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## Fourier Series

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**Fourier Series**主講者：虞台文**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**Fourier Series**Periodic Functions**The Mathematic Formulation**• Any function that satisfies where T is a constant and is called the period of the function.**Example:**Find its period. Fact: smallest T**Example:**Find its period. must be a rational number**Example:**Is this function a periodic one? not a rational number**Fourier Series**Fourier Series**A periodic sequence**f(t) t T 2T 3T Introduction • Decompose a periodic input signal into primitive periodic components.**Synthesis**T is a period of all the above signals Even Part Odd Part DC Part Let 0=2/T.**Orthogonal Functions**• Call a set of functions {k}orthogonal on an interval a < t < b if it satisfies**Orthogonal set of Sinusoidal Functions**Define 0=2/T. We now prove this one**Proof**m n 0 0**Proof**m = n 0**Define 0=2/T.**Orthogonal set of Sinusoidal Functions an orthogonal set.**Proof**Use the following facts:**f(t)**1 -6 -5 -4 -3 -2 - 2 3 4 5 Example (Square Wave)**f(t)**1 -6 -5 -4 -3 -2 - 2 3 4 5 Example (Square Wave)**f(t)**1 -6 -5 -4 -3 -2 - 2 3 4 5 Example (Square Wave)**Harmonics**T is a period of all the above signals Even Part Odd Part DC Part**Define , called the**fundamental angular frequency. Define , called the n-th harmonicof the periodic function. Harmonics**harmonic amplitude**phase angle Amplitudes and Phase Angles**Fourier Series**Complex Form of the Fourier Series**Complex Form of the Fourier Series**If f(t) is real,**amplitude**spectrum |cn| n phase spectrum Complex Frequency Spectra**f(t)**A t Example**A/5**0 -120 -80 -40 40 80 120 -150 -100 -50 50 100 150 Example**A/10**0 -120 -80 -40 40 80 120 -300 -200 -100 100 200 300 Example**f(t)**A t 0 Example**Fourier Series**Impulse Train**t**0 Dirac Delta Function and Also called unit impulse function.**Property**(t): Test Function**t**T 2T 0 T 2T 3T 3T Impulse Train**Fourier Series**Analysis of Periodic Waveforms**Waveform Symmetry**• Even Functions • Odd Functions**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**Example**Even Part Odd Part**T/2**T/2 T Half-Wave Symmetry and**T/2**T/2 T T/2 T/2 T Quarter-Wave Symmetry Even Quarter-Wave Symmetry Odd Quarter-Wave Symmetry**A**T T A/2 T T A/2 Hidden Symmetry • An asymmetry periodic function: • Adding a constant to get symmetry property.**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