Fourier Transforms of Special Functions

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Fourier Transforms of Special Functions

• ??????

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Content

• Introduction
• More on Impulse Function
• Fourier Transform Related to Impulse Function
• Fourier Transform of Some Special Functions
• Fourier Transform vs. Fourier Series

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Introduction

• Sufficient condition for the existence of a
  Fourier transform

• That is, f(t) is absolutely integrable.
• However, the above condition is not the necessary
  one.

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Some Unabsolutely Integrable Functions

• Sinusoidal Functions cos ?t, sin ?t,
• Unit Step Function u(t).
• Generalized Functions
• Impulse Function ?(t) and
• Impulse Train.

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Fourier Transforms of Special Functions

• More on
• Impulse Function

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Dirac Delta Function
and
Also called unit impulse function.
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Generalized Function

• The value of delta function can also be defined
  in the sense of generalized function

?(t) Test Function

• We shall never talk about the value of ?(t).
• Instead, we talk about the values of integrals
  involving ?(t).

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Properties of Unit Impulse Function
Pf)
Write t as t t0
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Properties of Unit Impulse Function
Pf)
Write t as t/a
Consider agt0
Consider alt0
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Properties of Unit Impulse Function
Pf)
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Properties of Unit Impulse Function
Pf)
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Properties of Unit Impulse Function
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Generalized Derivatives
The derivative f(t) of an arbitrary generalized
function f(t) is defined by
Show that this definition is consistent to the
ordinary definition for the first derivative of a
continuous function.
0
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Derivatives of the ?-Function
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Product Rule
Pf)
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Product Rule
Pf)
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Unit Step Function u(t)

• Define

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Derivative of the Unit Step Function

• Show that

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Derivative of the Unit Step Function
Derivative
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Fourier Transforms of Special Functions

• Fourier Transform Related to
• Impulse Function

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Fourier Transform for ?(t)
F
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Fourier Transform for ?(t)
Show that
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Fourier Transform for ?(t)
Show that
Converges to ?(t) in the sense of generalized
function.
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Two Identities for ?(t)
These two ordinary integrations themselves are
meaningless.
They converge to ?(t) in the sense of generalized
function.
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Shifted Impulse Function
Use the fact
F
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Fourier Transforms of Special Functions

• Fourier Transform of a Some Special Functions

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Fourier Transform of a Constant
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Fourier Transform of a Constant
F
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Fourier Transform of Exponential Wave
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Fourier Transforms of Sinusoidal Functions
F
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Fourier Transform of Unit Step Function
Let
F(j?)?
Can you guess it?
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Fourier Transform of Unit Step Function
Guess
0 B(?) must be odd
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Fourier Transform of Unit Step Function
Guess
0
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Fourier Transform of Unit Step Function
Guess
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Fourier Transform of Unit Step Function
F
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Fourier Transforms of Special Functions

• Fourier Transform vs. Fourier Series

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Find the FT of a Periodic Function

• Sufficient condition --- existence of FT

• Any periodic function does not satisfy this
  condition.
• How to find its FT (in the sense of general
  function)?

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Find the FT of a Periodic Function
We can express a periodic function f(t) as
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Find the FT of a Periodic Function
We can express a periodic function f(t) as
The FT of a periodic function consists of a
sequence of equidistant impulses located at the
harmonic frequencies of the function.
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ExampleImpulse Train
Find the FT of the impulse train.
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ExampleImpulse Train
cn
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ExampleImpulse Train
?0
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ExampleImpulse Train
F
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Find Fourier Series Using Fourier Transform
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Find Fourier Series Using Fourier Transform
Sampling the Fourier Transform of fo(t) with
period 2?/T, we can find the Fourier Series of f
(t).
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ExampleThe Fourier Series of a Rectangular Wave
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ExampleThe Fourier Transform of a Rectangular
Wave
F f(t)?