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It is said that a frequency domain equalizer is equivalent to an infinite length time domain equalizer, why is that?

And if this is the case, why hasn't frequency domain equalization been used earlier?

Has time domain equalization any advantages when compared with frequency domain equalization?

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Frequency domain equalization requires a certain block structure to be efficient. In OFDM, adding a cyclic prefix to the transmitted block, and using IDFT at the transmitter and DFT at the receiver allows the use of one-tap frequency equalizer. In the time domain, the equalizer length is proportional to the number of channel taps. We can use frequency domain equalization in single carrier systems as well, again with proper block structure, and operations.

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