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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

q = 1 L k = 0 L - 1 ( A ( ω k ) - A d ( ω k ) ) 2 = 1 L k = 0 L - 1 e ( ω k ) 2 0 π ( A ( ω ) - A d ( ω ) ) 2 d ω = 0 π e ( ω ) 2 d ω

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

ϵ = A - A d = C a - A d

where C is the matrix of cosines from Equation 48 from FIR Digital Filters , a is the vector of half of the filter coefficients from Equation 48 from FIR Digital Filters , and A 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

q = ϵ T ϵ

except for a scale factor of 1 L .

This could also be posed for the general phase problem by using H ( ω k ) rather than A ( ω k ) and h ( n ) , the actual impulse response, rather than a ( n ) , 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

q = 1 L k = 0 L - 1 | H ( ω k ) - H d ( ω k ) | 2

where H d ( ω k ) 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

ω k = 2 π k / L

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

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

q = 1 L k = 0 L - 1 | A ( 2 π k / L ) - A d ( 2 π k / L ) | 2

or with a simpler notation

q = 1 L k = 0 L - 1 | A k - A d k | 2

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

q = n = 0 L - 1 | h ( n ) - h d ( n ) | 2

where h d ( n ) is the length-L symmetric FIR filter that has the L frequency response amplitude samples A d k . 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 ( n ) with L frequency response samples A k .

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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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