1. Example I: F.T. Integration Property𝑡
𝑤(𝑡) 𝐴 −𝐴 𝑏 𝑎 −𝑏 −𝑎
Example I: F.T. Integration Property
𝐴 𝑏−𝑎
−𝐴 𝑎
𝑦 𝑡 = 𝐴 𝑏−𝑎 𝛿 𝑡+𝑏 − 𝑏𝐴 𝑎 𝑏−𝑎 𝛿 𝑡+𝑎 + 𝑏𝐴 𝑎 𝑏−𝑎 𝛿 𝑡−𝑎 𝑡−𝑎 − 𝐴 𝑏−𝑎 𝛿 𝑡−𝑏
𝑌 𝜔 = 𝐴 𝑏−𝑎 𝑒 𝑗𝜔𝑏 − 𝐴 𝑏−𝑎 𝑒 −𝑗𝜔𝑏 − 𝑏𝐴 𝑎 𝑏−𝑎 𝑒 𝑗𝜔𝑎 + 𝑏𝐴 𝑎 𝑏−𝑎 𝑒 −𝑗𝜔𝑎
𝑌(𝜔) = 𝑗𝜔2𝐴𝑏 𝑏−𝑎 sin 𝑏𝜔 𝑏𝜔 − 𝑗𝜔2𝐴𝑏 𝑏−𝑎 sin 𝑎𝜔 𝑎𝜔
𝑋 𝜔 = 𝑌 𝜔 𝑗𝜔 +𝜋 𝑌 0 𝛿(𝜔)= 2𝐴𝑏 𝑏−𝑎 𝑠𝑖𝑛𝑐 𝑏𝜔 −𝑠𝑖𝑛𝑐 𝑎𝜔
𝑊 𝜔 = 𝑋 𝜔 𝑗𝜔 +𝜋 𝑋 0 𝛿(𝜔)
𝑊(𝜔)= 2𝐴𝑏 𝑗𝜔(𝑏−𝑎) 𝑠𝑖𝑛𝑐 𝑏𝜔 −𝑠𝑖𝑛𝑐 𝑎𝜔
𝑊(𝜔)= 2𝐴𝑏 𝜔 𝑏−𝑎 [𝑠𝑖𝑛𝑐(𝑏𝜔)−𝑠𝑖𝑛𝑐(𝑎𝜔)] 𝑒 − 𝑗𝜋 2 𝑡
𝑥(𝑡) 𝑏 𝑎 −𝑏 −𝑎
𝐴 𝑏−𝑎
−𝐴 𝑎 𝑡
𝑦(𝑡) 𝑏 𝑎 −𝑏 −𝑎
𝑏𝐴 𝑎(𝑏−𝑎)
−𝐴 (𝑏−𝑎)
−𝑏𝐴 𝑎(𝑏−𝑎)
𝐴 (𝑏−𝑎)

2. Example II: F.T. Integration Property

3. Example: The F.T. of a periodic signal
𝑔 𝑡 =𝐴𝑟𝑒𝑐𝑡( 𝑡 𝑇 )
𝐺 𝜔 =𝐴𝑇𝑠𝑖𝑛𝑐( 𝑇𝜔 2 )
𝐹 𝜔 = 𝑛=−∞ ∞ 𝜔 0 𝐺 𝑛 𝜔 0 𝛿(𝜔−𝑛 𝜔 0 )
𝐹(𝜔)= 𝑛=−∞ ∞ 𝐴𝑇𝜔 0 𝑠𝑖𝑛𝑐 𝑛 𝜔 0 𝑇 2 𝛿(𝜔−𝑛 𝜔 0 )
The frequency spectra of periodic signals are made up of discrete frequency components in the form of impulses occurring at integer multiples (harmonics) of the fundamental frequency of the signal.
The weight of each impulse is found by multiplying the F.T of the generating function, evaluated at that harmonic frequency, by the fundamental frequency of the periodic signal.