Fourier Transform

The Fourier Transform or Fourier integral of a function f(t) is denoted by [tex]F(j\omega )[/tex] and is defined by,

[tex]F(j\omega ) = F[f(t)] = \int\limits_{ - \infty }^\infty  {f(t){e^{ - j\omega t}}} dt[/tex]

and the inverse Fourier transform is defined by,

[tex]f(t) = {F^{ - 1}}[F(j\omega )] = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty  {F(j\omega )} {e^{j\omega t}}d\omega  = \int\limits_{ - \infty }^\infty  {F(j2\pi f){e^{j2\pi f}}} df[/tex]

Explanation:

Consider the exponential Fourier series,

[tex]f(t) = \sum\limits_{ - \infty }^\infty  {{C_n}{e^{jn\omega t}}} [/tex]

where,[tex]{C_n} = \frac{1}{T}\int\limits_{ - \frac{T}{2}}^{\frac{T}{2}} {f(t){e^{ - jn\omega t}}} dt[/tex]

when f(t) is non periodic transient function,some changes are required. As T approaches infinity, [tex]\omega [/tex] approaches to zero and n becomes meaningless. [tex]{n\omega }[/tex] should be changed to [tex]\omega [/tex] only.

The following changes in notation are appropiate.

Any angular frequency,          [tex]n\omega  \to \omega [/tex]

Spacing between adjacent components, [tex]\omega  \to \Delta \omega [/tex]

Period, [tex]T \to \frac{{2\pi }}{{\Delta \omega }}[/tex]

Hence,[tex]{C_n}T \to \int\limits_{ - \infty }^\infty  {f(t){e^{ - j\omega t}}} dt[/tex]

This is the Fourier Transform of f(t), i.e.,[tex]F(j\omega )[/tex]

Now,[tex]f(t) = \sum\limits_{ - \infty }^\infty  {({C_n}T){e^{jn\omega t}}} (\frac{1}{T})[/tex]

As [tex]T \to \infty [/tex] , [tex]{C_n}T \to F(j\omega )[/tex], [tex]n\omega  \to \omega [/tex] and [tex]T \to \frac{{2\pi }}{{\Delta \omega }}[/tex] and [tex]\sum { \to \int {} } [/tex](summation approaches integration).

thus, [tex]f(t) = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty  {F(j\omega )} {e^{j\omega t}}d\omega [/tex]

Convergence of Fourier Transform:

When f(t) is singled valued function and is different from zero over an interval of time, the behavior of f(t) as

[tex]t \to  \pm \infty [/tex] determines the convergence of the Fourier transform.

The Fourier transform will exist if

[tex]\int\limits_{ - \infty }^\infty  {\left| {f(t)} \right|} dt < \infty [/tex]

Properties of Fourier Transform:

1.Linearity property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
y\left( t \right) \leftrightarrow Y(\omega )\\
z(t) \leftrightarrow Z(\omega )
\end{array}\]

For Time Domain; [tex]z(t) = Ax(t) + By(t)[/tex]

For Frequency Domain; [tex]Z(\omega ) = AX(\omega ) + BY(\omega )[/tex]

When a signal is linearly related in time domain it is also linearly related in frequency domain.

2. Time shifting property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
x\left( {t - {t_0}} \right) \leftrightarrow {e^{ - j\omega {t_0}}}X(\omega )\\
x(t + {t_0}) \leftrightarrow {e^{ + j\omega {t_0}}}X(\omega )
\end{array}\]

3. Time reversal property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
x( - t) \leftrightarrow X( - \omega )
\end{array}\]

case1: if x(t) be even function.

                                    [tex]x(t) = x( - t)[/tex]

and so,                        [tex]X(\omega ) = X( - \omega )[/tex]

If a function be even in time domain it's fourier transform is also even in frequency domain.

case2:if x(t) be odd function.

                                     [tex]x( - t) =  - x(t)[/tex]

and so,                         [tex]X( - \omega ) =  - X(\omega )[tex]

If a function is odd in time domain then it's fourier transform is also odd in frequency domain.

4. Scaling property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
x(at) \leftrightarrow \frac{1}{{\left| a \right|}}X\left( {\frac{\omega }{a}} \right)
\end{array}\]

5. Conjugate Symmetry Property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
{x^*}(t) \leftrightarrow {X^*}( - \omega )
\end{array}\]

case1: Real and Even function

For real: [tex]{x^*}(t) = x(t)[/tex]

For even: [tex]x(t) = x( - t)[/tex]

For real and even:[tex]{x^*}(t) = x( - t)[/tex]

After fourier transform; [tex]{X^*}( - \omega ) = X( - \omega )[/tex]

Hence, fourier transform of a real and even function will be also real and even.

case2: Real and Odd function

For real: [tex]{x^*}(t) = x(t)[/tex]

For odd: [tex]x( - t) =  - x(t)[/tex]

For real and odd:[tex]{x^*}(t) =  - x( - t)[/tex]

After fourier transform;[tex]{X^*}( - \omega ) =  - X( - \omega )[/tex]

Hence, fourier transform of a real and odd function will be imaginary and odd.

case3: Imaginary and Even functiion

For imaginary:[tex]{x^*}(t) =  - x(t)[/tex]

For even:For even: [tex]x(t) = x( - t)[/tex]

For imaginary and even:[tex]{x^*}(t) =  - x( - t)[/tex]

After fourier transform;[tex]{X^*}( - \omega ) =  - X(\omega )[/tex]

Hence, fourier transform of a imaginary and even function will be also imaginary and even.

case4:Imaginary and Odd function

For imaginary:[tex]{x^*}(t) =  - x(t)[/tex]

For odd: [tex]x( - t) =  - x(t)[/tex]

For imaginary and odd function:[tex]{x^*}(t) = x( - t)[/tex]

After fourier transform;[tex]{X^*}( - \omega ) = X( - \omega )[/tex]

Hence, fourier transform of a imaginary and odd function will be real and odd.

6. Differentiation in time domain:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
\frac{{dx(t)}}{{dt}} \leftrightarrow j\omega X(\omega )
\end{array}\]

7.Integration in time domain:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
\int\limits_{ - \infty }^t {x(t)dt \leftrightarrow \frac{{X(\omega )}}{{j\omega }}}  + \pi X(\omega )\partial (\omega )
\end{array}\]

The term,[tex]\pi X(\omega )\partial (\omega )[/tex] will exist if x(t) is finite.

8.Duality property:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
y(t) \leftrightarrow 2\pi X( - \omega )
\end{array}\]

9.Differentiation in frequency domain:

\[\begin{array}{l}
x(t) \leftrightarrow X(\omega )\\
tx(t) \leftrightarrow j\frac{{dX(\omega )}}{{d\omega }}
\end{array}\]