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### Least-SquaresLinear-PhaseFIR Filter Design

Another versatile, effective, and often-used case is the weighted least squares method, which is implemented in the matlab function firls and others. A good general reference in this area is [204].

Let the FIR filter length be samples, with even, and suppose we'll initially design it to be centered about the time origin (zero phase''). Then the frequency response is given on our frequency grid by

 (5.33)

Enforcing even symmetry in the impulse response, i.e., , gives a zero-phase FIR filter that we can later right-shift samples to make a causal, linear phase filter. In this case, the frequency response reduces to a sum of cosines:

 (5.34)

or, in matrix form:

 (5.35)

Recall from §3.13.8, that the Remez multiple exchange algorithm is based on this formulation internally. In that case, the left-hand-side includes the alternating error, and the frequency grid iteratively seeks the frequencies of maximum error--the so-called extremal frequencies.

In matrix notation, our filter-design problem can be stated as (cf. §3.13.8)

 (5.36)

where these quantities are defined in (4.35). We can denote the optimal least-squares solution by

 (5.37)

To find , we need to minimize
 (5.38)

This is a quadratic form in . Therefore, it has a global minimum which we can find by setting the gradient to zero, and solving for .5.14Assuming all quantities are real, equating the gradient to zero yields the so-called normal equations

 (5.39)

with solution

 (5.40)

The matrix

 (5.41)

is known as the (Moore-Penrose) pseudo-inverse of the matrix . It can be interpreted as an orthogonal projection matrix, projecting onto the column-space of [264], as we illustrate further in the next section.

Subsections
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