OE-8.5 IIR Filter Design & Filter Realization

What this is and why it exists

This design route borrows a century of analog filter theory and maps it into discrete time. That is why Butterworth and Chebyshev, which are analog approximations, appear in a digital course at all.

The mapping is the interesting part, and there are two ways to do it. One suffers aliasing. The other does not, and instead warps the frequency axis — which is a problem with a clean correction.

That correction is prewarping, and understanding *why* it is needed is the difference between applying a formula and knowing what you did.

The vocabulary

• Infinite impulse response — the response to an impulse never entirely ends.
• Butterworth approximation — flattest possible pass-band, gentle transition.
• Chebyshev approximation — sharper transition, bought with ripple.
• Impulse invariance — matching the digital impulse response to the analog one.
• Bilinear transform — mapping the whole analog frequency axis into the digital range.
• Warping — the frequency distortion the bilinear transform introduces.
• Prewarping — pre-distorting the target frequencies so warping lands them correctly.
• Direct form — a standard arrangement of delays, multipliers and adders.

The mental model

The two analog approximations differ in what they optimise. Butterworth is as flat as possible in the pass band and pays for that with a gentle transition. Chebyshev gets a sharper transition and pays with ripple — either in the pass band or the stop band, depending on which form you use. Neither is better. They are two answers to where you would rather be imperfect. Putting it that way makes the choice a design decision rather than a lookup.

Then the mapping into discrete time. Impulse invariance samples the analog impulse response. It preserves the shape of the response in time, and it inherits the sampling problem from the first topic. An analog filter's response extends beyond any frequency limit, so sampling it aliases. For a low-pass filter with good attenuation that may be tolerable. For a high-pass filter it is not.

The bilinear transform avoids aliasing entirely, and it does so by a trick worth appreciating. It compresses the *entire* infinite analog frequency axis into the finite digital range. Nothing is left over to fold back, so nothing aliases.

But squeezing an infinite axis into a finite one cannot be uniform. Frequencies are progressively compressed as they rise — that is warping. A filter designed for a particular edge frequency comes out with its edge somewhere else, and increasingly so at higher frequencies.

Prewarping is the correction, and it is exactly what the name says. Before designing the analog prototype, distort your target frequencies the opposite way. After the transform squeezes them, they land where you wanted. It is a one-line adjustment that people apply mechanically. Knowing the reason means you also know where it matters most, which is at the high end.

Realisation is the last step: turning coefficients into an arrangement of delays, multipliers and adders. Direct form one and direct form two compute the same thing with different amounts of storage, the second using fewer delay elements. And this is not a cosmetic choice. Finite word length means the structure you pick affects how coefficient rounding disturbs the response, which is why serious designs care about realisation at all.

What you should now be able to explain or do

Say what Butterworth and Chebyshev each optimise and what each pays. Explain why impulse invariance aliases and when that rules it out. Explain how the bilinear transform avoids aliasing, and why warping is the unavoidable consequence. Apply prewarping and say what it corrects. Draw both direct form realisations and say which uses less storage. Say why realisation matters beyond appearance.

Check yourself