 as shown from the following analysis. Firstly, only the difference or differential between the delays of the two channels is of interest. Suppose that the left and right channel delays are t(1) and t(r) respectively. New delays t'(1) and t'(r) are defined by adding any fixed delay t(a), such that:
The result is that the entire effect is heard a time t(a) later, or earlier where t(a) is negative. This general expression holds in the special case where t(a) =-t(r). Substituting:
By this transformation we can always reduce the delay in one channel to zero. In a practical implementation we must be careful to subtract out the smaller delay, so that the need for a negative delay never arises. It may be preferred to avoid this problem by leaving a fixed residual delay in one channel, and changing the delay in the other. If the fixed residual delay is of sufficient magnitude, the variable delay need not be negative.
Secondly, we need not control channel amplitudes independently. It is a common operation in audio engineering to change the amplitudes of signals either by amplification or attenuation. So long as both stereo channels are changed by the same ratio, there is no change in the positional information carried. It is the ratio or differential of amplitudes that is important and must be preserved. So long as this differential is preserved, all of the effects and illusions in this description are entirely independent of the overall sound level of reproduction. Accordingly, by an operation similar to that detailed above for timing or phase control, we may place all of the amplitude control in one channel, leaving the other at a fixed amplitude. Again, it may be convenient to apply a fixed residual attentuation to one channel, so that all required ratios are attainable by attenuation of the other. Full control is then available using a variable attenuator in one channel only.
We may thus specify all the required information by specifying the differential attentuation and delay as functions of frequency for a single channel. A fixed, frequency-independent attentuation and delay may be specified for the second channel; if these are left unspecified, we assume unity gain and zero delay.
Thus, for any one sound image position, and therefore any one left/right transfer function, the differential phase and amplitude adjusting (filtering) may be organized all in one channel or the other or any combination in between. One of sound processors 1500, 1501 can be simplified to no more than a variable impedance or to just a straight wire. It can not be an open circuit. Assuming that the phase and amplitude adjusting is performed in only one channel to provide the necessary differential between the two channels the transfer functions would then be represented as in FIGS. 17A and 17B.
FIGS. 17A represents a typical transfer function for the differential phase of the two channels, wherein the left channel is unaltered and the right channel undergoes phase adjustment on a frequency dependent basis over the audio spectrum. Similarly, FIG. 17B represents generally a typical transfer function for the differential amplitude of the two channels, wherein the amplitude of the left channel is unaltered and the right channel undergoes attentuation on a frequency dependent basis over the audio spectrum.
It is appreciated that the sound positioners: 1500, 1501 of FIG. 16, for example, can be analog or digital and may include some or all of the following circuit elements: filters, delays, inventors, summers, amplifiers, and phase shifters. These functional circuit elements can be organized in any fashion that results in the transfer function.
Several equivalent representations of this information are possible, and are commonly used in related arts.
For example, the delay may be specified as a phase change at any given frequency, using the equivalences:
Caution in applying this equivalence is required, because it is not sufficient to specify the principal value of phase; the full phase is required if the above equivalences are to hold.
A convenient repres