Complex Least Squares Approximation In the case of frequency domain FIR filter design, the error function given by :

$$E(\omega)=H(e^{j\omega})-D(e^{j\omega}) \tag{1}$$

is a linear function with respect to the unknown filter coefficients $h[n]$. Hence, the Least Squares frequency domain FIR filter design problem is a linear LS problem which can be solved by computing the solution to a system of linear equations.  (from this: Algorithms for the Constrained Design of Digital Filters with Arbitrary Magnitude and Phase Responses ).

I have a three questions :


How the error respect the unknown filter coefficient ?
Does equation (1) is the same : $E(\omega)=D(e^{j\omega})-H(e^{j\omega})~$?
Does this case is available with negative domain of frequency ?


Thank you in advance.
Let's assume we have $ m $ points in the Fourier Domain (Discrete Fourier Domain) for our reference frequency response. We'll designate their values by $ y \in \mathbb{C}^{m} $.

Also we need a filter of $ n $ coefficients to approximate the reference response as good as it can. We'll use $ h $ for that.

We need now the Fourier Matrix, $ F \in \mathbb{C}^{m \times n} $ in order to write the following linear complex least squares problem:

$$ \arg \min_{h} \frac{1}{2} {\left\| F h - y \right\|}_{2}^{2} $$

You could easily write this also in a Sum Form.
The nice thing about it is the solution is give by the classic Least Squares solution - $ \hat{h} = {\left( {F}^{H} F \right)}^{-1} {F}^{H} y $.

Of course, in practice, you need many point in frequency domain to add more and more restrictions. You could also write a continuous version of this problem using the DTFT.