Waveform considerations in space-variant optical processors.

Space-variant optical processors are the subject of considerable research.1 One of the most promising methods by which such systems can be realized is the application of coordinate transformations to the input data.2' 3 The extension of this technique to produce optical Mellin transforms4 and distortion-invariant pattern-recognition systems has been reported over the past three years.k7 Of particular interest in this Letter is the use of the Mellin transform in Doppler-invariant signal processors, as we described earlier.8' 9 However, because the Mellin transform is a space-variant operation, a time-sequential implementation of the required coordinate transformation would be required unless a fully parallel, real-time, coordinate transformation system can be developed. With such a properly configured system, a single-channel 1-D optical processor can be used to realize the necessary range/Doppler processing. At least two techniques have been reported by which the required parallel coordinate transformation can be realized. One scheme that is appropriate for 2-D inputs utilizes computer-generated holograms1 0 -12 but suffers from input space-bandwidth product limitations. A second method13'14 is of use with 1-D input functions and 1-D distortions only and results in large light losses. However, with refinements in these methods, our single-channel 1-D, real-time Mellin-transform Dopplerinvariant-processor concept8' 9 seems more realizable. Hence, in this Letter, we extend our earlier work8'9 to the case of coded waveforms, as are conventionally used in radar and communications. We now describe in more detail the waveform properties and operation of such space-variant optical processors with complex coded waveforms. Following a review of Mellin-transform-based Doppler-invariant signal processing we show that previously described space-variant signal processors8' 9' 15 cannot operate on conventional coded waveforms without a severe loss in system performance. Rather, proper operation of such processors requires transmission codes that are nonlinear. Such waveforms are then shown to result in a novel spread-spectrum coding that promises to provide new research avenues for waveform design in spread-spectrum, radar, and communications systems. As we have long advocated, optical signal processors should not be limited by the conventional waveform and system-design constraints introduced by existing processing technologies. The case at hand serves to demonstrate this point vividly. In future papers, we will describe the ambiguity functions, system-design parameters, and the advantageous noise, jammer, and interference-rejection features of such systems using nonlinear waveforms and space-variant optical processors. In the general formulation of space-variant pattern recognition, we represent the undistorted input signal (in 1-D for simplicity) by f(x) and a distorted version of it by