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Concept# Filter design

Summary

Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient degree to make it useful.
The filter design process can be described as an optimization problem where each requirement contributes to an error function that should be minimized. Certain parts of the design process can be automated, but normally an experienced electrical engineer is needed to get a good result.
The design of digital filters is a deceptively complex topic. Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of advanced research.
Typical requirements which are considered in the design process are:
The filter should have a specific frequency response
The filter should have a specific phase shift or group delay
The filter should have a specific impulse response
The filter should be causal
The filter should be stable
The filter should be localized (pulse or step inputs should result in finite time outputs)
The computational complexity of the filter should be low
The filter should be implemented in particular hardware or software
An important parameter is the required frequency response.
In particular, the steepness and complexity of the response curve is a deciding factor for the filter order and feasibility.
A first-order recursive filter will only have a single frequency-dependent component. This means that the slope of the frequency response is limited to 6 dB per octave. For many purposes, this is not sufficient. To achieve steeper slopes, higher-order filters are required.
In relation to the desired frequency function, there may also be an accompanying weighting function, which describes, for each frequency, how important it is that the resulting frequency function approximates the desired one. The larger weight, the more important is a close approximation.

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