Fourier Analysis: Signal Representation by Sinusoids

1.Frequency : rate of signal oscillation.
    Fundamental frequency is simply referred to the lowest frequency of a period signal.
    A harmonic of a periodic signal is any component frequency of the signal that is an iteger multiple of the fundamental frequency.
    A fast changing signal needs to be represented by high order harmonics.  
2.Fourier Series Representations
    Fourier analysis of signals and systems, including applications to image processing and computer vision, has been active research field for many years.
3.Fourier Series
   The Fourier Series is a signal analysis tool that allows for any periodic signal to be decomposed into an infinite sum of everlasting sinusoids.

   



    Coefficients can be computed as follows

    N Harmonics
    1-D Fourier Series Approximations
4.Signal Properties vs Fourier Series
    Various signal properties can translate into specific properties of the Fourier series
    The Fourier series of a sine or cosine wave contains a single harmonic because a sine or cosine wave cannot be decomposed into other sine or cosine waves.
5.Eulers' Formula
           






6.The Fourier Series in exponential form can be

7.Benefits of complex exponentials based Fourier Series representations
    Only need to perform one integration
    A single exponential can be manipulated more easily than a sum of sinusoids
    It provides a logical transition into a future introduction of the Fourier Transform.