S5-3.2 IIR Filter Design & Realization

What this is and why it exists

Analog filter design has a century of catalogue behind it. Digital filter design does not need to repeat that work.

Infinite impulse response design borrows the analog catalogue and maps it into discrete time. That is why Butterworth and Chebyshev appear in a digital course at all.

The vocabulary

• IIR filter — a filter whose output depends on past outputs as well as past inputs, so its impulse response never quite ends.
• Butterworth response — the approximation that is as flat as possible in the passband.
• Chebyshev response — an approximation that allows ripple in return for a sharper transition.
• Impulse invariance — mapping an analog design by sampling its impulse response.
• Bilinear transform — mapping an analog design by a substitution that squeezes the whole frequency axis into the available range.
• Frequency warping — the distortion of the frequency axis that the bilinear transform causes.
• Prewarping — moving the analog specification first, so the warped result lands where you wanted.
• Spectral transformation — converting one filter design into another shape.
• Direct form I — a structure with separate delay chains for input and output.
• Direct form II — a structure that shares one delay chain, using half the memory.

The mental model

Two classic responses head the catalogue. Butterworth is as flat as it can be in the passband, and its transition from pass to stop is gentle. Chebyshev allows a controlled ripple in the passband and buys a sharper transition at the same filter order. Choosing between them is the first real design decision here. Flatness or sharpness, at a fixed cost.

Then the design has to become digital, and there are two routes.

Impulse invariance samples the analog impulse response. That is intuitive and it has a flaw. Sampling causes aliasing, so any analog response that has not died away by half the sampling rate folds back and corrupts the design. It is usable only for filters that are already well confined in frequency.

The bilinear transform avoids that completely. It maps the entire infinite analog frequency axis into the finite digital one, so nothing is left over to alias. The price is that the mapping is not uniform. Frequencies are squeezed more and more as they rise, which is frequency warping. The fix is to distort the specification in the opposite direction first. That is prewarping, and knowing why the step exists is worth more than the formula for it.

Spectral transformation saves repeated work. Design one lowpass prototype well, then transform it into a highpass, bandpass or bandstop filter. One good design serves four needs.

The comparison with the other filter family is a trade, not a preference. IIR filters reach a given sharpness at a much lower order, because feedback is doing work for them. What they cannot generally offer is exactly linear phase. The other family can, at a higher order. Sharpness for a given cost, against phase behaviour, is the axis.

Realization is the last step, and it matters more than it first appears. Direct form I keeps separate delay chains for the input and output histories. Direct form II shares one, so it needs half the delay elements. On paper both compute the same response. On a processor with a fixed number of bits they do not behave identically, because rounding enters at different points. That is the subject of a later topic, and it is why structure is a design decision rather than a drawing choice.

Second order sections are the practical unit. A high-order filter is built by cascading them rather than as one large structure, because the small sections are far less sensitive to coefficient rounding.

What you should now be able to explain or do

Compare Butterworth and Chebyshev responses and choose between them for a stated requirement. Say why impulse invariance can alias and why the bilinear transform cannot. Explain frequency warping and what prewarping corrects. Use a spectral transformation to turn a lowpass prototype into another shape. Draw direct form I and direct form II and say why the choice matters in finite precision.

Check yourself