Damped solution

Equation (1) is the solution to an optimization problem that arises in many applications. Now that we know the solution, let us formally define the problem. First, we solve a simpler problem with real values. We choose to minimize the quadratic function of x  

$$Q(x) {=}(fx-y)^2 + \epsilon^2 x^2$$ (2)

The second term is called a damping factor because it prevents x from going to $\pm \infty$ when f=0. Set dQ/dx=0 getting  

$$0 {=}f(fx-y) + \epsilon^2 x$$ (3)

which yields the earlier answer $x=fy/(f^2+\epsilon^2)$.

With Fourier transforms, the signal X is a complex number at each frequency $\omega$.So we generalize equation (2) to  

$$Q(\bar X, X) {=} (\overline{FX-Y}) (FX-Y) + \epsilon^2 \bar X X {=} (\bar X \bar F - \bar Y) (FX-Y) + \epsilon^2 \bar X X$$ (4)

To minimize Q we could use a real-values approach where we express X=u+iv in terms of two real values u and v and then set $\partial Q/\partial u=0$ and $\partial Q/\partial v=0$.Alternately, we can use a complex-values approach where we set $\partial Q/\partial X=0$ and $\partial Q/\partial \bar X=0$.Let us examine $\partial Q/\partial \bar X$. 

$${\partial Q(\bar X, X)\over \partial \bar X} {=} \bar F (FX-Y) + \epsilon^2 X {=}0$$ (5)

The derivative $\partial Q/\partial X$ is the complex conjugate of $\partial Q/\partial \bar X$.So if one is zero, the other is too. Thus we don't need to specify both $\partial Q/\partial X=0$ and $\partial Q/\partial \bar X=0$because either one is enough. I usually take $\partial Q/\partial \bar X=0$.Solving (5) for X gives the answer (1).