Learning · Step 7

CIC filters

A CIC (cascaded integrator–comb) filter is the cheapest way to decimate or interpolate by a large factor. It is built entirely from adders and delays — no multipliers — so its fixed-point story is not about Q-formats or rounding. It is about one thing: register width.

A CIC decimator is N integrator stages running at the high rate, a rate change by R, then N comb stages at the low rate. The integrators each have a pole at z = 1 — they accumulate without bound. The whole filter's gain is (R·M)^N, where M is the comb's differential delay (1 or 2). To carry that gain without losing the top bits, every register must grow by ⌈N·log₂(R·M)⌉ bits over the input width. That is the Hogenauer bit-growth rule — the single most important number when you size a CIC.

Now the counterintuitive part: let the integrators overflow. They will wrap constantly — and that is correct. As long as every register is the full B_in + ⌈N·log₂(R·M)⌉ bits wide, two's-complement wraparound in the integrators is exactly undone by the comb subtractions downstream. So you must never saturate a CIC integrator — saturation clips the wrap and permanently corrupts the output. This is the opposite of the rule for an ordinary accumulator.

After the combs you have a full-width result with the (R·M)^N gain baked in. Scale it back down (a shift), then truncate to the output word — that final truncation is the filter's only quantization step. The explorer below sizes the registers for you.

Try it

Set the stage count, rate change, differential delay and input width — the register bit budget and the verdict update live.

Stages N 5 Rate change R 32

input bits bit growth

What to notice

Raise N — growth climbs by log₂(R·M) bits per stage. A fifth stage on a decimate-by-32 CIC adds another 5 bits.
Push R up — growth follows log₂R, so doubling the rate change adds N bits.
Switch M from 1 to 2 — that is one extra log₂2 = 1 bit per stage, i.e. N more bits.
Change the input width — the growth figure doesn't move. Input width just shifts the whole register width up; growth depends only on N, R and M.
No combination makes saturation safe. The width only decides how wide the registers are — the integrators must wrap either way.

Key fixed-point rules

Size every register to B_in + ⌈N·log₂(R·M)⌉ bits — integrators and combs alike. The widest stage sets the width for all of them.
Let the integrators wrap. Two's-complement overflow in the integrators is exactly cancelled by the comb subtractions downstream — provided the registers are full width. Saturating an integrator clips that wrap and corrupts the output for good.
No multipliers, no mid-filter rounding. A CIC is only adders and delays; the only quantization is the final truncation of the output word.
Compensate the (R·M)^N gain. The output carries the full filter gain — shift it back down (or fold it into a following gain stage) before truncating to the output width.
Prune only by the numbers. You may discard LSBs stage by stage (Hogenauer pruning) to shrink the hardware, but size each discard from the noise-variance bound — never by eye.

Step exam

Answer all 3 questions correctly to complete this step.

A CIC filter's register bit growth is:
The integrator stages of a CIC filter should be:
A CIC filter is built from: