumbnails/90.jpg)
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The block wave
Definition
Define the duty cycle of the block wave ba,T as ρ = aT .
TheoremFor the Fourier coefficients of the block wave ba,T we have
cn = ρ sinc(nρ)
−3ρ
−2ρ
−ρ 0 ρ
2ρ
3ρ
−7 −6 −5 −4 −3 −2 1 2 3 4 5 6 7
ρ
ρ sinc