Complex Numbers, Hermitian matrix

asked 2022-05-11 06:53:45 +0100

Todorova gravatar image

Hi guys, Can you please help with the following problem:

Given: β„Žπ‘ (𝑛) 𝑒^(βˆ’π‘—(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0))+πœ”_𝑏 (𝑛)

Required Result: |𝑦(𝑛)|^2=|β„Ž|^2 |𝑠(𝑛)|^2+|πœ”_𝑏 (𝑛)|^2+2β„œ[β„Žπ‘ (𝑛) 𝑒^(βˆ’π‘—(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0))+πœ”_𝑏^𝐻 (𝑛)]

My Solution: I used the following rule of complex numbers: |𝑧_1+𝑧_2 |^2=|𝑧_1 |^2+|𝑧_2 |^2+2𝑅(𝑧_1 (𝑧_2 )Β Μ… ) I assumed that 𝑧_1=β„Žπ‘ (𝑛) 𝑒^(βˆ’π‘—(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0)), and 𝑧_2=πœ”_𝑏 (𝑛). So, |𝑧_1 |^2 γ€–=|β„Žπ‘ (𝑛) 𝑒^(βˆ’π‘—(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0)) |γ€—^2=|β„Ž|^2 |𝑠(𝑛)|^2 𝑒^(βˆ’2𝑗(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0)) |𝑧_2 |^2=|πœ”_𝑏 (𝑛)|^2 𝑧_1 (𝑧_2 )Β Μ…=|β„Ž|^2 |𝑠(𝑛)|^2 γ€–|πœ”_𝑏 (𝑛)|^2 𝑒〗^(βˆ’2𝑗(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0)) |𝑧_1+𝑧_2 |^2=|𝑧_1 |^2+|𝑧_2 |^2+2β„œ(𝑧_1 (𝑧_2 )Β Μ… )=|β„Ž|^2 |𝑠(𝑛)|^2 𝑒^(βˆ’2𝑗(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0))+|πœ”_𝑏 (𝑛)|^2+2β„œ(|β„Ž|^2 |𝑠(𝑛)|^2 γ€–|πœ”_𝑏 (𝑛)|^2 𝑒〗^(βˆ’2𝑗(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0)) ) but I couldn’t reach the same result in Green. (1) How was the first term of the real part β€œβ„Žπ‘ (𝑛) 𝑒^(βˆ’π‘—(2πœ‹βˆ†_𝑓 𝑛+πœƒ_0))β€œ obtained? (2) How was the second term of the real part β€œHermitian matrix πœ”_𝑏^𝐻 (𝑛")β€œ obtained"? I appreciate your help.

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Comments

Please define objects one by one, using LaTeX is a good idea. Also, please try to isolate one sage-related question, best also showing sage code and the point where the coding became an issue. There is no result in Green. Which is the origin of the question, and why should it be related to sage?

dan_fulea gravatar imagedan_fulea ( 2022-05-17 21:53:03 +0100 )edit
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