Chapter 18, Problem 1RQ

To determine

Choose the correct option which does not have a Fourier transform.

Answer to Problem 1RQ

The correct option is $\underset{\_}{\left(\text{c}\right)\frac{1}{t}}$ that does not have a Fourier transform.

Explanation of Solution

Formula used:

Consider the general form of Fourier transform of $f\left(t\right)$ is represented as $F\left(\omega \right)$.

$F\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f\left(t\right)e^{-j\omega t}dt}$ (1)

Calculation:

The Fourier transform is defined as, it is an integral transformation of $f\left(t\right)$ from time domain to frequency domain.

Refer to Table 18.2 in the textbook for Fourier transform pairs.

Option (a):

Substitute $e^{t}u\left(-t\right)$ for $f\left(t\right)$ in equation (1).

$\begin{array}{c}F\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left(e^{t}u\left(-t\right)\right)e^{-j\omega t}dt}\\ =\frac{1}{1-j\omega }\end{array}$

Option (b):

Substitute $t{e}^{-3t}u\left(t\right)$ for $f\left(t\right)$ in equation (1).

$\begin{array}{c}F\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left(t{e}^{-3t}u\left(t\right)\right)e^{-j\omega t}dt}\\ =\frac{1}{{\left(3+j\omega \right)}^2}\end{array}$

Option (d):

Substitute $\left|t\right|u\left(t\right)$ for $f\left(t\right)$ in equation (1).

$\begin{array}{c}F\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left(\left|t\right|u\left(t\right)\right)e^{-j\omega t}dt}\\ =\frac{-2}{{\omega }^2}\end{array}$

Therefore options (a), (b) and (d) has Fourier transform and option (c) does not have a Fourier transform.

Conclusion:

Thus, the correct option is $\underset{\_}{\left(\text{c}\right)\frac{1}{t}}$ that does not have a Fourier transform.