2.5 Constrained approximation and mixed criteria  (Page 2/10)

where the ${\mu }_i$ are the necessary number of Langrange multipliers and $P$ is a scale factor that can be chosen for simplicity later. The first term in [link] is the integral squared error of the frequency response to be minimized and the second term will be zero when the equalityconstraints are satisfied at the frequencies, ${\omega }_i$ . The function $T\left(\omega \right)$ is the constraint function in that $A\left(\omega \right)$ must satisfy

Necessary conditions for the minimization of the integral squared error are that thederivative of the Lagrangian with respect to the filter parameters $a\left(n\right)$ defined in Equation 49 from FIR Digital Filters and to the Lagrange multipliers ${\mu }_i$ be zero [link] .

The derivatives of the Lagrangian with respect to $a\left(n\right)$ are

where from Equation 49 from FIR digital Filters we have for $n=1,2,\cdots ,M$

and for $n=0$

For $n=1,2,\cdots ,M$ this gives

and for $n=0$ gives

Using Equation 50 from FIR Digital Filters for $n=1,2,\cdots ,M$ , we have

and for $n=0$

Choosing $P=1/\pi$ gives

and

Writing [link] and [link] in matrix form gives

where $\mathbf{H}$ is a matrix with elements

except for the first row which is

because of the normalization of the $a\left(0\right)$ term. The $a_d\left(n\right)$ are the cosine coefficients for the unconstrained approximation to the idealfilter which result from truncating the inverse DTFT of $A_d\left(\omega \right)$ .

The derivative of the Lagrangian in [link] with respect to the Lagrange multipliers ${\mu }_i$ , when set to zero, gives

which is simply a statement of the equality constraints.

In terms of the filter's cosine coefficients $a\left(n\right)$ , from Equation 49 from FIR Digital Filters , this can be written"

and as matrices

where ${\mathbf{A}}_{\mathbf{c}}$ is the vector of frequency response values which are the desired response plus or minus the constraints evaluated at thefrequencies in the constraint set. The frequency response must interpolate these values. The matrix $\mathbf{G}$ is

except for the first column which is

Notice that if $K=1/\sqrt{2}$ , the first rows and columns are such that we have ${\mathbf{G}}^{\mathbf{T}}=\mathbf{H}$ .

The two equations [link] and [link] that must be satisfied can be written as a single matrix equation of the form

or, if $K=1/\sqrt{2}$ , as

which have as solutions

The filter corresponding to the cosine coefficients $a\left(n\right)$ minimize the $L_2$ error norm subject the equality conditions in [link] .

Notice that the term in [link] of the form $\mathbf{G}{\mathbf{a}}_{\mathbf{d}}$ is the frequency response of the optimal unconstrained filter evaluated at theconstraint set frequencies. Equation [link] could, therefore, be written

The constrained weighted least squares design of fir filters

Combining the weighted least squared error formulation with the constrained least squared error gives the general formulation of thisclass of problems.

We now modify the Lagrangian in [link] to allow a weighted squared error giving