filter design - 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.

Answer

Even though I think there is a lot of valuable general information in Stanley Pawlukiewicz's answer and in Royi's answer, I think that some specific questions have not been answered, at least as far as I understand the OP's questions.

Let me go through the 3 bullet points in the OP one by one:

1. If you mean how the error function depends on the filter coefficients, then for a length $N$ FIR filter you simply have $$H(e^{j\omega})=\sum_{n=0}^{N-1}h[n]e^{-jn\omega}=\mathbf{c}^H(\omega)\cdot \mathbf{h}\tag{1}$$

where $\mathbf{h}$ is the vector of filter coefficients, $\mathbf{c}(\omega)$ is given by $\mathbf{c}(\omega)=[1,e^{j\omega},\ldots,e^{j(N-1)\omega}]$, and $^H$ denotes the Hermitian conjugate. Clearly, $(1)$ is a linear function of the filter coefficients, hence the error function $E(\omega)$ is linear in $\mathbf{h}$.

2. 
   

   The sign of the error function is irrelevant, because you end up minimizing a squared error measure, which depends on $|E(\omega)|^2$, and which is, consequently, independent of the sign of $E(\omega)$.

   
   


3. 
   

   I did my best to understand what is meant by the third question but I failed. Maybe the OP can elaborate in a comment or by editing the question.