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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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5.3. Optimal approach of equal ripple in the stop-band and passband

Finding an optimal solution to the problem of the amplitude approximation of specifications is obtained by minimizing a distance criterion between the theoretical frequency responses and those brought about by synthesis.

To present this approach, we will consider a low-pass linear phase FIR filter of type II (see equations (5.30) and (5.31)); that is, with a symmetrical impulse response and a choice of N odd response.

We then introduce the quantities:

images

We have seen that in equations (5.30) and (5.31), the related transfer function satisfies the formula:

images

Taking into account equation (5.56), equation (5.57) is written as follows:

images

To simplify matters, we only consider the quantity Hr(f), which determines the amplitude:

images

The problem is in estimating the coefficients b(n) so that the frequency response is optimal by distributing the approximation error in the passband and the attenuated band.

images

Figure 5.18. Frequency response ...

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