Understanding Laplace and Fourier Transform Existence Conditions

This question asks about the conditions under which a function can possess a Laplace transform and a Fourier transform. These conditions are related to the integrability of the function or a modified version of it.

Laplace Transform Existence Condition Analysis

The unilateral Laplace transform \(F(s)\) of a function \(f(t)\) is defined as:

\[ L\{f(t)\} = F(s) = \int_{0^-}^{\infty} f(t)e^{-st} dt \]

For this integral to converge, which means for the Laplace transform to exist, the function \(f(t)\) must satisfy certain conditions. A common and sufficient condition for the existence of the Laplace transform (specifically, for absolute convergence) is that the integral of the absolute value of the function multiplied by an exponential term must be finite for some value of the real part of \(s\). Let \(s = \sigma + j\omega\). The condition for absolute convergence is:

\[ \int_{0^-}^{\infty} |f(t)|e^{-\sigma t} dt < \infty \]

for some real value \(\sigma\). This value \(\sigma\) is related to the Region of Convergence (ROC) of the Laplace transform.

Now let's evaluate statements A and B:

• Statement A: In order for a function to possess a Laplace transform, it must obey the condition \(\rm \displaystyle \int_{0^-}^\infty |f(t)|e^{-\alpha t}dt>\infty, \alpha\in Re^+\). This statement says the integral must diverge (be greater than infinity). This is the opposite of the condition for existence. Thus, statement A is incorrect.
• Statement B: In order for a function to possess a Laplace transform, it must obey the condition \(\rm \displaystyle \int_{0^-}^\infty |f(t)|e^{-\alpha t}dt<\infty, \alpha\in Re^+\). This statement says the integral must converge (be less than infinity) for some positive real \(\alpha\). This aligns with the condition for the existence of the Laplace transform for functions that are zero for \(t<0\) and grow no faster than an exponential \(e^{\sigma_0 t}\) as \(t \to \infty\). Thus, statement B is correct, representing a sufficient condition for existence for a class of functions.

Fourier Transform Existence Condition Analysis

The Fourier transform \(F(\omega)\) of a function \(f(t)\) is defined as:

\[ \mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t} dt \]

For this integral to converge, there are several conditions. A strong and commonly cited sufficient condition for the existence of the Fourier transform (specifically, for absolute integrability, which implies the integral converges) is that the integral of the absolute value of the function over the entire real line must be finite:

\[ \int_{-\infty}^{\infty} |f(t)| dt < \infty \]

Functions satisfying this condition are called absolutely integrable or belonging to the \(L_1\) space.

Now let's evaluate statements C and D:

• Statement C: For a function to have a Fourier transform, it must obey the condition \(\rm \displaystyle \int_{-\infty}^\infty |f(t)|dt<\infty, \). This statement says the integral of the absolute value of the function over the entire real line must converge (be less than infinity). This is the condition for absolute integrability, a sufficient condition for the existence of the Fourier transform. Thus, statement C is correct.
• Statement D: For a function to have a Fourier transform, it must obey the condition \(\rm \displaystyle \int_{-\infty}^\infty |f(t)|e^{-\alpha t}<\infty, \). This statement includes the term \(e^{-\alpha t}\) in the integral limits from \(-\infty\) to \(\infty\). This is not the standard condition for the Fourier transform's existence. The Fourier transform integral does not include this decaying exponential factor when defined over the entire real line. Thus, statement D is incorrect.

Conclusion on Correct Statements

Based on the analysis, statement B regarding the Laplace transform and statement C regarding the Fourier transform are the correct conditions for their existence.

The correct statements are B and C.

Let's look at the given options:

• Option 1: A and C only (Incorrect, A is wrong)
• Option 2: B and C only (Correct, B and C are correct)
• Option 3: B and D only (Incorrect, D is wrong)
• Option 4: A and D only (Incorrect, A and D are wrong)

Therefore, the correct option is the one that includes statements B and C only.

Revision Table: Transform Existence Conditions

Additional Information on Transform Existence

While the absolute integrability conditions (\(\int |f(t)|e^{-\sigma t} dt < \infty\) for Laplace and \(\int |f(t)| dt < \infty\) for Fourier) are sufficient for the transforms to exist (converge absolutely), they are not always necessary. For example, the Fourier transform of a constant or a sine wave does not converge absolutely, but their transforms exist in the sense of generalized functions (distributions), yielding Dirac delta functions. Similarly, for the bilateral Laplace transform, convergence can be conditional, meaning the integral \(\int f(t)e^{-st} dt\) converges but \(\int |f(t)|e^{-st} dt\) does not. However, for most introductory cases and the types of questions typically encountered, the absolute convergence conditions presented in statements B and C are the relevant criteria for existence.

The Region of Convergence (ROC) for the Laplace transform is defined by the range of \(\sigma = Re\{s\}\) for which the integral converges. The condition in statement B implies that there exists some \(\alpha\) (which corresponds to \(\sigma\) in \(s = \sigma + j\omega\)) for which the integral converges. The ROC is typically a strip in the s-plane (or a half-plane for causal signals).

For the Fourier transform, the condition \(\int |f(t)| dt < \infty\) corresponds to the ROC of the bilateral Laplace transform including the imaginary axis (\(\sigma=0\)). If the imaginary axis is within the ROC of the bilateral Laplace transform, the Fourier transform exists.