RF Design Lab. 1 AEM 2, 10장, JJEONG - 서강대학교 청년광장 - 홈home.sogang.ac.kr/sites/rfdesign/lect/advancedmath2_1… ·  · 2017-08-31RF Design Lab. 2 AEM 2, 10장,

AEM 2, 10장, JJEONGRF Design Lab. 1

AEM 2, 10장, JJEONGRF Design Lab. 2

cos

sin

cos

sin

그래프, -pi에서 pi까지 적분, 0에서 2pi까지 적분, 임의의 2pi 구간에서 적분, 0에서 pi까지 적분

cos 2

sin 2

AEM 2, 10장, JJEONGRF Design Lab.

White light through a prism

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Timedomain

Frequencydomain

Fouriertransform

AEM 2, 10장, JJEONGRF Design Lab.

Fourier series : infinite series representing periodic functions in terms of cosines and sines. Periodic function of period 0

(1)

If has period , it also has the period 2 , because 2

For any integer 1,2,3,⋯ ,

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AEM 2, 10장, JJEONGRF Design Lab.

Fourier series : representation of various function of period 2 in terms of the simple functions,

(3) 1, cos , sin , cos 2 , sin 2 ,⋯ , cos , sin ,⋯All these functions have the period 2 . They form the so-called trigonomeric system.

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AEM 2, 10장, JJEONGRF Design Lab.

Suppose that is a given function of period 2 and it can be represented by a series and has the sum

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(5) : Fourier series of Fourier coefficient of is given by the Euler formulas

(6)

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• : not defined (discontinuous) at 0• Sum of series at 0 equals 0

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(3) 1, cos , sin , cos 2 , sin 2 ,⋯ , cos , sin ,⋯

, : positive integers

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(6)

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AEM 2, 10장, JJEONGRF Design Lab.

Piece-wise continuity in Sec. 6.1

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is piecewise continuous on a finite interval ( ) if this interval canbe divided into finitely many subintervals in each of which is continuousand has finite limits as approaches either endpoint of such a subintervalfrom the interior. This then gives finite jumps as in Fig. 115 as the onlypossible discontinuities.

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HW:6,9,12,13 (Due: 1.5 week)

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Let’s set 2 . The transition from period 2 to 2 is effected by a suitablechange of scale. Let have period 2 . Then we can introduce a new variablesuch that , as a function of , has period 2 . If we set

Periodic functions of any period (not just 2 )

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AEM 2, 10장, JJEONGRF Design Lab.

Fourier series of periodic function of time

• with period 2

cos 2 sin 2

cos 2 sin 2

cos sin

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1

2cos

2sin

AEM 2, 10장, JJEONGRF Design Lab.

Applications of orthogonality• Quadrature modulation and demodulation in communications

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AEM 2, 10장, JJEONGRF Design Lab.

Applications of orthogonality

• OFDM (orthogonal frequency division multiplexing)

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12

1cos

1sin

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12

1cos

1sin

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A is defined on an interval 0, instead of , . We want to represent with a Fourier series. Thus, we extend to , 0 as either even or odd function.Then, we assume that is periodic with period of 2 . Therefore, we can represent it with Fourier series. The Fourier series gives correct on an interval 0, .

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