Page 456 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.7 Fourier Transforms 443

X { f ) real axis

- f -fn f Figure 13.7-2. FT of a ^ c o s l i r f ^ t .
In a similar manner, the FT of sin27r/„r is

X( f ) = - i j n [ S i f - f „ ) ~ S { f + f„)]

which is shown on the imaginary plane of Fig. 13.7-3.
X { f ) /-axis

jb{f+f„)

- f n
bn
2 Figure 13.7-3. FT of b ^ ú n l v f ^ t .
If we put the two FTs together in perpendicular planes, as shown in Fig. 13.7-4,
we obtain the complex conjugate coefficients - ib^ and C* = + ib^. Thus,
*

the product
C C* 1 *
4 4 ^
is the square of the magnitude of the Fourier series, which is generally plotted at ±f.

Figure 13.7-4. FT of fl„cos27r/,,/
+ b„sin2vf„t.

Example 13.7-3
We next determine the FT of a rectangular pulse, which is an example of an aperiodic
function. (See Fig. 13.7-5.) Its FT is

^ ( / ) = f dt = dt = At \^ )