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q = 1 π 0 π | A d ( ω ) - A ( ω ) | 2 d ω

where A d ( ω ) is the desired ideal amplitude response, A ( ω ) = n a ( n ) cos ( ω ( M - N ) n ) is the achieved amplitude response with the length h ( n ) related to h ( n ) by Equation 29 from FIR Digital Filters . This integral squared error is approximated by the discrete squared errordefined in [link] for L > > N which in some cases is much easier to minimize. However for some very useful cases, formulas can be foundfor h ( n ) that minimize [link] and that is what we will be considering in this section.

The unweighted least integral squared error approximation

If the error measure is the unweighted integral squared error defined in [link] , Parseval's theorem gives the equivalent time-domain formulation for the error to be

q = n = - | h d ( n ) - h ( n ) | 2 = 1 π 0 π | A d ( ω ) - A ( ω ) | 2 d ω .

In general, this ideal response is infinite in duration and, therefore, cannot be realized exactly byan actual FIR filter.

As was done in the case of the discrete error measure, we break the infinite sum in [link] into two parts, one of which depends on h ( n ) and the other does not.

q = n = - M M | h d ( n ) - h ( n ) | 2 + 2 n = M + 1 | h d ( n ) | 2

Again, we see that the minimum q is achieved by using h ( n ) = h d ( n ) for - M n M . In other words, the infinitely long h d ( n ) is symmetrically truncated to give the optimal least integral squared errorapproximation. The problem then becomes one of finding the h d ( n ) to truncate.

Here the integral definition of approximation error is used. This is usually what we really want, but in some cases the integrals can not becarried out and the sampled method above must be used.

Ideal constant gain passband lowpass filter

Here we assume the simplest ideal lowpass single band FIR filter to have unity passband gain for 0 < ω < ω 0 and zero stopband gain for ω 0 < ω < π similar to those in Figure 8a from FIR Digital Filters and [link] . This gives

A d ( ω ) = { 1 0 ω ω 0 0 ω 0 ω π

as the ideal desired amplitude response. The ideal shifted filter coefficients are the inverse DTFT from Equation 15 from Chebyshev or Equal Ripple Error Approximation Filters of this amplitude which for N odd are given by

h ^ d ( n ) = 1 π 0 π A d ( ω ) cos ( ω n ) d ω
= 1 π 0 ω 0 cos ( ω n ) d ω = ω 0 π sin ( ω 0 n ) ω 0 n

which is sometimes called a “sinc" function. Note h ^ d ( n ) is generally infinite in length. This is now symmetricallytruncated and shifted by M = ( N - 1 ) / 2 to give the optimal, causal length- N FIR filter coefficients as

h ( n ) = ω 0 π sin ( ω 0 ( n - M ) ) ω 0 ( n - M ) for 0 n N - 1

and h ( n ) = 0 otherwise. The corresponding derivation for an even length starts with the inverseDTFT in Equation 5 from Constrained Approximation and Mixed Criteria for a shifted even length filter is

h ^ d = 1 π 0 π A d ( ω ) cos ( ω ( n + 1 / 2 ) ) d ω = ω 0 π sin ( ω 0 ( n + 1 / 2 ) ) ω 0 ( n + 1 / 2 )

which when truncated and shifted by N / 2 gives the same formula as for the odd length design in [link] but one should note that M = ( N - 1 ) / 2 is not an integer for an even N .

Ideal linearly increasing gain passband lowpass filter

We now derive the design formula for a filter with an ideal amplitude response that is a linearly increasing function in the passband ratherthan a constant as was assumed above. This ideal amplitude response is given byand illustrated in [link] For N odd, the ideal infinitely long shifted filter coefficients are the inverse DTFT of this amplitude given by

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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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