In many physics fields, the radio emission of a composite system composed of a large number of randomly occurring but similar emission sources is encountered. In general, the composite system lasts longer than each individual component and individual source currents vary much more rapidly. This Letter presents a theory to understand the electromagnetic radiation spectrum of such a system. If the temporal distribution of the random occurrence of the component and the distribution to describe the relevant emission source properties are known, the spectrum of the composite system can be readily found from this theory. There are two main terms that define the spectrum: one term results from the coherent summation of the contributions from individual sources and is proportional to the square of the total number of the components in the system; the other term results from an incoherent summation and is proportional to the first power of that number. This can lead to drastically different spectral magnitudes in different spectral regions, typically with the spectral magnitude in the lower frequency region many orders of magnitude stronger than that in the higher frequency region.