Unless you generated the spectrum with the discrete time Fourier transform, that should be difficult indeed. This would be the only way I know...

You can read http://en.wikipedia.org/wiki/Nyquist_frequency on that.

The short answer seems to be no. The bandwidth B will be folded over on top of itself a number of times. Let folks know if this is not the case. What does work is if you are sampling lower than the carrier frequency, but that's not the question at hand.

Yes, I know this short answer :) But may be I should explain this problem more clearly. When there is a bank of analog filters dividing the signal band B to M subbands, then we can sample signal in each subband with M times lower sampling freq. But because there not exist ideal analog filters we obtain an aliasing in every subband. Thus I was wondering if one can estimate how 'big' this aliasing is? Personaly I think it is not possible but may be I'm wrong.

Thanks anyway

Perhaps, you may consider recently developed approaches based on the compressive sampling/sensing. CS enables one to acquire the

data at sub-Nyquist rates, and recover it accurately using, for example, convex optimization algorithms, but the signals are assumed to be sparse in certain transformation domains. Hence, maybe you can find the solution within the signal reconstruction algorithms.

EDIT20130416 (more on bandwidth, Shannon and sub-Nyquist extensions). Let me first comment the sentence "Lets say that a signal has a bandwidth B, thus the sampling frequency should be Fs>=2*B". This "version" of Shannon-Nyquist-Whittaker-Kotelnokov-Küpfmüller theorem (and Cauchy?,check for instance Sampling—50 Years After Shannon, M. Unser, http://bigwww.epfl.ch/publications/unser0001.html) is incorrect, albeit quite frequent. Fs>=2*B is a sufficient condition, and is not necessary. If the signals you are interested in have a maximal frequency at B, but are of narrow bandwidth (eg between 0.9B and B), there exists extensions or generalized sampling theorems, that may work for such signals. One typical algorithm is named after Papoulis and Gerchberg. Generally speaking, 2x"actual bandwidth" of the signal should suffice in theory, if the bandwidth is "well located". But there are cases where you need more.

But in general, if you want to sample "all the signals", then you will probably need a "general" sampling frequency above 2*B, and a bit more, to cope with "imperfect filters" and noise.

One alternative is to resort to a series of band-pass filters, each limiting your signals to a narrower bandwidth, in which your may sample at a lower rate, or downsample. Since traditional filters are non perfect, you may have some leakage or aliasing. This topic is well studied on the literature, under the study of generic filter banks (eg a series of band-pass filters). If your filters, altogether, span the whole spectrum ("general" sampling frequency above 2*B), then there is (almost always) a possibility to perfectly recover the input data, provided the filter bank satisfies certain "perfect reconstruction properties". So the aliasing may be canceled "after all". This possibility may depend on the number of filters, their design, and the down-sampling ratio (which need not be equal to the number of filters). For digital filters, you may find an account of generic "alias cancellation" for complex and oversampled filter banks in:

Optimization of Synthesis Oversampled Complex Filter Banks, Jérôme Gauthier, Laurent Duval and Jean-Christophe Pesquet, IEEE Transactions on Signal Processing, October 2009

http://www.laurent-duval.eu/Articles/Gauthier_J_2009_tsp_opt_socfb-optimization-synthesis-oversampled-filter-banks.pdf

If you have analog filters, i guess you may find synthesis digital filters that answer your problem (under the name "hybrid filter banks, for instance).

This type of analysis is for "every signal". Then, if you have more constraints on signal (finite support, reduced frequency bands, etc.) you may find alternative reconstructions, generally using optimization, that may cancel the aliasing as well. You have the compressive sensing approach, and for instance a different aspect called "Finite Rate of Innovation". They are well summarized in the recent "Sub-Nyquist Sampling: Bridging Theory and Practice", Moshe Mishali and Yonina C. Eldar

http://webee.technion.ac.il/people/YoninaEldar/Sampling.pdf

Look for instance at Figure 5, which gathers the subsampling allowed wrt bandwidth and band location (cf. 2x"actual bandwidth" alluded to above).

The compressed sensing techniques that I am familiar with require that you have some a priori knowledge of your signal. For example, it consists of sparse linear combinations of signals in some (possibly infinite) dictionary. If this applies to your problem, then it is probably possible, but if you have a lot of aliasing distortion, then I think that the distortion is going to distort your estimate (unless you have a significant amount of a priori information) which means you will underestimate the amount of the aliasing distortion in your signal.

It will be interesting if someone does have a solution...

I think that still you can avoid aliasing by using a higher order analog filter not ideal though but as u said that most of energy is contained in sub band which is M times lower then actual bandwidth, then if you apply a higher order analog filter to filter out that additional bandwidth and then sample as per Nyquist criterion.

I dont get what u mean by estimating how big the aliasing is? This is altogether a different question

This should certainly be possible by using a reconstruction filter not centered around DC -as is normally the case- but centered around the band where the initial energy was located. IF the (effective) band width of the original signal is lower than the Nyquist frequency of course. Obviously problems will arise if this (effective) band contains harmonics of the Nyquist frequency because in that case it will alias with itself and this cannot be repaired anymore.

