Complex FIR Filter Design

In linear-phase filter design, we assumed symmetry of our filter coefficients [ ] $\Rightarrow$

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

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

is complex
is complex
We still want (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:

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

where $A \in {\bf C}^{N\times M}$ , $b \in {\bf C}^{N\times 1}$ , and $x \in {\bf R}^{M\times 1}$

Hence we have,

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

Which can be written as:

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

$\displaystyle \min_x \left\vert \left\vert \left[ \begin{array}{c} {\cal{R}}(A... ... {\cal{R}}(b) \\ {\cal{I}}(b) \end{array}\right] \right\vert \right\vert _2^2$

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.

Subsections

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    FIR Digital Filter Design
       Optimal FIR Digital Filter Design
          Complex FIR Filter Design