Suppose the known frequency spectrum of a signal is: $$ G(n\Delta f)=9,4,3,5,1,6,... $$ where $n=0,1,2...$ and $\Delta f=1$ Hz

If, for some reason, we cannot sample $G$ at $n=0$ (i.e. $G(0)=9$ is not available), but can sample the frequencies everywhere else, including at finer $\Delta f$'s.

What can we do to minimize the error of the recovered time-domain signal $F(t)$?

For example, how fine $\Delta f$ should be? How many frequencies needed to be sampled? Can we use the discrete Fourier transform or should we use the continuous Fourier transform by interpolating the shape of $G$? etc.


When a signal is expressed as a linear superposition of orthogonal basis functions, like a DFT, when you remove one orthogonal component, that error is orthogonal to the remaining terms, so any perturbations to those remaining terms will not have an effect on the error due to the removal.

Nevertheless, missing "data" is an active area of research. If you can make some assumptions about what is missing, you can sometimes, with varying success recover that data.

There is a very popular family of algorithms known as the "EM algorithm"

Another more recent area is known as "Matrix Completion"

There is a mathematical property called "restricted isometry" that is used in sparse sensing that may of use in your problem.

When you google scholar the term "incomplete spectrum", you get hits like:

E. Serpedin, "Subsequence based recovery of missing samples in oversampled bandlimited signals," in IEEE Transactions on Signal Processing, vol. 48, no. 2, pp. 580-583, Feb 2000. doi: 10.1109/78.823989 Abstract: A new approach for recovering an arbitrary finite number of missing samples in an oversampled bandlimited signal is presented. This correspondence also proposes an approach for recovering the original signal's spectrum from the spectra of a certain number of subsequences, obtained by downsampling the original sequence. Closed-form expressions for the missing samples in terms of the known samples are obtained by exploiting the linear dependence relationship among the spectra of downsampled subsequences keywords: {bandlimited signals;signal reconstruction;signal sampling;spectral analysis;closed-form expressions;downsampled subsequences;downsampling;known samples;linear dependence;missing samples;oversampled bandlimited signals;spectrum;subsequence based recovery;Bandwidth;Closed-form solution;Equations;Fourier transforms;Frequency domain analysis;Iterative algorithms;Iterative methods;Numerical stability;Sampling methods;Signal restoration}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=823989&isnumber=17842

If you can express your problem in terms of any of these paradigms, you can arrive at solutions, and some have bounds on errors.

As was suggested in the question, an often useful simple heuristic is to interpolate the missing data. The most straightforward way to get an handle on error, is trial and error on actual data. A lot of times, acts of technical desperation work out.

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