Accuracy and frequency response of interpolating delay lines

Since they are in effect doing wavetable lookup, variable delay lines introduce distortion to the signals they operate on. Moreover, a subtler effect comes out in the case the delay line is not changing in length: the frequency response, in real situations, is never perfectly flat for a delay line whose length is not an integer.

If the delay time is changing from sample to sample, the distortion results of section 2.5 apply. To use them, we suppose that the delay line input can be broken down into sinusoids and consider separately what happens to each individual sinusoid. We can use table 2.5 to predict the RMS level of all the combined distortion products for an interpolated variable delay line.

We'll assume here that we want to use four-point interpolation. For sinusoids with periods longer than 32 samples (that is, for frequencies below 1/16 of the Nyquist frequency) the distortion is 96 dB or better--unlikely ever to be noticeable. At a 44 kHz. sample rate, these periods would correspond to frequencies up to about 1400 Hz. At higher frequencies the quality degrades, and above 1/4 the Nyquist frequency the distortion products, which are only down about 50 dB, should start to become readily audible.

The situation for a complex tone will depend primarily on the amplitudes and frequencies of its higher partials. Suppose, for instance, that a tone's partials above 5000 Hz. are at least 20 dB less than its strongest partial, and that above 10000 Hz they are down 60 dB. Then as a rough estimate, the distortion products from the range 5000-10000 will each be limited to about -68 dB and those from above 10000 Hz. will be limited to about -75 dB (because the worst figure in the table is about -15 dB and this must be added to the strength of the partial involved.)

If the high-frequency content of the input signal does turn out to give unacceptable distortion products, in general it is more effective to increase the sample rate than the number of points of interpolation. For periods greater than 4 samples, doubling the period (such as by doubling the sample rate) decreases distortion by about 24 dB.

The 4-point interpolating delay line's frequency response is nearly flat up to half the Nyquist frequency, but thereafter it can dive quickly. Suppose (to pick the worst case) that the delay is set halfway between two integers, say 1.5. Cubic interpolation gives:

$$x[1.5] = {{-x[0] + 9x[1] + 9 x[2] - x[3]} \over 8}$$

Now let $x[n]$ be a (real-valued) unit-amplitude sinusoid with angular frequency $\omega$, whose phase is zero at $1.5$:

$$x[n] = \cos(\omega \cdot (n - 1.5))$$

and compute $x[1.5]$ using the above formula:

$$x[1.5] = {{9 \cos(\omega/2) - \cos(3 \omega / 2)}\over4}$$

This is the peak value of the sinusoid that comes back out of the delay line, and since the peak amplitude going in was one, this shows the frequency response of the delay line. This is graphed in Figure 7.18. At half the Nyquist frequency ( $\omega = \pi / 2$) the gain is about -1 dB, which is a barely perceptible drop in amplitude. At the Nyquist frequency itself, however, the gain is zero.

Figure 7.18: Gain of a four-point interpolating delay line with a delay halfway between two integers. The DC gain is one.

As with the results for distortion, the frequency response improves radically with a doubling of sample rate. If we run our delay at a sample rate of 88200 Hz. instead of the standard 44100, we will get only about 1 dB of rolloff all the way up to 20000 Hz.