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### Filter Design Approach

We've been talking about the white-box approach in which every first-order element (mass, spring, ...) is explicitly modeled by a first-order finite-difference scheme. This is especially needed for elements that are time varying or pushed into nonlinear regimes of operation.

When a system is linear and time-invariant (LTI), there is no need for such fine-grained modeling, and we can take a a black-box approach, in which we need only model the frequency response from the input(s) to output(s) of the system using a digital filter.

Filter Design Approach to Ideal Integrators and Differentiators

Consider the following simple cases:

• Integrator:
(e.g., force-driven mass with a velocity output)
• Differentiator:
(e.g., force-driven spring with a velocity output)

The digital filter design formulation typically minimizes frequency-response error with respect to the filter coefficients

Ideal Frequency Responses

• Ideal Digital Integrator

• Ideal Digital Differentiator:

• Exact match is not possible in finite order
• Minimize where is the digital filter frequency response and denotes some norm of
• This is a digital filter design formulation

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