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Suppose that I have a sensor that can acquire samples $X[k]$ of the Fourier transform of an unknown signal $Y[t]$. An example is MRI, where the acquired data is in $k-$space. Now suppose that the unknown signal $Y[t]$ is known to be real and non-negative. My question is: is there a principled way to incorporate this knowledge into the spectral analysis algorithm that will estimate $Y[t]$ from $X[k]$, in order to produce an estimate with less bias or variance? I am thinking at non-parametric spectral estimation algorithms. A naive way of course would be to take the real part of $Y[t]$ and clip the negative values, but this does not seem to be optimal. I am looking for some sort of Cadzow's denoising method for spectral data.

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    $\begingroup$ Sounds like a good question, but off the top of my head, I'm not sure that knowing that the signal is nonnegative will help you very much. In the Fourier domain, that would just map to a large DC component (i.e. a large positive mean value); that condition wouldn't really affect the other spectral components. $\endgroup$ – Jason R May 21 '13 at 1:52
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To give a complete answer to this question you're going to need to provide more details about the kind of models you're considering in the first place. But yes, in many cases you can augment those models with a priori constraints on $Y[t]$, such as $0 \leq Y[t] \leq 1$.

For example, if the standard model has some sort of least-squares structure, then adding constraints of that type turns the problem into a bound-constrained least squares problem. There are a variety of approaches to solving such problems, and while they are more expensive than standard least squares, they are quite tractable. And it's very likely that such constraints will produce a better reconstruction.

Even without knowing more, though, I will say this: if your modeling approach does not produce real signals naturally, then you are almost certainly using the wrong modeling approach. It concerns me that you are even proposing taking the real part of the output of some other model. You should be searching the space of real signals if you know that to be the underlying structure.

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  • $\begingroup$ The model for the signal measurements that I am considering is $X[k]=x[k]+w[k], k=0,\ldots,N-1$ where $w[k]$ are i.i.d. samples from a circular complex normal distribution. Also, $\mathbf{x}=\mbox{DFT}(\mathbf{Y})$, $Y[t] \in \mathbb{R}$ and $Y[t] \geq 0$. $\endgroup$ – Arrigo May 21 '13 at 21:36
  • $\begingroup$ Can someone elaborate on this? So you're basically saying that if you know the Fourier transform of a signal, and you know that the original signal is real, then there is a method for reconstructing the original signal with less error than that of an inverse DFT? Is there a simple toy example of a reconstruction method that would clarify how this is possible? $\endgroup$ – endolith May 22 '13 at 13:46
  • $\begingroup$ Given a real $N$-point signal $X[t]$, the standard DFT $Y[k]$ will have complex symmetric structure: $Y[0]$ and $Y[N/2]$ are real, and $Y[k]=\overline{Y[N-k]}$ for $k=1,2,\dots,N/2-1$. Your method of sampling in frequency should be preserving that complex symmetric structure. If someone gives you the frequency domain representation $Y$, then yes the inverse DFT is the right way to recover it. But, if the underlying signal is real, then $Y$ had better have this complex symmetric structure. $\endgroup$ – Michael Grant May 22 '13 at 14:42

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