*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 correct answer to the question about the function performed by a Fast Fourier Transform (FFT) is:

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

### Explanation of the Correct Answer:
- The Fast Fourier Transform is an algorithm that efficiently computes the Discrete Fourier Transform (DFT) of a sequence. 
- The DFT transforms a sequence of time-domain samples into their frequency-domain representation, allowing us to analyze the frequency components of the signal.
- This is crucial in many applications, such as signal processing, communications, and audio analysis.

### Why the Other Choices are Incorrect:
- **A: Converting analog signals to digital form**  
  This describes the process of analog-to-digital conversion (ADC), which is not the function of an FFT. An FFT operates on digital signals, not directly on analog signals.

- **B: Converting digital signals to analog form**  
  This describes digital-to-analog conversion (DAC), which is the opposite of what an FFT does. An FFT does not convert digital signals back to analog; it analyzes the digital signal in the frequency domain.

- **D: Converting signals from the frequency domain to the time domain**  
  This describes the Inverse Fast Fourier Transform (IFFT), which is used to convert frequency-domain data back into the time domain. The FFT itself does not perform this function.

### Useful Formulas:
- The relationship between the time domain and frequency domain can be expressed as:
  
$$
X(f) = \sum_{n=0}^{N-1} x(n) e^{-j 2 \pi f n / N}
$$

where:
- $X(f)$ = frequency domain representation
- $x(n)$ = time domain samples
- $N$ = number of samples
- $j$ = imaginary unit

This formula illustrates how the FFT processes time-domain samples to produce frequency-domain data. Understanding this relationship is key for grasping the function of the FFT in signal processing.