《信号与系统 Signals and Systems》课程教学资料（英文版）lecture 6 CT Fourier Series Pairs

Signals and Systems Fall 2003 Lecture #6 23 September 2003 1. CT Fourier series reprise properties, and examples DT Fourier series 3. DT Fourier series examples and differences with CtFS

Signals and Systems Fall 2003 Lecture #6 23 September 2003 1. CT Fourier series reprise, properties, and examples 2. DT Fourier series 3. DT Fourier ser ies examples and differences with CTFS

cT Fourier series pairs 0 Re eview m()=∑ack=∑ 2Tkt/T' k= lte-jkwot dt T Skip it in future for shorthand k

CT Fourier Series Pairs Skip it in future for short hand

Another (important! )example: Periodic Impulse Train r(t)=∑6(t ampling funct important for sampling 2T T T 2T 1 T/2 1 2 k swot T T 6(t)e T/2 for all k All components ha () the same amplitude ()=7∑e (2) the same phase

Another (important!) example: Periodic I mpulse Train — All components have: (1) the same amplitude, & (2) the same phase

(A few of the) Properties of cT Fourier Series Linearity x(t)+) ak, g(t)+ bk =a.(t)+By(t)+aak+ Bbk Conjugate Symmetry alt) is rea l→a-k Refak+j imai lak Relak is even, Imak) is odd ak| is even,∠ ak is odd k Time shift qr(w k> 0 to =ake jk2to/T ntroduces a linear phase shift oc t

(A few of the) Properties of CT Fourier Series • Lineari t y Introduces a linear phase shift ∝ t o • Conjugate Symmetry • Time shift

Example: Shift by half period k丌 Ck已 k using e-jkwo T/2 y(t) 3727-12 m12372 k T FC.of∑(t-m) k

Example: Shift by half period

Parseval’ s Relation Power in the Average signal power Cth harmonic Energy is the same whether measured in the time-domain or the equency-domain Multiplication property x(t)→→ak,y(t)4bk( Both a(t)andy(t) are periodic with the same period T r(t)y()…ck=∑a Proof: ∑aco∑ 少Qbne1(1+m)ot+m=k ∑∑ a1bk-L swot e( t)

• Parseval’s Relation Energy is the same whether measured in the time-domain or the frequency-domain • Multiplication Property

Periodic convolution x(t),yt) periodic with period T -T-T/2 T22 T T -T/ 0 Tn T (G)y(t-r)dr not very meaningful E. g. If both a(t)and y(t) are positive, then c(t)xk y(

Periodic Convolution x ( t), y ( t) periodic with period T

Periodic convolution(continued) Periodic convolution: Integrate over any one period(e.g -T/2 to T/2) og(t-Td T(o)g(t-rdT T/2 where (t)-T/2<t<T/2 otherwise x(t

Periodic Convolution (continued) Periodic convolution: Integrate over any one period (e.g. - T/2 to T/2 )

Periodic Convolution(continued) Facts )E( is periodic with period T(why?) From Lecture #2, a()=a(t+T)g(t=y(t+r) for LTI systems In the convolution, treat g(t)as the input and r(t) as h(t 2) Doesnt matter what period over which we choose to integrate ()=/x(7)y(t-r)dr=a()8yt) Periodic 3)convolution (t+)→ak,y(t)→bk,(t)Ck in time T/a(te-jhwot =/(/m))- kwo(t-T) T 9八( Multiplication cT)e swoT dr= Tarok frequency

Periodic Convolution (continued) Facts 1) z ( t) is periodic with period T (why?) 2) Doesn’t matter what period over which we choose to integrate: 3) Periodic convolution in time Multiplication in frequency!

Fourier Series Representation of dt Periodic signals xn]-periodic with fundamental period N, fundamental frequency 2丌 n+N=an and wo Only e/o n which are periodic with period N will appear in the FS uN=k2丌分W=k0,k=0,±1,土2, There are only N distinct signals of this form j(k+Won joe nWO So we could just use e] wom, e] Iwon, eJ2wo n ,e(N-1)won However, it is often useful to allow the choice ofN consecutive values of k to be arbitrary

Fourier Series Representatio n of DT Periodic Signals • x[n] - periodic with fundamental period N, fundamental frequency • Only ejω n which are periodic with period N will appear in the FS • S o w e could just use • However, it is often useful to allow the choice of N consecutive values of k to be arbitrary. ⇓ • There are only N distinct signals of this form