I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$:

$$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$

where $\ast$ is the convolution. I know $A(f)$, $B(f)$, and $C(f)$. I measure $Y(f)$. Is there a way to retrieve $X(f)$? I know that if $C(f)$ is constant and $1$, then I can easily retrieve $X(f)$ except for frequencies with $A(f)=B(f)$. But how do I go about this for an arbitrary but known $C(f)$?

  • 1
    $\begingroup$ What's the order of operation? Is it $[X(f) \cdot B(f)]*C(f)$ or $X(f) \cdot [B(f)*C(f)]$ $\endgroup$ – Hilmar Oct 9 '19 at 19:44
  • $\begingroup$ I was about to ask the same question. What is the associativity? $\endgroup$ – Laurent Duval Oct 9 '19 at 19:49
  • $\begingroup$ First $X(f)B(f)$, then the convolution. $\endgroup$ – Test123 Oct 9 '19 at 19:52

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