How to interpret output of matched filter with complex input? I have implemented a matched filter based on the Fourier Transform approach. In the real numbers domain that means that I use as the coefficients of my filter (B) the inverted time-samples of the signal that I'm trying to find and I compute:

real_output = IFFT(FFT(A).*FFT(B))


Where, if we assume the moment of exact match, A is an input signal of real value samples and B is the signal made from inverted time-samples of A (or I can also conjugate the spectrum of B and I get the required time inversion because the signal is real). In order to avoid aliasing I perform both FFT with length(A)+length(B)-1 samples.
The output of this filter is always real and its maximum, points at the moment in which a match has occurred.

Now assume that instead of a real valued A we have a complex valued A. My questions are:


Is it possible to apply a matched filter for complex domain signals?
Would I have to invert the complex valued A to get the coefficients of my filter or should I do something else like conjugating them?
How do I interpret the output of the filter, should I say a match happens if the absolute value of the output is maximum or just when the real part is maximized?
In some experiments I have noticed that at the moment of a match the output of the filter is real if spectrum(B) is the complex conjugate of spectrum(A), is this correct?
Yes, it is possible (at least on paper or code, since complex signal don't exist physically) to apply a matched filter to complex signals.

This is one way to look at it that I think is illustrative. Assume that a pulse $p(t)$ is real and has energy 1, that is, $$\int_{-\infty}^\infty p^2(t) \, dt=1.$$ The filter matched to $p(t)$ has impulse response $p(-t)$. Then, if the filter's input is $ap(t)$ for real $a$, the matched filter's output (sampled at the appropriate time) is equal to $$ap(t)\star p(-t)=a.$$

If the pulse amplitude is complex, say $a+jb$ for real $a$ and $b$, then we can use the linearity of the convolution operation to calculate 
\begin{align}
(a+jb)p(t) \star p(-t) &amp;= \left(ap(t)+jbp(t)\right) \star p(-t) \\
&amp;=ap(t)\star p(-t) + jbp(t)\star p(-t) \\
&amp;= a+jb.
\end{align}

One way to interpret the result is that the filter is matching the real part and the imaginary part independently (but "at the same time"). Recall that $j=\sqrt{-1}$ is orthogonal to the real line; this means that $a$ and $b$ don't "interfere" with each other during the convolution.

To summarize, the answers to your questions are:


Yes.
If you can define the system as using a real pulse $p(t)$, then the matched filter is $p(-t)$. If you must use a complex pulse, then the matched filter is $p^\star(-t)$.
A "match" can be declared when the filter's ouput is close enough (in Euclidean distance sense) to the matched signal (see https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation for more details).
The output of the filter should be real only when $b$ is zero.