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I have a discrete-time system which can be described as:

$$ Y_m = \sum_{r=-N_g}^{R-1+N_g} c_r x[R(m-1) + r] $$

The unknowns are $c_k$ but I know that they have the following approximate behavior:

$$ c_r \approx \cases{1 \quad 0 \leq r < R \\ 0 \quad \mathrm{else}} $$

From this, one may recognize that this is nothing more than a discrete summer which obtains the sum of $R$ Nyquist rate samples of $x[n]$. However, in contrast to a normal FIR filter, the output sums are given at a rate $R$ times lower that the one of $x[n]$.

Now my issue is that I do not know $c_k$ exactly. For that reason I want to estimate them. Rewriting in vector matrix form, the Least Squares estimate looks like:

$$ \mathbf{Y}=\mathbf{X} \mathbf{c} \\ \mathbf{c} = (\mathbf{X}^{\#} \mathbf{X})^{-1} \mathbf{X}^{\#} \mathbf{Y} $$

The success of this method boils down to the conditioning of the matrix $\mathbf{X}$. If $x[n]$ is white, the matrix $\mathbf{X}$ has a low condition number and a rank that is equal to the number of unknowns: $\operatorname{rank}(\mathbf{X}) = R+2N_g$. This is called driving the unknown system with a persistently exciting input. Using simple MATLAB, I can see that the bandwidth of $x[n]$ must be at least $f_s/2 \cdot 0.9$ where $f_s$ is the Nyquist rate of the sequence $x[n]$. Under this condition, the estimation works as expected.

Due to physical constraints (ultimately, this is just a model of a physical system!), I cannot make $x[n]$ white, it needs to be bandlimited to $< f_s/2 \cdot 0.8$. But with such a signal, the rank of the matrix is much lower than the unknowns.

  • Is there any way to estimate $c_k$ with a non-white $x[n]$ sequence?
  • Potentially, only for a certain set of input signals (e.g., signals, that are also bandlimited, ignoring the behavior around $f_s/2$ ?
  • Can it be useful that I know the basic behavior of $c_r$ ?
divB
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    Your summer with a output rate of 1/R the input rate is what we call a decimating FIR filter, in particular a 1/R decimating moving average with a gain of R. – Marcus Müller Aug 12 '18 at 12:35
  • regarding the estimation problem, does https://dsp.stackexchange.com/a/50996/13320 help? – Marcus Müller Aug 12 '18 at 12:36
  • Your approximate behavior confuses me. I would think that some subset of c_k out of all possible k being nearly one and the rest near zero would make sense but you have a very specific sequence of k for the subset. Could you elaborate a bit –  Aug 12 '18 at 14:29
  • @StanleyPawlukiewicz: Where do you see $k$? I am not sure if I really understand your question but I assume you are understanding it right: It is a perfect summer if $c_0, \cdots, c_{R-1}=1$ and 0 otherwise. But for an actual implementation the zeroes are not really zeroes and the ones are not really ones (but follow, e.g., an exponential decay due to a dominant pole) – divB Aug 13 '18 at 09:59
  • @MarcusMüller: I am not sure ... I understand that $H^{-1}$ is the ZF of $H$ and I understand the MMSE solution (which requires additive iid Gaussian noise and knowledge of its variance). But in my case I am looking for Sys ID, not EQ. Do you want to say that the Least Squares (LS) solution I describe corresponds to a Zero-Forcing (ZF) solution and an alternative to LS (what?) could solve the issue? – divB Aug 13 '18 at 10:07
  • @MarcusMüller: In terms of linear algebra, the LS solution can be extended by something called Tikhonov or Ridge regularization in which an otherwise singular matrix can be made singular. Is this what you are trying to point me to? Then I am somewhat missing the link to the bandwidth of the underlying signal $x[n]$ – divB Aug 13 '18 at 10:10
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    @divB you're going much deeper than I wanted to point you :) I wanted to point you to the fact that you're doing system identification, in particular, of a FIR filter; and if you want to see how it's done in practice, gr-adapt (as linked there) has a few nice implementations. – Marcus Müller Aug 13 '18 at 11:10

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