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3: Complex Fourier Series

3: Complex Fourier Series•Euler's Equation

•Complex Fourier Series •Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 1 / 12

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2 sinθ=eiθ-e-iθ 2i

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2=1

2eiθ+1

2e-iθ

sinθ=eiθ-e-iθ 2i=-1

2ieiθ+1

2ie-iθ

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2=1

2eiθ+1

2e-iθ

sinθ=eiθ-e-iθ 2i=-1

2ieiθ+1

2ie-iθ

Most maths becomes simpler if you useeiθinstead ofcosθandsinθ

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2=1

2eiθ+1

2e-iθ

sinθ=eiθ-e-iθ 2i=-1

2ieiθ+1

2ie-iθ

Most maths becomes simpler if you useeiθinstead ofcosθandsinθExamples where usingeiθmakes things simpler:

Usingeiθ

Usingcosθandsinθ

ei(θ+φ)=eiθeiφ

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2=1

2eiθ+1

2e-iθ

sinθ=eiθ-e-iθ 2i=-1

2ieiθ+1

2ie-iθ

Most maths becomes simpler if you useeiθinstead ofcosθandsinθExamples where usingeiθmakes things simpler:

Usingeiθ

Usingcosθandsinθ

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Odd

Harmonics Only

•Symmetry Examples •Summary E1.10 Fourier Series and Transforms (2014-5543)Complex Fourier Series: 3 - 2 / 12

Euler's Equation:

eiθ= cosθ+isinθ [see RHB 3.3]

Hence:

cosθ=eiθ+e-iθ 2=1

2eiθ+1

2e-iθ

sinθ=eiθ-e-iθ 2i=-1

2ieiθ+1

2ie-iθ

Most maths becomes simpler if you useeiθinstead ofcosθandsinθExamples where usingeiθmakes things simpler:

Usingeiθ

Usingcosθandsinθ

ddθeiθ=ieiθ

Euler's Equation

3: Complex Fourier Series•Euler's Equation•Complex Fourier Series

•Averaging Complex

Exponentials

•Complex Fourier Analysis •Fourier Series↔

Complex Fourier Series

•Complex Fourier Analysis

Example

•Time Shifting •Even/Odd Symmetry •Antiperiodic?Oddquotesdbs_dbs17.pdfusesText_23