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