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In applications where large time-bandwidth products and high target speeds combine, linear FM pulse-compression systems used in radar or sonar suffer considerable losses in performance. This paper deals with the problem of wide-band signal optimizintion for the purpose of minimizing signal degradation resulting from Doppler distortion effects. The equation for the instantaneous frequency of a Doppler-transformed signal is derived. The optimum frequency-modulation law is then shown to be the linear period modulation (it can be with equal rights called hyperbolic frequency modulation or logarithmic phase modulation). It is interesting to note that some kinds of bats use this type of signal in their "sonar." Exact and approximate expressions for the spectrum of a linear-period-modulated rectangular pulse are given, and the shape of the compressed output pulse is considered. Signal degradation due to Doppler distortion is shown to be negligible in the case of linear-period-modulation pulses. The influence of pulse envelope on the compressed signal waveform is discussed, and envelope-optimization examples are presented. A possibility for active generation of the optimum signal is mentioned. Significant properties of linear FM and linear-period-modulation pulse compression systems are compared.