# Should I consider systems to be causal or non-causal by default in questions like these?

Could anyone explain to me when to consider one-sided Fourier transform and when to consider two-sided Fourier transform? Normally in a question, just an input signal is mentioned, without stating anything about the causality. Do we by default consider a system to be causal or non-causal? What's the norm?

For example, let us take this question:

For part (a), I was considering doing a Fourier transform of $v(t)$, and then doing the inverse Fourier transform, with the frequencies above $\omega_c=a$, and below $w_c=-a$ left out. If the inverse Fourier transform is $v'(t)$ (say), then I guess the answer should be either $\int_{-\infty}^{+\infty}(v'(t))^2dt$ or $\int_{0}^{+\infty}(v'(t))^2dt$. Not sure which.

So, should I do the Fourier transform like

$$\hat{V}(\omega)=\int_{-\infty}^{\infty}e^{-a|t|}e^{-jwt}dt$$

or

$$\hat{V}(\omega)=\int_{0}^{\infty}e^{-a|t|}e^{-jwt}dt$$ ?

• Whatever that integral from zero to infinity is, it's not a Fourier Transform. And the other one isn't a Fourier Transform, either. So, if you need to do a Fourier transform, neither one of these will be it. Commented Apr 21, 2018 at 19:46
• @MarcusMüller Ah, okay. That was a typo.
– user186505
Commented Apr 21, 2018 at 19:49

One way to solve this might be using Parseval's theorem, $$\int_{-\infty}^\infty | v(t) |^2 \, \mathrm{d}t = \frac{1}{2\pi} \int_{-\infty}^\infty | V(\omega) |^2 \, \mathrm{d}\omega,$$ and changing the limits of integration from $\infty$ to $a$.
The text passage discusses considerations for choosing between one-sided and two-sided Fourier transforms, particularly in the context of signal processing and system causality. It also includes a specific example problem involving the calculation of the energy of a signal using Parseval's theorem.
single_question_with_answer

the passage contains a single mathematical question about how to calculate the energy of a signal using the fourier transform, and it provides a potential method for solving it using parseval's theorem.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Should I consider systems to be causal or non-causal by default in questions like these?

Could anyone explain to me when to consider one-sided Fourier transform and when to consider two-sided Fourier transform? Normally in a question, just an input signal is mentioned, without stating anything about the causality. Do we by default consider a system to be causal or non-causal? What's the norm?

For example, let us take this question:

For part (a), I was considering doing a Fourier transform of $v(t)$, and then doing the inverse Fourier transform, with the frequencies above $\omega_c=a$, and below $w_c=-a$ left out. If the inverse Fourier transform is $v'(t)$ (say), then I guess the answer should be either $\int_{-\infty}^{+\infty}(v'(t))^2dt$ or $\int_{0}^{+\infty}(v'(t))^2dt$. Not sure which.

So, should I do the Fourier transform like

$$\hat{V}(\omega)=\int_{-\infty}^{\infty}e^{-a|t|}e^{-jwt}dt$$

or

$$\hat{V}(\omega)=\int_{0}^{\infty}e^{-a|t|}e^{-jwt}dt$$ ?

• Whatever that integral from zero to infinity is, it's not a Fourier Transform. And the other one isn't a Fourier Transform, either. So, if you need to do a Fourier transform, neither one of these will be it. Commented Apr 21, 2018 at 19:46
• @MarcusMüller Ah, okay. That was a typo.
– user186505
Commented Apr 21, 2018 at 19:49

One way to solve this might be using Parseval's theorem, $$\int_{-\infty}^\infty | v(t) |^2 \, \mathrm{d}t = \frac{1}{2\pi} \int_{-\infty}^\infty | V(\omega) |^2 \, \mathrm{d}\omega,$$ and changing the limits of integration from $\infty$ to $a$.
user
The text passage discusses considerations for choosing between one-sided and two-sided Fourier transforms, particularly in the context of signal processing and system causality. It also includes a specific example problem involving the calculation of the energy of a signal using Parseval's theorem.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question about how to calculate the energy of a signal using the Fourier transform, and it provides a potential method for solving it using Parseval's theorem.