Question

Which of the following statement is not correct?

• A. Laplace transform is a complex Fourier transform
• B. Fourier transform of a function can be obtained from its Laplace transform by replacing s by j\(\omega\)
• C. Fourier transform is the Laplace transform evaluated along the imaginary axis of the s-plane
• D. Convolution integrals cannot be evaluated using Fourier transform

Hint:

The Fourier Transform is a special case of the Laplace Transform evaluated on the \(j\omega\)-axis, provided the ROC includes it.
Convolution Theorem is a fundamental property of Fourier Transforms: \(f_1(t) * f_2(t) \longleftrightarrow F_1(j\omega) F_2(j\omega)\).

Answer 1

The Correct Option is D

(a) "Laplace transform is a complex Fourier transform": The Laplace transform \( F(s) = \int_0^\infty f(t)e^{-st}dt \) with \( s = \sigma + j\omega \). It can be seen as the Fourier transform of \( f(t)e^{-\sigma t}u(t) \). This statement is generally considered true in the sense that they are closely related and Laplace is a generalization. Correct 

(b) "Fourier transform of a function can be obtained from its Laplace transform by replacing \( s \) by \( j\omega \)": This is true if the Region of Convergence (ROC) of the Laplace transform includes the \( j\omega \)-axis. Conditionally True.

(c) "Fourier transform is the Laplace transform evaluated along the imaginary axis of the s-plane": Same as (b). Conditionally True.

(d) "Convolution integrals cannot be evaluated using Fourier transform": This is FALSE. The Convolution Theorem states that convolution in the time domain corresponds to multiplication in the frequency (Fourier) domain: \(\mathcal{F}\{f(t)*g(t)\} = F(j\omega)G(j\omega)\). This property is often used to simplify or evaluate convolutions. Incorrect

\[ \boxed{\text{Convolution integrals cannot be evaluated using Fourier transform}} \]