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YCor
YCor
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The Distributiondistribution of the power of the Sumsum of Inner Productsinner products of Two Independent Complex Normal Vectorstwo independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n|^2$$$$\Big|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n\Big|^2$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.

Thank you in advance.

The Distribution of the power of the Sum of Inner Products of Two Independent Complex Normal Vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n|^2$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.

Thank you in advance.

The distribution of the power of the sum of inner products of two independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$\Big|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n\Big|^2$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.

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William
William
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The Distribution of the power of the Sum of Inner Products of Two Independent Complex Normal Vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n|^2$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.

Thank you in advance.