I have two signals, $V\sin(\omega t)$ and $V\sin(\omega t+\theta)$.
I need to find the value of theta with an accuracy of $0.05$ degrees.
Can anybody suggest a method for phase difference determination without the use of PLL?
I have found several methods such as Hilbert, Kalman filters, etc. but not sure whether they would meet my accuracy requirement.
If the frequencies in both signals are phase locked (and not just nominally the same) you can just multiply them:
$$x(t) = V\sin(\omega t) \cdot V\sin(\omega t+\theta) = 0.5V^2 \cdot (cos(\theta) + cos (2\omega + \theta))$$.
The phase is simply the inverse cosine of twice the mean:
$$\theta = acos \left( \frac{2}{T\cdot V^2}\int_0^T x(t) dt \right) $$
This will only allow you to determine phases in two quadrants.
The longer you integrate the better the phase estimate will be so you can adjust the integration times to your noise condition and accuracy requirements.
If the two frequencies are NOT phase locked, you will need a PLL otherwise the phase will just drift around.
A phase accuracy of 0.05 degrees rms requires an SNR of 61.2 dB, as given by $20 Log_{10} (0.05 \times 2 \pi /360) $.
A simple and practical method is to multiply the signal with unknown phase with a reference cosine to get “I” and with a reference sine to yet “Q”, low pass filter the outputs of each to eliminate the sun term (multiplying real tones will give the sum and the difference of the frequencies , and the difference is the angle of interest). Then for accurate full four quadrant phase estimation use $atan2(Q, I)$. The sine and cosine are Hilbert Transform pairs, so if we know the frequency but not the phase we can simply generate the sine and cosine functions directly. If we do not know the frequency accurately but know the reasonable range of frequencies that it can be, then we can use the Hilbert Transform to create the sine and cosine needed easily enough.
To optimize SNR performance, the signal should be bandpass filtered first if there is a possibility of strong interference at other frequencies, and then hard-limited to remove all AM components. (Or AGC'd which is done when there is short term amplitude variation of interest such as modulated waveforms).
In the presence of white noise with stationary phase, the optimum strategy is to average the I and Q outputs prior to computing the arctangent, in which case the SNR will increase in power in proportion to the averaging time.
The 0.05 degree accuracy and 61.2 dB SNR requirement will impose practical challenges in that the phase measurement must certainly be done digitally (temperature sensitive dc offset issues preclude using analog mixers or multipliers), and even once digital all noise sources (quantization, drift, etc) must be carefully managed.
