The windows a real filter leaves

The clock that is too fast found that sampling a band on purpose turns the anti-alias requirement inside out. For a signal at baseband every sample rate in the requirement is a lower bound, and a faster clock is always at least as good. For a band from 100 to 120 MHz the allowed clocks are six disjoint windows — 240 MHz and above, 120 to 200, 80 to 100, 60 to 66.7, 48 to 50, and 40 exactly — with forbidden gaps between them, so 110 MHz fails while both 100 and 120 work.

Those windows assumed a band with hard edges: nothing outside 100 to 120 MHz at all. The essay named the first thing that assumption leaves out: doing them at, say, sixty decibels of rejection — with a real bandpass response in front — would give windows narrower than these by an amount that is itself a function of the filter’s order, which converts this essay’s clean arithmetic into that essay’s currency: a filter order priced in clock tolerance rather than in clock rate.

What a real filter lets through #

A real bandpass filter does not stop at the band’s edges. It rolls off, and between the band’s edges and the frequencies where it reaches the rejection asked for, it passes signal that is still strong enough to matter if it lands on the band. For a Butterworth bandpass filter of order $n$ built on the band’s own edges, the attenuation depends on the frequency through the bandpass transformation $\Omega = (f/f_0 - f_0/f)\,f_0/B$ with $f_0$ the band’s geometric centre and $B$ its width, and it reaches a rejection of $R$ decibels where $|\Omega| = (10^{R/10} - 1)^{1/2n}$. Solving that for $f$ gives the two frequencies, $f_L'$ below the band and $f_H'$ above it, between which the filter passes more than it should.

Everything in $[f_L', f_H']$ outside the band itself is signal the filter has let through, and a clock is good only if none of it folds onto the band. For a clock that puts the band in its $n$-th Nyquist zone, the unrejected signal just above the band folds about the zone’s upper edge and must land above the band, and the signal just below folds about the zone’s lower edge and must land below it. That gives

$$\frac{f_H + f_H'}{n} \;\le\; f_s \;\le\; \frac{f_L + f_L'}{n-1}$$

which is the hard-edged windows, $2f_H/n \le f_s \le 2f_L/(n-1)$, with each edge moved inward by half the transition band on its own side.

Six windows become three #

At 60 dB of rejection a twelfth-order filter passes more than it should between 93.2 and 128.8 MHz, and it keeps four of the six windows. An eighth-order filter’s span is 88.4 to 135.8 MHz and it keeps three; a sixth-order one, 82.4 to 145.6, three; a fourth-order one, 66.9 to 179.4 MHz, two. The windows that disappear are the low ones, the slow clocks, because a low window is narrow to begin with — the band’s own width is a large fraction of the zone — and the transition bands eat it first.

The window everybody would use, 120 to 200 MHz, survives every order drawn, but it shrinks: to 124.4–193.2 MHz at order twelve, 127.9–188.4 at eight, 132.8–182.4 at six and 149.7–166.9 at four. The top window, from 240 MHz up, loses its lower edge by the same arithmetic and never closes.

Every window is checked by the operation that defines it rather than by the formula that summarises it: the figure folds every frequency the filter passes above the rejection, at clocks just inside each window’s two edges and at its middle, and requires that none lands on the band; and it folds them at a clock one per cent below each window and requires that something does. The closed form and the fold agree at every window, including the narrowest — a twelfth-order filter’s fourth-zone window at 60 dB, seventy kilohertz wide.

Why the slow windows close first #

The low windows vanish before the high ones for a reason that is visible in the hard-edged formula. The $n$-th window runs from $2f_H/n$ to $2f_L/(n - 1)$, and its width is $2(f_H - nB)/(n(n-1))$: it shrinks roughly as the square of the zone number, and it reaches nothing at $n = f_H/B$, the lowest zone the band can occupy. The filter’s transition bands take $(f_H' - f_H)/n$ off the top edge and $(f_L - f_L')/(n - 1)$ off the bottom — an amount that shrinks only as the first power of the zone number.

So the filter’s bite is a larger share of the window the lower the clock. At the 120-to-200 MHz window, where $n = 2$, an eighth-order filter’s transition bands of 11.6 and 15.8 MHz take 7.9 MHz off the bottom edge and 11.6 off the top of an eighty-megahertz window. At the 80-to-100 MHz window, $n = 3$, the same transition bands take 5.3 and 5.8 MHz off a twenty-megahertz window. At 60 to 66.7, $n = 4$, they take more than the window has. The frequency a sample rate invents is the boundary that makes each zone a zone; the filter’s transition band is what fills the space between them.

