Suppose that there is a FIR filter F and a signal S. The filtered signal is the convolution of F and S, F * S.

The problem: how to calculate a signal S' such that F * S' = S' (the filtered version is the same as itself, up to a fixed time shift), and S' minimizes norm(S, S') (where norm is some kind of distance metric, for example sum of squared differences).

Loosely: how to calculate the most similar signal to a given signal which is not affected by a filter?

EDIT: The original problem may be over constrained because there may be very few (or no) signals for which F*S = S exactly. An alternate formulation is to find S' which minimizes the combined error alpha*norm( F*S' - S') + beta*norm( S' - S ), for some weights alpha and beta.

EDIT: The specific filter I have in this case is F=[ 0.00097656, -0.00976562, 0.06347656, -0.15625 , 0.18554688, 0.83203125, 0.18554688, -0.15625 , 0.06347656, -0.00976562, 0.00097656] which is a fairly modest lowpass filter. I was interested in the general case as well though.

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    $\begingroup$ A handwaving answer: It's easy to prove that $\mathcal{X} = \{S: F*S = S\}$ is a subspace. You are looking for a projection of an arbitrary signal onto $\mathcal{X}$. If you know more about F (eg. are there sinusoids that F filters out with gain 1 and zero phase shift?) then this projection simply the sum over appropriate inner products with such "unit-gain-no-phase-shift" signals. $\endgroup$
    – Atul Ingle
    Feb 10 '13 at 16:17
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    $\begingroup$ Are you truly looking for $F * S' = S'$? Seems to me that that would be impossible, unless $F$ is the trivial system where its output always equal to its input. I wasn't sure if you were looking for something like $F * S' = S$ instead. $\endgroup$
    – Jason R
    Feb 10 '13 at 19:12
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    $\begingroup$ If it's an LTI system, then any pure sinusoid not at a zero or pole in the frequency response will be unchanged except in amplitude and phase. If the frequency response of the FIR filter does not include a point with a gain of 1.0, then a solution does not exist. $\endgroup$
    – hotpaw2
    Feb 10 '13 at 21:06

If a filter with transfer function $H(f)$ passes a signal $x(t)$ with Fourier transform $X(f)$ unchanged, then $H(f)X(f) = X(f)$ for all $f$, and so $H(f)$ must have the property that $$H(f) = 1 ~ \text{for all frequencies}~ f~ \text{for which}~ X(f) \neq 0.$$ Thus, an arbitrary filter will not enjoy this property for very many signals, perhaps for no signal at all. Indeed, if $H(f_0) = H(-f_0) = 1$ for some $f_0$, then the filter will pass the sinusoid $A\cos(2\pi f_0t + \theta), -\infty < t < \infty$ unchanged, but it will not pass a sinusoidal pulse such as $$x(t) = \begin{cases} A\cos(2\pi f_0t + \theta), & -T < t < T,\\ 0, &\text{otherwise,}\end{cases}$$ without distorting it. Of course, it is possible that $H(f) \neq 1$ for all $f$ and so even passing a monochromatic sinusoid is not possible (cf. @hotpaw's comment). In short, the subspace referred to by @AtulIngle might well be the trivial subspace of zero dimension containing only the zero signal. It is also worth noting that if $H(f)$ has constant nonzero value over an interval, then the filter is physically unrealizable for essentially the same reason that an ideal low-pass filter is unrealizable.

So, the OP's question is likely to have no answer at all because the signals $S^\prime$ that he seeks to find may not exist at all for the given filter. Perhaps something else was intended to be asked?

  • $\begingroup$ Please modified version of problem above, to minimize both filter-induced distortion (which will likely be nonzero for the reasons you explain) and the difference from original signal. $\endgroup$
    – Alex I
    Feb 11 '13 at 0:52

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