SPFirst-L03-SK.pdf - Chapter 3 Spectrum Representation...

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Chapter 3 Spectrum Representation Sungoh Kwon
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Spectrum Definition: A compact representation of the frequency contents of a signal that is composed of sinusoids Example: 9/11/2013 2010 DSP by S.Kwon, University of Ulsan 2 0 100 250 100 250 f (in Hz) 3 / 7 j e 3 / 7 j e 2 / 4 j e 2 / 4 j e 10 ) 2 / 500 cos( 8 ) 3 / 200 cos( 14 10 ) ( t t t x
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Spectrum of a Sum of Sinusoids A general expression of the signal, which is a summation of sinusoids, is represented as where , X 0 is a DC component (2N+1) complex amplitudes and frequencies; two sided spectrum (f k , 1/2X k ): frequency domain representation {(0, 1/2X k ), (f 1 , 1/2X k ), (-f 1 , 1/2X * k ), … , {( f N , 1/2X N ), (-f N , 1/2X * N )} 9/11/2013 2010 DSP by S.Kwon, University of Ulsan 3 N k t f j k t f j k N k t f j k N k t f j j k N k t f j k N k k k k k k k k k k k e X e X X e X X e e A X e A X t f A A t x 1 2 * 2 0 1 2 0 1 2 0 1 ) 2 ( 0 1 0 2 1 2 1 Re Re Re ) 2 cos( ) ( k j k k e A a
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Let us define f 0 = 0, then 9/11/2013 2010 DSP by S.Kwon, University of Ulsan 4 k k k k k N N k t f j k N k t kf j k t f j k t f j N k t kf j k t f j k f f X X X a e a e X e X e X e X e X X t x k k k k k , 0 k for 2 1 0 k for 2 1 0 k for where 2 1 2 1 2 1 2 1 ) ( * 0 2 1 2 * 2 2 0 1 2 * 2 0 0
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9/11/2013 © 2003, JH McClellan & RW Schafer 5 Example of Spectrum t j j t j j t j j t j j e e e e e e e e t x ) 250 ( 2 2 / ) 250 ( 2 2 / ) 100 ( 2 3 / ) 100 ( 2 3 / 4 4 7 7 10 ) ( 0 100 250 100 250 f (in Hz) 3 / 7 j e 3 / 7 j e 2 / 4 j e 2 / 4 j e 10
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The process of analyzing the signal = to find its
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