Complex conjugate problem

[biometrix]

I'm reading about signal processing, and we want to approximate one signal from another.

Suppose the complex x(t) and y(t) are complex defined in [t1, t2] and c a constant. The error function is e(t) = x(t) - c y(t).

So the textbook continues in calculating the integral:

[MATH] \int_{t1}^{t2} |e(t)|^2dt = \int_{t1}^{t2} | x(t) - c\cdot y(t)|^2 dt= \int_{t1}^{t2} ( x(t) - c\cdot y(t) )(x(t) - c\cdot y(t))^* dt = [/MATH]
[MATH]\int_{t1}^{t2} ( x(t) - c\cdot y(t)) \cdot ( x^*(t) - c^* \cdot y^*(t))dt.[/MATH]
In the last integral we took the complex conjugate of e(t) but we applied the operation on the constant too. Is this correct ? How this affects the constant?

 

[Steven G]

Try looking at an example. Compute the conjugate of \(\displaystyle c\cdot y(t)\) Then compute \(\displaystyle c\cdot y(t)^* \ and \ c^*\cdot y(t)^*\) and see which is correct.

 

[pka]

I'm reading about signal processing, and we want to approximate one signal from another.
Suppose the complex x(t) and y(t) are complex defined in [t1, t2] and c a constant. The error function is e(t) = x(t) - c y(t).
So the textbook continues in calculating the integral:
[MATH] \int_{t1}^{t2} |e(t)|^2dt = \int_{t1}^{t2} | x(t) - c\cdot y(t)|^2 dt= \int_{t1}^{t2} ( x(t) - c\cdot y(t) )(x(t) - c\cdot y(t))^* dt = [/MATH][MATH]\int_{t1}^{t2} ( x(t) - c\cdot y(t)) \cdot ( x^*(t) - c^* \cdot y^*(t))dt.[/MATH]In the last integral we took the complex conjugate of e(t) but we applied the operation on the constant too. Is this correct ? How this affects the constant?

Using \(\overline{z}\) for conjugate we have: \(\overline{z\cdot w}\)=\(\overline{z}\cdot\overline{w}\); \(\overline{z+w}\)=\(\overline{z}+\overline{w}\).
Moreover, if \(\alpha\in\mathbb{R}\) then \(\overline{\alpha\cdot z}={\alpha}\cdot\overline{ z}\); and \(|z|^2=z\cdot\overline{ z}\).