Symmetry conditions for vestigial sideband systems

It is well known that in vestigial-sideband (v.s.b.) systems a flat overall amplitude response is ensured by imposing the symmetry conditions A(¿c+¿)+A(¿c¿¿)=1 and ß(¿c+¿)+ß(¿c¿¿)=2ß(¿c) on the amplitude response A(¿) and the phase response ß(¿) of the v.s.b. filtering arrangements between the transmit and the receive modulator of the v.s.b. system, together with the system condition ßc2¿ßc1=¿ß(¿c), where ¿c/2¿ is the common carrier frequency and ßc1 and ßc2 are the phase angles of the transmit and the receive carrier, respectively. In the letter, it is shown that these conditions, although sufficient, are not necessary, and a second set of sufficient conditions is presented: A2(¿c+¿)+A2(¿c¿¿)=1; ß(¿c+¿)+ß(¿c¿¿)=2ß(¿c); ßc2¿ßc1=±¿¿¿ß(¿c) or =±¿¿¿ß(¿c). The implications of the existence of two alternative sets of sufficient conditions are discussed.