frequency domain

1. The
Frequency
Domain

2. • French mathematician Joseph Fourier discovered that
 any periodic function can be expressed as a sum of
sines and cosines
• A periodic function has four important attributes:
 its amplitude, period, frequency, and phase.
• The frequency f is the inverse of the period (f = 1/P). It is
expressed in cycles per second, or hertz (Hz).
• To understand the concept of frequency domain, let’s
look at two simple examples.

3. • The function g(t) = sin(2πft) + (1/3)sin(2π(3f)t)
 is a combination of two sine waves with
 amplitudes 1 and 1/3,
 frequencies f and 3f
• Waves are periodic.
 Its sum is also periodic with frequency f that is...the
smaller of the two frequencies f and 3f.

4. Frequency domain
Time domain

5. • The frequency domain of g(t) is a function consisting of
just the two points
 (f, 1) and (3f, 1/3) shown in fig
• It indicates that the original function is made up of
 frequency f with amplitude 1, and
 frequency 3f with amplitude 1/3.
• This example is extremely simple, since it involves just
two frequencies.

6. • Consider the single square pulse
• Its time domain is
g(t) = { 1, −a/2 ≤ t ≤ a/2,
{ 0, elsewhere,
• but its frequency domain is
• It consists of all the frequencies from 0 to ∞, amplitudes
that drop continuously

7. • In general, a periodic function can be represented in the
frequency domain as the
 sum of sine waves with frequencies that are integer
multiples of some fundamental frequency.
• But the square pulse is not periodic
 Therefore frequency domain concepts can be applied to
nonperiodic functions too
• The spectrum of the frequency is the range of
frequencies it contains.
 Frequency spectrum is also called frequency content

8. • In the first example, the spectrum is
 the two frequencies f and 3f.
• In the second example,
 it is the entire range [0,∞].
• The bandwidth of the frequency domain is the width of
the spectrum.
 It is 2f in our first example
 and infinity in the second example.