17
Fourier Transform
Also, we can calculate the magnitude and phase of G(f) as follows:
1
G f
( ) =
1+ j f
2p
1
=
(2.5)
1 2
+ ( pf )
2
and
∠G f
( ) = −∠ (1 + j2pf )
= − tan
−1
( 2pf )
(2.6)
Example 2.2
Impulse function δ(t) (also known as Dirac function) plays an important role in
many areas of science and engineering such as physics, differential equations, and
signal processing. Before describing the mathematical definition of the impulse
function and calculating the FT of this function, we focus on the concept of the
impulse function and the need for such a mathematical entity.
Impulse function is a mathematical function that describes very fast bursts
of energy that are observed in some physical phenomena. As an example of
such burst pulses, consider the effect of smashing the ball in a volleyball game.
Volleyball players are not allowed to hold the ball in their hands; rather, they
are supposed to hit the ball. When smashing the ball, players apply a significant
amount of energy in a very short time. Such an impulse force applied to the ball
creates the most effective move in a volleyball game. The effects of such an action
can be modeled using an impulse function.
Impulse function plays an important role in the identification of unknown systems.
This role can be described through a simple example. Suppose you are given a black
box and you are asked to discover what is in the box. One quick way of guessing the
contents of the box is tapping on it and listening to the echoes. In a more scientific
world, you would apply some fast pressure impulses on the surface of the box and
observe the response of the contents in the box with respect to the impulse functions
you applied. If the box contains coins, you will hear a jingling sound, and if the box is
full of water, an entirely different sound and echo will be sensed. This type of identifying unknown systems is a fundamental technique in a field of science called “system
identification,” which focuses on modeling and describing unknown systems.
Now, we slowly approach a mathematical formulation of the impulse function. The burst-like concept of the impulse function implies that the mathematical
model must be zero for all points in time except for an infinitely small time interval
(as discussed earlier). In a more mathematical manner, assuming that the impulse
is applied at time t = 0, the mathematical representation of the impulse function
must be zero everywhere except for a small neighborhood around the origin. If
the impulse is assumed to be nonzeros for a very short period of time, then the
amplitude of impulse during this very short interval of time must be infinitely
large; otherwise, the total energy of the signal would become zero. In order to
see this more clearly, we focus on a mathematical model of the impulse function.
Consider function δ Δ (t) shown in Figure 2.1. Note that the energy of this signal
Fourier Transform
Also, we can calculate the magnitude and phase of G(f) as follows:
1
G f
( ) =
1+ j f
2p
1
=
(2.5)
1 2
+ ( pf )
2
and
∠G f
( ) = −∠ (1 + j2pf )
= − tan
−1
( 2pf )
(2.6)
Example 2.2
Impulse function δ(t) (also known as Dirac function) plays an important role in
many areas of science and engineering such as physics, differential equations, and
signal processing. Before describing the mathematical definition of the impulse
function and calculating the FT of this function, we focus on the concept of the
impulse function and the need for such a mathematical entity.
Impulse function is a mathematical function that describes very fast bursts
of energy that are observed in some physical phenomena. As an example of
such burst pulses, consider the effect of smashing the ball in a volleyball game.
Volleyball players are not allowed to hold the ball in their hands; rather, they
are supposed to hit the ball. When smashing the ball, players apply a significant
amount of energy in a very short time. Such an impulse force applied to the ball
creates the most effective move in a volleyball game. The effects of such an action
can be modeled using an impulse function.
Impulse function plays an important role in the identification of unknown systems.
This role can be described through a simple example. Suppose you are given a black
box and you are asked to discover what is in the box. One quick way of guessing the
contents of the box is tapping on it and listening to the echoes. In a more scientific
world, you would apply some fast pressure impulses on the surface of the box and
observe the response of the contents in the box with respect to the impulse functions
you applied. If the box contains coins, you will hear a jingling sound, and if the box is
full of water, an entirely different sound and echo will be sensed. This type of identifying unknown systems is a fundamental technique in a field of science called “system
identification,” which focuses on modeling and describing unknown systems.
Now, we slowly approach a mathematical formulation of the impulse function. The burst-like concept of the impulse function implies that the mathematical
model must be zero for all points in time except for an infinitely small time interval
(as discussed earlier). In a more mathematical manner, assuming that the impulse
is applied at time t = 0, the mathematical representation of the impulse function
must be zero everywhere except for a small neighborhood around the origin. If
the impulse is assumed to be nonzeros for a very short period of time, then the
amplitude of impulse during this very short interval of time must be infinitely
large; otherwise, the total energy of the signal would become zero. In order to
see this more clearly, we focus on a mathematical model of the impulse function.
Consider function δ Δ (t) shown in Figure 2.1. Note that the energy of this signal