I have never done this myself, but think you should digitally oversample the signal NOT using a digital low-pass filter but using a band-pass filter centered around the correct frequency. After this operation you have your signal at the higher rate, i.e. the rate you "should have been using in the first place" and then you can send this signal to appropriate DACs.

Good luck!

It appears you understand the relevant notion of "bandpass sampling" and that your worry is about the limited sharpness of any band-selecting analog filters. This is not really different from the notion of a limited sharpness of a low-pass analog filter with ordinary (low-pass) sampling. Just sketch the bandpass spectral shape and replicate it about all integer multiples of the final (lower) sampling rate. The overlap you see in your sketch is what you get.

Idea

First I want to thank all of you guys, and especially to Laurent Duval.

The case is that there is analog filter bank which splits the band B to M subbands. The output of the filter in every branch is sampled with frequency 2*B/M therefore, because of not ideal filter transfer function, I have some aliasing in digital signals in every subbband. The real problem is that I try to identify the transfer functions of this analog filter bank using only digital signals at the output. Of course I should assume the input signal i.e. the simplest one could be analog signal which has the spectrum equals one in the band B and zero elsewhere. But even for such simple input signal there is a problem of identifying filter bank in whole band B. I hope now the problem is clear.

You can try to test this filter bank using chirp signal with low chirp rate. On spectrograms of output signal alias would be easily identified. Instantaneous amplitude a[n] and frequency f[n] of the output signal at each filter in the bank can be used to estimate frequency response of the filter. Alias in instantaneous frequency should be relatively easy to resolve then |H(f'[n])| is approximately equal to a[n]. Analyzing f'[n], which for constant group delay filter should be changing just as instantaneous frequency of the input chirp, one can also measure group delay of the filter.

An unconditional no is my answer to this question given that you imply only limited additional information about the signal other than the down-sampled values.

This is not to say that re-sampling cannot be practical and effective in some situations (just promise not to mention it to a Bayesian statistician)

The proof is simple: For every down-sampled spectrum, you can always find multiple solutions with higher frequency components that down sample to the observed values; even if you assume some signal sub-band energies; you can always find multiple higher frequency sub-band solutions that still fit. Therefore it is never possible to recover the signal without additional assumptions about those higher frequency components except by the good luck that your assumption might happen to coincide with some intrinsic property of the equation that generates the samples.

The only means by which higher frequency spectrum are recovered require significant additional information about the process generating the samples; this additional information typically ranges from knowledge of the underlying physics behind the process, or to some assumption about the error model associated with the measurement, or to some assumption of smoothness in the underlying sample;

One approach is to measure at the higher rate under various conditions, sub-sample - and then find the transform that best regenerates the original data; this sort of sidesteps the intent of the original question I suspect - and even this cannot be exact unless the actual high frequency components are exact functions of the low frequency one.

I have in my time done lot of dynamic signal analysis of non-stationary signals. I am always amazed by how many empirical "models" of the data will exactly agree with all the sub-sampled observed values over extended periods - and yet have absolutely no predictive power whatsoever. - a-prior additional knowledge - (even if guessed at) of the signal is a must - and proof that it is correct relies on the scientific method- making a prediction and seeing if it still holds.

You can certainly obtain a high frequency spectrum by mean of a sparse (but irregular) sampling at a much low average frequency. If you do it well then the price you pay is a low uniform background noise in the frequency spectrum. (you may check "A Simple Method for Detecting Periodic Signals in Sparse Astronomical Event Data", Andres Cicuttin, A. Colavita, Alberto Cerdeira, Radu Mutihac, and Silvio Turrini, The Astrophysical Journal. 01/2009; 498(2):666. DOI:10.1086/305564).

What is important is that the sampling times are chosen following a Poisson process and are recorded with high time resolution, then you may do direct Fourier transformation of your samples and that is all.

The spectrum range and resolution are directly related to the total sampling time and sampling time resolution respectively, while the uniformity of the background noise in the frequency spectrum is granted by the Poissonian strategy to choose the sampling times

If the signal is spectrally sparse it can be sampled sub-Nyquist at the so called LANDAU rate - see "Necessary density conditions for sampling and interpolation of certain entire functions", by H.J. Landau at Bell Labs - classic paper.

If the positions of the non-zero sub-bands are not known then Prof. Yonina Eldar from my school concluded that it might be sampled at twice the Landau rate and still be reconstructed. The Landau rate is essentially the percentage of spectral occupancy times the sampling rate - but beware that some DSP must be done on the raw samples in any case - this is a problem in the so-called Compressed Sensing emerging area

You cannot reconstruct the spectral values outside the (-Fs/2, Fs/2) bandwidth, because you cannot accurately quantify any values outside this bandwidth with the information that you have. I further assume that you used an anti-aliasing filter in your setup. If so, responses are further filtered.