# Orthogonal Function Of Fourier Series

__ Orthogonal Function of Fourier Series:__ Basic Orthogonal Function used in Fourier Series are given as:

If we review the above example you will see that for each coefficients, if we wanted a_{k,} the coefficients of the k^{th} harmonic cosine, we multiplied the signals by cos ( 2πkf_{0}t ). which translate the components of interest to DC where it is extrated while discarding all other terms by integrating over exactly one cycle of the fundamental frequency. Similarly if we are interested in b_{k}, the coefficients of the k^{th} harmonic sine, we multiply the signal by sin ( 2πkf_{0}t ) which translate the components of interest to DC, where it is extracted while discarding all other terms by integrating over exactly one cycle of the fundamental frequency. This work because cos ( 2πkf_{0}t ) and sin ( 2πkf_{0}t ) are** orthogonal functions**, meaning that following orthogonality conditions are satiesfied by these functions.

Thus if we multiply a T - periodic signals by either cos ( 2πkf_{0}t ) or sin ( 2πkf_{0}t ) and integrate over one cycle, we isolate that components in the signals, contribution from all other components drop out of the results. Orthogonality implies that if there is a cos ( 2πkf_{0}t )components in a signal, it can only be represented by a_{k} in the Fourier series. In particular no other combination of coefficients could be used to represent that components. The same is true for a sin ( 2πkf_{0}t )component and b_{k}. Thus orthogonality implies that the Fourier series of a waveform is unique.