2.2 Least squared error design of fir filters

Because the integral of the square of a signal is a measure of its energy, there is some physical reason for minimizing the integral of the squared error [link] , [link] . Also, because of Parseval's theorem, a least squares approximation in the frequency domain is a least squaresapproximation in the time domain. However, minimizing the worst case squared error induces a minimum Chebyshev error problem in some formulations [link] .

Discrete frequency samples of error

If we approximate the integral squared error by the sum of the squared error as given by

where the approximation error as a function of frequency isdefined by $e\left(\omega \right)=A\left(\omega \right)-A_d\left(\omega \right)$ with $A\left(\omega \right)$ being the amplitude response of the filter and $A_d\left(\omega \right)$ being the desired amplitude response or the ideal response. The matrix statement for theerror vector becomes

where $\mathbf{C}$ is the matrix of cosines from Equation 48 from FIR Digital Filters , $\mathbf{a}$ is the vector of half of the filter coefficients from Equation 48 from FIR Digital Filters , and ${\mathbf{A}}_{\mathbf{d}}$ is the vector of samples of the ideal desired amplitude response. The number of samples of the amplitude response is $L$ which should be five to twenty times the length of the filter to give a good approximation of the integral in most cases. The error to beminimized is

except for a scale factor of $\frac{1}{L}$ .

This could also be posed for the general phase problem by using $H\left({\omega }_k\right)$ rather than $A\left({\omega }_k\right)$ and $h\left(n\right)$ , the actual impulse response, rather than $a\left(n\right)$ , a nomralized half of the impulse response.

Truncated frequency sampling design using the inverse fft or idct

The design problem is posed by defining an error measure $q$ as a sum of the squared differences between the actual and the desired frequencyresponse over a set of $L$ frequency samples. This error function is defined as

where $H_d\left({\omega }_k\right)$ are the $L$ samples of the desired response. This problem is easier to formulate and solve if the frequency samplesare equally spaced as in Equation 8 from FIR Filter Design by Frequency Sampling or Interpolation which gives

and the problem is restricted to linear-phase filters where the real-valued amplitude $A\left(\omega \right)$ can be approximated rather than the complex frequency response $H\left(\omega \right)$ . For approximations to a complex response, see "Complex and Minimum Phase Approximation" .

Linear phase and equally spaced samples cause [link] to become

or with a simpler notation

A very powerful property of the Fourier transform allows a straightforward design of least-squared-error FIR filters. Parseval's Theorem,which is based on the orthogonality of the DFT, states that the error defined by [link] in the frequency domain can also be calculated in the time domain by

where $h_d\left(n\right)$ is the length-L symmetric FIR filter that has the $L$ frequency response amplitude samples $A_{dk}$ . This may be calculated by the frequency sampling method in the section Four Types of Linear-Phase FIR Filters using the special formulas such as Equation 8 from FIR Filter Design by Frequency Sampling or Interpolation for length $L$ or the inverse DFT. The filter to be designed has a length-N symmetric impulse response $h\left(n\right)$ with $L$ frequency response samples $A_k$ .