The document discusses the frequency domain and its key concepts. The frequency domain represents periodic functions as a sum of sines and cosines with different amplitudes and frequencies. Any periodic function can be characterized by its amplitude, period, frequency (the inverse of period), and phase. The frequency domain shows the frequency spectrum, or range of frequencies that make up a function. It indicates the amplitudes of different frequency components.

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.