*Question E7F07:
What function is performed by a Fast Fourier Transform?*  

*Correct Answer is  C: Converting signals from the time domain to the frequency domain*  

### AI Response: (may contain errors)  

The Fast Fourier Transform (FFT) is a mathematical algorithm used to convert signals from the time domain to the frequency domain. This is why the correct answer is:

**C: Converting signals from the time domain to the frequency domain**

### Explanation:

1. **Correct Answer (C):**
   - The FFT takes a signal that varies over time and breaks it down into its constituent frequencies. This is useful for analyzing the frequency components of a signal, such as identifying different tones in a piece of music or analyzing the spectral content of a radio signal.

2. **Incorrect Answers:**
   - **A: Converting analog signals to digital form**
     - This process is known as Analog-to-Digital Conversion (ADC), not FFT. ADC involves sampling an analog signal at discrete intervals to create a digital representation.
   - **B: Converting digital signals to analog form**
     - This is known as Digital-to-Analog Conversion (DAC), which is the reverse of ADC. It involves converting digital data back into an analog signal.
   - **D: Converting signals from the frequency domain to the time domain**
     - This process is performed by the Inverse Fast Fourier Transform (IFFT), not the FFT. The IFFT takes frequency domain data and reconstructs the time domain signal.

### Useful Formula:

The FFT is a computationally efficient way to perform the Discrete Fourier Transform (DFT), which is defined as:

$$
X(k) = \sum_{n=0}^{N-1} x(n) \cdot e^{-j \frac{2\pi}{N} kn}
$$

- $X(k)$: Frequency domain representation
- $x(n)$: Time domain signal
- $N$: Number of samples
- $j$: Imaginary unit

The FFT algorithm significantly reduces the computational complexity of calculating the DFT, making it practical for real-time signal processing applications.