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**Content Periodic Functions Fourier Series**

Complex Form of the Fourier Series Impulse Train Analysis of Periodic Waveforms Half-Range Expansion Least Mean-Square Error Approximation

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Fourier Series Periodic Functions

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**The Mathematic Formulation**

Any function that satisfies where T is a constant and is called the period of the function.

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Example: Find its period. Fact: smallest T

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Example: Find its period. must be a rational number

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Example: Is this function a periodic one? not a rational number

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Fourier Series Fourier Series

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Introduction Decompose a periodic input signal into primitive periodic components. A periodic sequence T 2T 3T t f(t)

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**T is a period of all the above signals**

Synthesis T is a period of all the above signals Even Part Odd Part DC Part Let 0=2/T.

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Orthogonal Functions Call a set of functions {k} orthogonal on an interval a < t < b if it satisfies

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**Orthogonal set of Sinusoidal Functions**

Define 0=2/T. We now prove this one

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Proof m n

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Proof m = n

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**Orthogonal set of Sinusoidal Functions**

Define 0=2/T. an orthogonal set.

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Decomposition

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Proof Use the following facts:

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Example (Square Wave) 2 3 4 5 - -2 -3 -4 -5 -6 f(t) 1

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Example (Square Wave) 2 3 4 5 - -2 -3 -4 -5 -6 f(t) 1

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Example (Square Wave) 2 3 4 5 - -2 -3 -4 -5 -6 f(t) 1

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**T is a period of all the above signals**

Harmonics T is a period of all the above signals Even Part Odd Part DC Part

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**Harmonics Define , called the fundamental angular frequency.**

Define , called the n-th harmonic of the periodic function.

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Harmonics

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**Amplitudes and Phase Angles**

harmonic amplitude phase angle

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**Complex Form of the Fourier Series**

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Complex Exponentials

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**Complex Form of the Fourier Series**

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**Complex Form of the Fourier Series**

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**Complex Form of the Fourier Series**

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**Complex Form of the Fourier Series**

If f(t) is real,

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**Complex Frequency Spectra**

|cn| amplitude spectrum n phase spectrum

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Example t f(t) A

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**Example A/5 40 80 120 -40 -120 -80 50 100 150 -50 -100**

-120 -80 A/5 50 100 150 -50 -100 -150

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**Example A/10 40 80 120 -40 -120 -80 100 200 300 -100 -200**

-120 -80 A/10 100 200 300 -100 -200 -300

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Example t f(t) A

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Fourier Series Impulse Train

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Dirac Delta Function and t Also called unit impulse function.

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Property (t): Test Function

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Impulse Train t T 2T 3T T 2T 3T

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**Fourier Series of the Impulse Train**

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**Complex Form Fourier Series of the Impulse Train**

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**Analysis of Periodic Waveforms**

Fourier Series Analysis of Periodic Waveforms

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Waveform Symmetry Even Functions Odd Functions

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Decomposition Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t). Even Part Odd Part

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Example Even Part Odd Part

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Half-Wave Symmetry and T T/2 T/2

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**Quarter-Wave Symmetry**

Even Quarter-Wave Symmetry T T/2 T/2 Odd Quarter-Wave Symmetry T T/2 T/2

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**Hidden Symmetry The following is a asymmetry periodic function:**

Adding a constant to get symmetry property. A/2 A/2 T T

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**Fourier Coefficients of Symmetrical Waveforms**

The use of symmetry properties simplifies the calculation of Fourier coefficients. Even Functions Odd Functions Half-Wave Even Quarter-Wave Odd Quarter-Wave Hidden

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**Fourier Coefficients of Even Functions**

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**Fourier Coefficients of Even Functions**

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**Fourier Coefficients for Half-Wave Symmetry**

and T T/2 T/2 The Fourier series contains only odd harmonics.

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**Fourier Coefficients for Half-Wave Symmetry**

and

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**Fourier Coefficients for Even Quarter-Wave Symmetry**

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**Fourier Coefficients for Odd Quarter-Wave Symmetry**

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Example Even Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

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Example Even Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

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Example Odd Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

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Example Odd Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

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**Half-Range Expansions**

Fourier Series Half-Range Expansions

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**Non-Periodic Function Representation**

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Without Considering Symmetry**

T A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Expansion Into Even Symmetry**

T=2 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Expansion Into Odd Symmetry**

T=2 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Expansion Into Half-Wave Symmetry**

T=2 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Expansion Into Even Quarter-Wave Symmetry**

T/2=2 T=4 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Expansion Into Odd Quarter-Wave Symmetry**

T/2=2 T=4 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

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**Least Mean-Square Error Approximation**

Fourier Series Least Mean-Square Error Approximation

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**Approximation a function**

Use to represent f(t) on interval T/2 < t < T/2. Define Mean-Square Error

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**Approximation a function**

Show that using Sk(t) to represent f(t) has least mean-square property. Proven by setting Ek/ai = 0 and Ek/bi = 0.

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**Approximation a function**

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Mean-Square Error

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Mean-Square Error

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Mean-Square Error

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