Oversampling is the window that never closes #

The top window — every clock above $f_H + f_H'$ — survives every filter at every rejection, and it is ordinary oversampling: the band in the first Nyquist zone, as if it were at baseband. It has no upper edge, so its tolerance is unlimited, and a filter of any order leaves it open. What the filter decides for it is where it begins: 240 MHz with hard edges, 255.8 MHz behind an eighth-order filter at 60 dB, 299 MHz behind a fourth-order one.

That is the baseband problem again, and the earlier essays on it priced the filter in clock rate. The floor every filter has found that an all-pole design keeps falling past its stopband while an elliptic one does not, so a faster clock always helps an all-pole filter; two-thirds of a bit for a factor of twenty-eight priced the order’s own noise against the clock rate it saves. Bandpass sampling adds the lower windows as an alternative to paying for the top one, and the price of taking one of them is paid in the filter’s order at the exchange rate above.

The window as a clock tolerance #

A window’s width is what a clock is allowed to wander by. A crystal oscillator is good to parts per million and never leaves a window it starts in; a clock derived from a synthesiser that must also serve other converters, or a sample rate chosen from a list by a system that did not know about the band, has to fit a window with its own tolerance. So the useful measure of a window is its half-width over its centre: the clock tolerance it allows.

In the second zone — the 120-to-200 MHz window — a brick wall allows ±25 per cent. At 60 dB of rejection a filter of order four allows ±5.4 per cent, five ±12.2, six ±15.7, eight ±19.1, twelve ±21.7, and thirty ±23.9. The third zone, 80 to 100 MHz with hard edges and ±11 per cent, opens only from order six; the fourth, ±5.3 per cent with hard edges, only from order ten.

That is the conversion the earlier essay asked for: a filter’s order is bought in clock tolerance, and the exchange rate is steep at low orders and flat at high ones. Going from four to six buys ten points of tolerance; going from twelve to thirty buys two. And a lower zone — a slower clock, and a cheaper converter — needs more order for the same tolerance, because its window starts narrower.

One order per ten decibels #

The rejection enters the edges only through $(10^{R/10} - 1)^{1/2n} \approx 10^{R/(20n)}$, which depends on the ratio of the rejection to the order. So every pair of rejection and order with the same ratio gives the same windows.

At 40 dB the second zone allows ±15.7 per cent at order four; at 60 dB, at order six; at 80 dB, at order eight. Ten decibels of rejection per order is the exchange rate, and it holds at every tolerance and every zone to the accuracy of the approximation, which for these rejections is better than the figures print. A designer who needs twenty decibels more rejection of what folds back pays two orders of filter to keep the same clock tolerance, or pays in tolerance at the same order.

At 80 dB a fourth-order filter keeps only the top window — sampling at 240 MHz or above, which is ordinary oversampling and gives up the whole point of bandpass sampling. The 120-to-200 MHz window opens at order five. The eighth-order filter at 80 dB leaves the same three windows the sixth-order filter left at 60, which is the exchange rate again.

The hard-edged windows, for comparison #

The hard-edged picture is the limit every row of the hero figure approaches as the order rises, and it is worth keeping in view because it says what a filter can never buy: no filter widens a window beyond its hard-edged width, and no filter opens a window the hard edges forbid. The filter’s order only decides how much of each hard-edged window survives, and a design that finds its clock in a gap of the hard-edged picture cannot be rescued by any filter at all.

What the order costs elsewhere #

The exchange rate prices order in tolerance, and order has costs of its own that a designer weighs against the tolerance it buys.

Delay, and its flatness. A bandpass filter’s group delay across its band grows with its order and peaks at the band’s edges, so an eighth-order filter delays the edges of the 100–120 MHz band more than its centre by an amount that a fourth-order filter does not. For a band carrying a modulated signal that delay variation is distortion, and flat delay bought with more delay is what it costs to remove it: an all-pass equaliser, more order again.

Component spread. A high-order bandpass filter’s element values spread over a wider range and its response is more sensitive to each of them. What a steep skirt costs counts that sensitivity for a lowpass, and the bandpass transformation inherits it: the transition bands this page depends on move when the parts do, and a window computed for the nominal filter is narrower for the worst-case one.

