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Problem 10.16. Show that if T is a multiple of the period of fc, then the terms sin(2 )cf tπ and cos(2 )cf tπ are orthogonal over the interval [ ]0 0,t T t+ . Solution [ ] [ ] 0 0 0 0 0 0 0 0 0 1sin(2 )cos(2 ) sin(4 ) 2 1 cos(4 ) 8 1 cos(4 ( )) cos(4 ) 8 1 sin(4 2 ) sin(2 ) 4 T t T t c c ct t T t c t c c c c c c c c f t f t dt f t dt f t f f t T f t f f t f T f T f π π π ππ π ππ π π ππ + + + = = − = − + − −= + ⋅ ∫ ∫ where we have used the equivalence cosA - cosB = 2sin[(A+B)/2]sin[(B-A)/2)]. If T is a multiple of the period of fc, then fcT = integer, and 0)2sin( =Tfcπ . Therefore, . That is, ∫ + =Ttt cc dttftf00 0)2cos()2sin( ππ )2sin( tf cπ and )2cos( tf cπ are orthogonal over the interval [t0, t0+T]. Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
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