# Fourier Analysis

This involves two operations:

- The evaluation of the co-efficient
*a*_{0}*, a*_{n}_{ }and*b*._{n} - Truncation of the infinite series after a finite number of terms so that
*f**t*) is represented within allowable error (-Done later).

# Evaluation of Fourier Coefficients

Following symmetries are considered:

- Odd or Rotation Symmetry,
- Even or Mirror Symmetry,
- Half-Wave or, Alternation Symmetry, and
- Quarter-Wave Symmetry.

**Odd Symmetry**

A function

*f**(**x*) is said to be odd if,*f*

*(*

*x*) = â€“

*f*

*(â€“*

*x*)

*Odd function*Hence, for odd functions

*a*_{0}_{}= 0 and*a**= 0 and*_{n}Thus, the

*Fourier series expansion of an odd function contains only the sine terms, the constant and the cosine terms being zero.***Even Symmetry**

A function

*f**(**x*) is said to be even, if*f*

*(*

*x*) =

*f*

*(â€“*

*x*)

âˆ´

*a*_{0}_{}=*âˆ´*

*a*=

_{n}and

*b**= 0*_{n}

*Even function*Thus, the

*Fourier series expansion of an even periodic function contains only the cosine terms plus a constant, all sine terms being zero.***Half â€“Wave or Alternation Symmetry**

A periodic function

*f**(**t*) is said to have half wave symmetry if it satisfies the condition,*f*

*(*

*t*) = â€“

*f*

*(*

*t*Â±

*T*/2), where

*T*â€“ time period of the function

**Quarterâ€“Wave Symmetry**

The symmetry may be regarded as a combination of first three kinds of symmetry provided that the origin is properly chosen.

# Truncating Fourier Series

When a periodic function is represented by a Fourier series, the series is truncated after a finite number of terms.

So, the periodic function is approximated by a trigonometric series of (2

*N*+ 1) terms as,*S*

*(*

_{N}*t*) =

such that the co-efficients

*a*_{0}_{},*a**and*_{n}*b**are chosen to give the least mean square error.*_{n}The truncation error is,

*e*

_{N}*(*

*t*) =

*f*(

*t*) â€“

*S*

*(*

_{N}*t*)

So, the mean square error/figure of merit/the cost criterion for optimal minimal error is,

*E*

*=*

_{N}where,

*E**is a function of*_{N}*a*_{0}_{},*a**and*_{n}*b**, but not of*_{n}*t*.