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Fourier Series 主講者：虞台文

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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

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Introduction Decompose a periodic input signal into primitive periodic components. A periodic sequence T2T3T t f(t)f(t)

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Synthesis DC Part Even Part Odd Part T is a period of all the above signals 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 0 m n 0

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Proof 0 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)f(t) 1

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

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

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

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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 amplitudephase angle

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Fourier Series 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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If f(t) is real,

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Complex Frequency Spectra |cn||cn| amplitude spectrum nn phase spectrum

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Example t f(t)f(t) A

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40 80 120 -40 -80 A/5 5050 10 0 15 0 -5 0

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Example 40 80 120 -40 -80 A/10 10 0 20 0 30 0

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Example t f(t)f(t) A 0

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Fourier Series Impulse Train

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Dirac Delta Function and 0 t Also called unit impulse function.

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Property (t): Test Function

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Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T

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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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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 f e (t) and an odd function f o (t). Even Part Odd Part

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Example Even Part Odd Part

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Half-Wave Symmetry and TT/2 T/2

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Quarter-Wave Symmetry Even Quarter-Wave Symmetry TT/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 T TT A/2 A/2 T TT

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

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Fourier Coefficients for Odd Quarter-Wave Symmetry T T/2 T/2

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Example Even Quarter-Wave Symmetry T T/2 T/2 1 11 TT T/4 T/4

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Example Even Quarter-Wave Symmetry T T/2 T/2 1 11 TT T/4 T/4

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Example T T/2 T/2 1 11 TT T/4 T/4 Odd Quarter-Wave Symmetry

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Example T T/2 T/2 1 11 TT T/4 T/4 Odd Quarter-Wave Symmetry

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

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

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

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

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

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

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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 S k (t) to represent f(t) has least mean-square property. Proven by setting E k / a i = 0 and E k / b i = 0.

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Approximation a function

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Mean-Square Error

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