OE-8.5 IIR Filter Design & Filter Realization
You can design IIR filters from Butterworth and Chebyshev approximations using impulse invariance and the bilinear transform with prewarping, and draw direct form realizations.
IIR design borrows a century of analog filter theory and maps it to discrete time, which is why Butterworth and Chebyshev appear here at all. The bilinear transform avoids the aliasing that impulse invariance suffers, at the cost of frequency warping - and prewarping is the correction, which is exactly why the step exists. Realization structures matter because mathematically identical filters behave differently in finite precision, and direct form II uses fewer delay elements.
Work through these
Butterworth approximation
The first classic response, flat in the passband and gentle in transition. It is the starting point of the design route this topic follows.
Chebyshev approximation
The alternative, trading ripple for a sharper transition at the same order. Choosing between the two is the first real design decision here.
Impulse invariance method
The first way of mapping an analog design into discrete time, and the one with a flaw: it can alias. That flaw is exactly what the next item avoids.
Bilinear transformation method and prewarping
The preferred route, because it cannot alias. The price is frequency warping, and prewarping is the correction; knowing why the step exists is worth more than the formula.
Realization of IIR filters: direct form I
The straightforward structure, using separate delays for the two halves of the difference equation. It is the reference the next item improves on.
Realization of IIR filters: direct form II
The same response with half the delay elements, by sharing them. In finite precision that structural choice changes the filter's real behaviour.
Realization of FIR filters: direct form and linear phase structures
The structures for the other filter family, including the one that exploits coefficient symmetry to halve the multipliers. A nice case of mathematics buying hardware.
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