Derivation(continued) Thus. for T <t< T X(jkwo)e kwot k= k ∑nX(ko) AsT→∞,∑0→∫du, we get the Ct Fourier Transform pair weSt dw Synthesis equation T X(j)= co e(t)e-jut dt Analysis equation
Derivation (continued)
For what kinds of signals can we do this? (1) It works also even if x()is infinite duration, but satisfies a) Finite energy a(t) dt In this case there is zero energy in the error (t)=(t) lewT dw Then b) dirichlet conditions 1 a(t) at points of continuity (i)2 oo X(w)ejt dw= midpoint at discontinuity (iii) Gibb's phe c) By allowing impulses in x(t)or in Xgo), we can represent even more signals E. g. It allows us to consider Ft for periodic signals
a) Finite energy In this case, there is zero energy in the error For what kinds of signals can we do this? (1) It works also even if x(t) is infinite duration, but satisfies: E.g. It allows us to consider FT for periodic signals c) By allowing impulses in x(t) or in X(jω), we can represent even more signals b) Dirichlet conditions