Fourier Series and Fourier Transform

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1 Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved Spring 2007 Fourier Series and Fourier ransform, Slide 1

2 he Complex Exponenial as a Vecor Q Noe: sin(ω) e jω ω cos(ω) I Euler s Ideniy: Consider I and Q as he real and imaginary pars As explained laer, in communicaion sysems, I sands for in-phase and Q for quadraure As increases, vecor roaes counerclockwise We consider e jw o have posiive frequency Spring 2007 Fourier Series and Fourier ransform, Slide 2

3 he Concep of Negaive Frequency Q Noe: cos(ω) ω I -sin(ω) e -jω As increases, vecor roaes clockwise We consider e -jw o have negaive frequency Noe: A-jB is he complex conjugae of A+jB So, e -jw is he complex conjugae of e jw Spring 2007 Fourier Series and Fourier ransform, Slide 3

4 Add Posiive and Negaive Frequencies Q Noe: e jω 2cos(ω) I e -jω As increases, he addiion of posiive and negaive frequency complex exponenials leads o a cosine wave Noe ha he resuling cosine wave is purely real and considered o have a posiive frequency Spring 2007 Fourier Series and Fourier ransform, Slide 4

5 Subrac Posiive and Negaive Frequencies Q Noe: 2sin(ω) -e -jω I e jω As increases, he subracion of posiive and negaive frequency complex exponenials leads o a sine wave Noe ha he resuling sine wave is purely imaginary and considered o have a posiive frequency Spring 2007 Fourier Series and Fourier ransform, Slide 5

6 Fourier Series he Fourier Series is compacly defined using complex exponenials Where: Spring 2007 Fourier Series and Fourier ransform, Slide 6

7 From he Previous Lecure he Fourier Series can also be wrien in erms of cosines and sines: Spring 2007 Fourier Series and Fourier ransform, Slide 7

8 Compare Fourier Definiions Le us assume he following: hen: So: Spring 2007 Fourier Series and Fourier ransform, Slide 8

9 Square Wave Example A /2 -A Spring 2007 Fourier Series and Fourier ransform, Slide 9

10 Graphical View of Fourier Series As in previous lecure, we can plo Fourier Series coefficiens Noe ha we now have posiive and negaive values of n Square wave example: An 3 Bn n Spring 2007 Fourier Series and Fourier ransform, Slide n

11 Indexing in Frequency A given Fourier coefficien,,represens he weigh corresponding o frequency nw o I is ofen convenien o index in frequency (Hz) Af f Spring 2007 Fourier Series and Fourier ransform, Slide Bf f

12 -A A he Impac of a ime (Phase) Shif /2 A -A /4 /4 Consider shifing a signal in ime by d Define: Which leads o: Spring 2007 Fourier Series and Fourier ransform, Slide 12

13 -A A Square Wave Example of ime Shif /2 A -A /4 /4 o simplify, noe ha excep for odd n Spring 2007 Fourier Series and Fourier ransform, Slide 13

14 -A A Graphical View of Fourier Series /2 A -A /4 /4 Af Af Bf f f Spring 2007 Fourier Series and Fourier ransform, Slide Bf f f

15 Magniude and Phase We ofen wan o ignore he issue of ime (phase) shifs when using Fourier analysis Unforunaely, we have seen ha he A n and B n coefficiens are very sensiive o ime (phase) shifs he Fourier coefficiens can also be represened in erm of magniude and phase where: Spring 2007 Fourier Series and Fourier ransform, Slide 15

16 -A A Graphical View of Magniude and Phase /2 A -A /4 /4 Xf Xf f f Φf Φf / f -/ f Spring 2007 Fourier Series and Fourier ransform, Slide 16

17 Does ime Shifing Impac Magniude? Consider a waveform along wih is Fourier Series We showed ha he impac of ime (phase) shifing on is Fourier Series is We herefore see ha ime (phase) shifing does no impac he Fourier Series magniude Spring 2007 Fourier Series and Fourier ransform, Slide 17

18 Parseval s heorem he squared magniude of he Fourier Series coefficiens indicaes power a corresponding frequencies Power is defined as: Noe: * means complex conjugae Spring 2007 Fourier Series and Fourier ransform, Slide 18

19 he Fourier ransform he Fourier Series deals wih periodic signals he Fourier ransform deals wih non-periodic signals Spring 2007 Fourier Series and Fourier ransform, Slide 19

20 Fourier ransform Example A - Noe ha is no periodic Calculaion of Fourier ransform: Spring 2007 Fourier Series and Fourier ransform, Slide 20

21 Graphical View of Fourier ransform A - X(j2f) his is called a sinc funcion f Spring 2007 Fourier Series and Fourier ransform, Slide

22 Summary he Fourier Series can be formulaed in erms of complex exponenials Allows convenien mahemaical form Inroduces concep of posiive and negaive frequencies he Fourier Series coefficiens can be expressed in erms of magniude and phase Magniude is independen of ime (phase) shifs of he magniude squared of a given Fourier Series coefficien corresponds o he power presen a he corresponding frequency he Fourier ransform was briefly inroduced Will be used o explain modulaion and filering in he upcoming lecures We will provide an inuiive comparison of Fourier Series and Fourier ransform in a few weeks Spring 2007 Fourier Series and Fourier ransform, Slide 22

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