And the band’s frequency, which no window changes. A clock’s jitter costs a phase error proportional to the frequency being sampled, and that is 110 MHz in every window — the second zone and the fifth alike. So a low window does not raise the jitter penalty, but it does mean paying a 110 MHz jitter penalty with a clock that is a fraction of that rate, which a baseband design at the same clock never pays. The saving in converter speed that a low window buys is partly spent on a cleaner clock.

What this means for choosing a clock #

Choose the zone first, with the filter’s order in mind. A second-zone clock with a sixth-order filter at 60 dB allows ±15.7 per cent; a third-zone clock needs at least order six to exist and allows far less. The slower clock is cheaper in the converter and more expensive in the filter, and the exchange is steep.

Put the clock at the window’s centre, not the band’s. A window’s centre in sample rate is not where any arithmetic about the band would put it, and the tolerance is symmetric only about the window’s own centre; the figures give it.

And write the rejection into the specification of the clock. A clock tolerance quoted without the rejection it was computed for is a tolerance for a brick wall, and every real filter narrows it. What the filter in front costs priced the baseband filter in clock rate; for a band, the same filter is priced in clock tolerance, and what a steep skirt costs is where the order’s other costs — ripple, delay, sensitivity — are counted against what it buys here.

How the windows were found and checked #

The filter’s edges are the two solutions of $(f/f_0 - f_0/f)\,f_0/B = \pm(10^{R/10} - 1)^{1/2n}$, in closed form. The windows are $(f_H + f_H')/n \le f_s \le (f_L + f_L')/(n - 1)$ for every zone $n$ whose window is non-empty and whose clock is at least twice the band’s width.

Each window is checked by folding: every frequency between $f_L'$ and $f_H'$ outside the band is reduced to the first Nyquist zone at the stated clock, three thousand of them across the span, and none may land on the band’s own image at clocks two per cent inside each window’s edges and at its middle; at a clock one per cent below each window, one must. The figure also requires that a gentler filter never has more windows than a steeper one, and that each zone’s tolerance rises with the order towards, and never reaches, the brick wall’s.

Other filter shapes, a jittered clock and the way back out #

The filter is a Butterworth on the band’s edges. A Chebyshev or elliptic filter of the same order has a steeper transition and leaves wider windows; the exchange rate of ten decibels an order is the Butterworth’s, and a filter with zeros in its stopband buys more rejection per order.

The clock is exact inside its window. A clock’s jitter moves each sampling instant, and the phase error it produces on a signal at 110 MHz is proportional to that frequency — so sampling in a high zone multiplies the jitter’s cost by the zone number, which the window arithmetic does not see.

And the conversion runs one way. The same windows exist on the way out: a converter that generates a band by producing one of its images in a high zone needs a reconstruction filter that selects that image and rejects its neighbours, and the neighbours are spaced by the clock exactly as the aliases are here. The staircase on the way out is the image arithmetic at baseband; for a deliberately high-zone output the filter’s order is again bought in clock tolerance, by the same ten decibels an order, with the output’s sinc roll-off added to the bill.

And the band’s own images are not examined. Odd zones fold the band through with its spectrum intact and even zones reverse it; the second-zone window delivers a mirrored band, free to undo digitally and not free to forget.

Still open: the jitter the zone multiplies, the elliptic filter, and the reversed spectrum #

The jitter the zone number multiplies. A clock’s rms jitter costs a signal at frequency $f_\mathrm{in}$ a phase error of $2\pi f_\mathrm{in}$ times it, whatever the clock rate. Sampling the 110 MHz band in the second zone at 150 MHz pays the jitter penalty of a 110 MHz signal, where a baseband design at the same rate would pay that of a 20 MHz one — a resolution cost that sets a floor under the tolerance these windows allow.

A filter with zeros. An elliptic bandpass filter reaches 60 dB in a fraction of a Butterworth’s transition band, and its exchange rate is not ten decibels an order. How much of the hard-edged window an elliptic filter of order four keeps, against the Butterworth’s ±5.4 per cent, is the measurement that would say whether the steeper filter’s ripple and delay are worth its tolerance.

The spectrum reversed. Half the windows deliver the band mirrored, and a design that has to change clocks between windows — because its tolerance does not fit one — also changes whether its spectrum is reversed. Which windows do, and what a system that hops between them must track, is a parity the window arithmetic reports as a zone number and does not draw.