This paper introduces an efficient radar transmitter with improved spectral confinement, enabled by a pulse waveform that contains both amplitude and frequency modulation. The theoretical behavior of the Class-B power amplifier (PA) under a Gaussian envelope is compared to that of a Class-A PA. Experimental validation is performed on a 4-W 10-GHz GaN monolithic microwave integrated circuit PA, biased in Class B with a power-added efficiency (PAE) of 50%. When driven with a Gaussian-like pulse envelope with a 5-MHz linear frequency modulation (LFM), the PA demonstrates a 31% average efficiency over the pulse duration. To improve the efficiency, a simple resonant supply modulator with a peak efficiency of 92% is used for the pulse Gaussian amplitude modulation, supplemented by pre-distortion on the PA input, while the LFM is provided through the PA input. This case results in a five-point improvement in system efficiency with an average ${\rm PAE}=40\hbox{\%}$ over the pulse duration for the PA alone, and with simultaneous 40-dB reduction in spectral emissions relative to a rectangular pulse with the same energy.