The bandwidth noise sees

Noise out of a system is √(density × bandwidth). The density is a property of the source and the previous essay computes it. The bandwidth is a property of the response, and it is not the number almost everybody uses.

The −3 dB point is where a response has fallen by thirty per cent in voltage. It is a useful marker for a signal, and for noise it is the wrong quantity entirely: a filter does not stop at its corner, it continues to pass a diminishing amount above it, and all of that contributes.

What is being integrated, and what π/2 is #

The equivalent noise bandwidth is ∫|H(f)|²df divided by |H|² in the passband: the width of the brick wall that would pass the same noise power.

For a single pole, $|H|^2 = 1/(1 + (f/f_c)^2)$, and the integral of that from zero to infinity is $f_c \times \pi/2$. So the noise bandwidth is 1.5708 times the corner, exactly, and using the corner instead of it understates the noise power by that factor and the noise voltage by its square root, which is 21%.

The figure measures it. It sweeps the solved response of a real network — a resistor and a capacitor, assembled and inverted at every frequency on the grid — squares the magnitude, integrates trapezoidally in linear frequency on a logarithmic grid, and divides by the −3 dB point found by bisection on the same response. The answer is 1.5706, against π/2 = 1.5708.

Two decisions in that sentence are worth defending, because both were got wrong first.

The grid is logarithmic and the rule is linear. The response is logarithmic in frequency, so a linear grid would spend all its points in the last decade and none where the response is changing. The integral, though, is a linear one — ∫|H|²df — so the trapezoidal rule has to be applied in linear f on a logarithmic grid. Pairing them the other way is a silent factor of ln 10.

The stretch below the grid is not empty. A logarithmic axis cannot reach zero, so the integral starts at some small frequency and the region below it is missed. That region is not negligible: starting at $f_c/1000$ leaves out one hertz of a 1,571 Hz integral, which is 0.064% — and it was exactly the size of the discrepancy that led to it being found. The response there is flat by construction, since the grid starts deep in the passband, so the missing piece is a rectangle and can be added exactly.

With both corrected the measured ratio is 1.5706 and the residual is the truncation at the top of the sweep, which falls as the sweep is extended: 1.5697 at a thousand times the corner, 1.5706 at ten thousand, 1.5706 at a hundred thousand.

What the ratio does as poles are added #

The single pole’s π/2 is the extreme case, and the ratio falls towards one as the skirt steepens — which is what a brick wall is, in the limit.

Measured on realised networks at a common −3 dB point: 1.5706 for one pole, 1.1107 for two, 1.0472 for three, 1.0262 for four, 1.0167 for five, 1.0115 for six.

That sequence is worth reading as a practical statement. A two-pole filter already gets most of the benefit — the excess over a brick wall falls from 57% to 11% — and beyond about three poles the noise bandwidth is within five per cent of the corner and there is very little left to gain. Filters are made steeper for reasons other than noise, and the noise argument stops paying almost immediately.

The shape of the diminishing return is worth a moment too. The excess over unity goes 0.571, 0.111, 0.047, 0.026, 0.017, 0.012 for one through six poles, and the ratios between consecutive terms settle towards about 0.7 — so each additional pole removes roughly a third of what is left. That is slow enough that the first two poles do most of the work and fast enough that nothing beyond the fourth is worth having, which is a narrower conclusion than “more poles are better” and a more useful one.

The family that comes out below one #

The three families this collection builds — Butterworth, Chebyshev and Bessel — are all normalised to the same measured −3 dB point, so their noise bandwidths are directly comparable. They are not the same, and one of them is surprising.

At order three: Butterworth 1.0472, Bessel 1.0736, Chebyshev 0.9992. At order five: 1.0167, 1.0385, 0.9637. At order six: 1.0115, 1.0386, 1.0687.

Two things in that table need explaining and both are real.

Chebyshev, at odd order, is below one. Its noise bandwidth is less than its own −3 dB point, which sounds impossible for a filter whose response extends above the corner. It is not: the response is already falling inside the band, because an odd-order Chebyshev starts at unity, dips into its ripple band, and comes back — so the area lost inside the corner exceeds the area gained outside it. The −3 dB point of a Chebyshev sits beyond the ripple band edge rather than at it, and that gap is where the deficit comes from.

