3.1 Properties of the fourier transform

Properties of the fourier transform

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

1. Linearity:
   $$\alpha x_1\left(t\right)+\beta x_2\left(t\right)\leftrightarrow \alpha X_1\left(j\Omega \right)+\beta X_2\left(j\Omega \right)$$
   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:
   $$x(t-\tau )\leftrightarrow e^{-j\Omega \tau }X\left(j\Omega \right)$$
   To derive this property we simply take the FT of $x(t-\tau )$
   $${\int }_{-\infty }^{\infty }x(t-\tau )e^{-j\Omega t}dt$$
   using the variable substitution $\gamma =t-\tau$ leads to
   $$t=\gamma +\tau$$
   and
   $$d\gamma =dt$$
   We also note that if $t=\pm \infty$ then $\tau =\pm \infty$ . Substituting [link] , [link] , and the limits of integration into [link] gives
   $$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(\gamma \right)e^{-j\Omega (\gamma +\tau )}d\gamma & =& e^{-j\Omega \tau }{\int }_{-\infty }^{\infty }x\left(\gamma \right)e^{-j\Omega \gamma }d\gamma \hfill \\ & =& e^{-j\Omega \tau }X\left(j\Omega \right)\hfill \end{array}$$
   which is the desired result.
3. Frequency shift:
   $$x\left(t\right)e^{j{\Omega }_{0}t}\leftrightarrow X\left(j(\Omega -{\Omega }_0)\right)$$
   Deriving the frequency shift property is a bit easier than the time shift property. Again, using the definition of FT we get:
   $$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(t\right)e^{j{\Omega }_{0}t}{e}^{-j\Omega t}dt& =& {\int }_{-\infty }^{\infty }x\left(t\right)e^{-j(\Omega -{\Omega }_0)t}dt\hfill \\ & =& X\left(j(\Omega -{\Omega }_0)\right)\hfill \end{array}$$
4. Time reversal :
   $$x(-t)\leftrightarrow X(-j\Omega )$$
   To derive this property, we again begin with the definition of FT:
   $${\int }_{-\infty }^{\infty }x(-t)e^{-j\Omega t}dt$$
   and make the substitution $\gamma =-t$ . We observe that $dt=-d\gamma$ and that if the limits of integration for $t$ are $\pm \infty$ , then the limits of integration for $\gamma$ are $\mp \gamma$ . Making these substitutions into [link] gives
   $$\begin{array}{ccc}\hfill -{\int }_{\infty }^{-\infty }x\left(\gamma \right)e^{j\Omega \gamma }d\gamma & =& {\int }_{-\infty }^{\infty }x\left(\gamma \right)e^{j\Omega \gamma }d\gamma \hfill \\ & =& X(-j\Omega )\hfill \end{array}$$
   Note that if $x\left(t\right)$ is real, then $X(-j\Omega )=X{\left(j\Omega \right)}^{*}$ .
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
   $$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h(t-\tau )d\tau$$
   The convolution property is given by
   $$Y\left(j\Omega \right)\leftrightarrow X\left(j\Omega \right)H\left(j\Omega \right)$$
   To derive this important property, we again use the FT definition:
   $$\begin{array}{ccc}\hfill Y\left(j\Omega \right)& =& {\int }_{-\infty }^{\infty }y\left(t\right)e^{-j\Omega t}dt\hfill \\ & =& {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }x\left(\tau \right)h(t-\tau )e^{-j\Omega t}d\tau dt\hfill \\ & =& {\int }_{-\infty }^{\infty }x\left(\tau \right)\left[{\int }_{-\infty }^{\infty },h,(t-\tau ),e^{-j\Omega t},d,t\right]d\tau \hfill \end{array}$$
   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):
   $$w\left(t\right)=x\left(t\right)y\left(t\right)\leftrightarrow \frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j(\Omega -\Theta )\right)Y\left(j\Theta \right)d\Theta$$
   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]
   $$\begin{array}{ccc}\hfill w\left(t\right)& =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j(\Omega -\Theta )\right)Y\left(j\Theta \right)e^{j\Omega t}d\Theta d\Omega \hfill \\ & =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right)\left[\frac{1}{2\pi },{\int }_{-\infty }^{\infty },X,\left(j(\Omega -\Theta )\right),e^{j\Omega t},d,\Omega \right]d\Theta \hfill \end{array}$$
   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,
   $$2\pi x(-t)={\int }_{-\infty }^{\infty }X\left(j\Omega \right)e^{-j\Omega t}d\Omega$$
   Now if we swap the $t$ and the $\Omega$ in [link] , we arrive at the desired result
   $$2\pi x(-\Omega )={\int }_{-\infty }^{\infty }X\left(jt\right)e^{-j\Omega t}dt$$
   The right-hand side of [link] is recognized as the FT of $X\left(jt\right)$ , so we have
   $$X\left(jt\right)\leftrightarrow 2\pi x(-\Omega )$$

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