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I try to understand the 'arbitrary' what does it meant? I had read many references ,such a one it is 'begin the process with a transfer function of your choice' ,in other reference :related by the starting point,

In other hand i read about Chebyshev approximation with arbitrary magnitude, how can i find the better design with arbitrary magnitude ? any method of design have a criteria ,the 'arbitrary' related by the method of design or related by the transfer function?

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The term "arbitrary" in this context refers to the desired frequency response, and what it means is that the respective design method accepts any desired response. So unlike required by many standard design routines, the magnitude response does not need to be piecewise constant or linear, and, more importantly, the phase response need not be linear or minimum-phase. The corresponding design problem is a complex approximation problem because the desired frequency response which must be approximated is a complex-valued function.

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https://en.m.wiktionary.org/wiki/arbitrary

see the third definition for mathematics.

you the filter designer are free to choose what the magnitude response is desired to be and the filter design algorithm ( a numerical optimization) will give you the “best” it can do.

The term “best” in the context of what you quoted in your question is the Chebychev error. This tries to minimize the worst case error and produces an “equal ripple” magnitude frequency response.

The Parks McClellen (sometimes called Remez) uses this minimax or also called Chebychev norm criteria.

in summary the word “arbitrary” in filter design isn’t arbitrary. There are many words in DSP that have a common meaning that ordinary people use in evert day life that has a twist in DSP.

One example is “so-called” which is usually an insult in common English but in DSP means that it is a common or frequently used name for something .

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