I want to estimate time delay between two complex narrow band signals. I can use any correlation method. There are several methods in the literature about sub-sample time delay estimation. Generally this subs-sample delay estimation is done through phase estimation. My question is: are these sub-sampling methods using phase requires interpolation or not? I was expecting that we are interpolating in Fourier domain and using phase information to estimate time delay. Any comment would be appreciated.
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$\begingroup$ Frequency domain interpolation is directly meaningful if your signals are sinusoidal (or complex exponential), as you need the phase of the sinusoid at its true frequency, not at that of the nearest bin. For general signals phase is not so helpful as it would need to be unwrapped for comparison between the two signals, and that is difficult. $\endgroup$– Olli NiemitaloSep 15, 2015 at 6:25
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$\begingroup$ yes, our interest is narrow band. The question is about interpolation. Is it interpolation, any comment please. $\endgroup$– CreatorSep 15, 2015 at 6:54
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$\begingroup$ If the signal is sinusoidal and you know the frequency in advance (or somehow track it) then you can use something like the Goertzel algorithm to directly calculate the Fourier transform at the frequency. There's no frequency domain interpolation there. $\endgroup$– Olli NiemitaloSep 15, 2015 at 18:44
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$\begingroup$ @OlliNiemitalo I am interested in the sub-sample time delay estimation not frequency.According to wikipedia Goertzel algorithm is related to frequency estimation. $\endgroup$– CreatorSep 15, 2015 at 19:35
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$\begingroup$ There are forms of the (complex) Goertzel algorithm that can be used for phase estimation as well. $\endgroup$– hotpaw2Sep 15, 2015 at 23:08
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1 Answer
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For finite length signals, interpolation (both for frequency and phase) in the frequency domain is helpful if the frequency is not at a DFT/FFT bin center. Relative phase estimation is required for delay estimation in the frequency domain. This relative phase estimation can produce a sub-time-domain-sample delay estimation result. This delay estimation can have an uncertainty related to an unknown integer multiple of the signal's period due to phase unwrapping errors, unless there is some other time domain envelope information available.
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$\begingroup$ Thank you for your interest. Without interpolation can we get more information from phase of the Fourier transform which is not possible from correlation in the time domain? I wanted to ask this question originally? $\endgroup$– CreatorSep 15, 2015 at 21:03
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$\begingroup$ Without interpolation can we get better time delay estimate from Fourier domain compare to time domain? The same question but I am asking different to make it clearer. $\endgroup$– CreatorSep 15, 2015 at 21:05
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$\begingroup$ Without interpolation, the error(s) depends on the delay and the frequency. Thus, could go either way for the "better" estimate. $\endgroup$– hotpaw2Sep 15, 2015 at 21:12
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$\begingroup$ Is there a proof or mathematical justification? I can understand it from intuition but a mathematical analysis would be helpful. $\endgroup$– CreatorSep 15, 2015 at 21:57
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$\begingroup$ Your question may be far too broad for a short mathematical justification. $\endgroup$– hotpaw2Sep 15, 2015 at 23:10