Fourier-Series - Electrical Circuits SAITM AIT Program...

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Electrical Circuits SAITM AIT Program B.Sc. in Engineering in (ICT/ Mechatronics/ Electronics/Telecommunications) Department of Mechatronics Engineering, Faculty of Engineering, South Asian Institute of Technology and Management (SAITM)
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2 Fourier Series
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3 Background In the early nineteenth century, Joseph Fourier (1768-1830), while studying the problem of heat flow, developed a cohesive theory of such series. Consequently, they were named after him A function f ( x ) is said to have a period T or to be periodic with period T if for all x, f ( x + T) = f ( x ), where T is a positive constant. The least value of T > 0 is called the least period or simply the period of f ( x ).
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4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 0 20 40 60 80 100 120 0 50 100 150 200 250 300 Yes Since Periodic No Can you apply Fourier Series to the following signals
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5 Let f ( x ) be defined in the interval (–L, L) and outside of this interval by f ( x + 2L) = f ( x ); i.e., f ( x ) is 2L periodic. It is through this avenue that a new function on an infinite set of real numbers is created from the image on (–L, L). The Fourier series or Fourier expansion corresponding to f ( x ) . ) sin cos ( 2 ) ( 1 0 n n n L x n b L x n a a x f
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3 , 2 , 1 , 0 sin ) ( 1 cos ) ( 1 n dx L x n x f L b dx L x n x f L a L L n L L n . ) sin cos ( 2 ) ( 1 0 n n n L x n b L x n a a x f Where 6
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7
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8 Orthogonality Conditions for the Sine and Cosine Functions
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9 Odd Symmetry Functions A function f ( x ) is called odd if f (–x ) = f ( x ). x 3 , x 5 – 3x 3 + 2x, sin x, and tan 3x Even Symmetry Functions A function f (x) is called even if f (–x) = f (x). x 4 , 2x 6 – 4x 2 + 5, cos x, and e x + e –x
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10 Odd Symmetry Functions Even Symmetry Functions
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Tan function 11
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12 In the Fourier series corresponding to an odd function, only sine terms can be present . In the Fourier series corresponding to an even function , only cosine terms can be present (and possibly a constant, which we shall consider a cosine term).
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Half-Wave Symmetry ) ( ) ( T t f t f and   2 / ) ( T t f t f T T /2 T /2 T 13
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Fourier Coefficients for Half-Wave Symmetry ) ( ) ( T t f t f and   2 / ) ( T t f t f ) sin cos ( ) ( 1 0 0 n
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This note was uploaded on 10/30/2011 for the course MECHATRONI EN11ME2050 taught by Professor Perera during the Spring '11 term at Asian Institute of Technology.

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Fourier-Series - Electrical Circuits SAITM AIT Program...

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