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The generation of second-harmonic radiation in a non-linear dielectric by an arbitrary broad-band optical signal is analyzed and application of the theory is made to harmonic generation by a short pulse. The spectrum and time dependence of the harmonic radiation is calculated in terms of the corresponding quantities for the fundamental pulse. Several types of pulses, including a frequency-swept pulse, are treated. Experimental results on the second-harmonic spectrum are presented and compared with the theory, as are previously reported results on the time dependence.