39 Continuous and Discrete time complex exponential signal | Signals and Systems | AL AMIN
Complex exponential signals
Complex exponential signal continuous time .
Let us define Complex continuous time exponential signal as x(t)=𝑒^(𝑗𝜔_𝜊 𝑡)
An important property of complex exp signal is periodicity. i e
𝑒^(𝑗𝜔_𝜊 𝑡)= 𝑒^(𝑗𝜔_𝜊 (𝑡+𝑇))= 𝑒^(𝑗𝜔_𝜊 𝑡). 𝑒^(𝑗𝜔_𝜊 𝑇)= 𝑒^(𝑗𝜔_𝜊 𝑡).1 because 𝑒^(𝑗𝜔_𝜊 𝑇)=cos 𝜔_𝜊 T+jsin𝜔_𝜊T
=cos 𝜔_𝜊2𝜋+𝑗𝑠𝑖𝑛𝜔_𝜊 2𝜋
=1
If 𝜔_𝜊=0 ,then x(t)=1,which is periodic for any value of t
If 𝜔_𝜊≠0 𝑡ℎ𝑒𝑛 the fundamental frequency is 𝜔_𝜊.i.e 𝜔_𝜊=2𝜋/𝑇_𝜊