R19 - ENEE 241 02 * READING ASSIGNMENT 19 Tue 04/21 Lecture...

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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 19 Tue 04/21 Lecture Topics: orthogonality of Fourier sinusoids; determination of Fourier series coefficients Key Points: • The inner product of two continuous-time periodic signals f ( t ) and g ( t ) with common period T is defined by h f , g i = 1 T Z T f * ( t ) g ( t ) dt • The Fourier sinusoids v ( k ) ( t ) = e jk Ω t , where k ∈ Z , are pairwise orthogonal and have unit norm: h v ( k ) , v ( ‘ ) i = ‰ 1 , k = ‘ ; , k 6 = ‘ . • The coefficients of the Fourier series s ( t ) = ∞ X k =-∞ S k e jk Ω t can be obtained from S k = h v ( k ) , s i = 1 T Z T s ( t ) e- jk Ω t dt Theory and Examples: 1 . Most “nice” periodic signals in continuous time have a Fourier series expansion, i.e., they can be expressed as s ( t ) = ∞ X k =-∞ S k e jk Ω t ( ♠ ) where Ω is the fundamental frequency. The proof of this fact—including important details such as the class of signals for which this representation holds and the manner in which the infinite sum converges—is beyond the scope of this course. A more accessible, and also practical, question is the following: given a signal s ( t ) which has the Fourier series representation shown above, how can we obtain the coefficients { S k } ? Put differently, if ( ♠ ) is the synthesis equation which constructs the signal s ( t ) as a sum of Fourier sinusoids with coefficients { S k } , what analysis equation produces each S k from s ( t )? As we will soon see, the answer involves the inner product of the k th Fourier sinusoid and the signal s ( t ). 2 . Let f ( t ) and g ( t ) be complex-valued signals which are both periodic with period T . Their inner product h f , g i is defined as h f , g i def = 1 T Z T f * ( t ) g ( t ) dt We use boldface symbols to emphasize that, as was the case with vectors earlier, the inner...
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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R19 - ENEE 241 02 * READING ASSIGNMENT 19 Tue 04/21 Lecture...

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