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Let $X$ be a tall $M\times N$ matrix with complex elements, i.e. $M >> N$, and $h$ an $N\times 1$ complex vector. Furthermore, $c$ is an $M\times 1$ vector, $\Sigma_h$ an $N\times N$ diagonal matrix with positive elements, $C_X$ an $MN\times MN$ positive semidefinite matrix and $vec(X)$ denotes the vectorization of $X$. I am wondering if there is any closed form expression for the following integral

$$\int_X \int_{h \in A_{\epsilon}(\tau)} e^{-(1/2)(c - Xh)^H (c-Xh) - (1/2)h^H\Sigma_h h - (1/2) vec(X)^H C_Xvec(X)} dX dh, $$ where $A_{\epsilon}(\tau) = \{ h | \sum_{j=\tau}^N |h_j|^2 < \epsilon\}$.

For example, I manage to evaluate the integral across the matrix $X$, but the final integral across $h$ then seems intractable. On the other hand, if I start by evaluating across $h$ first, then it seems that that integral has no closed form solution. Even if it can be expressed by some complicated functions, the final integral across $X$ then seems intractable.

Any comments or ideas are welcome.

Thanks!

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  • $\begingroup$ Just some comments. It seems that you haven't defined $c$. The $h^H \Sigma_h h$ term in the exponent seems rather ill adapted to get a closed form result for the $h$-integral over your domain. What is most important is that you have to be mentally prepared for the non-existence of any closed form for this integral. If a closed form didn't exist, what would you still really want to know about the result? $\endgroup$ Sep 21, 2015 at 22:38
  • $\begingroup$ Hi! Thank you for your thoughts! In the problem description, it now says $c$ instead of $d$, as it should be. So $c$ is just some given constant complex vector. I also have a feeling a closed form does not exist. However, you brought up an interesting question. Indeed, I am not really after a closed form here. Instead, I am interested in finding the $\tau$, $1\leq\tau\leq N$, that maximizes that expression for a given vector $c$. However, since I believe that one can get any $tau$ by changing $c$, I feel I must have a closed form expression to solve this maximization. $\endgroup$
    – DzeKap
    Sep 22, 2015 at 8:02

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