3.1 Properties of the fourier transform

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Several of the most important properties of the Fourier transform are derived.

Properties of the fourier transform

The Fourier Transform (FT) has several important properties which will be useful:

1. Linearity:
$\alpha {x}_{1}\left(t\right)+\beta {x}_{2}\left(t\right)↔\alpha {X}_{1}\left(j\Omega \right)+\beta {X}_{2}\left(j\Omega \right)$
where $\alpha$ and $\beta$ are constants. This property is easy to verify by plugging the left side of [link] into the definition of the FT.
2. Time shift:
$x\left(t-\tau \right)↔{e}^{-j\Omega \tau }X\left(j\Omega \right)$
To derive this property we simply take the FT of $x\left(t-\tau \right)$
${\int }_{-\infty }^{\infty }x\left(t-\tau \right){e}^{-j\Omega t}dt$
using the variable substitution $\gamma =t-\tau$ leads to
$t=\gamma +\tau$
and
$d\gamma =dt$
We also note that if $t=±\infty$ then $\tau =±\infty$ . Substituting [link] , [link] , and the limits of integration into [link] gives
$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \left(\gamma +\tau \right)}d\gamma & =& {e}^{-j\Omega \tau }{\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \gamma }d\gamma \hfill \\ & =& {e}^{-j\Omega \tau }X\left(j\Omega \right)\hfill \end{array}$
which is the desired result.
3. Frequency shift:
$x\left(t\right){e}^{j{\Omega }_{0}t}↔X\left(j\left(\Omega -{\Omega }_{0}\right)\right)$
Deriving the frequency shift property is a bit easier than the time shift property. Again, using the definition of FT we get:
$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(t\right){e}^{j{\Omega }_{0}t}{e}^{-j\Omega t}dt& =& {\int }_{-\infty }^{\infty }x\left(t\right){e}^{-j\left(\Omega -{\Omega }_{0}\right)t}dt\hfill \\ & =& X\left(j\left(\Omega -{\Omega }_{0}\right)\right)\hfill \end{array}$
4. Time reversal :
$x\left(-t\right)↔X\left(-j\Omega \right)$
To derive this property, we again begin with the definition of FT:
${\int }_{-\infty }^{\infty }x\left(-t\right){e}^{-j\Omega t}dt$
and make the substitution $\gamma =-t$ . We observe that $dt=-d\gamma$ and that if the limits of integration for $t$ are $±\infty$ , then the limits of integration for $\gamma$ are $\mp \gamma$ . Making these substitutions into [link] gives
$\begin{array}{ccc}\hfill -{\int }_{\infty }^{-\infty }x\left(\gamma \right){e}^{j\Omega \gamma }d\gamma & =& {\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{j\Omega \gamma }d\gamma \hfill \\ & =& X\left(-j\Omega \right)\hfill \end{array}$
Note that if $x\left(t\right)$ is real, then $X\left(-j\Omega \right)=X{\left(j\Omega \right)}^{*}$ .
5. Time scaling: Suppose we have $y\left(t\right)=x\left(at\right),a>0$ . We have
$Y\left(j\Omega \right)={\int }_{-\infty }^{\infty }x\left(at\right){e}^{-j\Omega t}dt$
Using the substitution $\gamma =at$ leads to
$\begin{array}{cc}\hfill Y\left(j\Omega \right)& =\frac{1}{a}{\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \gamma /a}d\gamma \hfill \\ & =\frac{1}{a}X\left(\frac{\Omega }{a}\right)\hfill \end{array}$
6. Convolution: The convolution integral is given by
$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau$
The convolution property is given by
$Y\left(j\Omega \right)↔X\left(j\Omega \right)H\left(j\Omega \right)$
To derive this important property, we again use the FT definition:
$\begin{array}{ccc}\hfill Y\left(j\Omega \right)& =& {\int }_{-\infty }^{\infty }y\left(t\right){e}^{-j\Omega t}dt\hfill \\ & =& {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right){e}^{-j\Omega t}d\tau dt\hfill \\ & =& {\int }_{-\infty }^{\infty }x\left(\tau \right)\left[{\int }_{-\infty }^{\infty },h,\left(t-\tau \right),{e}^{-j\Omega t},d,t\right]d\tau \hfill \end{array}$
