Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I am reading a paper which contains a lot of matrices. Anyway, there is this equation:

[tex]\|\mathbf{H}_3\mathbf{H}_1\mathbf{e}\|^2=\mathbf{e}^{H}\mathbf{H}_1^{H}\mathbf{H}_3^{H}\mathbf{H}_3\mathbf{H}_1\mathbf{e}[/tex]

where superscript H means conjugate transpose, and boldface Hs are N-by-N circulant and toeplitz matrices, where the first column is defined as:

[tex]\begin{array}{ccccccc}h_i(0)&h_i(1)&\cdots &h_i(L)&\mathbf{0}_{1\times N-L-1}\end{array}[/tex]

and e is some N-by-1 vector. It is claimed that the above equation can be approximated as:

[tex]\|\mathbf{H}_3\|^2\|\mathbf{H}_1\mathbf{e}\|^2[/tex]

but the authors did not say how and why? They just claimed that in simulation the mean square error between both of them is tolerable and small. Further it is said that:

[tex]\|\mathbf{H}_3\|^2=N\sum_{m=0}^L|h_3(m)|^2[/tex]

are all of that justifiable? and how?

Thanks in advance

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# About Circulant Matrices

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