Even orders behave differently. At order six Chebyshev is back above one, at 1.0687. An even-order Chebyshev starts at the bottom of its ripple band and comes up, so its passband shape relative to its corner is different and the sign of the effect reverses. The figure’s assertion is therefore made for odd orders only, and the restriction is the interesting part rather than a convenience: the claim is about the shape, so it is asserted where the shape holds.

Where the odd-order deficit comes from, precisely #

The Chebyshev result is odd enough to deserve more than a sentence, and it is entirely explicable once the two frequencies involved are separated.

A Chebyshev filter is defined by a ripple band: its response oscillates between one and 1/√(1 + ε²) up to a frequency, and falls monotonically after it. That frequency — the ripple band edge — is where the definition ends, and it is not the −3 dB point. For a 0.5 dB ripple the band edge is where the response is 0.5 dB down, and the −3 dB point is further out, by an amount that grows with the order.

So when three families are normalised to a common −3 dB point, as they are here and in the filters field, the Chebyshev’s ripple band is pushed inside that point. Its response is already well below unity for a substantial part of the interval from zero to the corner, and the noise it fails to pass there exceeds the noise it does pass beyond the corner.

The deficit therefore grows with order, because the gap between the ripple band edge and the −3 dB point grows: 0.9992 at order three, 0.9637 at order five. And it reverses for even orders, because an even-order Chebyshev’s response starts at the bottom of the ripple band rather than the top — the same normalisation then places its band differently and the arithmetic goes the other way.

None of that makes Chebyshev a quiet filter in any absolute sense. It makes the ratio to its own −3 dB point less than one, which is a statement about a comparison and not about a component. Two filters compared at their ripple band edges would rank differently again, and the reason the comparison here uses the −3 dB point is that it is the number a datasheet gives.

Why any of this matters #

The consequence is one sentence and it is the reason this essay exists rather than a paragraph in the previous one.

Two filters with the same corner frequency admit different amounts of noise, and the difference is up to fifty per cent in power. A designer who chooses a filter for its skirt, and then computes the noise from its corner frequency, has made an error whose size depends on a property of the filter that the corner frequency does not report.

Put concretely: the same source through a one-pole filter at 10 kHz and through a five-pole Chebyshev at 10 kHz produces noise voltages in the ratio √(1.5706/0.9637) = 1.28. Twenty-eight per cent, from a choice made for entirely unrelated reasons.

The trade this adds to #

What a steep skirt costs measures the price of a sharper cutoff in delay variation and in ringing: at order five, Chebyshev is 16 dB further down at three times the corner than Butterworth and its passband deviation is the same, and what it pays is 49% of delay variation against Bessel’s 0.06%.

Noise is a fourth axis on that comparison and it does not agree with the other three. Chebyshev, which is the worst of the three for delay, is the best of the three for noise at every odd order. Bessel, which is the best for delay, is the worst for noise at every order tested.

There is no ordering that makes one family best, which is the honest state of the subject and the reason all three exist. What this page contributes is that the noise column is computable and usually is not computed — the table it would come from is in some textbooks and not in most, and the integration takes a few lines given a solved response.

What the measurement rests on #

The whole page is one integral repeated, so it is worth being explicit about what makes it trustworthy.

The response being integrated is a solved network rather than a polynomial. That matters here for a specific reason: the first version used the polynomial recovered by sampling the determinant, and for the higher-order families that recovery quietly returned a different filter — a fifth-order cascade’s roots are spread far enough that the trim discards a genuine coefficient, and the −3 dB point came back at 657 Hz for a filter normalised to 1 kHz. Nothing about the resulting numbers looked wrong; they were simply about a filter nobody had asked for.

Solving the network directly at each frequency is slower and cannot fail that way. Every point on every curve here is an assembled matrix, inverted, with current law rebuilt from the element relations and the energy counted twice before the answer is returned — the same check every other figure on this site runs, applied a few thousand times per figure.

A last observation about π/2 #

There is something worth noticing about the single-pole result that is easy to walk past.

π/2 is not an approximation, a fitted constant or a convention. It is $\int_0^\infty dx\,/$(1 + x²), which is arctan evaluated at infinity, and it comes out of the shape of a one-pole response and nothing else. Any single-pole system — a resistor and a capacitor, a thermal time constant, a mechanical damper — passes exactly π/2 times its corner frequency’s worth of white noise power. The constant belongs to the shape rather than to electronics.

