The envelope that leaves early

The delay that is a centre of mass found that a low-pass filter’s group delay at zero frequency is exactly the balance point of its impulse response, and that the theorem says nothing at all about a band-pass, whose impulse response integrates to zero and has no balance point. A band-pass still has a group delay — $-d\phi/d\omega$ is perfectly well defined at its centre frequency, and flat magnitude, unflat delay drew it across three families of low-pass. The question is what that number is the delay of.

The textbook answer is that it is the delay of the envelope of a narrow-band signal, as distinct from the delay of its carrier, which is the phase delay $-\phi/\omega$. That answer has two consequences that are usually stated and rarely watched. At a band-pass’s centre the phase is zero, so the carrier is not delayed at all while the envelope carrying it is delayed by $2Q/\omega_0$. And at a notch’s centre the group delay is negative, so the envelope of a burst should come out before the envelope that went in.

This essay marches tone bursts through both and watches.

An envelope without a demodulator #

Measuring an envelope’s delay usually means demodulating the output — rectifying and filtering it, or computing an analytic signal — and every such step adds a delay and a distortion of its own. The measurement here avoids that entirely, using the one property that makes it possible: the network is linear and real.

The burst is a Gaussian envelope of standard deviation σ on a 1 kHz carrier. It is marched through the network twice: once on a sine carrier and once on a cosine carrier, with the identical envelope. A linear real network responds to the complex burst $g(t)e^{j\omega t}$ with a complex output whose real and imaginary parts are its responses to $g(t)\cos\omega t$ and $g(t)\sin\omega t$, so the output’s envelope is the modulus of the two marched outputs, exactly, at every sample. No filter, no rectifier, no Hilbert transform.

The march is the trapezoidal rule on the netlist, forty steps a carrier cycle, and the envelope’s peak is located by a parabola through the three highest samples. The band-pass is a series resonator read across its resistance, so its response at resonance is exactly one with zero phase; the group delay there is $2Q/\omega_0$, 3.183 milliseconds for a Q of ten at one kilohertz.

The burst is made long against the band-pass’s response time — its standard deviation is forty milliseconds at Q = 10, scaled with Q at other settings — because the theory being tested is a statement about narrow-band signals, and the last section measures exactly how narrow.

The carrier is not delayed #

The envelope comes out 3.171 milliseconds late, within 0.4 per cent of the group delay; at Q = 20 it is 6.312 against 6.366. The carrier inside it does something quite different.

The output’s zero crossings fall within 6.4 microseconds of the input’s — about two degrees of carrier phase, which is the march’s own resolution — while the envelope those crossings sit under is three milliseconds late. Three milliseconds is three whole carrier cycles. The output at the middle of the burst is, cycle for cycle, in phase with the input at the middle of the burst; it is simply smaller than it should be there and larger later.

That is the clearest possible picture of the difference between the two delays. Phase delay is about where the carrier’s zero crossings are; group delay is about where the energy is. A resonator at its centre frequency passes the carrier with no phase shift, and it does so while storing energy and releasing it slowly — the same energy whose decay time resonance, and the bandwidth it sets exactly read as a bandwidth. The burst’s energy has to fill the resonator before it comes out, and that filling time is the group delay.

The envelope reads the group delay at its own carrier #

The theory says the envelope’s delay is the group delay at the carrier frequency, wherever the carrier is. Moving the carrier across the band-pass’s response tests that against the whole curve.

At nine carriers from 880 to 1120 hertz the measured envelope delays follow the group-delay curve to within four per cent, falling from 3.17 milliseconds at the centre to about half that at the half-power frequencies, 950 and 1050 hertz, and to under half a millisecond well outside the band. The two routes have nothing in common: one is a derivative of a phase computed from the network at one frequency, the other is the peak of an envelope computed from two time-domain marches.

The agreement is closest at the centre and loosest at the edges, where the group delay changes most steeply with frequency. That is the first hint of where the reading fails: group delay is a derivative at one frequency, and a burst occupies a band of frequencies around its carrier; where the derivative is not constant across that band, the burst’s different parts are delayed by different amounts and its envelope is reshaped rather than simply moved.

