Filtering
One of the reasons to transform a time-domain signal to a frequency-domain signal is for the purpose of filtering. A filter is an operation that changes a signal. Much like filters in your kitchen sink, filters let some frequencies pass (water), while stopping other frequencies (large food remains). Filters are used in a variety of applications, ranging from audio to radar systems.
Filters are categorized by their behavior. A filter that lets through low frequencies and stops high frequencies is called a low-pass filter (LPF). Similarly, a high-pass filter (HPF) will allow only high frequencies to pass. There are also other categorizations such as band-pass filters (allows only a specific band of frequencies), band-stop filters (allows anything but a specific band of frequencies), and notch filters (suppresses very few frequencies).
Filters are further categorized by their behavior to an impulse input—that is, the output of the filter as a function of time assuming you were to input a short spike to the filter. Filters that eventually will forget the impulse are known as finite-impulse-response (FIR) filters, and filters that never forget are known as infinite-impulse-response (IIR) filters. From a very simplistic approach, if a filter does not rely on previous outputs (no feedback), it is considered an FIR; otherwise, it's an IIR.
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