This collection has a habit of finding relations that look exact and are not, so it is worth marking one that is. Resonance and its bandwidth finds a half-power width of exactly f₀/Q — and finds, in the same figure, that the band is not centred on the resonance, which every textbook picture draws as though it were. The pattern there was an exact relation sitting beside a false one.

Here the exact relation sits beside a widespread substitution: the −3 dB point used where the noise bandwidth belongs. The substitution is not an approximation to π/2; it is a different number, twenty-one per cent away, and the reason it survives is that it is close enough to look like a rounding. An error of a factor of two would have been caught decades ago. An error of twenty-one per cent gets absorbed into a margin.

Measuring it — integrating the response rather than reading the corner — takes about ten lines given a solved network, and this page’s whole contribution is that nothing about it needs to be quoted.

What the integral says about measurement time #

There is a use for the noise bandwidth that is not about filters at all, and it is worth ending on because it is where the quantity is most often needed and least often recognised.

Any measurement that averages is a filter. Averaging N samples taken at intervals of T is a low-pass filter whose noise bandwidth is 1/(2NT) — the reciprocal of twice the total measurement time — and that expression contains no reference to any corner frequency because a boxcar average does not have one in any useful sense.

So the noise on an averaged measurement is √(density/(2t)) where t is how long it took, and the resolution improves as the square root of the time. Resolving a microvolt against a ten nanovolt-per-root-hertz floor at a signal-to-noise ratio of ten needs a noise bandwidth of 100 Hz, which is five milliseconds of averaging. Ten times the resolution needs a hundred times the time — half a second — and a hundred times the resolution needs fifty seconds.

That relation is the corner where averaging stops working’s subject and it is the practical form of everything on this page. A bandwidth is not only a property of a filter someone chose; it is a property of how long a measurement was allowed to take, and the two are the same quantity computed the same way — with the qualification that essay adds, which is that the identity holds for white noise and collapses for flicker: a white sequence’s block mean falls as the −0.510 power of the block length and a pink one’s as the −0.087, so a thousand-sample average buys a factor of 36.9 on one and 2.1 on the other.

The twenty-one per cent, where it is spent #

A noise voltage computed from a −3 dB point is twenty-one per cent low, and that is a larger error than most of the refinements a designer would reach for first. Three measurements elsewhere in this collection are stated against a bandwidth and inherit it directly.

Two thresholds because there is a floor measures hysteresis in units of the noise’s own standard deviation — 3.57 of them — which is exactly the quantity this essay’s factor multiplies. Twenty-one per cent of σ is twenty-one per cent of the hysteresis a design needs, and the requirement is one that has to be met with margin rather than approached.

The floor a converter sets crosses a quantiser’s floor against a source’s Johnson noise in a bandwidth stated as a number, and says so: the honest version needs an equivalent noise bandwidth computed on the anti-alias filter in front, which would move every crossing by a few tenths of a bit in a direction that depends on the family.

And a floor and a ceiling quotes 66.3 decibels of range in a ten-kilohertz measurement, where the ten kilohertz is a stated band rather than a realised filter — so the floor half of that range carries this essay’s correction and the ceiling half does not, which means the number is a slightly optimistic statement of what a real chain has.

The five-pole Chebyshev’s 0.964 is the reason none of that can be fixed by a single fudge factor. The correction is above one for a gentle roll-off and below one for a steep one, so it has to be computed on the response that is actually there. Which is a small argument for having a solver: the integral that produces 1.5706 is the same integral for every response, it needs no closed form, and the table of ratios that a designer would otherwise look up covers the few families somebody tabulated. A real front end is rarely one of those families — it is a filter, an amplifier’s own roll-off and a stray capacitance in series — and its ratio is neither π/2 nor 0.964 but whatever the product of the three integrates to.

Where the factor is spent #

A noise bandwidth that is π/2 times a first-order corner is the factor every voltage in this field is computed with. The floor a resistor sets is where it turns a density into a voltage, measured two ways. The floor a circuit has is where it decides an optimum source resistance, and The total that has no resistor in it is the one floor in the field where it cancels out entirely. Three families, one corner is where the responses being integrated come from.