Through a notch, the envelope peaks early #

A notch filter’s group delay at its centre is negative. For a series resonator to ground behind a source resistance, with a small resistance $r$ in the resonator so that the null is finite, it is $-2L\left(\frac1r - \frac{1}{R + r}\right)$: for 0.1 henries, 50 ohms and a kilohm of source, −3.810 milliseconds. If the envelope theory is right, a burst at the notch’s centre should come out with its envelope peaking 3.8 milliseconds before the input’s.

It does. The output envelope peaks 3.746 milliseconds early, within two per cent of the negative group delay. The output is small — the notch passes 4.8 per cent of the carrier at its centre — but its shape is the input’s shape, shifted earlier in time.

This looks like a violation of causality and is not one. Nothing comes out before something has gone in: the burst’s Gaussian envelope has been rising, very slowly, since long before its peak, and the output at every instant depends only on the input up to that instant. What the notch does is subtract. It removes most of the signal, and the part it removes is weighted toward the burst’s later portion, because the notch’s resonator takes time to build up the cancelling current and so cancels best what has been present longest. What survives the subtraction is weighted toward the earlier portion, and its peak is earlier. A smooth envelope’s later half is predictable from its earlier half, and a network can, in that sense, anticipate it.

The anticipation fails the moment the envelope is not smooth. A burst that switches on abruptly carries no information about its own future before the switch, and no causal network’s output can precede the switch. The negative delay is a property of slowly varying envelopes, and it holds only to the extent that the input is one.

How long is long enough #

Both readings — the positive delay of the band-pass and the negative one of the notch — rest on the burst being narrow-band. That condition can be measured directly by shortening the burst.

The reading is wrong by 60.6 per cent for a burst whose spread is 0.6 of the band-pass’s inverse bandwidth, $Q/\omega_0$, and within one per cent only beyond about seventeen. In time terms the condition is that the burst must be seven or eight times longer than the group delay it is measuring: the burst’s spectrum must fit inside the region where the phase is nearly linear, and a resonator’s phase is linear only near its centre.

So group delay is honest about envelopes that are slow, and it says nothing reliable about one that is fast. A short burst through a narrow band-pass is not delayed by $2Q/\omega_0$ — it is smeared out to the band-pass’s own impulse response, a decaying ringing whose length is the resonator’s, and its peak arrives at a time set by that ringing rather than by any derivative of phase.

Where the two delays are spent #

The distinction between phase and group delay is not academic in any system that sends information on a carrier. A radio receiver’s intermediate-frequency filter is a band-pass, and a modulated signal’s information is in its envelope, so the filter’s group delay is the delay the information experiences and its variation across the band is the distortion the information suffers — the band-pass counterpart of the delay variation flat delay, bought with more delay equalised in a low-pass. A carrier-recovery loop, by contrast, locks to the carrier’s zero crossings and sees the phase delay.

In a resonant sensor the two delays can be used separately. The phase at the centre tells whether the excitation is on resonance — zero phase, as the carrier figure shows — while the group delay tells how much energy the resonator stores, which is its Q. Measuring a burst’s envelope delay is therefore a way of measuring Q that needs no amplitude calibration at all: $Q = \omega_0 \tau_g/2$. With the 3.171 milliseconds measured here, it gives a Q of 9.96 for a resonator built with a Q of ten, and the discrepancy is the 0.4 per cent the finite burst length costs.

The notch’s negative delay has a use too, and a warning. Negative-group-delay circuits are used to cancel the positive delay of other stages over a narrow band, so that the envelope of a slowly varying signal passes a chain with less total delay. The warning is that the cancellation holds only for envelopes as smooth as the ones measured here, and that the price is the depth of the notch: a negative delay of 3.8 milliseconds came with 26 decibels of attenuation, and a larger negative delay needs a deeper notch, which then has to be made up with gain.

The same two delays on a transmission line #

The distinction between the two delays is not special to resonators; it appears wherever the phase is not proportional to frequency, and a lossy circuit-board track is the most common place a digital designer meets it. On an ideal line the phase is exactly proportional to frequency, the phase delay and the group delay are the same number at every frequency, and both equal the length over the velocity. On a real one, the dielectric’s permittivity falls with frequency and the conductor’s internal inductance adds a frequency-dependent term, so the phase is not linear and the two delays part.

