# Problem using MMSE estimation of channel frequency response

I need to model the minimum mean square error (MMSE) performance in estimating channel frequency response. I have channel's power delay profile (PDP) as a table with tap delays and powers.

The channel frequency response (FR) is computed in frequency domain via $$H(f) = \sum_i a_i \exp\left(j2\pi\tau_i f\right),$$ where $\tau_i$ are delays from the PDP and $a_i$ are independent complex gaussian variables with variances $\sigma^2_i$, taken from the same PDP. MMSE is given by $$\hat{H}_{MMSE} = R_{HH}\left(R_{HH}+\frac{\beta}{SNR}I\right)^{-1}\hat{H}_{LS}.$$ To compute covariance matrix $R_{HH}$ I compute a vector with components $$r_k = \sum_i \sigma_i^2 \exp\left(j2\pi\tau_i f\right)$$ and construct a Toeplitz matrix with it.

To model MSE I compute an $H(f)$ realization, add AWGN and apply an MSSE. In this case $\beta=1$ since transmitted signal is all 1's. The process is repeated N times and MSE is averaged.

The problem is the estimation error is so big that it beats simple LS estimate only on very low SNR and then comes to an error floor.

If instead I compute $R_{HH}$ as $HH^*$, i.e. as an ideal momentary statistic, everything works fine.

Some additional notes. Numerical computation of $R_{HH}$ via lots of channel realizations yields the same result as using the construction with $r_k$, so it's correct and consistent. If the overall concept is fine then the mistake should be in the code. I will post MWE if needed.

• Question: do you add AWGN at H(f)? Normally in problems like this, AWGN is added in the time domain, not in frequency domain. – JohnMarvin Mar 24 '16 at 14:58
• Yes, I add AWGN at H(f). Fourier transform is linear and unitary, so AWGN in time domain transformed to AWGN in frequency domain. And lots of papers on estimation also have it this way. – Zvord Mar 24 '16 at 18:04
• @JohnMarvin True, but the discrete Fourier transform of white Gaussian noise is... White Gaussian noise. :-) – Peter K. Mar 24 '16 at 18:04
• I think this question could be greatly improved by defining the abbreviations when first used. As it is, it requires very specific domain knowledge to recognize the problem being addressed. – Jacob Mar 24 '16 at 19:19