# Minimum SNR Requirements for Maximum Likelihood Frequency Estimation

In certain applications, you have enough SNR available to, for example, perform an FFT and identify peak location and hence the signal frequency. If my understanding is correct, parameter estimation techniques, such as maximum likelihood, will only be useful when your SNR is so low that you can't run a basic peak search to identify frequency.

However, I have seen literature where people plot performance of various estimation algorithms vs SNRs upto 40~50dB. It seems like a waste of processing power to perform ML on a high SNR (40dB is quite a high value) signal and extract the already obvious information?

Is there a 'rule of thumb' to say, use statistical algorithms below such SNR levels?

The FFT is really a bank of matched filters - each FFT bin corresponds to the output of a matched filter. The criterion for a matched filter is to maximize the SNR at the output of the filter - Note SNR at the output is measured differently than at the input.

While the FFT can be used for frequency estimation, it isn't great. The resolution of the FFT is $1/T$ , where $T$ is the duration of the signal.

Now, there are several methods for estimating the frequency (parametric methods, Eigenvalue analysis etc). In measuring the performance of these estimators you are interested in looking at the variance of the estimator at a particular SNR (assuming your estimator is unbiased). You will usually find that the variance of the estimate decreases as the SNR increases - that's why you see people looking at such high SNRs.

At these high SNRs you can use the FFT as an estimator, but the variance of that estimate is probably higher than if you used the other techniques.

• thanks. This was useful. are those statistical techniques any good for small number of samples? Most of the papers on the subject seem to make an implicit assumption that a lot of data is available. – user4673 Jan 23 '14 at 14:03
• what I feel missing is another set of performance curves plotting variance of an estimator vs number of samples acquired?? – user4673 Jan 23 '14 at 14:05
• I would add to your answer, that a matched filter IS a maximum likelihood estimate given that the noise is AWGN. – Tarin Ziyaee Jan 23 '14 at 14:36
• @user4673 If you have enough samples to do an FFT you can use the other techniques, but these techniques have their issues too. Autoregressive models - how many degrees of freedom should you use. Eigenvectors - form an estimate of the correlation matrix - again how many DOF? If you truly have a small number of samples, you could look at compressive sensing or more precisely sparse representation. – David Jan 23 '14 at 15:25
• @David, If the model is an harmonic signal + AWGN the peak of the DFT is the ML estimator. Could you please refer to other methods you talked about (Parametric methods, Eigenvalue analysis etc...). Thank You. – Royi Jan 26 '14 at 8:42

I doubt that there is such a rule because your assumption- the negative of computational work more than offsets the benefits of lower error probabilities- assumes that computations are expensive and errors are not. The "costs" of computational work and errors depends entirely on the application.