RF Filter Yield: Component Tolerances vs Chebyshev

The Scenario: An Anti-Alias Filter for a 300 MS/s ADC

You're digitizing 0–100 MHz with an ADC sampling at 300 MS/s. Anything the ADC sees between 200 and 300 MHz folds back into 0–100 MHz ($f_{alias} = 300 - f$), and so does anything between 300 and 400 MHz ($f - 300$). So the front-end filter has two jobs:

• Flat passband: at most 1.0 dB of variation from DC to 100 MHz.
• Stopband: at least 40 dB of rejection from 200 MHz upward, so folded signals land 40 dB down.

You pick a 5th-order Chebyshev low-pass with 0.5 dB ripple and a 100 MHz cutoff. The ideal design rejects 42.2 dB at 200 MHz. Built from E24 standard values — 130 nH, 39 pF, 200 nH, 39 pF, 130 nH in a 50 Ω system — it rejects 41.2 dB: rounding to real part values already cost 1 dB of the 2 dB margin.

That still passes on paper. The question is what happens when the parts arrive with their tolerances.

Monte Carlo Setup

The RF Filter Monte Carlo tool builds hundreds or thousands of virtual filters, each with every inductor and capacitor drawn from its tolerance distribution around the standard value, and simulates each one exactly with ABCD matrices. It then counts the builds that meet your spec.

A build passes when its peak-to-peak |S21| variation from DC to the cutoff is within 1.0 dB and its rejection from 2× the cutoff upward is at least 40 dB. Leave either limit blank and the tool uses a default derived from the standard-value build — 0.5 dB more variation, 3 dB less rejection. The limits never move with the tolerance you are testing.

What the Results Show

The ± is the 95 % confidence interval of the estimate from 2,000 trials. With Gaussian 5 % parts the design is safe; with 10 % parts, one board in six fails. The failures split almost evenly: 175 of the 2,000 builds miss the 40 dB stopband — the 1 dB of margin left after E24 rounding is what the tolerance eats — and 155 exceed 1.0 dB of passband variation, from ripple peaks near the cutoff.

Gaussian or Uniform?

"±5 %" on a reel label says how far a part may be from nominal, not how the values are distributed inside that band. The tool offers two models:

The uniform model is the conservative one: it treats a value at the edge of the band as just as likely as nominal. Real distributions are often closer to Gaussian — but parts that were binned (5 % parts from which the 1 % parts were sorted out) can be depleted in the middle, which is worse than either. If you don't know your supplier's distribution, design for the uniform number.

Why Chebyshev Is Tolerance-Sensitive

A Chebyshev filter places its poles close to the $j\omega$ axis to get its equiripple passband and steep skirt. Each pole does more work than in a Butterworth design, so a small change in an element value moves the response more. The tool's sensitivity ranking shows where: the centre inductor (L3, 200 nH) affects the passband about twice as much as any other element. If you can only afford one tight-tolerance part, make it that one.

The sensitivity to element values can be written

but the Monte Carlo is more useful than the formula: the five elements interact, and a capacitor high in its band combined with an inductor low in its band moves the response by more than either would alone.

Buying the Yield Back

With 5 % Gaussian parts and the same 1 dB / 40 dB limits:

Lower ripple is not a free fix. Dropping to 0.1 dB of ripple flattens the passband but costs 6 dB of stopband: the design no longer meets 40 dB at all. Ripple is what buys the Chebyshev its selectivity. One more order buys margin. A 7th-order design rejects 62 dB at 200 MHz, so tolerances have 22 dB to work with: with 10 % parts it yields 98.2 %, against 83.5 % for the 5th-order one. The price is two more components and a little more insertion loss. Watch even orders. A 6th-order Chebyshev cannot be equally terminated: its prototype needs a 99.2 Ω load in a 50 Ω system. The tool designs and simulates with that load and says so. Into 50 Ω the passband would not be equiripple. Tighter parts. Going from 10 % to 5 % raises yield from 83.5 % to 99.5 %; 2 % parts take it to 100 %. The sensitivity ranking says where to spend first: the centre inductor.

What About Butterworth?

A Butterworth filter's cutoff is its −3 dB point, so its passband variation up to the cutoff is 3 dB by definition — it can't meet a 1.0 dB flatness limit to 100 MHz with a 100 MHz cutoff. Moving the cutoff up to keep 1 dB at 100 MHz pushes the skirt out too, and a 7th-order Butterworth then no longer reaches 40 dB at 200 MHz. For this spec the Chebyshev is the right family; the Monte Carlo tells you how much margin it needs.

Reading the Yield Histogram

The tool plots the distribution of passband variation across the trials, alongside the 5th–95th percentile envelope of the response. For the 10 % case the envelope is under 0.2 dB wide below 50 MHz but opens to 3–4 dB between 90 and 190 MHz, around the cutoff and down the skirt — exactly where both kinds of failure come from. Incoming inspection therefore needs two measurements, not one: flatness near the cutoff and rejection at 200 MHz.

Use the RF Filter Monte Carlo tool with your own limits before committing to a component order.

Updated 2026-09-19. The tool now judges yield against limits that stay fixed as the tolerance changes, uses the high-pass bands for high-pass filters, and honours the Gaussian distribution. An earlier version of this article described pass criteria the tool did not use and quoted yields from before those fixes.