# OFDM time vs. frequency domain channel estimation/equalization

In OFDM, the majority of Equalizers are used in frequency domain. I mean the signal is transmitted in time domain (after performing iFFT), then to estimate the channel we should perform FFt to use any type of equalization such that LS or MMSE.

My question, what's about if we perform the equalization in time domain? it means at receiver side, we perform any type of estimation/equalization before performing fft (estimate and equalize in time domain)? What's the advantages of using frequency domain OR using time domain estimation and equalization?

Thank you

Traditionally, OFDM became popular in WiFi and LTE because the channel model consisted of multi-path. That is, the radio signal transmitted in 1-6GHz frequencies bounced from various obstacles (walls, trees, cars, humans) at the receiver. Of course this is time varying because obstacle position or transmitter/receiver position also changes. But to simplify the computation, if we consider a multi-path channel at time $$t$$, its baseband model for $$L$$ tap multipath is

$$h[n] = \sum_{k=0}^{L-1}a_k\delta[n-k]$$ where $$a_k = r_ke^{j\phi_k}$$ is the complex tap for path $$k$$. If you plot the FFT of this channel, you will find it is not a flat but rather having different gains at different frequencies. To equalize this in time domain, you would use adaptive filtering techniques. The problem is complexity and time taken for adaptive filtering to estimate the channel. By the time adaptive filter error tapers down, channel would have changed (due to movement of transmitter/receiver/obstacle). So for each OFDM burst, there is a need to immediately equalize to demodulate the packet and ultimately show the data on the device! This is where advantage of OFDM comes into picture. Even though the complexity of OFDM is higher (overhead of preamble, pilot symbols, need to maintain orthogonality), the equalization simplifies to a single tap equalizer in frequency domain as the whole frequency selective channel is split into small flat-fading channel due to IFFT -> Cyclic Prefix -> FFT method. You make sure the circular convolution happens by inserting cyclic prefix so in frequency domain it turns into a point-by-point equalization $$Y[k]/H_{eq}[k]$$ where $$Y[k]$$ is the received symbol at subcarrier $$k$$, and $$H_{eq}[k]$$ is the equalizer tap you computed using pilots.

EDIT: I understand the channel model given above is very simplified view of multi-path model. The simplification is explained in the famous book by Prof. David Tse and Prof. Pramod V (https://web.stanford.edu/~dntse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter2.pdf)

• OK, got it. thank you – Fatima_Ali Mar 29 '20 at 4:52
• When you say pilots, isn't the multipath fading estimated by the preamble because then it can estimate every subcarrier? I thought pilots were for CFO – Lewis Kelsey Jun 22 at 18:18