Most efficient way to find single dominant frequency (without amplitude) in analog signal

Question

I want to construct a reference locked sine oscillator. The multibit sine wave is calculated using DDS. The multibit reference signal is typically not sinusoidal, can have offset and substantial noise, but is expected to contain one dominant frequency.

If this dominant frequency is nearly known, a standard PLL approach will work to keep track of the dominant frequency in the reference signal.

However, I now suppose this cannot work to acquire initial reference lock if the dominant frequency is far away from the oscillator frequency. In this case, the PLL gets stuck on a locally dominant frequency.

What is the common approach to resolve this ? Calculating an DFT seems like overkill, since it produces a lot of unnecessary information. A recursive 4 sample Zoom FFT looks a bit more efficient. But maybe there is a way that directly modifies the PLL filter to capture further off frequencies ?

Please let me know if I missed crucial info.

Answer 1

One approach is to use a phase/frequency detector as the circuit determining error and to ensure that the initial loop bandwidth is wide enough to “see” both the weaker signal that would otherwise be a false acquisition as well as the strongest signal desired. The loop will lock onto the strongest signal within the loop bandwidth as long as it is 6 dB above the total power of the other tones (a phase vector diagram illustrates well the need for 6 dB margin as anything less causes abrupt 0/180 degree changes In phase such that a phase lock loop can’t converge to zero error). Once locked, the bandwidth can then be tightened for better SNR performance on the locked signal. This approach also leads to much faster acquisition time in general and is often referred to as “gear-shifting”.

If the signals are much wider than the acquisition bandwidth then more elaborate search and assessment methods are needed prior to setting the center frequency for acquisition. A separate frequency lock loop with wider bandwidth can be employed or other techniques that sweep over the entire bandwidth and access signal strength (such as the FFT or if processing time is available and resources scarce a linear frequency ramp can be employed). Note that the FFT approach can be significantly simplified since the number of bins (samples) needed in the FFT computation only need to be enough to ensure one bin for each step size equal to the loop acquisition bandwidth or less (so if the entire bandwidth of interest is ten times the loop bandwidth in its widest setting for acquisition or a real signal, then a 20 point FFT could be used to determine the location of the strongest bin).

Answer 2

The multibit reference signal is typically not sinusoidal, can have offset and substantial noise, but is expected to contain one dominant frequency.

Against offset, you'd practically always start your processing with a high-pass filter.

If you know your dominant frequency to be sufficiently less than Nyquist, a fixed low-pass filter would also be a cheap way to increase your SNR before even doing anything complicated.

What you're looking for is the answer to the question

What's $f_d$ of the dominant frequency in the signal?

and that's a parametric estimation problem. So, I'd recommend using a parametric estimator.

I'm going to go ahead and recommend something that will seem quite like overkill (due to the scary complex algorithms mentioned therein), but it's really quite straightforward in the sizes of problem you deal with.

Do ROOT-MUSIC with a signal subspace size of $m=2$ and a noise subspace size of $n-m=1$ or $2$.

ROOT-MUSIC roughly works like the following:

1. You estimate a autocovariance matrix (and that you do by taking a vector $X$ of $n$ samples, and multiplying it with its transpose conjugate, so that you get a $n\times n$ matrix $\hat R_{XX}$. You typically accumulate and average a few of these to get an averaged $\bar R_{XX}$. Not that $n$ is "really small".
2. you do an Eigenvalue Decomposition. Because $\bar R_{XX}$ is real and symmetrical by construction, that becomes really trivial.
3. You take the two largest Eigenvalues (of your 3 to 5 eigenvalues...), and the matching eigenvector. These column vector $S$ spans your signal space! Yay! (Pick two, because your sine is real, and that means it's composed of two inherently linked complex sinusoids of $\pm f_d$).
4. You, however, consider the other eigenvectors. These span the noise space matrix $G$ (which has dimension 1 or 2 , depending on your choice of $n-m$), and by how eigenvalue decompositions of hermitian matrices like $\bar R_{XX}$ happen, that space is orthogonal to the signal space. If you calculated the "pure" sinusoid of the right frequencies $f_d$, their projection into the noise space would be zero. You exploit that. Let $z=e^{j2\pi ft}$, and define $a=\left[ z^{-0}, z^{-1}\right]$. Then, finding the zeros ("roots") of $$a^* GG^* a$$ (which is a very boring 1st or second degree polynomial) gives you your frequencies.