Teleseismic Broadband P-wave seismograms radiated by May 1990 Southern Sudan and December, 2005 Lake Tanganyika earthquakes at different azimuths have been analyzed on the basis of magnitude spectra. The two earthquakes represent the largest shocks in the East African Rift System and its extension in southern Sudan. Both events have the same surface wave magnitude (M_S=7.2).  The maximum average spectral magnitude for the first event was determined as 6.79 at 4 seconds period compared to 6.33 at 4 seconds period estimated from the second event. The other source parameters for the two earthquakes were also estimated. The first event had a seismic moment over fourth that of the second event. The two events are radiated from patches of faults having radii of 13.05 and 7.85 km respectively. The average displacement and stress drop are estimated to be 0.56 m and 1.65 MPa for the first event and 0.43m and 2.20 MPa for the second one. The source parameters, which describe inhomgeneity of the fault, are also determined from the magnitude spectra. These additional parameters are complexity, asperity radius, displacements across the asperity and ambient stress-drop. Both events produce moderate rupture complexity. Compared to the second event, the first event is characterized by relatively higher complexity, a low average stress-drop and a high ambient stress. A reasonable explanation for the relative variations in these parameters is that the source for each event is located at different tectonic environments. The focal mechanism solutions supplement such observation.