This paper treats some bandwidth properties of modulated signals of the formq(t) \cos [\omega_{c}t + \varphi(t)]where bothq(t)and\varphi(t)are modulating time functions. For such simultaneous amplitude and angle modulation (AAM), relationships are given connecting the bandwidth of the modulated signal with the properties of the modulating time functions. Some useful bounds on the bandwidth are found and a relationship is derived between the amplitude and the angle modulating functions which results in a minimum bandwidth. Several examples of such minimum bandwidth signals are given.