Fourier series formula

That pulse is half the period of the square wave above. It has a Fourier transform of X(f)=10−3sinc(10−3f) X ( f ) = 10 − 3 s i n c ( 10 − 3 f ) The figure below shows the amplitude spectrum of the square pulse, |X(f)| as a blue line, and the amplitude of the Fourier series of the square wave, |Xk| as red stars.

Fourier analysis The ideal square wave contains only components of odd-integer harmonic frequencies (of the form 2π(2k − 1)f ). Sawtooth waves and real-world signals contain all integer harmonics. A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs phenomenon.

Integration of a Square Wave The integrator has converted the square wave input to a triangular wave at the output , the slope of this wave describes the increase in area beneath the square wave (moving from left to right).

Forward Fourier Transform: Analysis Equation. X(ω)=+∞∫−∞x(t)e−jωtdt . Inverse Fourier Transform: Synthesis Equation. x(t)=12π+∞∫−∞X(ω)ejωtdω