Chapter 3 Fourier series 53.3.2 Determination of Fourier Series Representation x()=∑ Synthesis equation 综合公式 k T J<To x(t e kobo dt Analysis equation 0 分析公式 a k Fourier series coefficients Spectral Coefficients 6
6 Chapter 3 Fourier Series §3.3.2 Determination of Fourier Series Representation ( ) jk t k k x t a e 0 + =− = ( ) 0 0 0 1 jk t k T a x t e dt T − = Synthesis equation 综合公式 Analysis equation 分析公式 k a ——Fourier Series Coefficients Spectral Coefficients
Chapter 3 Fourier series Example 3.5 Periodic square wave defined over one period as xlt < x 0T<t<T/2 TT/2-T10T1T/2 T +T/2 2T, x(t d T 2 ak=sinhoo k≠0 Defining sinc(x) SInd 2T sinc(koF) T 7
7 Chapter 3 Fourier Series Example 3.5 Periodic square wave defined over one period as ( ) = 0 T t / 2 1 t 1 1 T T x t 1 x(t) -T -T/2 –T1 0 T1 T/2 T t x(t)dt T a T T + = / 2 - /2 0 1 Defining ( ) x x c x sin sin = ( ) 0 1 1 sin 2 c k T T T ak = 0 1 sin 0 k k T a k k = T T1 2 =
Chapter 3 Fourier series 08 sin cloT 丌 05 sin c(oT) 0.4 17;2z Isin (oT) 0.2 0 -0.2 -0.4 -1.5 0.5 0.5 T固定,a的包络2Tsnc7定 8
8 Chapter 3 Fourier Series ( ) 1 1 sin T = c T ( ) 2 1 1 sin T = c T ( ) 3 1 1 sin T = c T T1固定，Tak 的包络 2T1 sin c(T1 )固定
Chapter 3 Fourier series Figure 4.2 2 4 (b)T=8T TIIDeanID MIID T=16T T↑→an=2丌/Tψ谱线变密
9 Chapter 3 Fourier Series T 0 = 2 /T 谱线变密 Figure 4.2 ( ) 4 1 a T = T ( ) 8 1 b T = T ( ) 16 1 c T = T
Chapter 3 Fourier series Example Periodic Impulse Trains(周期冲激串) x()=∑8(t-k7 k=-∞ 2T-TL0T 2T T/2T/2 k k=0,±1,±2,… T k 元 x()=∑e"r T k: 0002 ()=∑H/2n)m e
10 Chapter 3 Fourier Series Example Periodic Impulse Trains (周期冲激串) x(t) − 2T −T 0 T 2T t (1) ( ) 2 1 jk t T k x t e T + =− = ak T 1 -ω0 0 ω0 2ω0 −T / 2 T/ 2 0, 1, 2, 1 = k = T ak ( ) 2 1 2 jk t T k y t H jk e T T + =− = ( ) ( ) k x t t kT + =− = −