The response of a mismatched filter which will reduce the effects of radar clutter energy on signal detection is considered in this paper. Clutter is assumed to result from scatterers having a range-velocity distribution different from that of a desired target. A tradeoff between signal-to-clutter ratio (SCR) and signal-to-noise ratio (SNR) is made. The mathematical treatment involves controlling the distribution of the volume of the squared magnitudet/fcross-correlation function associated with a given transmitted waveform and the response of the mismatched receiving filter. A measure of this volume is minimized for variations in mismatched filter response under suitable constraints on the output signal and noise powers. The essential result is contained in a linear integral equation. A method of solution is illustrated which enables one to choose a "best" tradeoff between SCR and SNR.

The uniqueness theorem for ambiguity functions states that ff waveformsu(t)andv(t)have the same ambiguity function, i.e.,\chi_{u}(\tau, \Delta) = X_{\upsilon , \Delta), thenu(t)andv(t)are identical except for a rotation, i.e.,v(t) = e^{i\lambda}u(t), where\lambdais a real constant. Through the artifice of treating the even and odd parts of the waveforms, denotede(t)ando(t), respectively, correlative results have been obtained for the real and imaginary parts of ambiguity functions. Thus, ifRe \{\chi_{u}(\tau, \Delta)\} = Re \{\chi_{\upsilon}(\tau, \Delta)\}, thene_{\upsilon}(t) = e^{i \lambda e}e_{u}(t)ando_{\upsilon}(t) = e^{i \lambda o}o_{u}(t). FromRe \{\chi_{u}(\tau, \Delta)\}, the waveform classu(t) = e^{i\lambda} [e_{u}(t) + e^{ik}o_{u}(t)]may be constructed, but because of the arbitrary rotation,e^{ik}, a unique\chi_{u}-function is not determinable, in general. An important exception to this statement is the case when\chi_{u}(\tau, \Delta)is real, andRe \{\chi_{u}\} = \chi_{u}determines a unique waveform (within a rotation) and this waveform can only be even or odd. IfIm \{\chi_{u}(\tau, \Delta)\} = Im \{\chi_{\upsilon(\tau, \Delta)\}thene_{\upsilon}(t) =ae^{i \gamma}e_{u}(t)ando_{\upsilon}(t) = 1/ae^{ir}o_{u}(t). IfIm \{\chi_{u}(\tau, \Delta)\}is given, {\em and}u(t)is known to have unit energy, then within rotations of the forme^{i \lambda}, only two possible waveform choices are possible foru(t). If it also is known which ofe_{u}(t)ando_{u}(t)has the greater energy, the functionIm \{\chi_{u}(\tau, \Delta)\}uniquely determinesu(t)(within a rotation) and the complete\chi_{u}-function. The results on magnitude/phase relationships include a formula which enables one to compute the squared magnitude of an ambiguity function as an ordinary two-dimensional correlation function. Self-reciprocal two-dimensional Fourier transforms are demonstrated for the product of the squared-magnitude function and either of the first partial derivatives of the phase function.

Mequally probable symbols may be encoded as continuous waveforms whose sample values are the coordinates of the vertices of a regular simplex inNdimensions. These vertices or code points, therefore, are equally spaced, and the resulting waveforms are adapted particularly to low signal-to-noise communication systems. An upper bound for the information rate in an additive Gaussian noise channel, based on the use of regular-simplex coded waveforms, has been calculated. The results indicate that for signal-to-noise energy ratios less than approximately unity and for error probabilities ranging from10^{-2}tol0^{-8}, this rate is a sizable percentage of the ideal rate which has been derived by Shannon and Tuller.

Woodward's result for the ambiguity function, that the volume associated with its squared magnitude over the time-shift and frequency-shift plane is a constant, has been shown to be true also for a cross-ambiguity function for two time functions. If complex time functions have been obtained by means of a Hilbert transformation from real time functions, it is found for the cross-ambiguity function that the volumes under the squared real part and under the squared imaginary part are constant and contribute equally to the volume under the squared magnitude function. A "distance" function for two time functions is defined to be the integrated squared difference between these functions. The relation for the squared real part of the ambiguity function readily yields an invariant relation for the volume associated with this distance function in the case of Hilbert-derived complex time functions. An especially simple invariant relation for the "mean" distance, as computed over the time-shift and frequency-shift plane, exists for such time functions having finite energy and finite mean value.