## Transformations for radar ambiguity functionsThis investigation sheds new light on the area of radar ambiguity functions and their ability to improve signal design. The problem is approached from two directions. First, a number of signal transformations are presented along with their corresponding ambiguity functions. These include both linear transformations induced by processing the signal through a linear time-variant system and nonlinear transformations generated by functional transformations upon the signal. Three simple linear signal transformations are shown to generate the allowable class of linear transformations of the ambiguity plane for signals that have rational or entire functions as their spectra. Hence, for the class of time-limited signals the complete class is presented. It is proven that nonlinear transformations of the ambiguity plane are restricted to only unrealistic signals. Second, the ambiguity function is perturbed in a number of ways in an effort to improve some quality of the signal for radar purposes. To this end tests based upon existing theorems are developed to determine the applicability of any particular perturbation. The class of linear transformations of the ambiguity function are presented in two forms: for ambiguity functions given in analytic and in digital form. |

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### Contents

STATISTICAL DERIVATION OF THE AMBIGUITY | 12 |

PROPERTIES AND FOURIER TRANSFORMS OF AMBIGUITY | 19 |

TRANSFORMATIONS OF THE AMBIGUITY PLANE | 47 |

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### Common terms and phrases

97 Figure ambiguity func ambiguity plane amplitude analytic signal assume assumption axis becomes oo calculate change of variables chirp signal complete convolution cross-ambiguity function DeClaris denote density function determine Dirac delta function Dispersive Delay Line Doppler Doppler shift entire functions example expansion coefficients Figure 13 find that oo Fourier transform G-function Gersho H-Separable System Hence Hilbert transform impulse responses inner product kernel Klauder linear integral transformation linear transformation m=l n=l magnitude magnitude-squared minimum phase modulation necessary and sufficient noise Nonlinear Chirp obtain oo and oo oo oo oo V(t orthonormal Output Pulse Trains radar ambiguity function radar signals radar system range and range-rate range resolution rational function resolution properties return signal sample values shown Siebert signal transformations signal u(t simply spectra substituting sufficient condition Sussman 15 targets Theory time-variant system tion transmitted waveform uniqueness theorem vector space Wilcox zero