RF Design Lab. 1 AEM 2, 10장, JJEONG - 서강대학교 청년광장 -...
Transcript of RF Design Lab. 1 AEM 2, 10장, JJEONG - 서강대학교 청년광장 -...
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
3
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,⋯ ,
4
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.
5
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
6
(5) : Fourier series of Fourier coefficient of is given by the Euler formulas
(6)
AEM 2, 10장, JJEONGRF Design Lab. 11
• : not defined (discontinuous) at 0• Sum of series at 0 equals 0
AEM 2, 10장, JJEONGRF Design Lab. 12
(3) 1, cos , sin , cos 2 , sin 2 ,⋯ , cos , sin ,⋯
, : positive integers
AEM 2, 10장, JJEONGRF Design Lab.
Piece-wise continuity in Sec. 6.1
17
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.
AEM 2, 10장, JJEONGRF Design Lab. 20
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 )
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
26
1
2cos
2sin
AEM 2, 10장, JJEONGRF Design Lab.
Applications of orthogonality• Quadrature modulation and demodulation in communications
27
AEM 2, 10장, JJEONGRF Design Lab.
Applications of orthogonality
• OFDM (orthogonal frequency division multiplexing)
28
AEM 2, 10장, JJEONGRF Design Lab. 34
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, .