Time-Delay Filters  373

Figure 13.26: Distortion plot for the 2nd-order stage output (A2) of the 3rd-order 80 usec allpass filter in Figure 13.22, when it is loaded by the RC or CR final stage, and also with no load (NL). Polypropylene capacitors, 5532s, input 9 Vrms.

is now peaking in delay, the MFB filter has a higher Q and therefore a higher passband gain, here 0 dB. Thus R3 is half the value of R4, R5 instead of a quarter. Therefore the noise gain in the subtractor stageA2 is 3.6 dB lower at 12.0 dB.

This is another reason why a 3rd-order allpass is better than a 2nd-order allpass delay filter; it is not only flatter, it is quieter. Is this an original observation? I suspect so.

Higher-Order Allpass Filters

If 3rd-order filters do not provide a sufficiently extended flat-delay characteristic, then higher-order filters might be considered. We have just seen that a 3rd-order allpass filter provides the most extended flat-delay response for a given component count and also has usefully lower noise. If greater delays or flatter delay characteristics than this configuration can provide are required, higher-order filters may be the answer.

The basic principle, as we saw for the 3rd-order filter, is to take one or more 2nd-order stages with a high Q, giving a peak in the delay response, and then cancel out the peak by cascading it with one or

374  Time-Delay Filters

Figure 13.27: Distortion plot for the 2nd-order-stage output of the 3rd-order 80 usec allpass filter in Figure 13.22 loaded only by the CR final stage. Adding the 3k3 output biasing resistor reduces distortion below 6 kHz. Polypropylene capacitors, 5532s, input 9 Vrms.

more stages, either 1stor 2nd-order, with a low Q. Figure 13.28 shows how higher-order filters are built up. For each stage the Q is that of the overall allpass function and not the Q of the MFB filter, which is a different quantity. For example, in a 2nd-order allpass filter the Q for maximal flatness is 0.58, but the Q of the MFB filter is 0.50. Here the 1st-order sections have been put at the end of the chain; this makes no difference, as all the stages have unity gain.

Designing high-order filters is not something to try by hand, and the usual design procedure is to consult a table of allpass filter coefficients, as in Table 13.4. This assumes you are using the 1–2BP approach to making allpass filters.

To design your filter, pick an order and the cutoff frequency f required. For each stage of the complete filter, first select the value of C = C1 = C2 in the MFB filter (here I am using the component designations in the first stage of Figure 13.20); if this results in resistor values that are undesirably low or high, then you will have to go back and rethink the value of C. R1 and R2 are then calculated using the coefficients ai and bi thus:

378  Time-Delay Filters

Figure 13.31: Fourth-order allpass filter built by cascading two 2nd-order stages. Delay = 80 usec.

Figure 13.32: Fifth-order allpass filter built by cascading two 2nd-order stages and one 1st-order stage. Delay = 80 usec.

allowed to fall below 500 Ω if 5532 or similar opamp types are being used. In the first stage here this means that R4 and R5 have to be 2 kΩ to get R4 up to 544.8 Ω. In the second stage the Q is higher and the MFB has more gain, so R9, R10 can be reduced to 1 kΩ for better noise performance, while R8 is a respectably high 648.9 Ω.

The values of C1, C2 and C3, C4 can be selected independently for each 2nd-order stage, and C3, C4 have therefore been set lower in value, at 4n7, to keep R6, R7 above 1 kΩ in value.

Figure 13.32 shows a 5th-order allpass filter made by cascading two 2nd-order allpass stages and one 1st-order stage. Because of the higher cutoff frequency, both C1, C2 and C3, C4 have been set to 4n7 to keep the associated resistors above 1 kΩ. C5 can be set independently and has been made 10 nF to give a suitable value for R11: as low as possible for noise purposes, but not so low that it puts an excessive load on A4.

Figure 13.33 shows a 6th-order allpass filter made by cascading three 2nd-order allpass stages. In the third high-Q stage it has been necessary to reduce C5, C6 to 3n3 to keep the associated resistors above 1 kΩ.

The final 2nd-order section has a Q of 1.02, which means a passband gain of 2.095 times in the MFB filter. This could cause headroom problems. It is another reason not to embrace higher-order allpass filters without careful consideration.