frequency domain one. The filter amplitude and phase response, determined by one of the above methods, is then applied to the frequency domain representation of the signal by complex multiplication. Finally, an inverse Fourier transform is applied, bringing the signal back to the time domain for digital to analog conversion.
Alternatively, we may specify the response directly in the time domain as a real impulse response. This response is mathematically equivalent to the frequency domain amplitude and phase response, and may be obtained from it by application of an inverse Fourier transform. We may apply this impulse response directly in the time domain by convolving it with the time domain representation of the signal. It may be demonstrated that the operation of convolution in the time domain is mathematically identical with the operation of multiplication in the frequency domain, so that the direct convolution is entirely equivalent to the frequency domain operation detailed in the preceding paragraph.
Since all digital computations are discrete rather than continuous, a discrete notation is preferred to a continuous one. It is convenient to specify the response directly in terms of the coefficients which will be applied in a recursive direct convolution digital filter, and this is readily done using a z-plane notation that parallels the s-plane notation. Thus, if T(z) is s time domain response equivalent to T(s) in the frequency domain: ##EQU2## Where N(z) and D(z) have the form:
N(z)=c.sub.o +c.sub.1 z.sup.-1 +c.sub.2 z.sup.-2 +. . . +c.sub.n z.sup.-n(9)
D(z)=d.sub.o +d.sub.1 z.sup.-1 +d.sub.2 z.sup.-2 +. . . +d.sub.m a.sup.-m(10)
In this notation the coefficients c and d suffice to specify the function as the a and b coefficients did in the s-plane, so equal compactness is possible. The z-plane filter may be implemented directly if the operator z is interpreted such that
z.sup.-1 is a delay of n sampling intervals.
Then the specifying coefficients c and d are directly the multiplying coefficients in the implementation. We must restrict the specification to use only negative powers of z, since these corresponds to positive delays. A positive power of z would correspond to a negative delay, that is a response before a stimulus was applied.
With these notations in hand we may described equipment to allow placement of images of broad and sounds such as speech and music. For these purposes the sound processor of the present invention, for example, processor 802 of FIG. 8, may be embodied as a variable two-path analog filter with variable path coupling attenuators as in Fig. 18A.
In FIG. 18A, a monophonic or monaural input signal 1601 is input to two filters 1610, 1630 and also to two potentiometers 1651, 1652. The outputs from filters 1610, 1630 are connected to potentiometers 1653, 1654. The four potentiometers 1651-1654 are arranged as a so-called joystick control such that they act differentially. One joystick axis allows control of potentiometers 1651, 1652; as one moves such as to pass a greater proportion of its input to its output, the other is mechanically reversed and passes a smaller proportion of its input to its output. Potentiometers 1653, 1654 are similarly differentially operated on a second, independent joystick axis. Output signals from potentiometers 1653, 1654 are passed to unity gain buffers 1655, 1656 respectively, which in turn drive potentiometers 1657, 1658, respectively, that are coupled to act together; they increase or decrease the proportion of input passed to the output in step. The output signals from potentiometers 1657, 1658 pass to a reversing switch 1659, which allows the filter signals to be fed directly or interchanged, to first inputs of summing elements 1660, 1670.
Each responsive summing element 1660, 1670 receives at its second input an output from potentiometers 1651, 1652. Summing element 1670 drives inverter 1690, and switch 1691 allows selection of the direct or inverted signal to drive input 1684 of attenuator 1689. The output of attenuator 1689 is the so-called right-channel