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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) = – f (–x)
Description: 1666.png
Odd function
Hence, for odd functions a0 = 0 and an = 0 and Description: 3377.png
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 (–x)
∴ a0Description: 3383.png
 anDescription: 3389.png
and bn = 0
Description: Description: 904.png
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 –Wave or Alternation Symmetry
A periodic function f (t) is said to have half wave symmetry if it satisfies the condition,
f (t) = – f (t ± T/2), where T – time period of the function
  1. Quarter–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) = Description: 3472.png
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) – SN(t)
So, the mean square error/figure of merit/the cost criterion for optimal minimal error is,
ENDescription: 3478.png
where, EN is a function of a0 , an and bn, but not of t.

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