entation commonly used in electronic engineering is the complex s-plane representation. All filter characteristics realizable using real analog components (any many that are not) may be specified as a ratio of two polynomials in the Laplace complex frequency variable s. The general form is: ##EQU1##
Where T(s) is the transfer function in the s plane, Ein(s) and Eout(s) are the input and output signals respectively as functions of s, and the numerator and denominator functions N(s) and D(s) are of the form:
N(s)=a.sub.o +a.sub.1 s+a.sub.2 s.sup.2 +a.sub.3 s.sup.3 +. . . +a.sub.n s.sup.n (6)
N(s)=b.sub.o +b.sub.1 s+b.sub.2 s.sup.2 +b.sub.3 s.sup.3 +. . . +b.sub.n s.sup.n (7)
The attraction of this notation is that it may be very compact. To specify the function completely at all frequencies, without need of interpolation, we need only specify the n+1 coefficients a and the n+1 coefficients b. With these coefficients specified, the amplitude and phase of the transfer function at any frequency may readily be derived using well-known methods. A further attraction of this notation is that it is the form most readily derived from analysis of an analog circuit, and therefore, stands as the most natural, compact, and well-accepted method of specifying the transfer function of such a circuit.
Yet another representation convenient for use in describing the present invention is the z-plane representation. In the preferred embodiment of the present invention, the signal processor will be implemented as digital filters in order to obtain the advantage of flexibility. Since each image position may be defined by a transfer function, we need a form of filter in which the transfer function may be readily and rapidly realized with a minimum of restrictions as to which functions may be achieved. A fully programmable digital filter is appropriate to meet this requirement.
Such a digital filter may operate in the frequency domain, in which case, the signal is first Fourier transformed to move it from a time domain representation to a 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 imp