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During this time, it was well known that the best filters contain an equiripple characteristic in their frequency response magnitude and the elliptic filter or Cauer filter was optimal with regards to the Chebyshev approximation.

When the digital filter revolution began in the s, researchers used a bilinear transform to produce infinite impulse response IIR digital elliptic filters. They also recognized the potential for designing FIR filters to accomplish the same filtering task and soon the search was on for the optimal FIR filter using the Chebyshev approximation.

Several attempts to produce a design program for the optimal Chebyshev FIR filter were undertaken in the period between and Otto Herrmann, for example, proposed a method for designing equiripple filters with restricted band edges.

Another method introduced at the time implemented an optimal Chebyshev approximation, but the algorithm was limited to the design of relatively low-order filters.

This has become known as the Maximal Ripple algorithm. The Maximal Ripple algorithm imposed an alternating error condition via interpolation and then solved a set of equations that the alternating solution had to satisfy.

History[ edit ] In August , James McClellan entered graduate school at Rice University with a concentration in mathematical models of analog filter design and enrolled in a new course called "Digital Filters" due to his interest in filter design. At that time, DSP was an emerging field and as a result lectures often involved recently published research papers. The following semester, the spring of , Thomas Parks offered a course called "Signal Theory," which McClellan took as well.

He brought the paper by Hofstetter, Oppenheim, and Siegel, back to Houston, thinking about the possibility of using the Chebyshev approximation theory to design FIR filters. This ultimately led to the Parksâ€”McClellan algorithm, which involved the theory of optimal Chebyshev approximation and an efficient implementation. By the end of the spring semester, McClellan and Parks were attempting to write a variation of the Remez exchange algorithm for FIR filters.

It took about six weeks to develop and some optimal filters had been designed successfully by the end of May. McClellan joined Schlumberger in , where he worked for five years. Kilby Signal Processing Medal Parks joined the faculty at Rice University. He was a faculty member from to , when he joined the faculty at Cornell University.

Make sure that the error alternates on the ordered set of frequencies as described in 4 and 5. Return to Step 2 and iterate. If the alternation theorem is not satisfied, then we go back to 2 and iterate until the alternation theorem is satisfied. If the alternation theorem is satisfied, then we compute h n and we are done.

To gain a basic understanding of the Parksâ€”McClellan Algorithm mentioned above, we can rewrite the algorithm above in a simpler form as: Guess the positions of the extrema are evenly spaced in the pass and stop band.

Perform polynomial interpolation and re-estimate positions of the local extrema. Move extrema to new positions and iterate until the extrema stop shifting. Explanation[ edit ] The picture above on the right displays the various extremal frequencies for the plot shown. The extremal frequencies are the maximum and minimum points in the stop and pass bands.

The stop band ripple is the lower portion of ripples on the bottom right of the plot and the pass band ripple is the upper portion of the ripples on the top left of the plot.

In contrast to the Maximum Ripple approach, the band edges could now be specified ahead of time. To achieve an efficient implementation of the optimal filter design using the Parks-McClellan algorithm, two difficulties have to be overcome: Defining a flexible exchange strategy, and Implementing a robust interpolation method.

Two faces of the exchange strategy were taken to make the program more efficient: Allocating the extremal frequencies between the pass and stop bands, and Enabling movement of the extremals between the bands as the program iterated. The movement between bands was controlled by comparing the size of the errors at all the candidate extremal frequencies and taking the largest. The second element of the algorithm was the interpolation step needed to evaluate the error function.

They used a method called the Barycentric form of Lagrange interpolation, which was very robust. The optimal frequency response will barely reach the maximum ripple bounds. The derivative of a polynomial of degree L is a polynomial of degree L-1, which can be zero at most at L-1 places.

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## Digital Filter Design

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## DIGITAL FILTER DESIGN PARKS AND BURRUS PDF

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