Page 243 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

Thus, from Eqs. (5.96) and (5.97) we conclude that

which shows that the set {eJkwo': k = 0, +_ 1, f 2,. . . ) is orthogonal on any interval over a period
To.

5.2. Using the orthogonality condition (5.98), derive Eq. (5.5) for the complex Fourier
coefficients.
From Eq. (5.4)

Multiplying both sides of this equation by e-imwo' and integrating the result from to to
(to + To), we obtain

Then by Eq. (5.98) Eq. (5.99) reduces to

Changing index m to k, we obtain Eq. (5.51, that is,

We shall mostly use the following two special cases for Eq. (5.101): to = 0 and to = - T0/2,
respectively. That is,

5.3. Derive the trigonometric Fourier series Eq. (5.8) from the complex exponential
Fourier series Eq. (5.4).
Rearranging the summation in Eq. (5.4) as

and using Euler's formulas