# Fourier Analysis

This involves two operations:
1. The evaluation of the co-efficient a0, an and bn.
2. 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:
1. Odd or Rotation Symmetry,
2. Even or Mirror Symmetry,
3. Half-Wave or, Alternation Symmetry, and
4. Quarter-Wave Symmetry.
1. Odd Symmetry
A function f (x) is said to be odd if,

f (x) = â€“ f (â€“x)
Odd function

Hence, for odd functions a0 = 0 and an = 0 and

Thus, the Fourier series expansion of an odd function contains only the sine terms, the constant and the cosine terms being zero.
1. Even Symmetry
A function f (x) is said to be even, if
f (x) = f (â€“x)
âˆ´ a0
âˆ´ an
and bn = 0
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.
1. 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
1. 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 (2N + 1) terms as,

SN (t) =

such that the co-efficients a0an and bn are chosen to give the least mean square error.
The truncation error is,

eN (t) = f(t) â€“ SN(t)

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

EN

where, EN is a function of a0 , an and bn, but not of t.