A new in-phase/quadrature (I/Q) complex modulation technique to realize single-sideband time-modulated phase-only weighting on an array antenna is proposed. Based on modulating rectangular pulses and trapezoidal pulses, the theories of generating a scanning beam at the single positive sideband are presented. It was found that using trapezoidal pulse modulation can obtain excellent performance in power efficiency and undesired sideband level than using rectangular pulse modulation. Unlike conventional time-modulated antenna array where each array element is connected to a RF switch, in the proposed scheme, each element is connected to an I/Q channel modulator which is composed of two Wilkinson power dividers, two RF switches, two $0/\pi $ phase shifters in RF channels, and one $\pi/2 $ fixed-phase shifter in the control circuit. By properly controlling both the phase shifters and the switches in time domain, the single-sideband modulation can be realized with the uniform amplitude and variable phase in frequency domain. The I/Q modulator acts as a phase shifter and can be used to form a phased array. In this paper, the main efforts have been paid to find the pulse sequences to suppress the spurious bands. To validate the proposed technique, simulated results of an eight-element single-sideband time-modulated linear array obtained from the array factor and full-wave analysis are reported. In the case of using rectangular pulses, the power efficiency of the scanning beam is 91.47% and the peak level of the highest undesired harmonic is $-13.98\;\mathrm{dB} $ . In the case of using trapezoidal pulses, the best values can reach 99.79% and $-27.98\;\mathrm{dB} $ .