Using the time shift property, the quantity in the brackets is ${e}^{-j\Omega \tau }H\left(j\Omega \right)$ , giving
$\begin{array}{ccc}\hfill Y\left(j\Omega \right)& =& {\int }_{-\infty }^{\infty }x\left(\tau \right){e}^{-j\Omega \tau }H\left(j\Omega \right)d\tau \hfill \\ & =& H\left(j\Omega \right){\int }_{-\infty }^{\infty }x\left(\tau \right){e}^{-j\Omega \tau }d\tau \hfill \\ & =& H\left(j\Omega \right)X\left(j\Omega \right)\hfill \end{array}$
Therefore, convolution in the time domain corresponds to multiplication in the frequency domain.
7. Multiplication (Modulation):
$w\left(t\right)=x\left(t\right)y\left(t\right)↔\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)Y\left(j\Theta \right)d\Theta$
Notice that multiplication in the time domain corresponds to convolution in the frequency domain. This property can be understood by applying the inverse Fourier Transform [link] to the right side of [link]
$\begin{array}{ccc}\hfill w\left(t\right)& =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)Y\left(j\Theta \right){e}^{j\Omega t}d\Theta d\Omega \hfill \\ & =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right)\left[\frac{1}{2\pi },{\int }_{-\infty }^{\infty },X,\left(j\left(\Omega -\Theta \right)\right),{e}^{j\Omega t},d,\Omega \right]d\Theta \hfill \end{array}$
The quantity inside the brackets is the inverse Fourier Transform of a frequency shifted Fourier Transform,
$\begin{array}{ccc}\hfill w\left(t\right)& =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right)\left[x,\left(t\right),{e}^{j\Theta t}\right]d\Theta \hfill \\ & =& x\left(t\right)\frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right){e}^{j\Theta t}d\Theta \hfill \\ & =& x\left(t\right)y\left(t\right)\hfill \end{array}$
8. Duality: The duality property allows us to find the Fourier transform of time-domain signals whose functional forms correspond to known Fourier transforms, $X\left(jt\right)$ . To derive the property, we start with the inverse Fourier transform:
$x\left(t\right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\Omega \right){e}^{j\Omega t}d\Omega$
Changing the sign of $t$ and rearranging,
$2\pi x\left(-t\right)={\int }_{-\infty }^{\infty }X\left(j\Omega \right){e}^{-j\Omega t}d\Omega$
Now if we swap the $t$ and the $\Omega$ in [link] , we arrive at the desired result
$2\pi x\left(-\Omega \right)={\int }_{-\infty }^{\infty }X\left(jt\right){e}^{-j\Omega t}dt$
The right-hand side of [link] is recognized as the FT of $X\left(jt\right)$ , so we have
$X\left(jt\right)↔2\pi x\left(-\Omega \right)$

The properties associated with the Fourier Transform are summarized in [link] .

 Property $y\left(t\right)$ $Y\left(j\Omega \right)$ Linearity $\alpha {x}_{1}\left(t\right)+\beta {x}_{2}\left(t\right)$ $\alpha {X}_{1}\left(j\Omega \right)+\beta {X}_{2}\left(j\Omega \right)$ Time Shift $x\left(t-\tau \right)$ $X\left(j\Omega \right){e}^{-j\Omega \tau }$ Frequency Shift $x\left(t\right){e}^{j{\Omega }_{0}t}$ $X\left(j\left(\Omega -{\Omega }_{0}\right)\right)$ Time Reversal $x\left(-t\right)$ $X\left(-j\Omega \right)$ Time Scaling $x\left(at\right)$ $\frac{1}{a}X\left(\frac{\Omega }{a}\right)$ Convolution $x\left(t\right)*h\left(t\right)$ $X\left(j\Omega \right)H\left(j\Omega \right)$ Modulation $x\left(t\right)w\left(t\right)$ $\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)W\left(j\Theta \right)d\Theta$ Duality $X\left(jt\right)$ $2\pi x\left(-\Omega \right)$

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