80Z [an os(nat) +b si(not) f(t) = ao +n=1For given f(t), how can we find out what amount ofeach harmonic is required?6
𝑓 𝑡 = 𝑎0 + 𝑛=1 ∞ [𝑎𝑛 cos 𝑛𝜔𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑡 ] For given 𝑓(𝑡), how can we find out what amount of each harmonic is required? 6
Complete orthonormal setSines and cosines form an orthonormal set,for m + n2元71sin(mwt) sin(nwt) dt =sin(mwt) sin(nwt) d(wt) = 0wJo02sin(mwt) cos(nwt) dt = 0Jocos(mwt) cos(nwt) dt = 0OAnd~2元T1sin(mwt)dt =sin?(mwt) d(wt) =2w JoTTTcos*(mat) dt =2Jo
Complete orthonormal set Sines and cosines form an orthonormal set, for 𝑚 ≠ 𝑛 න 0 𝑇 sin 𝑚𝜔𝑡 sin(𝑛𝜔𝑡)𝑑𝑡 = 1 𝜔 න 0 2𝜋 sin 𝑚𝜔𝑡 sin 𝑛𝜔𝑡 𝑑(𝜔𝑡) = 0 න 0 𝑇 sin 𝑚𝜔𝑡 cos(𝑛𝜔𝑡) 𝑑𝑡 = 0 න 0 𝑇 cos 𝑚𝜔𝑡 cos(𝑛𝜔𝑡) 𝑑𝑡 = 0 And න 0 𝑇 sin2 𝑚𝜔𝑡 𝑑𝑡 = 1 𝜔 න 0 2𝜋 sin2 𝑚𝜔𝑡 𝑑(𝜔𝑡) = 𝑇 2 න 0 𝑇 cos2 𝑚𝜔𝑡 𝑑𝑡 = 𝑇 2 7