Signals and Systems - Electrical Engineering

4.3 Complex Exponential Fourier Series 245

since

H(j 0 )= 1

H(j10,000)≈

1

j 104

=

−j

10,000

H(−j10,000)≈

1

−j 104

=

j

10,000

Thus, this circuit acts like a low-pass filter by keeping the DC component (with the low frequency

=0) and essentially getting rid of the high-frequency(=10,000)component of the signal.

Notice that the frequency response can also be obtained by considering the phasor ratio for a

generic frequency, which by voltage division is

Vc

Vs

=

1 /j

1 + 1 /j

=

1

1 +j

which for=0 is 1 and for=10,000 is approximately−j/10,000 (i.e., corresponding toH(j 0 )

andH(j10,000)=H∗(j10,000)). n

Fourier and Laplace

French mathematician Jean-Baptiste-Joseph Fourier (1768–1830) was a contemporary of Laplace with whom he shared

many scientific and political experiences [2, 7]. Like Laplace, Fourier was from very humble origins but he was not as

politically astute. Laplace and Fourier were affected by the political turmoil of the French Revolution and both came in close

contact with Napoleon Bonaparte, French general and emperor. Named chair of the mathematics department of the Ecole

Normale, Fourier led the most brilliant period of mathematics and science education in France. His main work was “The

Mathematical Theory of Heat Conduction” where he proposed the harmonic analysis of periodic signals. In 1807 he received

the grand prize from the French Academy of Sciences for this work. This was despite the objections of Laplace, Lagrange,

and Legendre, who were the referees and who indicated that the mathematical treatment lacked rigor. Following Galton’s

advice of “Never resent criticism, and never answer it,” Fourier disregarded these criticisms and made no change to his

1822 treatise in heat conduction. Although Fourier was an enthusiast for the Revolution and followed Napoleon on some

of his campaigns, in the Second Restoration he had to pawn his belongings to survive. Thanks to his friends, he became

secretary of the French Academy, the final position he held.

4.3 Complex Exponential Fourier Series

The Fourier series is a representation of a periodic signalx(t)in terms of complex exponentials or
sinusoids of frequency multiples of the fundamental frequency ofx(t). The advantage of using the
Fourier series to represent periodic signals is not only the spectral characterization obtained, but in
finding the response for these signals when applied to LTI systems by means of the eigenfunction
property.

Mathematically, the Fourier series is an expansion of periodic signals in terms of normalized orthog-
onal complex exponentials. The concept of orthogonality of functions is similar to the concept of