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?
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?