Recirculating delay networks

It is sometimes desirable to connect the outputs of one or more delays in a network back into their own or each others' inputs. Instead of getting one or several echos of the original sound as in the example above, we can potentially get an infinite number of echos, each one feeding back into the network to engender yet others.

The simplest example of a recirculating network is the
*recirculating comb filter*
whose block diagram is shown in Figure 7.7. As with the
earlier, simple comb filter, the input signal is sent down a delay line whose
length is samples. But now the delay line's
output is also fed back to its input; the delay's input is the sum of
the original input and the delay's output. The output is
multiplied by a number before feeding it back into its input.

The time domain behavior of the recirculating comb filter is shown in Figure
7.8. Here we consider the effect of sending an impulse into the
network. We get back the original impulse, plus a series of echos, each
in turn samples after the previous one, and multiplied each time by the
gain . In general, a delay network's output given an impulse as input is
called the network's
*impulse response*.

Note that we have chosen a gain that is less than one in absolute value.
If we chose a gain greater than one (or less than -1), each echo would have
a larger magnitude than the previous one. Instead of falling exponentially
as they do in the figure, they would grow exponentially. A recirculating
network whose output eventually falls toward zero after its input terminates
is called
*stable*;
one whose output grows without bound is called *unstable*.

We can also analyse the recirculating comb filter in the frequency domain. The situation is now quite hard to analyze using real sinusoids, and so we get the first big payoff for having introduced complex numbers, which greatly simplify the analysis.

If, as before, we feed the input with the signal,

with , we can write the output as

Here the terms in the sum come from the series of discrete echos. It follows that the amplitude of the output is:

This is a geometric series; we can sum it using the standard technique. First multiply both sides by to give:

and subtract from the original equation to give:

Then solve for :

A faster (but slightly less intuitive) method to get the same result is to
examine the recirculating network itself to yield an equation for , as
follows. We named the input and the output . The signal going
into the delay line is the output , and passing this through the delay
line and multiplier gives

This plus the input is just the output signal again, so:

and dividing by and using gives:

This is equivalent to the earlier equation for .

Now we would like to make a graph of the frequency response (the gain as a
function of frequency) as
we did for non-recirculating comb filters in
Figure 7.6. This
again requires that we make a preliminary picture in the complex plane. We
would like to estimate the magnitude of equal to:

where we used the multiplicative property of magnitudes to conclude that the magnitude of a (complex) reciprocal is the reciprocal of a (real) magnitude. Figure 7.9 shows the situation graphically. The gain is the reciprocal of the length of the segment reaching from the point 1 to the point . Figure 7.10 shows a graph of the frequency response as a function of the angular frequency .

Figure 7.9 can be used to analyze how the frequency response
should
behave
qualitatively as a function of . The height and bandwidth of the peaks
both depend on . The maximum value that can attain is
when

This occurs at the frequencies as in the simple comb filter above. At these frequencies the gain reaches

The next important question is the bandwidth of the peaks in the frequency response. So we would like to find sinusoids , with frequency , giving rise to a value of that is, say, 3 decibels below the maximum. To do this, we return to Figure 7.9, and try to place so that the distance from the point 1 to the point is about times the distance from 1 to (since :1 is a ratio of approximately 3 decibels).

We do this by arranging for the imaginary part of to be roughly or its negative, making a nearly isosceles right triangle between the points 1, , and . (Here we're supposing that is at least 2/3 or so; otherwise this approximation isn't very good). The hypotenuse of a right isosceles triangle is always times the leg, and so the gain drops by that factor compared to its maximum.

We now make another approximation, that the imaginary part of
is approximately the angle in radians it cuts from the real axis:

So the region of each peak reaching within 3 decibels of the maximum value is about

(in radians) to either side of the peak. The bandwidth narrows (and the filter peaks become sharper) as approaches its maximum value of .

As with the non-recirculating comb filter of Section 7.3, the teeth of the comb are closer together for larger values of the delay . On the other hand, a delay of (the shortest possible) gets only one tooth (at zero frequency) below the Nyquist frequency (the next tooth, at , corresponds again to a frequency of zero by foldover). So the recirculating comb filter with is just a low-pass filter. Delay networks with one-sample delays will be the basis for designing many other kinds of digital filters in Chapter 8.