lecture 8.pptx - Signals and Systems Lecture 8 Fourier series Introduction A Fourier series is an expansion of a periodic function f(t in terms of an

lecture 8.pptx - Signals and Systems Lecture 8 Fourier...

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Signals and Systems Lecture 8
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A Fourier series is an expansion of a periodic function f ( t ) in terms of an infinite sum of cosines and sines Introduction 1 0 ) sin cos ( 2 ) ( n n n t n b t n a a t f Fourier series
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In other words, any periodic function can be resolved as a summation of constant value and cosine and sine functions: 1 0 ) sin cos ( 2 ) ( n n n t n b t n a a t f ) sin cos ( 1 1 t b t a 2 0 a ) 2 sin 2 cos ( 2 2 t b t a ) 3 sin 3 cos ( 3 3 t b t a
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The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually , and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.
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= + + + + + Periodic Function 2 0 a t a cos 1 t a 2 cos 2 t b sin 1 t b 2 sin 2 f(t) t
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1 0 ) sin cos ( 2 ) ( n n n t n b t n a a t f where T dt t f T a 0 0 ) ( 2 frequency l Fundementa 2 T T n tdt n t f T a 0 cos ) ( 2 T n tdt n t f T b 0 sin ) ( 2 *we can also use the integrals limit . 2 / 2 / T T
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Example 1 Determine the Fourier series representation of the following waveform.
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Solution First, determine the period & describe the one period of the function: T = 2 2 1 , 0 1 0 , 1 ) ( t t t f ) ( ) 2 ( t f t f
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Then, obtain the coefficients a 0 , a n and b n : 1 0 1 0 1 ) ( 2 2 ) ( 2 2 1 1 0 2 0 0 0 dt dt dt t f dt t f T a T Or, since y = f ( t )
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