Complex FIR Filter Design

In linear-phase filter design, we assumed symmetry of our filter coefficients [ $h(n) = h(-n)$ ] $\Rightarrow$

• The filter frequency response became a sum of cosines (``zero phase'')
• The matrix $A$ was real
• The desired magnitude response $b$ was real
• The final zero-phase filter $\hat{x}$ could be right-shifted $L/2$ samples to get a corresponding causal linear-phase FIR filter

Now we would like to specify a complex frequency response. This means that:

• $b$ is complex
• $A$ is complex
• We still want $x$ (our filter coefficients) to be real

If we try to use ' $\backslash$ ' or pinv in Matlab, we will generally get a complex result for $\hat{x}$

Summarizing our problem:

$$\min_x \left\Vert\,Ax-b\,\right\Vert _2$$

where, $A \in \mathbb{C}^{NxM}$ , $b \in \mathbb{C}^{Nx1}$ , and $x \in \mathbb{R}^{Mx1}$

Hence we have,

$$\min_x \left\Vert \left[{\cal{R}}(A)+j{\cal{I}}(A)\right]x - \left[ {\cal{R}}(b)+j{\cal{I}}(b) \right] \right\Vert _2^2$$

which can be written as:

$$\min_x \left\Vert\, {\cal{R}}(A)x- {\cal{R}}(b) +j \left[ {\cal{I}}(A)x+{\cal{I}}(b) \right] \,\right\Vert _2^2$$

or

$$\min_x \left\Vert \left[ \begin{array}{c} {\cal{R}}(A) \\ {\cal{I}}(A) \end{array} \right] x - \left[ \begin{array}{c} {\cal{R}}(b) \\ {\cal{I}}(b) \end{array}\right] \right\Vert _2^2$$

which is written in terms of only real variables.

Hence, we can use the standard least squares solvers in Matlab and end up with a real solution.

Related paper

``Design of Fractional Delay Filters Using Convex Optimization'' (Mohonk-97, Music 421 handout):

http://ccrma.stanford.edu/~jos/eps/mohonk97.ps.gz