Recognizing even,odd and half symmetry in signals and their fourier series co-efficient

To find symmetry in signal, you have to flip it and check if you get the original signal back or
inverted form of original signal. if you get original signal back then signal has even symmetry and if signal get inverted by flipping, then signal has odd symmetry otherwise no symmetry.
If the periodic signal x(t ) exhibits some symmetry, then the computation of the CTFS coefficients is simplified considerably. note that fourier series is sum of frequency harmonic of even and odd symmetry of signal

Below, we list the properties of the trigonometric coefficients of the CTFS for symmetrical signals.

•  If x(t ) is an even function, then bn = 0 for all n. In other words, an even signal is represented by its dc component and a linear combination of a cosine function since cosine is an even signal/function as shown below

main property of even function is

because integrating from –T/2 to 0 is same as 0 to T/2. utilizing these, fourier series co-efficient become

note: if x(t) is real and even, then exponential co-efficient is real and even.

•  If x(t ) is an odd function, then a0 = an = 0 for all n. In other words, an odd signal can be represented by a linear combination of a sine function since sine wave is an odd signal as shown below

An odd function fo(t) has this major characteristic:

because integration from −T/2 to 0 is the negative of that from 0 to T/2. With this property, the Fourier coefficients for an odd function become

note: if x(t) is real and odd, then exponential co-efficient is imaginary and odd.

•  If x(t ) is a real function, then the trigonometric CTFS coefficients a0, an, and bn are also real-valued for all n.
• A function is half wave (odd) symmetric if
  

Notice that for each function, one half-cycle is the inverted, version of the adjacent half-cycle.The Fourier coefficients become