Fourier Series Representation
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This document discusses Fourier series representation of signals. It defines orthogonal sets and explains that any signal can be represented as a combination of orthogonal basis functions using Fourier series. There are two common ways to represent a signal using Fourier series: using trigonometric or exponential basis functions. An example signal is represented using both trigonometric and exponential Fourier series. The magnitude and phase spectra of the example signal are also calculated from the Fourier series coefficients.
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Fourier Series Representation
- 1. Prepared By: Preet Aghera 140110111002 Denish Thummar 140110111061 G H Patel College of Engineering and Technology - V V Nagar Electronics and Communication (2151004) Guided By:Prof. Samir D. Trapasiya
- 3. /*Complete Set of orthogonal may create any signal from it.*/ , condition for orthogonal : then two signal , are orthogonal. )(1 tg )(2 tg 0)()( 2 0 1 =•∫ ∞ tgtg )(1 tg )(2 tg
- 4. Two way to Represent : • Trigonometric Exponential fourier series fourier series Cos + Sin θj e
- 5. Orthogonal sets Trigonomatric Fourier Sereis 0)()( 2 0 1 =•∫ ∞ tgtg 0)()( 2 0 1 0 =•∫ tgtg T nmattmwtnw T ≠=•∫ ,0)cos()cos( 0 0 0 0 nmT == ,2 0 )3sin(),2sin(),sin(),0sin( )3cos(),2cos(),cos(),0cos( 000 000 twtwtw twtwtw
- 6. Fourier series )]sin()cos([)( 00 1 0 tnwbtnwaatg nn ++= ∑ ∞ ∫ ∫ ∫ = = = 0 0 0 0 0 0 0 0 0 0 0 0 0 )sin()( 2 )cos()( 12 1 T n T n T dttnwtg T b dttnwtg T a dta T a where
- 7. Example 1 g(t) QUE:Represent g(t) in Trigonomatric fourier series and also find magnitude and phase spectra ∫= 0 0 0 0 0 1 T dta T a 2 π 2 π − ππ− 1 2 2 ,2)(, 00 ===−−= π π πππ wTPeriod ∫ − = 2 2 2 1 π ππ dt 2 1 =
- 8. ∫= 0 0 0 0 )cos()( 2 T n dttnwtg T a ∫ − ⋅= 2 2 )cos(1 )2( 2 π ππ dtntan ) 2 sin( 2sin1 2 2 π ππ π π n nn nt an = = − ,...15,11,7,3, 2 ,....13,9,5,1, 2 ,...6,4,2,0 =−= == == ifn n a ifn n a ifna n n n π π
- 9. Fourier Series Magnitude Spectra Phase Spectra Magnitude Freuency Phase Freuency )]sin()cos([)( 00 1 0 tnwbtnwaatg nn ++= ∑ ∞ )]cos() 2 sin( 2 [ 2 1 )( 0 1 tnw n n tg n ⋅+= ∑ ∞ = π π ...])5cos( 5 1 )3cos( 3 1 )[cos( 2 2 1 )( 000 ++−+= twtwtwtg π
- 10. By putting value if an and bn in g(t) na nb nc nθ 22 nnn bac += n n n c a =θcos n n n c b =θsin nnn ca θcos= nnn cb θsin= )]sin(sin)cos(cos[)( 00 1 0 tnwctnwcatg nnnn θθ ++= ∑ ∞ )]cos([)( 0 1 0 n n n tnwcctg θ−+= ∑ ∞ = 00 ac =
- 11. Magnitude Spectra Phase Spectra May be zero… nc 0nw0w 02w 03w 03w 0c 1c 2c 3c nθ 0nw0w 02w 03w 03w 0θ 1θ 2θ 3θ 0c
- 13. Example Represent g(t) in terms of exponential fourier series and also find magnitude and Phase spectra 1 2 t e − dtetg T D tjnw T n 0 0 00 )( 1 − ∫ ⋅= 2 22 0 0 === π ππ T w dteD tjn n )2( 0 1 1 − ∫ ⋅= π π dteD tjn n )2 2 1 ( 0 1 +− ∫= π π
- 14. ))2 2 1 (( 1 0 )2 2 1 ( tjn e D tjn n +− = +− π π − + = +− π π )2 2 1 ( 1 )41( 2 jn n e jn D ⋅− + − = − − π π π jn n ee jn D 22 1 )41( 2 − + − = − 2 1 )41( 2 π π e jn Dn )41( 504.0 jn Dn + = π
- 15. Fourier series ,......2,1,0,1,2..... )41( 504.0 )( 2 −−= + = ∑ ∞ −∞= n e jn tg jnt n π tjnw neDtg 0 )( ∑ ∞ ∞− = jtjt jtjt n e j e j e j e j tg 42 24 )81( 504.0 )41( 504.0 504.0 )41( 504.0 )81( 504.0 )( + + + + + − + − = −− ∞ −∞= ∑ ππ ππ
- 16. )81( 504.0 2 j D − =− π 0625.0 65 504.0 )81( 504.0 2 == − =− j D π )41( 504.0 1 j D − = π 87.82 1 8 tan 1 2 = −=∠ − −D 1222.0 17 504.0 )41( 504.0 1 == − =− j D π 83.75 1 4 tan 1 1 = −=∠ − −D 0,504.0 00 =∠= DD 83.75)4(tan,122.0 1 11 −==∠= − DD 87.82,0625.0 22 −=∠= DD
- 17. Magnitude Spectra Phase Spectra 0w 02w0w−02w− 0w 02w0w−02w− 0 0nw 0nw 504.0 1222.0 0625.0 0625.0 1222.0 87.82− 87.82 83.75− 83.75
- 18. THANK YOU
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