Fourier Transform

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

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

and the inverse Fourier transform is defined by,

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

Explanation:

Consider the exponential Fourier series,

$f(t) = \sum\limits_{ - \infty }^\infty  {{C_n}{e^{jn\omega t}}}$

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

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

The following changes in notation are appropiate.

Any angular frequency,          $n\omega  \to \omega$

Spacing between adjacent components, $\omega  \to \Delta \omega$

Period, $T \to \frac{{2\pi }}{{\Delta \omega }}$

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

This is the Fourier Transform of f(t), i.e.,$F(j\omega )$

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

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

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

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

$t \to  \pm \infty$ determines the convergence of the Fourier transform.

The Fourier transform will exist if

$\int\limits_{ - \infty }^\infty  {\left| {f(t)} \right|} dt < \infty$

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; $z(t) = Ax(t) + By(t)$

For Frequency Domain; $Z(\omega ) = AX(\omega ) + BY(\omega )$

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.

                                    $x(t) = x( - t)$

and so,                        $X(\omega ) = X( - \omega )$

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.

                                     $x( - t) =  - x(t)$

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: ${x^*}(t) = x(t)$

For even: $x(t) = x( - t)$

For real and even:${x^*}(t) = x( - t)$

After fourier transform; ${X^*}( - \omega ) = X( - \omega )$

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

case2: Real and Odd function

For real: ${x^*}(t) = x(t)$

For odd: $x( - t) =  - x(t)$

For real and odd:${x^*}(t) =  - x( - t)$

After fourier transform;${X^*}( - \omega ) =  - X( - \omega )$

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

case3: Imaginary and Even functiion

For imaginary:${x^*}(t) =  - x(t)$

For even:For even: $x(t) = x( - t)$

For imaginary and even:${x^*}(t) =  - x( - t)$

After fourier transform;${X^*}( - \omega ) =  - X(\omega )$

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

case4:Imaginary and Odd function

For imaginary:${x^*}(t) =  - x(t)$

For odd: $x( - t) =  - x(t)$

For imaginary and odd function:${x^*}(t) = x( - t)$

After fourier transform;${X^*}( - \omega ) = X( - \omega )$

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,$\pi X(\omega )\partial (\omega )$ 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:

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