Time Domain To Frequency Domain Calculator Formula

Fourier Transform Formula:

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

Frequency Domain Result F(ω):

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1. What is the Fourier Transform?

The Fourier Transform is a mathematical operation that transforms a time-domain function into its frequency-domain representation. It decomposes a function into its constituent frequencies, revealing the frequency spectrum of the original function.

2. How Does the Calculator Work?

The calculator uses the Fourier Transform formula:

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

Where:

\( F(\omega) \) — Frequency domain representation
\( f(t) \) — Time domain function
\( \omega \) — Angular frequency (rad/s)
\( t \) — Time (seconds)
\( i \) — Imaginary unit

Explanation: The transform integrates the product of the time function and a complex exponential over all time, revealing the frequency content.

3. Importance of Time-Frequency Conversion

Details: The Fourier Transform is fundamental in signal processing, communications, audio analysis, and many areas of physics and engineering where frequency analysis is needed.

4. Using the Calculator

Tips: Enter a time-domain function (like "sin(t)" or "exp(-t)"), angular frequency in rad/s, and time in seconds. The calculator will compute the frequency domain representation.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be transformed?
A: The calculator can handle common functions like sine, cosine, exponential, and polynomial functions.

Q2: What is the difference between angular frequency and regular frequency?
A: Angular frequency (ω) is measured in radians per second, while regular frequency (f) is in Hertz. They are related by ω = 2πf.

Q3: What does the complex result mean?
A: The result includes both magnitude (amplitude) and phase information of the frequency components.

Q4: Are there limitations to this calculator?
A: The calculator provides symbolic results for standard functions. For complex or piecewise functions, specialized software may be needed.

Q5: What's the inverse operation?
A: The Inverse Fourier Transform converts frequency domain back to time domain: \( f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega \).