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I think the term 'signal reconstruction' means to obtain an analog signal from the discrete-time sampled signal. Then, some filtering is to be performed in the frequency domain or some convolution is to be performed in the time domain. These concepts are well described in most DSP books.
the nequiest theorem tells that the sampling rate will be double than the highest frequency of the signal to be sampled. this condition is applied due ti the fact that the signal can be reconstruct from the samples.
coming to your question, the interpolation can be used for the reconstruction of the discrete time signal.
In signal processing, reconstruction usually means the determination of an original continuous signalPerhaps the most widely used reconstruction formula is as follows. Let {ek} is a basis of L2 in the Hilbert space sense; for instance, one could use the canonical
,
although other choices are certainly possible. Note that here the index k can be any integer, even negative.
Then we can define a linear map R by
for each , where (dk) is the basis of given by
(This is the usual discrete Fourier basis.)
The choice of range is somewhat arbitrary, although it satisfies the dimensionality requirement and reflects the usual notion that the most important information is contained in the low frequencies. In some cases, this is incorrect, so a different reconstruction formula needs to be chosen.
A similar approach can be obtained by using wavelets instead of Hilbert bases. For many applications, the best approach is still not clear today.
from a sequence of equally spaced samples
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