Which method mathematically converts the time function into a sine wave of different frequencies?

The Fast Fourier Transform (FFT) is the correct method for mathematically converting a time function into a sine wave of different frequencies. This is a critical process in signal processing, especially in the analysis of EEG data. The FFT takes a time-domain signal, which varies over time, and transforms it into the frequency domain, revealing the different frequencies that compose the original signal.

This transformation is particularly useful in EEG studies, where brain activity can be characterized by various frequency bands such as delta, theta, alpha, beta, and gamma waves. By using the FFT, EEG technologists can analyze these frequencies to understand more about the brain's electrical activity, identify abnormalities, and diagnose conditions.

In contrast, the other options serve different functions and do not perform this specific transformation. An impedance meter measures the electrical resistance in a circuit, an A/D converter changes analog signals to digital for processing, and a differential amplifier amplifies the difference between two input signals. None of these methods directly involve converting time-domain data into a frequency-domain representation like the FFT does.