The delay that is not one number measured that parting on a track: the phase delay and the group delay of the same line differing by an amount that grows with length. For a digital edge the relevant delay is neither exactly — an edge is a broadband signal, not a narrow-band envelope — but for a clock or a narrow-band serial signal it is the group delay at the signal’s fundamental that sets when the bits arrive, and the phase delay that sets where the carrier’s zero crossings fall. The skew that moves with frequency found the skew between two conductors of a pair changing with frequency for the same reason: when the phase of each is not proportional to frequency, there is no single delay to compare.

The resonator and the line are therefore the same lesson at two scales. A delay is a statement about a derivative of phase when it is about energy or information, and about the phase itself when it is about a carrier’s zero crossings; the two coincide only where the phase is a straight line through the origin, and nothing with a resonance or a loss in it has one.

The same theorem, seen from the low-pass side #

The low-pass result and the band-pass result are one theorem, and seeing them together explains why the low-pass case was exact and this one is not. A low-pass’s zero-frequency delay is the centre of mass of its impulse response because a step’s spectrum is concentrated at zero frequency, where the delay is read; any slowly varying signal through a low-pass is delayed by exactly $\tau_g(0)$, and the delay of a slow signal is the first moment. Shift that whole statement up to a carrier frequency, by modulating, and the low-pass becomes the band-pass’s equivalent baseband response: the envelope sees a low-pass whose zero-frequency delay is the band-pass’s group delay at the carrier.

So the envelope delay is the centre of mass — of the envelope’s response rather than of the impulse response — and the “how long is long enough” curve is the band-pass version of the half-way time differing from the centre of mass. A Gaussian envelope’s peak coincides with its centre of mass, so a slow Gaussian burst’s peak moves by exactly the centre-of-mass delay; a short burst’s output envelope is no longer Gaussian, its peak and its centre of mass part, and the peak drifts away from the group delay in just the way a ringing low-pass step’s half-way time drifted from its centroid. The selectivity that is not free counts the cost of a narrow band in another currency; this is the same cost counted in time.

What the measurement leaves out #

The resonators are ideal second-order sections. A practical band-pass of several sections has a group delay that is flatter across its centre and steeper at its edges, so its narrow-band condition is easier to meet at the centre and harder at the edges, and its envelope distortion is concentrated where the delay curve turns. The Q the components allow sets the limit on how narrow, and therefore how long a delay, a real resonator can give.

The envelope is Gaussian, which is the smoothest envelope there is and the kindest to the theory. A burst with a rectangular envelope has a spectrum with slowly decaying sidelobes that never fit inside a narrow band, and its output shows the resonator’s own ringing at both edges; the notch’s early peak in particular would be swamped by the transient at switch-on. The early arrival is real, and it is a property of smooth envelopes only.

And the march has its own error: forty steps a carrier cycle with the trapezoidal rule introduces a phase error of a fraction of a degree a cycle, which over a burst of hundreds of cycles is the 6.4 microseconds of zero-crossing mismatch the carrier figure shows. It is small against the milliseconds of envelope delay, and finer steps reduce it in proportion to the square of the step.

The number to keep #

A band-pass’s group delay at its centre is the delay of an envelope, not of a carrier: through a Q of ten at one kilohertz a slow burst’s envelope comes out 3.171 milliseconds late against $2Q/\omega_0 = 3.183$, while its carrier comes out in phase. A notch’s negative group delay is real for smooth envelopes: −3.810 milliseconds predicted, the envelope peaking 3.746 milliseconds early, at 4.8 per cent of the input’s size. Both readings need the burst to be long against the inverse bandwidth — 60.6 per cent wrong at 0.6 of it, under one per cent beyond about seventeen.

Still open: the burst with sharp edges, the chain of resonators, and the delay that carries information #

A rectangular burst. The resonator’s transient at each edge of a sharply switched burst carries its own timing, and it is what a pulsed radar or a keyed tone actually meets. Measuring how much of the envelope’s delay survives as a statement about edges would bound the theory from the other side.

A chain of resonators. Coupled resonators make a band-pass with a flatter delay; the envelope of a burst through one should hold its shape over a wider band of carriers. How the narrow-band condition widens with the number of coupled sections is the band-pass analogue of the Bessel filter’s flat delay.

Information that cannot arrive early. A negative group delay moves a smooth envelope’s peak earlier, but it cannot move a change in the envelope — the moment information is added — earlier than it happens. Marching a burst whose envelope changes abruptly partway would show the peak arriving early and the change arriving on time.