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As I know that in case of one tap OFDM equalizer, (ZF equalizer), we should have $N$ division complexity since we divide every subcarriers by its correspondent channel tap in frequency domain.

But I couldn't calculate how much exactly the computational complexity in case if we used MMSE equalizer for the same OFDM system.

Could you please help me how to calculate that?

Thank you

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For a SISO channel, assuming you have estimated channel coefficients for each sub-carrier $k$, the MMSE equalizer is $$ \hat{h}_k=\frac{h_k^*}{|h_k|^2 + \frac{\sigma_x^2}{N_0}} $$ So you can see already there is a multiplication in the numerator ($h_k^*$), and then there is a division by a term in the denominator. So for all $N$ subcarrier, this itself will be $2O(N) \rightarrow O(N)$ for large $N$. There is additional $|h_k|^2$ computation which is basically multiplication with conjugate. So if you have precomputed SNR ($\frac{\sigma_x^2}{N_0}$) and $h_k$ for all sub-carriers, complexity should still be linear on $N$. But this is a very simplified assumption because in most practical OFDM packets, there a pilot sequence periodically present so you may need to update $h_k$ using available pilot sequence.

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  • $\begingroup$ Note that this is only correct if the noise samples are uncorrelated. $\endgroup$ – AlexTP May 23 at 12:15
  • $\begingroup$ Thank you for replying. You mean complexity is $N$ ? So why I usually notice the MMSE equalizer provides good results but it's more complex compared to ZF equalizer ?? $\endgroup$ – Fatima_Ali May 24 at 1:15
  • $\begingroup$ @Fatima_Ali MMSE Equalizer is complex in the sense it has to compute the SNR along with additional multiplications compared to ZF Equalizer which is just a single division. So it is about $3N$ computations roughly if you have done precomputations. But if $N$ is large like $1024$, this is still linear in $N$. But instead of $3$ if it was $\log_2 (N) \times N = 10 \times N$, the difference in complexity would be apparent. Afraid to say, computational time order is another vast topic itself (en.wikipedia.org/wiki/Time_complexity). $\endgroup$ – jithin May 24 at 4:42

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