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Figure 13.6: The general form of an allpass filter. The path bypassing

T(s) must be inverted and have half the gain.

frequency. It will be either a simple RC 1st-order filter or a 2nd-order bandpass filter; its output is amplified by two and subtracted from the original signal.

It is important to be aware that allpass filters do not inherently have an absolutely flat frequency response. Due to the finite gain-bandwidth-product of the opamps and components that do not have mathematically exact values, the magnitude response is likely to show small deviations from perfect flatness. These deviations are however usually very small and of no consequence compared with speaker unit tolerances.

First-Order Allpass Filters

Figure 13.7 shows the two versions of the basic 1st-order allpass filter. This deceptively simple circuit is analogous to a 1st-order RC filter, and so gives a slow phase change as frequency changes.

The first version in Figure 13.7a is non-inverting at low frequencies; in other words the phase-shift is 0°.As the frequency rises, the phase-shift increases until it approaches 180° (inverting) at high frequencies, as shown in Figure 13.10. This changing phase-shift is equivalent to a delay. The sharpeyed reader will note that in Figure 13.10 the phase has actually reached 180° and is clearly headed for more. This plot was produced by simulating the circuit with a model of a real opamp, rather than just evaluating a mathematical equation, and the extra phase-shift has accumulated because of the finite open-loop bandwidth of that opamp. This is what will happen when you build real crossovers, but with any modern opamp the effects are negligible. Looking at Figure 13.10 and Figure 13.11 we can see that the phase is still changing quite quickly in the 10 kHz–20 kHz range (from −135° to −158°, i.e. 21°) when the delay has fallen to a quarter of its maximum value. This is not going to give us a minimum-phase crossover.

When the positions of the resistor and capacitor are exchanged, the same phase change is obtained but phase inverted, so the second version in Figure 13.7b inverts at low frequencies but has an in-phase output at high frequencies.An allpass filter gives twice the maximum phase-shift of an ordinary filter of the same order. The non-inverting version of Figure 13.7a could also be called the RC version,

as the resistor comes before the capacitor, forming a lowpass filter. This means the common-mode voltage will fall with frequency. The inverting version of Figure 13.7b is correspondingly the CR version, and the capacitor-resistor arm now acts as a highpass filter.

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Figure 13.7: Two versions of the 1st-order allpass filter: (a) non-inverting or RC type (b) inverting or CR type.

A1st-order allpass filter has only one parameter to set: the RC time-constant; the delightfully simple calculation for the low-frequency delay is shown in Equation 13.4. There is no such thing as the Q of a 1st-order allpass. The output of an allpass filter does not have a “turnover” as such.According to some authors its operation is defined by the frequency at which the group delay has fallen to 1/√2 of its low frequency value; while this corresponds in a way to “−3 dB” for an amplitude-frequency filter, it has no real significance in itself.Amore useful description is the frequency f90 at which the phase-shift reaches 90°, halfway between the extremes of 0° and 180°, because it is derived very easily from the circuit values, as in Equation 13.5. Note that these are not the same frequencies.

You will recall that in our example loudspeaker, the tweeter signal had to be delayed by 80 usec. The resulting RC allpass circuit is shown in Figure 13.8.

If you trustingly feed a positive-going step waveform into Figure 13.8, what emerges is an immediate negative spike followed by a long slow approach to the steady input voltage, as shown in Figure 13.9.“Doesn’t look much like a delay to me!” I hear you cry, which was pretty much what I cried when I first tried this experiment.

The reason for this unhelpful-looking output is that allpass filters do not give a constant time delay with frequency, and this one is no exception. Figure 13.11 shows how as the frequency goes up the group delay is indeed 80 usec initially, but begins to decrease early as the frequency increases.

Figure 13.10 gives the phase response for comparison. The delay is down by 10% at 2.5 kHz and down to 50% at 9.3 kHz, slowly approaching zero above 100 kHz. This is clearly not much use for equalising the delay in the HF path. Since there is only one variable—the time-constant R3, C1—the only way to keep the delay constant to higher frequencies would be to reduce its value, which would make the filter useless for implementing the required delay compensation.

The fall-off in delay with frequency explains Figure 13.9; the high frequencies that make up the edge of the input step function are hardly delayed at all, and since they are subject to a 180° phase-shift, give the immediate inverted spike. The lower frequencies are delayed but get through eventually,

358  Time-Delay Filters

Figure 13.8: A non-inverting RC 1st-order allpass filter designed for a group delay of 80 usec.

Figure 13.9: The disconcerting response of the 80 usec 1st-order allpass filter to a 1 V step input.

causing the slow rise. The mistake we have made is applying a stimulus waveform with a full frequency range.

Figure 13.12 shows the rather more convincing result obtained if the 1 V step input is band-limited before it is applied to our allpass filter. The step input (Trace 1) has been put through a 4th-order Bessel lowpass filter with a −3 dB frequency of1.2 kHz, the Bessel characteristic being chosen to prevent overshoot in the filtered waveform; a Butterworth filter would have given a bit of overshoot and confused matters. The Bessel filter output is the leisurely-rising waveform of Trace 2. When this goes through the allpass filter it emerges as Trace 3 with almost exactly the same shape, but delayed by 80 usec. That looks a bit more like a delay, eh?

If you look closely at Figure 13.12 you will see that the output of the allpass filter does show a very small amount of undershoot before it rises. This could have been further suppressed by reducing the

360  Time-Delay Filters

Figure 13.12: The 1 V step input is Trace 1; putting it through a 4th-order Bessel-Thomson filter gives Trace 2, and the 80 usec 1st-order allpass filter delays it to give Trace 3.

cutoff frequency of the lowpass filter.You will also note that the output of the 4th-order Bessel lowpass filter, Trace 2, has already been delayed a good deal compared with the step input Trace 1, in fact far more than by the simple 1st-order allpass filter, because the Bessel is 4th-order. The use of Bessel lowpass filters to delay signals is examined further at the end of this chapter.

The circuit in Figure 13.7a must be non-inverting at low frequencies, because when C1 is effectively an open circuit, we get a non-inverting stage because of the direct connection to the non-inverting input. Likewise, in Figure 13.7b, when C1 is effectively open circuit, the configuration is clearly a unity-gain inverter.

The input impedance of the RC version in Figure 13.8 is 2.7 kΩ at 10 Hz, falling to 646 Ω at 1 kHz, whereafter it remains flat (it happens to be 666 Ω at 100 Hz, but I don’t think you should try to read too much into that). The equivalent CR version (with all component values the same) has an input impedance that is flat at 1 kΩ from 10 Hz to 1 kHz.Above that the impedance rises slowly until it levels off at 1.8 kΩ around 30 kHz. It is pretty clear that the CR version will be an easier load for the preceding stage, especially at high audio frequencies, and this will have its effect on the distortion performance of that stage. There is more on that in the later section on the performance of 3rd-order allpass filters.

Figure 13.11 shows an elegantly sinuous curve, quite unlike the tidy straight-line approximation roll-offs we are used to when we look at filter amplitude responses. These latter of course are plotted logarithmically with dB on the vertical axis, whereas here we have group delay time as a linear vertical axis. This is how it is normally done; dB are useful for measuring amplitude, not least because they